Point Process Calculus in Time and Space [1 ed.] 9783030627522, 9783030627539
This book provides an introduction to the theory and applications of point processes, both in time and in space. Present
257 17 3MB
English Pages 556 [562] Year 2020
Preface
Contents
Chapter 1 Generalities
1.1 Point Processes as Random Measures
1.2 Campbell’s Formula and Moment Measures
1.3 The Distribution of a Point Process
1.4 Convergence in Distribution and Variation
1.5 Cluster Point Processes
1.6 The Stieltjes–Lebesgue Calculus
1.7 Exercises
Chapter 2 Poisson Processes on the Line
2.1 Counting Process and Interval Sequence
2.2 The Smoothing Formula
2.3 Poisson Martingales and Stochastic Integrals
2.4 Watanabe’s Characterization
2.5 HMCs and Stochastic Differential Equations
2.6 HMCs and Time-Scaled HPPs
2.7 Exercises
Chapter 3 Spatial Poisson Processes
3.1 Sampling a Poisson Process
3.2 The Covariance and Exponential Formulas
3.3 Marked Spatial Poisson Processes
3.4 Operations on Poisson Processes
3.5 Change of Probability
3.6 Exact Sampling of Cluster Point Processes
3.7 The Boolean Model
3.8 Exercises
Chapter 4 Renewal and Regenerative Processes
4.1 Renewal Point processes
4.2 The Renewal Theorem
4.3 Blackwell’s Theorem and its Refinements
4.4 Regenerative Processes
4.5 Multivariate Renewal Equations
4.6 Semi-Markov Processes
4.7 Exercises
Chapter 5 Point Processes with a Stochastic Intensity
5.1 The Smoothing Formulas
5.2 Regenerative Form of the Intensity Kernel
5.3 Martingales as Stochastic Integrals
5.4 Time Scaling
5.5 Continuous Change of Probability
5.6 Extension of the Theory of Stochastic Intensity
5.7 Grigelionis’ Representation
5.8 Origin and Motivation of the Martingale Approach
5.9 Exercises
Chapter 6 Exvisible Intensity of Finite Point Processes
6.1 The Janossy Density
6.2 The Spatial Smoothing formula
6.3 Exvisibility and Predictability
6.4 Finite Markov Point Processes
6.5 Spatial Birth-and-Death Point Processes
6.6 An Alternative Model
6.7 Exercises
Chapter 7 Palm Probability on the Line
7.1 Stationary Point Processes
7.2 A First Look at Palm Probability
7.3 Palm Theory on the Line: Basic Formulas
7.4 From Palm Probability to Stationary Probability
7.5 Local Interpretation of Palm Probability
7.6 The Cross-ergodic Theorem
7.7 Palm Probability and Stochastic Intensity
7.8 Exercises
Chapter 8 Palm Probability in Space
8.1 The Voronoi Cell and the Inversion Formula
8.2 The Local Interpretation
8.3 Ergodicity
8.4 The Mecke Measure
8.5 The Reduced Mecke Measure
8.6 Exercises
Chapter 9 The Power Spectral Measure
9.1 The Covariance Measure
9.2 The Bartlett Spectral Measure
9.3 Long-range Dependence
9.4 Transformations of the Spectral Measure
9.5 Exercises
Chapter 10 The Information Content of Point Processes
10.1 Filtering
10.2 Separation of Detection and Filtering
10.3 Hiding Information in a Point Process
10.4 Noisy Points
10.5 Random Sampling
10.6 Exercises
Chapter 11 Point Processes in Queueing
11.1 A Review of Markovian Queueing Theory
11.2 Poisson Systems
11.3 The G/G/1/∞ Queue
11.4 PASTA, Little, etc.
11.5 Selected Applications
11.6 Exercises
Chapter 12 Hawkes Point Processes
12.1 As a Branching Point Process
12.2 Rates of Extinction and of Installation
12.3 The Bartlett Spectrum of the Hawkes Process
12.4 Exact Sampling of Hawkes Processes
12.5 Branching Point Processes Without Ancestor
12.6 Kerstan Point Processes
12.7 Exercises
Appendix
A.1 Measurability and Measure
A.2 Stochastic Processes
A.3 Martingales
A.4 Internal History of a Marked Point Process
A.5 Local vs. Global Absolute Continuity
Bibliography
Index
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Probability Theory and Stochastic Modelling 98
Pierre Brémaud
Point Process Calculus in Time and Space An Introduction with Applications
Probability Theory and Stochastic Modelling Volume 98
Editors-in-Chief Peter W. Glynn, Stanford, CA, USA Andreas E. Kyprianou, Bath, UK Yves Le Jan, Orsay, France Advisory Editors Søren Asmussen, Aarhus, Denmark Martin Hairer, Coventry, UK Peter Jagers, Gothenburg, Sweden Ioannis Karatzas, New York, NY, USA Frank P. Kelly, Cambridge, UK Bernt Øksendal, Oslo, Norway George Papanicolaou, Stanford, CA, USA Etienne Pardoux, Marseille, France Edwin Perkins, Vancouver, Canada Halil Mete Soner, Zürich, Switzerland
The Probability Theory and Stochastic Modelling series is a merger and continuation of Springer’s two well established series Stochastic Modelling and Applied Probability and Probability and Its Applications. It publishes research monographs that make a significant contribution to probability theory or an applications domain in which advanced probability methods are fundamental. Books in this series are expected to follow rigorous mathematical standards, while also displaying the expository quality necessary to make them useful and accessible to advanced students as well as researchers. The series covers all aspects of modern probability theory including • • • • • •
Gaussian processes Markov processes Random Fields, point processes and random sets Random matrices Statistical mechanics and random media Stochastic analysis
as well as applications that include (but are not restricted to): • Branching processes and other models of population growth • Communications and processing networks • Computational methods in probability and stochastic processes, including simulation • Genetics and other stochastic models in biology and the life sciences • Information theory, signal processing, and image synthesis • Mathematical economics and finance • Statistical methods (e.g. empirical processes, MCMC) • Statistics for stochastic processes • Stochastic control • Stochastic models in operations research and stochastic optimization • Stochastic models in the physical sciences
More information about this series at http://www.springer.com/series/13205
Pierre Brémaud
Point Process Calculus in Time and Space An Introduction with Applications
123
Pierre Brémaud Paris, France
ISSN 2199-3130 ISSN 2199-3149 (electronic) Probability Theory and Stochastic Modelling ISBN 978-3-030-62752-2 ISBN 978-3-030-62753-9 (eBook) https://doi.org/10.1007/978-3-030-62753-9 Mathematics Subject Classification: 60-xx, 90-xx, 93-xx, 94-xx © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_1
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A7 E = Rm M/ B7 ν Bb Q7 i?2 7Q`K ν(C) = C λ(x)dx 7Q` bQK2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM λ : Rm → R- i?2 SQBbbQM T`Q+2bb N Bb b B/ iQ /KBi i?2 BMi2MbBiv 7mM+iBQM λ(x)X A7 BM //BiBQM λ(x) ≡ λ- N Bb + HH2/ ?QKQ;2M2Qmb SQBbbQM T`Q+2bb U?TTV QM Rm rBi? BMi2MbBiv Q` ` i2 λX A7 λ ≡ 1- Bi Bb bQK2iBK2b + HH2/ bi M/ `/ SQBbbQM T`Q+2bbX *Qt S`Q+2bb2b .Qm#Hv biQ+? biB+ SQBbbQM T`Q+2bb2b- Q` *Qt T`Q+2bb2b-j `2 +QMbi`m+i2/ BM irQ bi2TbX 6B`bi QM2 /` rb ` M/QK BMi2MbBiv K2 bm`2 T`Q+2bb- i? i Bb HQ+ HHv }@ MBi2 ` M/QK K2 bm`2 ν QM i?2 K2 bm` #H2 bT +2 (E, E)- M/ ? pBM; /QM2 bQ- QM2 ;2M2` i2b SQBbbQM T`Q+2bb N rBi? i?2 BMi2MbBiv K2 bm`2 νX 6Q`K HHv, .2}MBiBQM RXRXRy G2i ν #2 HQ+ HHv }MBi2 ` M/QK K2 bm`2 QM EX G2i G #2 σ@}2H/ +QMi BMBM; i?2 σ@}2H/ F ν ;2M2` i2/ #v νX TQBMi T`Q+2bb N QM E bm+? i? i- ;Bp2M G- N Bb SQBbbQM T`Q+2bb QM E rBi? i?2 BMi2MbBiv K2 bm`2 ν Bb + HH2/ /Qm#Hv biQ+? biB+ SQBbbQM T`Q+2bb rBi? `2bT2+i iQ G rBi? i?2 +QM/BiBQM H BMi2MbBiv K2 bm`2 ν- Q` *Qt T`Q+2bb /B`2+i2/ #v i?2 ` M/QK K2 bm`2 νX A7 E = Rm M/ ν(dx) = λ(x) dx 7Q` bQK2 MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 biQ+? biB+ T`Q+2bb {λ(x)}x∈Rm - i?2 TQBMi T`Q+2bb N Bb + HH2/ /Qm#Hv biQ+? b@ iB+ SQBbbQM T`Q+2bb rBi? `2bT2+i iQ G rBi? i?2 U+QM/BiBQM HV BMi2MbBiv 7mM+iBQM {λ(x)}x∈Rm X A7 KQ`2Qp2` λ(x) ≡ Λ- MQM@M2; iBp2 ` M/QK p `B #H2- N Bb + HH2/ KBt2/ SQBbbQM T`Q+2bb Q7 BMi2MbBiv ΛX _2M2r H SQBMi S`Q+2bb G2i {Sn }n≥1 #2 M BB/ b2[m2M+2 Q7 TQbBiBp2 ` M/QK p `B #H2b rBi? +QKKQM +mKm@ H iBp2 /Bbi`B#miBQM 7mM+iBQM F X h?2 TQBMi T`Q+2bb QM R+ /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib n Tn = Sk (n ≥ 1) k=1
Bb + HH2/ `2M2r H TQBMi T`Q+2bbX am+? TQBMi T`Q+2bb2b rBHH #2 bim/B2/ BM *? Ti2` 9X SQBMi S`Q+2bb2b QM :`B/b .2}MBiBQM RXRXRR ;`B/ QM Rm Um ≥ 1V Bb +QHH2+iBQM Q7 TQBMib Q7 i?2 ivT2 {(n1 T1 , . . . , nm Tm ) ; n1 , . . . , nm ∈ Z} , r?2`2 T1 , . . . , Tm `2 TQbBiBp2 `2 H MmK#2`bX j
(*Qt- RN88)X
URXRV
e
*>Sh1_ RX :1L1_GAhA1a
1t KTH2 RXRXRk, _ M/QK SQBMib QM i?2 :`B/X G2i {Xn1 ,n2 }n1 ,n2 ∈Z #2 /Qm#Hv BM/2t2/ b2[m2M+2 Q7 {0, 1}@p Hm2/ ` M/QK p `B #H2bX TQBMi T`Q+2bb QM i?2 ;`B/ URXRV UrBi? m = 2V Bb +QMbi`m+i2/ 7`QK i?Bb b2[m2M+2 #v TH +BM; TQBMi i (n1 T1 , n2 T2 ) B7 M/ QMHv B7 Xn1 ,n2 = 1X G2i pn1 ,n2 := P (Xn1 ,n2 = 1)X h?2 +QM/BiBQM pn1 ,n2 < ∞ n1 ,n2 ∈Z
Bb bm{+B2Mi 7Q` i?Bb TQBMi T`Q+2bb iQ #2 }MBi2 Ui? i Bb- iQ ? p2 HKQbi@bm`2Hv }MBi2 MmK#2` Q7 TQBMibV bBM+2 i?2 bmK BM i?2 H27i@? M/ bB/2 2[m Hb i?2 p2` ;2 iQi H MmK#2` Q7 TQBMibX AM i?2 + b2 i? i {Xn1 ,n2 }n1 ,n2 ∈Z Bb M BM/2T2M/2Mi b2[m2M+2- Bi Bb HbQ M2+2bb `v U#v i?2 "Q`2HĜ* Mi2HHB H2KK 7Q` BM/2T2M/2Mi b2[m2M+2bVX A7 Xn1 ,n2 = 1 7Q` HH n1 , n2 ∈ Z- i?2 +Q``2bTQM/BM; U/2i2`KBMBbiB+V TQBMi T`Q+2bb Bb + HH2/ +QKTH2i2 ;`B/ UQM R2 VX P7 +Qm`b2 bBKBH ` /2}MBiBQMb `2 p BH #H2 7Q` i?2 ;`B/ QM Rd M/ Mv d ≥ 1X 1t KTH2 RXRXRj, ai iBQM `v :`B/X +QKTH2i2 ;`B/ QM R2 Bb MQi bi iBQM@ `v TQBMi T`Q+2bbX >Qr2p2`- i?2 ` M/QKHv b?B7i2/ p2`bBQM Q7 Bi(n1 T1 + V1 , n2 T2 + V2 )
(n1 , n2 ∈ Z) ,
r?2`2 V1 M/ V2 `2 BM/2T2M/2Mi ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, T1 ) M/ [0, T2 ) `2bT2+iBp2Hv- Bb bi iBQM `vX h?Bb + M #2 T`Qp2/ /B`2+iHv Q` #v mbBM; i?2 G TH +2 7mM+iBQM H U1t KTH2 RXjXReVX .2}MBiBQM RXRXR9 G2i N #2 ` M/QK K2 bm`2 QM (E, E)X h?2 b2i 7mM+iBQM C → ν(C) := E [N (C)]
(C ∈ E)
mMB[m2Hv /2}M2b K2 bm`2 QM (E, E) + HH2/ i?2 BMi2MbBiv K2 bm`2 Q7 N X .2}MBiBQM RXRXR8 AM i?2 + b2 Q7 M HX+X/X#X bT +2 E- HQ+ HHv }MBi2 TQBMi T`Q+2bb N Bb + HH2/ }`bi@Q`/2` TQBMi T`Q+2bb B7 E [N (C)] < ∞ 7Q` HH `2H iBp2Hv +QKT +i b2ib C ∈ EX J `F2/ SQBMi S`Q+2bb2b hQ 2 +? TQBMi Q7 TQBMi T`Q+2bb K v #2 ii +?2/ M ii`B#mi2- 7Q` BMbi M+2 i?2 ?2B;?i Q7 i?2 i`22 ;`QrBM; i?2`2 Q` i?2 `2[mB`2/ b2`pB+2 iBK2 Q7 +mbiQK2` T`2@ b2MiBM; ?BKb2H7 i b2`pB+2 /2bF i i?Bb iBK2X h?Bb ii`B#mi2 Bb `2T`2b2Mi2/ #v ` M/QK 2H2K2Mi- + HH2/ i?2 K `F Q7 i?2 TQBMiX h?Bb K `F + M #2 Q7 p2`v ;2M2` H M im`2- ` M;BM; 7`QK MmK#2` iQ 7mM+iBQM Q` TQBMi T`Q+2bb- b rBHH i?2 + b2 BM i?2 /2}MBiBQM Q7 S HK T`Q# #BHBiv U*? Ti2` dVX .2}MBiBQM RXRXRe G2i N M/ {Xn }n∈N #2 b BM G2KK RXRX8X G2i (K, K) #2 bQK2 K2 bm` #H2 bT +2 M/ H2i {Zn }n∈N #2 ` M/QK b2[m2M+2 rBi? p Hm2b BM KX h?2 b2[m2M+2 {(Xn , Zn )}n∈N Bb + HH2/ K `F2/ TQBMi T`Q+2bb QM E rBi? K `Fb BM KX h?2 b2[m2M+2 {Zn }n∈N Bb i?2 K `F b2[m2M+2 M/ N Bb i?2 # b2 TQBMi T`Q+2bbX
RXRX SPALh S_P*1aa1a a _L.PJ J1al_1a
d
E
L
C
R
6B;m`2 RXRXkX K `F2/ TQBMi T`Q+2bbX >2`2 N (C × L) = 2X .2}M2 TQBMi T`Q+2bb N UBM i?2 b2Mb2 Q7 .2}MBiBQM RXRXkV QM E × K #v N (C × L) := 1C (Xn )1L (Zn ) (C ∈ B(E) , L ∈ K) . URXkV n∈N
LQi2 i? i bBM+2 Δ ∈ / C- i?2 TQBMib Xn ∈ {Δ} /Q MQi TT2 ` BM i?2 bmK #Qp2 UdzTQBMib i BM}MBiv `2 2t+Hm/2/ǴVX q2 b? HH Q++ bBQM HHv mb2 i?2 MQi iBQM NZ BMbi2 / Q7 N X h?2 7QHHQrBM; T?` b2b `2 i?2M +QMbB/2`2/ 2[mBp H2Mi, dzi?2 K `F2/ TQBMi T`Q+2bb {(Xn , Zn )}n∈N Ǵdzi?2 K `F2/ TQBMi T`Q+2bb N Ǵdzi?2 K `F2/ TQBMi T`Q+2bb NZ Ǵ- M/ dzi?2 K `F2/ TQBMi T`Q+2bb (N, Z)ǴX UAM i?2 H bi irQ MQi iBQMb- Z Bb bvK#QHB+ `2T`2b2Mi iBQM Q7 i?2 b2[m2M+2 Q7 K `FbXV .2}MBiBQM RXRXRd A7 BM //BiBQM i?2 b2[m2M+2 {Zn }n∈N Bb BB/ M/ BM/2T2M/2Mi Q7 N - i?2M N Bb + HH2/ K `F2/ TQBMi T`Q+2bb rBi? BM/2T2M/2Mi BB/ K `FbX h?2 /2}MBiBQM Q7 i?2 BMi2MbBiv K2 bm`2 Q7 K `F2/ TQBMi T`Q+2bb Bb bBKBH ` iQ i? i Q7 M mMK `F2/ TQBMi T`Q+2bb U.2}MBiBQM RXRXR9VX .2}MBiBQM RXRXR3 h?2 BMi2MbBiv K2 bm`2 Q7 i?2 K `F2/ TQBMi T`Q+2bb NZ := {(Xn , Zn )}n∈N Q7 .2}MBiBQM RXRXRe Bb i?2 K2 bm`2 νZ mMB[m2Hv /2}M2/ QM (E × K, E ⊗ K) #v 1D ((Xn , Zn )) (D ∈ E ⊗ K) . νZ (D) := E n∈Z
1t KTH2 RXRXRN, AMi2MbBiv J2 bm`2 Q7 J `F2/ SQBMi S`Q+2bb rBi? AM/2T2M/2Mi BB/ J `FbX h?2 BMi2MbBiv K2 bm`2 Q7 i?2 K `F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd Bb i?2 T`Q/m+i K2 bm`2 ν × QZ - r?2`2 ν Bb i?2 BMi2MbBiv K2 bm`2 Q7 i?2 # b2 TQBMi T`Q+2bb U1t2`+Bb2 RXdX9VX G2i E M/ K #2 HX+X/X#X bT +2b rBi? i?2B` "Q`2H σ@}2H/b B(E) M/ K := B(K) := E × K Bb M HX+X/X#X bT +2 r?Qb2 "Q`2H σ@}2H/ B(E) Bb i?2 `2bT2+iBp2HvX h?2M E T`Q/m+i B(E) ⊗ KX
3
*>Sh1_ RX :1L1_GAhA1a
.2}MBiBQM RXRXky h?2 + MQMB+ H bT +2 Q7 HQ+ HHv }MBi2 TQBMi T`Q+2bb2b QM E rBi? p (E) Q7 HQ+ HHv K `Fb BM K Bb- #v /2}MBiBQM- i?2 + MQMB+ H bT +2 Mp (E × K) := M }MBi2 TQBMi K2 bm`2b QM EX TQBMi K2 bm`2 μ ∈ Mp (E × K) Bb `2T`2b2Mi2/ #v b2[m2M+2 { xn }n∈N = {(xn , zn )}n∈N Q7 TQBMib BM E × KX ++Q`/BM; iQ i?2 mbm H i2`KBMQHQ;v- μ := {xn }n∈N Bb i?2 # b2 TQBMi K2 bm`2 M/ i?2 b2[m2M+2 {zn }n∈N Bb i?2 b2[m2M+2 Q7 K `Fb- zn ∈ K #2BM; i?2 K `F Q7 i?2 # b2 TQBMi xn ∈ EX
RXk * KT#2HHǶb 6Q`KmH M/ JQK2Mi J2 bm`2b SQBMi T`Q+2bb UbiQ+? biB+V BMi2;` Hb `2 BMi2;` Hb rBi? `2bT2+i iQ ` M/QK TQBMi K2 bm`2X aBM+2 i?Bb ` M/QK K2 bm`2 Bb BM i?Bb T `iB+mH ` + b2 +QmMiBM; K2 bm`2i?2b2 BMi2;` Hb `2 BM 7 +i bmKbX G2i (E, E) #2 M `#Bi` `v K2 bm` #H2 bT +2 M/ N HQ+ HHv }MBi2 TQBMi T`Q+2bb Q` TQBMi T`Q+2bb BM i?2 b2Mb2 Q7 .2}MBiBQM RXRXkX G2i μ #2 K2 bm`2 QM i?2 K2 bm` #H2 bT +2 (E, E) M/ H2i ϕ : (E, E) → (R, B(R)) #2 K2 bm` #H2 7mM+iBQM 7Q` r?B+? i?2 BMi2;` H E ϕ dμ Bb r2HH /2}M2/X h?Bb BMi2;` H rBHH HbQ #2 /2MQi2/ #v μ(ϕ)X q?2M ϕ : E → R Bb K2 bm` #H2 7mM+iBQM M/ N Bb TQBMi T`Q+2bb- i?2 7QHHQrBM; MQi iBQMb `2T`2b2Mi i?2 b K2 K i?2K iB+ H Q#D2+i UB7 i?2 H ii2` Bb r2HH /2}M2/V, ϕ(Xn ) , ϕ(x)N (dx) , N (ϕ) . n∈N
E
h?2 H bi irQ MQi iBQMb TTHv iQ Mv ` M/QK K2 bm`2X AM i?2 }`bi MQi iBQM- r?B+? +QM+2`Mb QMHv TQBMi T`Q+2bb2b- r2 mb2 i?2 +QMp2MiBQM i? i i?2 bmK 2ti2M/b QMHv iQ i?Qb2 BM/B+2b n bm+? i? i Xn ∈ E- 2t+Hm/BM; i?2 TQBMib dz i BM}MBivǴ UBM 7 +iϕ(Δ) Bb MQi /2}M2/VX AM i?2 bBim iBQM Q7 .2}MBiBQM RXRXRe- Q#b2`p2 i? i ϕ(x, z)N (dx × dz) = ϕ(Xn , Zn ), E×K
n∈N
rBi? i?2 b K2 +QMp2MiBQM b i?2 QM2 Dmbi ;`22/ mTQM +QM+2`MBM; TQBMib i BM}MBivX h?2Q`2K RXkXR G2i N #2 ` M/QK K2 bm`2 QM i?2 K2 bm` #H2 bT +2 (E, E) rBi? BMi2MbBiv K2 bm`2 νX h?2M- 7Q` HH K2 bm` #H2 7mM+iBQMb ϕ : E → R r?B+? `2 2Bi?2` MQM@M2; iBp2 Q` BM L1R (ν)- i?2 BMi2;` H N (ϕ) Bb r2HH /2}M2/ UTQbbB#Hv BM}MBi2 r?2M ϕ Bb QMHv bbmK2/ iQ #2 MQM@M2; iBp2V M/ E [N (ϕ)] = ν(ϕ) . AM T `iB+mH `- N (ϕ) Bb XbX }MBi2 B7 ϕ ∈ L1R (ν)X
URXjV
RXkX *JS"1GGǶa 6P_JlG L. JPJ1Lh J1al_1a
N
h?2 7Q`KmH URXjV Bb FMQrM b i?2 U}`biV * KT#2HH 7Q`KmH X9 S`QQ7X 6B`bi- bmTTQb2 i? i ϕ Bb bBKTH2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM- i? i Bb- Q7 i?2 7Q`K L α h 1 Ch , h=1
r?2`2 L ∈ N- αh ∈ R+ M/ C1 , . . . , CL `2 /BbDQBMi K2 bm` #H2 bm#b2ib Q7 EX h?2M L L E[N (ϕ)] = E ah N (Ch ) = ah ν(Ch ) = ν(ϕ) . h=1
h=1
LQr H2i ϕ #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM M/ {ϕn }n∈N MQM@/2+`2 bBM; b2[m2M+2 Q7 bBKTH2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb rBi? HBKBi ϕX G2iiBM; n i2M/ iQ ∞ BM E[N (ϕn )] = ν(ϕn ) vB2H/b i?2 MMQmM+2/ `2bmHi- #v KQMQiQM2 +QMp2`;2M+2X AM i?2 + b2 r?2`2 ϕ ∈ L1R (ν)- bBM+2 E [N (ϕ± )] = ν(ϕ± ) < ∞- i?2 ` M/QK p `B #H2b N (ϕ± ) `2 P @ XbX }MBi2- M/ i?2`27Q`2 N (ϕ) = N (ϕ+ ) − N (ϕ− ) Bb r2HH /2}M2/ M/ }MBi2- M/ E[N (ϕ)] = E[N (ϕ+ )] − E[N (ϕ− )] = ν(ϕ+ ) − ν(ϕ− ) = ν(ϕ) . 1t KTH2 RXkXk, * KT#2HHǰb 6Q`KmH 7Q` J `F2/ SQBMi S`Q+2bb2b rBi? AM/2T2M/2Mi BB/ J `FbX *QMbB/2` i?2 K `F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd- i? i Bb- K `F2/ TQBMi T`Q+2bb N QM E rBi? BM/2T2M/2Mi BB/ K `Fb {Zn }n∈N i FBM; i?2B` p Hm2b BM bQK2 K2 bm` #H2 bT +2 (K, K) M/ Q7 +QKKQM /Bbi`B#miBQM QZ X Aib BMi2MbBiv K2 bm`2 Bb νZ (dx × dz) = ν(dx)QZ (dz) U1t KTH2 RXRXRNVX * KT@ #2HHǶb i?2Q`2K i?2M `2 /b b 7QHHQrbX A7 i?2 K2 bm` #H2 7mM+iBQM ϕ : Rm × K → R Bb 2Bi?2` MQM@M2; iBp2 Q` BM L1R (ν × QZ )- i?2M i?2 bmK ϕ(Xn , Zn ) n∈N
Bb P @ XbX r2HH /2}M2/ UTQbbB#Hv BM}MBi2 B7 ϕ Bb QMHv bbmK2/ MQM@M2; iBp2V M/ E ϕ(Xn , Zn ) = E [ϕ(x, Z)] ν(dx) , n∈N
E
r?2`2 Z Bb Mv K@p Hm2/ ` M/QK p `B #H2 rBi? /Bbi`B#miBQM QZ X h?2 7QHHQrBM; p i ` Q7 * KT#2HHǶb 7Q`KmH Bb bQK2iBK2b M22/2/ Ub22 i?2 M2ti i?2Q`2K 7Q` BMbi M+2V, 9
(* KT#2HH- RNRy)X
Ry
*>Sh1_ RX :1L1_GAhA1a
*Q`QHH `v RXkXj G2i N #2 TQBMi T`Q+2bb QM i?2 K2 bm` #H2 bT +2 (E, E) rBi? BMi2MbBiv K2 bm`2 ν- /2}M2/ QM i?2 T`Q# #BHBiv bT +2 (Ω, F, P )X G2i G #2 bm#@σ@ }2H/ Q7 F- BM/2T2M/2Mi Q7 F N X h?2M- 7Q` HH K2 bm` #H2 7mM+iBQMb ϕ : (E × Ω, E ⊗ G) → (R, B(R)) r?B+? `2 2Bi?2` MQM@M2; iBp2 Q` BM L1R (ν × P )- i?2 BMi2;` H ϕ(x, ω) N (ω, dx) N (ϕ)(ω) := E
Bb XbX r2HH /2}M2/ UTQbbB#Hv BM}MBi2 r?2M ϕ Bb QMHv bbmK2/ MQM@M2; iBp2V M/ E [N (ϕ)] = ν(E [ϕ]). AM T `iB+mH `- B7 ϕ ∈
L1R (ν
× P )- N (ϕ) Bb XbX }MBi2X
S`QQ7X h?2 bb2`iBQM Bb i`m2 7Q` ϕ(x, ω) = u(x)v(ω) , r?2`2 u : E → R+ Bb E@K2 bm` #H2 M/ v : Ω → R+ Bb G@K2 bm` #H2- bBM+2 BM i?Bb + b2E [N (ϕ)] = E[v]E [N (u)] = E[v]ν(u) = ν(E[v]u) = ν(E [ϕ]) . h?2 `2bi Bb bi` B;?i7Q`r `/ TTHB+ iBQM Q7 .vMFBMǶb 7mM+iBQM H i?2Q`2K Uh?2Q`2K XRXeVX _2+ HH i? i K2 bm`2 μ QM i?2 K2 bm` #H2 bT +2 (E, E) Bb + HH2/ MQM@ iQKB+Q` /Bzmb2- B7 μ({a}) = 0 7Q` Mv bBM;H2iQM {a}X h?2Q`2K RXkX9 hrQ BM/2T2M/2Mi TQBMi T`Q+2bb2b N1 M/ N2 QM i?2 K2 bm` #H2 bT +2 (E, E) rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b ν1 M/ ν2 ? p2 XbX MQ TQBMib BM +QKKQM B7 M/ QMHv B7 ν1 ({x}) ν2 (dx) = 0 , M/ i?2M Q7 +Qm`b2
URX9V
E
E
ν2 ({x}) ν1 (dx) = 0X
h?Bb Bb i?2 + b2 7Q` BMbi M+2 B7 QM2 Q7 i?2K ? b MQM@ iQKB+ BMi2MbBiv K2 bm`2X S`QQ7X "v *Q`QHH `v RXkXj N1 ({x}) N2 (dx) = ν1 ({x}) ν2 (dx) . E E
E
()
h?2`27Q`2- B7 E ν1 ({x}) ν2 (dx) = 0- i?2M E E N1 ({x}) N2 (dx) = 0 M/ BM T `iB+mH ` E N1 ({x}) N2 (dx) = 0- P @ XbX- r?B+? BKTHB2b i? i N1 M/ N2 ? p2 P @ XbX MQ +QKKQM TQBMibX HbQ- B7 N1 M/ N2 ? p2 XbX MQ +QKKQM TQBMibP ( E N1 ({x}) N2 (dx) = 0) = 1 M/ BM T `iB+mH ` E E N1 ({x}) N2 (dx) = 0i? i Bb- #v *Q`QHH `v RXkXj- E ν1 ({x}) ν2 (dx) = 0X
RXkX *JS"1GGǶa 6P_JlG L. JPJ1Lh J1al_1a
RR
JQK2Mi J2 bm`2b _2+ HH i? i TQBMi T`Q+2bb N QM i?2 HX+X/X#X bT +2 E bm+? i? i E [N (C)] < ∞ 7Q` HH `2H iBp2Hv +QKT +i b2ib C ⊂ E Bb + HH2/ }`bi@Q`/2` TQBMi T`Q+2bb M/ i? i i?2 K2 bm`2 ν QM (E, B(E)) /2}M2/ #v ν(C) = E [N (C)] Bb i?2M HQ+ HHv }MBi2 K2 bm`2- i?2 BMi2MbBiv K2 bm`2- HbQ + HH2/ i?2 }`bi KQK2Mi K2 bm`2 Q7 N X h?2 TQBMi T`Q+2bb N Bb + HH2/ b2+QM/@Q`/2` TQBMi T`Q+2bb B7 7Q` HH `2H iBp2Hv +QKT +i b2ib C ⊂ E E N (C)2 < ∞ . URX8V AM T `iB+mH `- b2+QM/@Q`/2` TQBMi T`Q+2bb Bb HbQ }`bi@Q`/2` TQBMi T`Q+2bbX 6Q` b2+QM/@Q`/2` TQBMi T`Q+2bb N - i?2 b2+QM/ KQK2Mi K2 bm`2 M2 QM E × E Bb r2HH M/ mMB[m2Hv /2}M2/ #v i?2 7Q`KmH M2 (A × B) = E [N (A) N (B)]
(A, B ∈ B(E))
M/ i?Bb K2 bm`2 Bb HQ+ HHv }MBi2 U1t2`+Bb2 RXdXRRVX .2MQi2 #v N 2 = N × N i?2 TQBMi T`Q+2bb QM E × E r?Qb2 TQBMib `2 i?2 Q`/2`2/ T B`b (Xn , Xk ) Un, k ∈ NVX P#b2`pBM; i? i (N 2 )(A × B) = N (A)N (B)r2 b22 i? i M2 Bb i?2 BMi2MbBiv K2 bm`2 Q7 N × N M/ i?2`27Q`2- #v * KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : E × E → R g(Xn , Xk ) = g(t, s) M2 (dt × ds) , E E
n∈N k∈N
E
M/ i?Bb 7Q`KmH Bb HbQ i`m2 7Q` Mv K2 bm` #H2 7mM+iBQM g : E × E → C bm+? i? i |g(t, s)| M2 (dt × ds) < ∞ . E
E
h?2 #Qp2 /2}MBiBQMb M/ `2bmHib 2ti2M/ M im` HHv iQ ?B;?2` KQK2Mi K2 @ bm`2bX G2i n ≥ 1 #2 TQbBiBp2 BMi2;2`X bBKTH2 HQ+ HHv }MBi2 N QM E bm+? i? i E [N (C)n ] < ∞ 7Q` HH `2H iBp2Hv +QKT +i b2ib C ∈ B(E) Bb + HH2/ M n@i? Q`/2` TQBMi T`Q+2bbX h?2 K2 bm`2 Mn QM E n /2}M2/ #v Mn (A1 × · · · × An ) := E [N (A1 ) · · · N (An )] Bb i?2 n@i? Q`/2` KQK2Mi K2 bm`2X h?Bb HQ+ HHv }MBi2 K2 bm`2 Bb i?2 BMi2MbBiv K2 @ bm`2 Q7 i?2 TQBMi T`Q+2bb N (n) r?Qb2 TQBMib `2 i?2 Q`/2`2/ n@imTH2b (Xi1 , . . . , Xin ) Ui1 , . . . , in ∈ NVX h?2`27Q`2- #v * KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2; iBp2 K2 bm`@ #H2 7mM+iBQMb g : E n → R E ··· g(Xi1 , . . . , Xin ) = · · · g(x1 , . . . , xn ) Mn (dx1 × · · · × dxn ) . i1 ∈N
in ∈N
E
E
P7 +Qm`b2 M1 = ν- i?2 BMi2MbBiv K2 bm`2X G2i N #2 M n@i? Q`/2` TQBMi T`Q+2bb QM EX *QMbB/2` MQr i?2 TQBMi T`Q+2bb N !(n) r?Qb2 TQBMib `2 i?2 Q`/2`2/ n@imTH2b (Xi1 , . . . , Xin ) Q7 /BbiBM+i TQBMibX h?2 HQ+ HHv }MBi2 K2 bm`2 Mn! QM E n /2}M2/ #v
Rk
*>Sh1_ RX :1L1_GAhA1a Mn! (A1 × · · · × An ) = E N !(n) (A1 × · · · × An ) ,
+ HH2/ i?2 n@i? 7 +iQ`B H KQK2Mi K2 bm`2- Bb i?2 BMi2MbBiv K2 bm`2 Q7 i?2 TQBMi T`Q+2bb N !(n) X h?2`27Q`2- #v * KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : E n → R = E g(Xi1 , . . . , Xin ) = · · · g(x1 , . . . , xn ) Mn! (dx1 × · · · × dxn ) , URXeV E
i1 ,...,in ∈N
E
r?2`2 i?2 bvK#QH = bB;MB}2b i? i i?2 bmK Bb `2bi`B+i2/ iQ i?2 Q`/2`2/ n@imTH2b i1 , . . . , in ∈ N Q7 T B`rBb2 /BbiBM+i BMi2;2`bX P7 +Qm`b2 M1! = M1 = ν Bb i?2 BMi2MbBiv K2 bm`2 Q7 N X HbQ MQi2 i? i Mn! (An ) = E [N (A)(N (A) − 1) · · · (N (A) − n + 1)] , r?B+? 2tTH BMb i?2 TT2HH iBQM dz7 +iQ`B HǴ K2 bm`2X h?2Q`2K RXkX8 6Q` b2+QM/ Q`/2` TQBMi T`Q+2bb N iQ #2 bBKTH2 Bi Bb M2+2bb `v M/ bm{+B2Mi i? i i?2 M2! @K2 bm`2 Q7 i?2 /B ;QM H Q7 E × E #2 MmHHX S`QQ7X h?Bb 7QHHQrb 7`QK i?2 2[m HBiv 1{x1 =x2 } M2! (dx1 × dx2 ) = E E
E
= i1 ,i2 ∈N
1{Xi1 =Xi2 } .
1t KTH2 RXkXe, R@+HQb2 S B`b- i F2 RX T B` (x, y) Q7 TQBMib Q7 Rd Bb + HH2/ R@+HQb2 B7 i?2 1m+HB/2 M /Bbi M+2 ||x − y|| Bb H2bb i? M Q` 2[m H iQ RX h?2 MmK#2` Q7 R@+HQb2 T B`b BM i?2 b[m `2 [0, 1] × [0, 1] Q7 bBKTH2 b2+QM/@Q`/2` TQBMi T`Q+2bb N QM R2 2[m Hb 1 1[0,1]2 (Xn )1[0,1]2 (Xk )1{||Xn −Xk ||≤R} , 2 n k;k=n M/ i?2`27Q`2 Bib K2 M 2[m Hb- #v URXeV 1 1{||x−y||≤R} M2! (dx × dy) . 2 [0,1]2 [0,1]2
q2 MQr HBbi 72r `2H iBQMb i? i `2 2Bi?2` Q#pBQmb Q` 2 bBHv /2`Bp2/X A7 A1 , . . . , An `2 Kmim HHv /BbDQBMi Mn! (A1 × · · · × An ) = Mn (A1 × · · · × An ) . 6Q` n = 2-
M2! (A1 × A2 ) = M2 (A1 × A2 ) − ν(A1 ∩ A2 ) .
RXkX *JS"1GGǶa 6P_JlG L. JPJ1Lh J1al_1a
Rj
_2K `F RXkXd JQ`2 rBHH #2 b B/ QM i?2 b2+QM/ KQK2Mi K2 bm`2b BM *? Ti2` N /2pQi2/ iQ i?2 +QKTmi iBQM Q7 TQr2` bT2+i` H K2 bm`2b Q7 rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb2bX 1t KTH2 RXkX3, JQK2Mi K2 bm`2b Q7 i?2 SQBbbQM T`Q+2bb "v i?2 BM/2@ T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b- r2 ? p2 i?2 7QHHQrBM; `2bmHib +QM+2`MBM; Bib KQK2Mi K2 bm`2b U1t2`+Bb2 RXdXRRVX M2 (A1 × A2 ) = ν(A1 )ν(A2 ) + ν(A1 ∩ A2 ) M/ M2! (A1 × A2 ) = ν(A1 )ν(A2 ) . A7 N Bb M ?TT QM Rm Q7 BMi2MbBiv λ- i?2 K2 bm`2 Mn! /KBib /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 n ;Bp2M #v ρ!n (x1 , . . . , xn ) = λn .
1t KTH2 RXkXN, R@+HQb2 S B`b- h F2 kX "v bT2+B HBxBM; i?2 `2bmHi Q7 1t KTH2 RXkXe- r2 Q#i BM i? i 7Q` ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 Q7 BMi2MbBiv λi?2 p2` ;2 MmK#2` Q7 R@+HQb2 T B`b BM i?2 b[m `2 [0, 1] × [0, 1] Bb 1 2 λ 1{||x−y||≤R} dx dy . 2 [0,1]2 [0,1]2
h?2 *QBM+B/2M+2 6mM+iBQM A7 i?2 n@i? 7 +iQ`B H K2 bm`2 /KBib /2MbBiv ρ!n (x1 , . . . , xn ) rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 n - i?Bb /2MbBiv Bb bQK2iBK2b + HH2/ i?2 Un@i?V +QBM+B/2M+2 7mM+@ iBQM BM pB2r Q7 i?2 7QHHQrBM; BM7Q`K H BMi2`T`2i iBQM, dzi?2 T`Q# #BHBiv i? i i?2`2 Bb QM2 TQBMi BM 2 +? QM2 Q7 i?2 Kmim HHv /BbDQBMi BM}MBi2bBK H #Qt2b dx1 - Ę- dxn Bb T`QTQ`iBQM H iQ ρ!n (x1 , . . . , xn )ǴX h?2 BMi2`2bi Q7 /2}MBM; TQBMi T`Q+2bb #v K2 Mb Q7 Bib +QBM+B/2M+2 7mM+iBQM `2bB/2b BM Bib `2 / #BHBiv BM i2`Kb Q7 ii` +iBQM Q` `2TmHbBQMX h?2`2 2tBbi H `;2 +H bb2b Q7 TQBMi T`Q+2bb2b r?B+? `2 /2}M2/ pB i?2B` +Q@ BM+B/2M+2 7mM+iBQMbX h?2 K BM i?2Q`2iB+ H T`Q#H2K bbQ+B i2/ rBi? i?Bb ivT2 Q7 /2b+`BTiBQM Bb iQ T`Qp2 i?2 2tBbi2M+2 M/ mMB+Biv Q7 TQBMi T`Q+2bb /Bbi`B#miBQM rBi? i?2 ;Bp2M + M/B/ i2 +QBM+B/2M+2 7mM+iBQMX h?Bb T`Q#H2K ? b #22M iQ H `;2 2ti2Mi `2bQHp2/ 7Q` i?2 /2i2`KBM Mi H M/ T2`K M2Mi H TQBMi T`Q+2bb2bX 1t KTH2 RXkXRy, .2i2`KBM Mi H M/ T2`K M2Mi H TQBMi T`Q+2bb2bX G2i K : E × E → C #2 K2 bm` #H2 7mM+iBQM M/ m #2 HQ+ HHv }MBi2 K2 bm`2 QM EX h?2 TQBMi T`Q+2bb N QM E Bb + HH2/ /2i2`KBM Mi H TQBMi T`Q+2bb rBi? # +F;`QmM/ K2 bm`2 m M/ F2`M2H K B7 7Q` HH k ≥ 1
R9
*>Sh1_ RX :1L1_GAhA1a M!k (dx1 · · · dxn ) = det{K(xi , xj )}1≤i,j≤k m(dx1 ) · · · m(dxk ) .
h?2 7mM+iBQM ρ!n (x1 , . . . , xk ) := det{K(xi , xj )}1≤i,j≤k Bb + HH2/ i?2 k@i? 7 +iQ`B H KQK2Mi /2MbBiv rBi? `2bT2+i iQ m⊗k X AM T `iB+mH `- i?2 BMi2MbBiv K2 bm`2 Q7 N Bb ν(dx) = K(x, x) m(dx). E
h?2 /2i2`KBM Mi H TQBMi T`Q+2bb2b `2 bBKTH2 TQBMi T`Q+2bb2b U1t2`+Bb2 RXdXNVX M BMi2`2biBM; bi #BHBiv T`QT2`iv Q7 i?2B` /Bbi`B#miBQM Bb i? i i?BMMBM; /2i2`KBM Mi H TQBMi T`Q+2bb `2bmHib BM /2i2`KBM Mi H TQBMi T`Q+2bb U1t2`+Bb2 RXdX3VX .2i2`KB@ M Mi H TQBMi T`Q+2bb2b `2 `2TmHbBp2 #2+ mb2 i?2 /2i2`KBM Mi TT`Q +?2b x2`Q r?2M irQ `Qrb Q` +QHmKMb `2 +HQb2 iQ #2BM; HBM2 `Hv BM/2T2M/2MiX AM i?2 2H2K2Mi `v T `iB+H2 +QMi2ti- i?Bb MiB@#mM+?BM; 2z2+i Bb +? ` +i2`BbiB+ Q7 i?2 72`KBQMb U2H2+@ i`QMbVX h?2 /2}MBiBQM Q7 T2`K M2Mi H TQBMi T`Q+2bb2b Bb 7Q`K HHv i?2 b K2 b i? i Q7 /2i2`KBM Mi H TQBMi T`Q+2bb2b- 2t+2Ti i? i BMbi2 / Q7 i?2 /2i2`KBM Mi QM2 mb2b i?2 T2`K M2Mi Q7 i?2 K i`B+2b {K(xi , xj )}1≤i,j≤k X8 AM +QMi` bi rBi? i?2 /2i2`KBM Mi H TQBMi T`Q+2bb2b- i?2 T2`K M2Mi H TQBMi T`Q+2bb2b `2 ii` +iBp2- M/ i?Bb T`QT2`iv Bb +? ` +i2`BbiB+ Q7 i?2 #QbQMb UT?QiQMbVX Hi?Qm;? i?2b2 T`Q+2bb2b `2 Q7 ;`2 i BMi2`2bi BM bT2+B}+ `2 b- BM+Hm/BM; S?vbB+b-e i?2B` M HvbBb Bb mbm HHv BM M HviB+ H i2`Kbd M/ i?2`27Q`2 /Q2b MQi [mBi2 }i rBi? i?2 KQ`2 T`Q# #BHBbiB+ TT`Q +? Q7 i?Bb #QQF M/ Bib HBKBi2/ ;Q H +2Mi2`2/ `QmM/ biQ+? biB+ + H+mHmbX
RXj h?2 .Bbi`B#miBQM Q7 SQBMi S`Q+2bb .2}MBiBQM RXjXR G2i E #2 M HX+X/X#X bT +2 rBi? "Q`2H σ@}2H/ B(E) M/ H2i N #2 ` M/QK K2 bm`2 QM (E, B(E)) /2}M2/ QM i?2 T`Q# #BHBiv bT +2 (Ω, F, P )X h?2 T`Q# #BHBiv PN := P ◦ N −1 QM (M (E), M(E)) Bb + HH2/ i?2 /Bbi`B#miBQM Q7 N X 1tT2+i iBQM rBi? `2bT2+i iQ PN rBHH #2 /2MQi2/ #v EPN X 72r 2t KTH2b rBHH 7 KBHB `Bx2 i?2 `2 /2` rBi? i?2 MQi iBQMX 6Q` C ∈ B(E) M/ a ∈ RPN ({μ ; μ(C) ≥ a}) = P (N (C) ≥ a) . 6Q` MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM f : E → R μ; f (x) μ(dx) ≤ b =P f (x) N (dx) ≤ b . PN E 8
E
_2+ HH i? i i?2 T2`K M2Mi Q7 K i`Bt Bb Q#i BM2/ 7`QK i?2 7Q`KmH ;BpBM; i?2 /2i2`KBM Mi #v +? M;BM; i?2 − bB;Mb iQ + bB;MbX e (J ++?B- RNdR - RNdR# M/ RNd8)- ("2M `/ M/ J ++?B- RNdj)X d (a?B` B M/ h F ? b?B- kyyj)X
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa
R8
6Q` MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM g : M (E) → R+ EPN [g] = E [g(N )] Q`- 2[mBp H2MiHv
g(μ) PN (dμ) =
M (E)
g(N (ω)) P (dω) . Ω
h?2 M2ti /2}MBiBQM ;Bp2b i?2 T`2+Bb2 K2 MBM; Q7 i?2 2tT`2bbBQM dzb KTHBM; Ui?2 /Bbi`B#miBQM Q7V TQBMi T`Q+2bbǴX .2}MBiBQM RXjXk a KTHBM; /Bbi`B#miBQM P QM (Mp (E), Mp (E)) Bb iQ +QMbi`m+i dzb KTH2Ǵ N (ω) Q7 bQK2 TQBMi T`Q+2bb rBi? i?2 /Bbi`B#miBQM PN = PX JQ`2 ;2M2` HHv- b KTHBM; T`Q# #BHBiv /Bbi`B#miBQM Q QM K2 bm` #H2 bT +2 (U, U) +QMbBbib- #v /2}MBiBQM- BM ;2M2` iBM; ` M/QK 2H2K2Mi X(ω) rBi? p Hm2b BM (U, U ) r?Qb2 /Bbi`B#miBQM Bb QX AM Q`/2` iQ ;2M2` i2 X Bi Bb bmTTQb2/ i? i QM2 ? b i /BbTQbBiBQM Mv MmK#2` Q7 +QTB2b Q7 dz2 bBHv ;2M2` i2/Ǵ ` M/QK 2H2K2Mib- bm+? b 7Q` BMbi M+2- BB/ ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM mMBi BMi2`p H Q` SQBbbQM p `B #H2bX AKTHB+Bi BM i?2 /2}MBiBQM Q7 b KTHBM; Bb i?2 M2+2bbBiv i? i i?2 ;2M2` iBQM Q7 X b?QmH/ `2[mB`2 QMHv }MBi2 Umbm HHv ` M/QKV MmK#2` Q7 QT2` iBQMbX a KTHBM; Bb Q7i2M + HH2/ 2t +i b KTHBM; BM Q`/2` iQ BMbBbi QM i?2 /Bz2`2M+2 rBi? TT`QtBK i2 b KTHBM;- r?2`2 i?2 b KTH2 Q#i BM2/ ? b /Bbi`B#miBQM dz+HQb2Ǵ iQ i?2 i `;2i /Bbi`B#miBQM- r?2`2 i?2 +HQb2M2bb Bb mbm HHv K2 bm`2/ BM i2`Kb Q7 i?2 p `B iBQM /Bbi M+2X h?2 b KTHBM; Bbbm2 rBHH #2 2t KBM2/ i p `BQmb TH +2b BM i?Bb #QQFX h?2 /Bbi`B#miBQM Q7 TQBMi T`Q+2bb K v #2 +? ` +i2`Bx2/ BM i H2 bi i?`22 r vb, #v Bib }MBi2@/BK2MbBQM H /Bbi`B#miBQMb- #v Bib G TH +2 7mM+iBQM H M/ #v Bib pQB/@ M+2 T`Q# #BHBiv 7mM+iBQMX "27Q`2 T`Q+22/BM; iQ i?2 +Q``2bTQM/BM; +? ` +i2`Bx iBQMb 72r T`2HBKBM `v `2bmHib `2 M22/2/X 6B`bi- r2 b?Qr i? i i?2 +QHH2+iBQM Q7 b2ib C ∈ B(E) 7Q` r?B+? r2 ? p2 iQ +?2+F i? i N (C) Bb ` M/QK p `B #H2 BM Q`/2` 7Q` i?2 K TTBM; N : Ω → M (E) iQ #2 K2 bm` #H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(E) M/ M(E) + M #2 +QMbB/2` #Hv bK HH2` i? M B(E)X am+? +QHH2+iBQM Bb /2b+`B#2/ BM i?2 7QHHQrBM; /2}MBiBQM, .2}MBiBQM RXjXj .2}M2 E0 iQ #2 Mv U}t2/V +QHH2+iBQM Q7 `2H iBp2Hv +QKT +i bm#@ b2ib Q7 E bm+? i? i UBV E0 Bb π@bvbi2K Ui? i Bb, Bb bi #H2 #v }MBi2 BMi2`b2+iBQMbVUBBV σ(E0 ) = B(E)- M/ UBBBV i?2`2 2tBbib b2[m2M+2 {En }n≥1 BM E0 bm+? i? i 2Bi?2` U V, 7Q`Kb T `iBiBQM Q7 E- Q` U#V, BM+`2 b2b iQ EX
Re
*>Sh1_ RX :1L1_GAhA1a
1t KTH2 RXjX9, h?2 `2+i M;H2bX G2i E := Rd X h?2 +H bb E0 Q7 b2ib Q7 i?2 7Q`K d I= (aj , bj ] ([aj , bj ] ⊆ R) j=1
b iBb}2b i?2 `2[mB`2K2Mib Q7 .2}MBiBQM RXjXjX
h?2Q`2K RXjX8 X 6Q` i?2 K TTBM; N : Ω → M (E) U`2bTX Mp (E)V iQ #2 K2 bm` #H2- Bi bm{+2b i? i N (C) : Ω → R+ #2 ` M/QK p `B #H2 7Q` HH C ∈ E0 X "X
M(E) = σ {μ ; μ(C) ∈ A} ; C ∈ E0 , A ∈ B(R+ ) .
()
AM T `iB+mH `- H2iiBM; I := ∩kj=1 {μ ; μ(Cj ) ∈ Aj } ; Cj ∈ E0 , Aj ∈ B(R+ ) (1 ≤ j ≤ k) , k ∈ N+ , U π@bvbi2KV- r2 ? p2 i? i σ(I) = M(E)X S`QQ7X X h?2 T`QQ7 Bb ;Bp2M 7Q` i?2 + b2 BBBU#V Q7 .2}MBiBQM RXjXj- i?2 Qi?2` + b2 #2BM; bBKBH `X 6Bt nX h?2 +QHH2+iBQM G := {C ∈ B(E) ; ω → N (ω, C ∩ Kn ) Bb K2 bm` #H2 7`QK (Ω, F) iQ (R+ , B(R+ ))} Bb d@bvbi2KX Uh?2 7 +i i? i K2 bm`2 μ ∈ M (E) Tmib }MBi2 K bb QM `2H iBp2Hv +QKT +i b2i- ?2`2 Kn - TH vb `QH2 BM i?2 T`QQ7 Q7 bi #BHBiv Q7 G mM/2` T`QT2` /Bz2`2M+2XV aBM+2 G ⊃ F0 - r2 ? p2 i? i G ⊃ σ(E0 ) = B(E)X h?2`27Q`2 N (C ∩ Kn ) Bb ` M/QK p `B #H2 7Q` HH C ∈ B(E) M/ HH n ≥ 1X 6BM HHv- 7Q` HH C ∈ B(E)N (C) Bb ` M/QK p `B #H2- #2BM; i?2 HBKBi b n ↑ ∞ Q7 i?2 ` M/QK p `B #H2b N (C ∩ Kn )X "X AM - i F2 Ω = M (E) M/ H2i N #2 i?2 B/2MiBiv K TTBM; QM M (E)X "v /2}MBiBQM- M(E) = σ{{μ ; μ(C) ∈ A}, C ∈ B(E), A ∈ B(R+ )}X h?2 +QM+HmbBQM Q7 T `i BKTHB2b i? i r2 + M `2TH +2 BM i?2 H bi B/2MiBiv C ∈ B(E) #v C ∈ E0 X P#pBQmbHv- i?2 `B;?i@? M/ bB/2 Q7 UV Bb 2[m H iQ σ(I)X
_2K `F RXjXe h?2 b K2 `2bmHi TTHB2b rBi? i?2 b K2 T`QQ7 r?2M `2TH +BM; M (E) #v Mp (E)X AM i?Bb + b2- bBM+2 TQBMi K2 bm`2 i F2b Bib p Hm2b BM N ∪ {∞}- H2iiBM; i?Bb iBK2 I := ∩kj=1 {μ ; μ(Cj ) = nj } ; Cj ∈ E0 , nj ∈ N+ , (1 ≤ j ≤ k) , k ∈ N+ U π@bvbi2KV- r2 ? p2 i? i σ(I) = Mp (E)X
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa
Rd
6BMBi2@/BK2MbBQM H .Bbi`B#miBQMb h?2Q`2K RXjXd G2i E0 #2 b BM .2}MBiBQM RXjXjX h?2 /Bbi`B#miBQM PN Q7 ` M/QK K2 bm`2 N QM (E, B(E)) Bb +QKTH2i2Hv +? ` +i2`Bx2/ #v i?2 /Bbi`B#miBQMb Q7 i?2 p2+iQ`b (N (C1 ), . . . , N (Cm )) 7Q` HH BMi2;2`b m ≥ 1 M/ HH C1 , . . . , Cm ∈ E0 X S`QQ7X q2 ? p2 b22M BM h?2Q`2K RXjX8 i? i i?2 +H bb I Q7 bm#b2ib Q7 M (E) Q7 i?2 7Q`K {μ; μ(C1 ) ∈ A1 , . . . , μ(Ck ) ∈ Ak } r?2`2 Cj ∈ E0 M/ Aj ∈ B(E) U1 ≤ j ≤ kV Bb bi #H2 #v }MBi2 BMi2`b2+iBQM M/ ;2M2` i2b M(E)X h?2`27Q`2- irQ T`Q# #BHBiv K2 bm`2b ;`22BM; QM I- ;`22 QM σ(I) Uh?2Q`2K XRXdVX M/- b r2 Dmbi `2+ HH2/- σ(I) = M(E)X .2}MBiBQM RXjX3 h?2 }/B U}MBi2@/BK2MbBQM HV /Bbi`B#miBQM Q7 TQBMi T`Q+2bb N Bb- #v /2}MBiBQM- i?2 +QHH2+iBQM Q7 i?2 T`Q# #BHBiv /Bbi`B#miBQMb Q7 i?2 p2+iQ`b (N (C1 ), . . . , N (Cm )) (m ∈ N+ - C1 , . . . , Cm ∈ B(E))X h?2Q`2K RXjXd i2HHb mb BM T `iB+mH ` i? i i?2 }/B /Bbi`B#miBQM `2bi`B+i2/ iQ i?2 +QHH2+iBQM Q7 b2ib I /2}M2/ BM h?2Q`2K RXjXd +? ` +i2`Bx2b i?2 /Bbi`B#miBQM Q7 HQ+ HHv }MBi2 TQBMi T`Q+2bbX h?2 G TH +2 6mM+iBQM H .2}MBiBQM RXjXN G2i N #2 ` M/QK K2 bm`2 QM i?2 HX+X#X/X bT +2 EX h?2 G TH +2 7mM+iBQM H Q7 N Bb i?2 K TTBM; LN bbQ+B iBM; rBi? MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM ϕ : E → R+ i?2 MQM@M2; iBp2 `2 H MmK#2` LN (ϕ) := E e−N (ϕ) . AM i2`Kb Q7 i?2 /Bbi`B#miBQM PN Q7 N LN (ϕ) = e− E ϕ(x) μ(dx) PN (dμ) . M (E)
1t KTH2 RXjXRy, G TH +2 6mM+iBQM H Q7 i?2 "BMQKB H SQBMi S`Q+2bbX h?Bb TQBMi T`Q+2bb QM Rm ? b U}MBi2V MmK#2` T Q7 TQBMib- r?2`2 T Bb #BMQKB H ` M/QK p `B #H2 Q7 bBx2 n M/ T ` K2i2` p ∈ (0, 1), P (T = k) =
n k p (1 − p)n−k k
(0 ≤ k ≤ n) .
:Bp2M T = k- i?2 k TQBMib `2 HQ+ i2/ BM/2T2M/2MiHv Q7 QM2 MQi?2` QM Rm +@ +Q`/BM; iQ i?2 b K2 T`Q# #BHBiv /Bbi`B#miBQM QX Ai Bb bBKTH2 B7 M/ QMHv B7 Q Bb MQM@ iQKB+X
R3
*>Sh1_ RX :1L1_GAhA1a T LN (ϕ) = E e− j=1 ϕ(Xj ) n k − j=1 ϕ(Xj ) =E 1{T =k} e k=1
=
n
k P (T = k)E e− j=1 ϕ(Xj )
k=1
=
n
k P (T = k)E e−ϕ(X1 )
k=1
=
n n k=1
k pk (1 − p)n−k E e−ϕ(X1 )
k
k pE e−ϕ(X1 ) = (1 − p)n k 1 − p k=1 n −ϕ(X1 ) n −ϕ(x) = 1+p e = 1−p e Q(dx) . n n
E
1t KTH2 RXjXRR, G TH +2 6mM+iBQM H Q7 SQBbbQM S`Q+2bbX MiB+BT iBM; `2bmHi Q7 *? Ti2` j Uh?2Q`2K jXkXk i?2`2BMV- i?2 G TH +2 7mM+iBQM H Q7 SQBbbQM T`Q+2bb QM E rBi? BMi2MbBiv K2 bm`2 ν Bb
LN (ϕ) = exp
e
−ϕ(x)
− 1 ν(dx)
.
E
1t KTH2 RXjXRk, G TH +2 6mM+iBQM H Q7 *Qt S`Q+2bbX G2i N #2 *Qt T`Q+2bb QM E rBi? /B`2+iBM; K2 bm`2 M Q7 G TH +2 i` Mb7Q`K LM X q2 +QKTmi2 Bib G TH +2 i` Mb7Q`K, LN (ϕ) = E e−N (ϕ) = E e− E ϕ(x) N (dx) = E E e− E ϕ(x) N (dx) | F M
= E exp (e−ϕ(x) − 1) M (dx) . E
h?2`27Q`2 LN (ϕ) = LM (1 − e−ϕ ) .
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa
RN
1t KTH2 RXjXRj, G TH +2 6mM+iBQM H Q7 *QMi` +i2/ SQBMi S`Q+2bbX G2i N #2 bBKTH2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ H2i α > 0X .2}M2 i?2 dz+QMi` +i2/Ǵ3 TQBMi T`Q+2bb Nc,α /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib {αXn }n∈N X Aib G TH +2 7mM+iBQM H Bb LNc,α (ϕ) = E exp − ϕ(αXn ) = LN (ϕ(α ·)) , n∈N
r?2`2 α · /2MQi2b i?2 7mM+iBQM x → αxX h?2 G TH +2 7mM+iBQM H TH vb 7Q` TQBMi T`Q+2bb2b `QH2 M HQ;Qmb iQ i? i Q7 i?2 mbm H G TH +2 i` Mb7Q`K 7Q` ` M/QK p2+iQ`bX 6Q` BMbi M+2, h?2Q`2K RXjXR9 h?2 G TH +2 7mM+iBQM H Q7 HQ+ HHv }MBi2 ` M/QK K2 bm`2 N QM i?2 HX+X#X/X bT +2 E +? ` +i2`Bx2b Bib /Bbi`B#miBQMX S`QQ7X Ai bm{+2b iQ b?Qr i? i i?2 G TH +2 7mM+iBQM H Q7 TQBMi T`Q+2bb N +? ` +@ i2`Bx2b Bib }MBi2@/BK2MbBQM H /Bbi`B#miBQMX 6Q` i?Bb- Dmbi Q#b2`p2 i? i 7Q` HH K ≥ 1 M/ HH /BbDQBMi K2 bm` #H2 b2ib C1 - Ę- CK BM B(E)- i?2 G TH +2 i` Mb7Q`K Q7 i?2 p2+iQ` (N (C1 ), . . . , N (CK ))- i? i Bb- i?2 7mM+iBQM −t1 N (C1 )−···−t N (C ) K K (t1 , . . . , tK ) ∈ RK , + → E e −N (ϕ) Bb Q7 i?2 7Q`K E e r?2`2 ϕ = t1 1C1 + · · · + tK 1CK X h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb H27i 7Q` i?2 `2 /2`X *Q`QHH `v RXjXR8 TQBMi T`Q+2bb N QM Rm Bb bi iBQM `v B7 M/ QMHv B7 Bib G TH +2 i` Mb7Q`K LN Bb bm+? i? i LN (ϕ) = LN (Sa ϕ) 7Q` HH MQM@M2; iBp2 7mM+iBQMb ϕ 7`QK Rm iQ R M/ HH a ∈ Rm - r?2`2 Sa ϕ Bb i?2 7mM+iBQM /2}M2/ #v Sa ϕ : t → ϕ(t − a)X 1t KTH2 RXjXRe, ai iBQM `v :`B/- h F2 kX AM Q`/2` iQ T`Qp2 i?2 bi @ iBQM `Biv Q7 i?2 b?B7i2/ ;`B/ Q7 1t KTH2 RXRXRj- Bi bm{+2b iQ b?Qr i? i 7Q` Mv MQM@M2; iBp2 7mM+iBQM ϕ 7`QK R2 iQ R- i?2 [m MiBiv E e n,m∈Z ϕ(nT1 +V1 +α,nT2 +V2 +β) Bb BM/2T2M/2Mi Q7 α- β ∈ RX h?Bb [m MiBiv 2[m Hb T1 T2 e n,m∈Z ϕ(nT1 +v1 +α,nT2 +v2 +β) dv2 dv1 . 0
0
h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK i?2 b?B7i@BMp `B M+2 Q7 i?2 G2#2b;m2 K2 bm`2X h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb H27i b M 2t2`+Bb2X 3
P7 +Qm`b2- B7 α > 1- Bi Bb BM 7 +i /BH i2/XXX
ky
*>Sh1_ RX :1L1_GAhA1a
h?2Q`2K RXjXRd G2i Ni (i ∈ I) #2 +QHH2+iBQM Q7 TQBMi T`Q+2bb2b QM i?2 HX+X/X#X bT +2 E- r?2`2 I Bb M `#Bi` `v BM/2t b2iX A7 7Q` Mv }MBi2 bm#b2i J ⊆ I- Mv +QHH2+iBQM ϕi (i ∈ J) Q7 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb 7`QK E iQ R −Ni (ϕi ) E e− i∈J Ni (ϕi ) = , E e
URXdV
i∈J
i?2M Ni (i ∈ I) Bb M BM/2T2M/2Mi 7 KBHv Q7 TQBMi T`Q+2bb2bX h?BMM2/ SQBMi S`Q+2bb2b G2i N #2 bBKTH2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT +2 E rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2 M BB/ b2[m2M+2 Q7 BM/2T2M/2Mi K `Fb Q7 N - 2 +? Zn i FBM; Bib p Hm2b BM {0, 1}- rBi? i?2 T`Q# #BHBiv p ∈ (0, 1) 7Q` i?2 p Hm2 1X h?2 TQBMi T`Q+2bb Nthin,p /2}M2/ #v Nthin,p (C) :=
1C (Xn )Zn
n∈N
Bb + HH2/ i?2 p@i?BMMBM; Q7 N X 1 +? TQBMi Q7 N Bb `2i BM2/ BM Nthin,p rBi? T`Q# #BHBiv p- BM/2T2M/2MiHv Q7 2p2`vi?BM; 2Hb2X 1t KTH2 RXjXR3, G TH +2 6mM+iBQM H Q7 h?BMM2/ SQBMi S`Q+2bb2bX q2 +QKTmi2 i?2 G TH +2 7mM+iBQM H Q7 i?2 i?BMM2/ TQBMi T`Q+2bb, LNthin,p (ϕ) = E exp − ϕ(Xn )Zn =E
n∈N
exp (−ϕ(Xn )Zn )
n∈N
=E E =E
exp (−ϕ(Xn )Zn ) |F
N
n∈N
n∈N
n∈N
=E
= E exp
E exp (−ϕ(Xn )Zn ) |F
N
{p exp(−ϕ(Xn )) + (1 − p)}
log (p exp(−ϕ(Xn )) + (1 − p))
n∈N
= LN − log pe−ϕ(·) + 1 − p .
h?2 BMi2`K2/B `v `2bmHi Q#i BM2/ BM i?2 T2MmHiBK i2 HBM2 Q7 i?2 + H+mH iBQM Q7 i?2 T`2pBQmb 2t KTH2 rBHH #2 `2+Q`/2/ 7Q` 7mim`2 `272`2M+2 BM i?2 7Q`K,
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa LNthin,p (ϕ) = E exp
Rm
− log 1 − p(1 − e−ϕ(x) ) N (dx)
kR
.
URX3V
h?2 p@i?BMMBM; QT2` iBQM Dp U0 ≤ p ≤ 1V +ib QM T`Q# #BHBiv /Bbi`B#miBQMb QM Mp (E), Mp (E), B7 P Bb bm+? TQBMi T`Q+2bb /Bbi`B#miBQM- Dp P Bb i?2 /Bbi`B#miBQM Q7 i?2 TQBMi T`Q+2bb Q#i BM2/ 7`QK TQBMi T`Q+2bb rBi? /Bbi`B#miBQM P #v p@i?BMMBM;X h?2Q`2K RXjXRN h?2 QT2` iBQM Dp Bb QM2@iQ@QM2X S`QQ7X q2 ? p2 iQ b?Qr i? i i?2`2 + MMQi #2 irQ /Bbi`B#miBQMb P M/ P bm+? i? i Dp P = Dp P X AM/22/- B7 7Q` HH MQM@M2; iBp2 7mM+iBQMb ϕ LN − log pe−ϕ(·) + 1 − p = LN − log pe−ϕ(·) + 1 − p i?2M LN (f ) = LN (f ) 7Q` HH MQM@M2; iBp2 7mM+iBQMb f X hQ b22 i?Bb- Bi bm{+2b iQ Q#b2`p2 i? i 7Q` Mv a ≥ 0 i?2`2 2tBbib QMHv QM2 b ≥ 0 bm+? i? i e−a = 1 − p 1 − e−b () M/- +QMp2`b2Hv- 7Q` Mv b ≥ 0 i?2`2 2tBbib QMHv QM2 a ≥ 0 b iBb7vBM; UVX
h?2 `2bmHi Q7 1t KTH2 RXjXR3 + M #2 bHB;?iHv 2ti2M/2/X .2}MBiBQM RXjXky G2i p : E → [0, 1] #2 K2 bm` #H2 7mM+iBQMX G2i N #2 bBKTH2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT +2 E rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Un }n∈N #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X h?2 TQBMi T`Q+2bb Nthin,p(·) /2}M2/ #v Nthin,p(·) (C) := 1C (Xn )1{Un ≤p(Xn )} n∈N
Bb + HH2/ i?2 p(·)@i?BMMBM; Q7 N X h?2Q`2K RXjXkR h?2 G TH +2 7mM+iBQM H Q7 Nthin,p(·) Bb ;Bp2M #v i?2 7Q`KmH LN (ϕ) = LN − log p(·)e−ϕ(·) + 1 − p(·) . h?2 T`QQ7 Bb bBKBH ` iQ i?2 QM2 ;Bp2M BM 1t KTH2 RXjXR3X Ai Bb `2[mB`2/ BM 1t2`+Bb2 RXdXRjX 1t KTH2 RXjXkk, h?BMMBM; i?2 SQBbbQM S`Q+2bbX A7 N Bb SQBbbQM T`Q+2bb rBi? i?2 HQ+ HHv BMi2;` #H2 BMi2MbBiv K2 bm`2 ν
−ψ(x)
−1)ν(dx) , LNthin,p(·) (ϕ) = LN (ψ) = e− E (e −ϕ(x) −ψ(x) r?2`2 ψ(x) := − log p(x)e + 1 − p(x) X h?2`27Q`2 e = p(x)e−ϕ(x) + 1 − p(x) = p(e−ϕ(x) − 1) + 1 M/ }M HHv
−ϕ(x) −1 p(x)ν(dx) ) LNthin,p(·) (ϕ) = e− E (e .
h?2`27Q`2- p(·)@i?BMMBM; SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2 bm`2 ν(·) `2bmHib BM SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2 bm`2 νp(·) (dx) := p(x)ν(dx)X
kk
*>Sh1_ RX :1L1_GAhA1a
h?2 S`Q# #BHBiv :2M2` iBM; 6mM+iBQM H G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM Rm X G2i V (Rm ) #2 i?2 +QHH2+iBQM Q7 K2 bm` #H2 7mM+iBQMb h : Rm → R p MBb?BM; QmibB/2 #QmM/2/ b2i M/ bm+? i? i 0 ≤ h(x) ≤ 1 7Q` HH x ∈ Rm X h?2 T`Q# #BHBiv ;2M2` iBM; 7mM+iBQM H Q7 N Bb i?2 K TTBM; GN : V (Rm ) → R /2}M2/ #v GN (h) := LN (log h) = E e Rm log h(x) N (dx) . 1[mBp H2MiHv-
GN (h) = E
URXNV
h(Xn ) ,
n∈N
r?2`2 Bi Bb mM/2`biQQ/ i? i i?2 bmK 2ti2M/b iQ HH TQBMib Xn BM i?2 bmTTQ`i Q7 h M/ i? i Bi Bb 2[m H iQ 1 B7 i?2`2 Bb MQ TQBMi BM i?2 bmTTQ`i Q7 hX h?2 /2}MBiBQM Q7 GN (h) m 2ti2M/b iQ Mv 7mM+iBQM h : R → R bm+? i? i | log h(x)|ν(dx) < ∞- bBM+2 + Rm i?2 BMi2;` H Rm log h(x) N (dx) BM URXNV Bb i?2M r2HH /2}M2/ M/ }MBi2 Uh?2Q`2K RXkXRVX h?2 pQB/ M+2 S`Q# #BHBiv 6mM+iBQM >2`2 Bb v2i MQi?2` +? ` +i2`Bx iBQM Q7 i?2 /Bbi`B#miBQM Q7 TQBMi T`Q+2bb i? i Bb bQK2iBK2b mb27mHX .2}MBiBQM RXjXkj G2i N #2 TQBMi T`Q+2bb QM i?2 K2 bm` #H2 bT +2 (E, E)X h?2 7mM+iBQM v : E → [0, 1] /2}M2/ #v v(B) = P (N (B) = 0)
(B ∈ E)
Bb + HH2/ i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM Q7 N X v(B) Bb i?2`27Q`2 i?2 T`Q# #BHBiv i? i N dz pQB/bǴ i?2 b2i BX 1t KTH2 RXjXk9, pQB/ M+2 S`Q# #BHBiv 6mM+iBQM Q7 SQBbbQM S`Q@ +2bbX 6Q` SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 ν- v(B) = exp(−λν(B))X
h?2Q`2K RXjXk8 G2i (E, d) #2 +QKTH2i2 b2T ` #H2 K2i`B+ bT +2X G2i E #2 i?2 "Q`2H σ@}2H/ QM EX h?2 /Bbi`B#miBQM Q7 bBKTH2 TQBMi T`Q+2bb N QM (E, E) Bb +? ` +i2`Bx2/ #v Bib pQB/ M+2 T`Q# #BHBiv 7mM+iBQMX
_2K `F RXjXke LQi2 i? i i?2 bbmKTiBQM Q7 bBKTHB+Biv Bb M2+2bb `v 7Q` i?2 #Qp2 `2bmHiX 6Q` BMbi M+2- /Qm#HBM; i?2 KmHiBTHB+Biv Q7 i?2 TQBMib Q7 ;Bp2M TQBMi T`Q+2bb /Q2b MQi Hi2` i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQMX
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa
kj
S`QQ7X h?Bb `2bmHiN b vb i? i B7 irQ bBKTH2 TQBMi T`Q+2bb2b ? p2 i?2 b K2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM- i?2v ? p2 i?2 b K2 /Bbi`B#miBQMX hQ b?Qr i?Bb- Bi bm{+2b iQ b?Qr i? i i?2 }MBi2@/BK2MbBQM H /Bbi`B#miBQMb Q7 bBKTH2 TQBMi T`Q+2bb N + M #2 Q#i BM2/ 7`QK Bib pQB/ M+2 T`Q# #BHBiv 7mM+iBQM HQM2X 6Q` i?Bb- Bi Bb 2MQm;? iQ T`Qp2 i? i 7Q` HH BMi2;2`b k ≥ 1- HH K2 bm` #H2 b2ib A1 , . . . , Ak M/ HH BMi2;2`b n 1 , . . . , nk P (N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk ) + M #2 2tT`2bb2/ BM i2`Kb Q7 v HQM2X h?Bb Bb /QM2 BM 7Qm` bi2TbX ai2T RX 6Q` Mv "Q`2H b2ib A1 , . . . , Ak , B- r2 ? p2 P (N (A1 ) > 0, . . . , N (Ak ) > 0, N (B) = 0) = P (N (A1 ) > 0, . . . , N (Ak−1 ) > 0, N (B) = 0) − P (N (A1 ) > 0, . . . , N (Ak−1 ) > 0, N (Ak ∪ B) = 0) , M/ 7Q` k = 1P (N (A1 ) > 0, N (B) = 0) = P (N (B) = 0) − P (N (B ∪ A1 ) = 0) = v(B) − v(B ∪ A1 ) . h?Bb b?Qrb i? i P (N (A1 ) > 0, . . . , N (Ak ) > 0, N (B) = 0) + M #2 `2+m`bBp2Hv +QKTmi2/ 7`QK i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM vX n Q7 M2bi2/ T `iBiBQMb ai2T kX amTTQb2 i? i r2 ? p2 b2[m2M+2 Kn = {Kn,i }ki=1 Q7 E bm+? i? i 7Q` Mv /BbiBM+i x- y ∈ E- i?2`2 2tBbib M n bm+? i? i x M/ y #2HQM; iQ irQ /BbiBM+i b2ib Q7 i?2 T `iBiBQM Kn UBM Qi?2` rQ`/b- i?2 b2[m2M+2 Q7 T `iBiBQMb {Kn }n≥0 2p2Mim HHv b2T ` i2b i?2 TQBMib Q7 EVX G2i 7Q` n ≥ 1 M/ A ∈ B(E)-
Hn (A) :=
kn
H(A ∩ Kn,i ) ,
i=0 n X aBM+2 i?2 b2[m2M+2 r?2`2 H(C) := 1{N (C)>0} UC ∈ EVX G2i Kn ∩A := {A ∩ Kn,i }ki=0 Q7 T `iBiBQMb {Kn ∩A}n≥1 Q7 A 2p2Mim HHv b2T ` i2b i?2 TQBMib Q7 A- M/ bBM+2 Hn (A) +QmMib i?2 MmK#2` Q7 b2ib Q7 Kn ∩ A i? i +QMi BM i H2 bi QM2 TQBMi Q7 i?2 TQBMi T`Q+2bb- r2 ? p2 BM pB2r Q7 i?2 bbmK2/ bBKTHB+Biv Q7 i?2 TQBMi T`Q+2bb
lim Hn (A) = N (A) ,
n↑∞
XbX
ai2T jX h?2 T`Q# #BHBiv P (Hn (A) = l) + M #2 2tT`2bb2/ BM i2`Kb Q7 i?2 pQB/@ M+2 T`Q# #BHBiv 7mM+iBQM v HQM2 bBM+2 UrBi? An,i = A ∩ Kn,i V P (H(An,0 ) = i0 , . . . , H(An,kn ) = ikn ) P (Hn (A) = l) = i0 ,...,ik ∈{0,1} n k Σ n ij =l j=1
M/ 7Q` i0 , . . . , ikn ∈ {0, 1} N L K2/ 7i2` _ûMvB- r?Q BMi`Q/m+2/ i?2 MQiBQM M/ TTHB2/ Bi iQ SQBbbQM T`Q+2bb2b (_ûMvBRNed)X 6Q` i?2 ;2M2` H + b2, (JƺM+?- RNdR)X
k9
*>Sh1_ RX :1L1_GAhA1a P (H(An,0 ) = i0 , . . . , H(An,kn ) = ikn ) = P (∩l;il =1 {N (An,l ) > 0} ∩ {N (∪m;im =0 An,m ) = 0}) ,
[m MiBiv r?B+? + M #2 2tT`2bb2/ BM i2`Kb Q7 i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM v HQM2- b r2 b r BM ai2T RX JQ`2 ;2M2` HHv- 7Q` HH l1 , . . . , lk ∈ NP (Hn (A1 ) = l1 , . . . , Hn (Ak ) = lk ) Bb 2tT`2bbB#H2 BM i2`Kb Q7 i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM- M/ i?2 b K2 Bb i`m2 Q7 P (Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk ) (n1 , . . . , nk ∈ N) . ai2T 9X 6BM HHv- Q#b2`p2 i? i {Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk } ↓ {N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk } M/ i?2`27Q`2 lim P (Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk ) = P (N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk ) .
n↑∞
h?2 T`QQ7 Bb MQr HKQbi /QM2X Ai Dmbi `2K BMb iQ +QMbi`m+i i?2 b2[m2M+2 Q7 T `iB@ iBQMb {Kn }n≥1 X .2MQi2 #v B(a, r) i?2 +HQb2/ # HH Q7 +2Mi2` a M/ ` /Bmb rX aBM+2 (E, d) Bb b2T ` #H2- i?2`2 2tBbib +QmMi #H2 b2i {a1 , a2 , . . . } i? i Bb /2Mb2 BM EX h?2 }`bi T `iBiBQM K1 +QMbBbib Q7 irQ b2ib K11 := B(a1 , 1) ,
K10 := E\K11 .
amTTQb2 i? i r2 ? p2 +QMbi`m+i2/ Kn−1 X h?2 M2ti T `iBiBQM Kn Bb +QMbi`m+i2/ b 7QHHQrbX G2iiBM; Bn,i := B ai , 2−(n−i)
(i = 1, . . . , n) M/ Bn,0 := E\
n
Bn,i ,
i=1
/2}M2 T `iBiBQM Cn = {Cn,i }ni=0 #v Cn,0 := Bn,0 ;
Cn,1 := Bn,1 ;
Cn,i := Bn,i \
i−1
Bn,j
(i = 2, . . . , n) .
j=1
AM Q`/2` iQ Q#i BM i?2 T `iBiBQM Kn M2bi2/ BM Kn−1 - r2 BMi2`b2+i Cn M/ Kn−1 - i? i Bb Kn := {Cn,i ∩ Kn−1,j (j = 0, . . . , kn−1 , i = 0, . . . , n)} . _2K `F RXjXkd AM i?2 + b2 E = Rm - bBKTH2 b2[m2M+2 Q7 M2bi2/ T `iBiBQMb +QmH/ #2 i?2 7QHHQrBM;- b v rBi? m = 1 7Q` MQi iBQM H +QMp2MB2M+2, Kn = (i2−n , (i + 1)2−n ] ; i ∈ Z .
RXjX h>1 .Aah_A"lhAPL P6 SPALh S_P*1aa
k8
.2}MBiBQM RXjXk3 h?2 + T +Biv 7mM+iBQM H Q7 TQBMi T`Q+2bb N bbQ+B i2b rBi? +QKT +i b2i C i?2 p Hm2 T (C) := P (N (C) > 0) . P7 +Qm`b2- i?Bb 7mM+iBQM H Bb BMiBK i2Hv `2H i2/ iQ i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM- bBM+2 T (C) = 1 − v(C)X AM T `iB+mH `- i?2 + T +Biv 7mM+iBQM H Q7 bBKTH2 TQBMi T`Q+2bb +? ` +i2`Bx2b Bib /Bbi`B#miBQMX MQi?2` +HQb2Hv `2H i2/ MQiBQM- i?Bb iBK2 7Q` bi iBQM `v TQBMi T`Q+2bb2b- Bb i?2 7QHHQrBM;, .2}MBiBQM RXjXkN h?2 +QMi +i /Bbi`B#miBQM 7mM+iBQM Q7 bi iBQM `v TQBMi T`Q+2bb N ≡ {Xn }n≥1 QM Rm Bb i?2 7mM+iBQM F : R+ → [0, 1] /2}M2/ #v F (r) := P (d(u, N ) ≤ r) ,
URXRyV
r?2`2 7Q` u ∈ E- d(u, N ) := inf n≥1 d(u, Xn )X h?2 bi iBQM `Biv Q7 N ;m ` Mi22b i? i i?2 `B;?i@? M/ bB/2 Q7 URXRyV Bb BM/2T2M/2Mi Q7 uX q2 ? p2 i? i F (r) = P (N (B(u, r) > 0) = T (B(u, r)) = T (B(0, r)) . h?2`27Q`2- BM i?2 bi iBQM `v + b2- i?2 +QMi +i /Bbi`B#miBQM 7mM+iBQM /2i2`KBM2b i?2 + T +Biv 7mM+iBQM H 7Q` +QKT +i b2ib i? i `2 +HQb2/ /BbFbX >Qr2p2` Bi Bb MQi bm{+B2Mi iQ /2i2`KBM2 i?2 + T +Biv 7mM+iBQM H 7Q` HH +QKT +i b2ibX 1t KTH2 RXjXjy, *QMi +i .Bbi`B#miBQM 6mM+iBQM Q7 >QKQ;2M2Qmb SQBbbQM S`Q+2bbX A7 N Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv λF (r) = P (N (B(u, r)) > 0) = 1 − P (N (B(u, r)) = 0) = 1 − e−λ
m (B(u,r))
= 1 − e−λκd r , m
r?2`2 κm Bb i?2 m@pQHmK2 Q7 i?2 mMBi # HH B(0, 1) ⊂ Rm X G2i W #2 +QKT +i bm#b2i Q7 Rm Q7 }MBi2 G2#2b;m2 K2 bm`2X h?2 7QHHQrBM; 2biBK iQ` Q7 F (r) 1 1{d(u,N )≤r} du , () F (r) := d
(W ) W Bb mM#B b2/ b i?2 7QHHQrBM; 2H2K2Mi `v +QKTmi iBQM b?Qrb,
1 1 E E F (r) = d 1{d(u,N )≤r} du = d E 1{d(u,N )≤r} du
(W )
(W ) W W 1 1 = d P ({d(u, N ) ≤ r) du = d F (r) du = F (r) .
(W ) W
(W ) W
ke
*>Sh1_ RX :1L1_GAhA1a
_2K `F RXjXjR AM TTHB+ iBQMb- W Bb i?2 Ub v- +QKT +iV Q#b2`p iBQM rBM/Qri? i Bb- QMHv TQBMib Q7 N HvBM; BM W `2 Q#b2`p2/X AM T `iB+mH `- F (r) + MMQi #2 +QKTmi2/ r?2M i?2`2 `2 TQBMib Q7 N QmibB/2 W i? i `2 i /Bbi M+2 7`QK W H2bb i? M rX M Q#pBQmb +Q``2+iBQM Q7 i?Bb 2/;2 2z2+i Bb iQ `2TH +2 BM i?2 `B;?i@? M/ bB/2 Q7 UV W #v Wr := {x ∈ W ; d(x, W ) ≤ r} . h?2 +Q``2bTQM/BM; 2biBK i2 F(r) :=
1 m
(Wr )
1{d(u,N )≤r} du
()
Wr
rBHH i?2M #2 mM#B b2/X >Qr2p2`- BMimBiBp2Hv- i?2 p `B M+2 Q7 i?Bb 2biBK iQ` rBHH #2 H `;2 7Q` H `;2 p Hm2b Q7 r BM +QKT `BbQM rBi? i?2 bBx2 Q7 W UHQQF i i?2 2ti`2K2 + b2 Q7 p Hm2b Q7 r H `;2` i? M i?2 ` /Bmb Q7 ;Bp2M # HH +QMi BMBM; i?2 +QKT +i rBM/Qr W VX
RX9 *QMp2`;2M+2 BM .Bbi`B#miBQM M/ o `B iBQM h?Bb b2+iBQM ;Bp2b b?Q`i BMi`Q/m+iBQM iQ i?2 i?2Q`B2b Q7 r2 F +QMp2`;2M+2Ry M/ Q7 +QMp2`;2M+2 BM p `B iBQM Q7 TQBMi T`Q+2bb2bX *QMp2`;2M+2 BM .Bbi`B#miBQM q2 F +QMp2`;2M+2 Q7 TQBMi T`Q+2bb2b Bb BM i2`Kb Q7 #QmM/2/ M/ +QMiBMmQmb 7mM+@ iBQMb f : Mp (E) → R+ X aBM+2 +QMiBMmBiv Bb /2}M2/ rBi? `2bT2+i iQ iQTQHQ;v- Qm` }`bi i bF Bb i?2`27Q`2 iQ /2}M2 bm+? iQTQHQ;v QM Mp (E)- M/ KQ`2 ;2M2` HHv QM M (E)X h?Bb iQTQHQ;v Bb i?2 p ;m2 iQTQHQ;vX b2[m2M+2 {μn }n≥1 Q7 K2 bm`2b Q7 M (E) Bb b B/ iQ +QMp2`;2 p ;m2Hv iQ μ ∈ M (E) B7 7Q` HH MQM@M2; iBp2 +QMiBMmQmb 7mM+iBQMb f : E → R rBi? +QK@ T +i bmTTQ`i- limn↑∞ E f dμn = E f dμX h?Bb Bb /2MQi2/ #v v
μn → μ . h?2Q`2K RX9XR h?2 7QHHQrBM; `2 2[mBp H2Mi, v
UBV μn → μX UBBV lim supn μn (K) ≤ μ(K) 7Q` HH `2H iBp2Hv +QKT +i b2ib K ∈ B(E)X UBBBV lim inf n μn (G) ≥ μ(G) 7Q` HH QT2M b2ib G ∈ B(E)X o ;m2 +QMp2`;2M+2 Bb BM 7 +i +QMp2`;2M+2 rBi? `2bT2+i iQ iQTQHQ;v U+ HH2/ i?2 p ;m2 iQTQHQ;vV QM M (E) i? i + M #2 K2i`Bx2/ b b2T ` #H2 K2i`B+ bT +2X h?2 σ@ H;2#` M(E) + M #2 b?QrM iQ #2 i?2 "Q`2H }2H/ QM M (E) 7Q` i?Bb iQTQHQ;v, M(E) ≡ B(M (E))X h?2 b K2 /2}MBiBQMb TTHv M/ i?2 b K2 `2bmHib ?QH/ rBi? Ry JQbiHv rBi?Qmi T`QQ7bc b22 (L2p2m- RNdd) UBM 6`2M+?V- (. H2v M/ o2`2@CQM2b- kyyj- kyy3) M/ (E HH2M#2`;- kyRd)X (_2bMB+F- RN3d) +QMi BMb bm++BM+i ++QmMi Q7 i?2 i?2Q`vX
RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL
kd
M (E) `2TH +2/ #v Mp (E)X AM T `iB+mH ` Mp (E) ≡ B(Mp (E))X Ai im`Mb Qmi i? i Mp (E) Bb p ;m2Hv +HQb2/ M/ i?2`27Q`2 K2 bm` #H2 bm#b2i Q7 M (E) M/ i? i Mp (E) Bb i?2 i` +2 Q7 M(E) QM Mp (E)X o ;m2 +QMp2`;2M+2 BM Mp (E) ? b ` i?2` bBKTH2 M/ M im` H BMi2`T`2i iBQM BM i2`Kb Q7 i?2 TQBMib Q7 i?2 TQBMi K2 bm`2b BMpQHp2/X h?2Q`2K RX9Xk h?2 b2[m2M+2 {μn }n≥1 Q7 K2 bm`2b Q7 Mp (E) +QMp2`;2b p ;m2Hv iQ μ ∈ Mp (E) Bz 7Q` Mv +QKT +i bm#b2i K Q7 E bm+? i? i μ(∂K) = 0 U∂K /2MQiBM; i?2 #QmM/ `v Q7 KV- i?2`2 2tBbib M n(K) ∈ N bm+? i? i 7Q` n ≥ n(K)i?2`2 2tBbib H #2HHBM; {xi (n); 1 ≤ i ≤ q} Q7 i?2 TQBMib Q7 μn BM K M/ H #2HHBM; {xi ; 1 ≤ i ≤ q} Q7 i?2 TQBMib Q7 μ BM K bm+? i? i μn (· ∩ K) =
q
εxi (n) (·) M/ μ(· ∩ K) =
i=1
q
εxi (·)
()
i=1
M/ (1 ≤ i ≤ q) .
lim xi (n) = xi
n↑∞
()
h?2 #Qp2 +? ` +i2`Bx iBQM Q7 p ;m2 +QMp2`;2M+2 Bb mb27mH 7Q` T`QpBM; i?2 +QM@ iBMmBiv Q7 7mM+iBQM f : Mp (Rm ) → RX 1t KTH2 RX9Xj, hrQ 2t KTH2b Q7 +QMiBMmQmb K TTBM;bX 6Q` Mv #QmM/2/ K2 bm` #H2 b2i C ∈ Rm - i?2 K TTBM; μ → μ(C) 7`QK Mp (E) iQ R Bb +QMiBMmQmbX v hQ T`Qp2 i?Bb- r2 ? p2 iQ b?Qr i? i B7 μn → μ- i?2M μn (C) → μ(C)X b C Bb #QmM/2/ M/ μ Bb σ@}MBi2 TQBMi K2 bm`2- QM2 + M Hr vb }M/ +QKT +i b2i K ⊃ C bm+? i? i μ(∂K) = 0X lbBM; QMHv UV Q7 h?2Q`2K RX9Xk- r2 b22 i? i 7Q` HH n ≥ n(K) U b /2}M2/ BM i?2 bi i2K2Mi Q7 i?2 i?2Q`2KV- μn (C) = μ(C)X bBKBH ` `;mK2Mi Ui?Bb iBK2 mbBM; #Qi? UV M/ UVV rBHH b?Qr i? i i?2 K TTBM; μ → μ(g) = Rm g(x) μ(dx) Bb +QMiBMmQmb r?2M2p2` g : Rm → R Bb +QMiBMmQmb 7mM+iBQM rBi? +QKT +i bmTTQ`iX .2}MBiBQM RX9X9 G2i {Nn }n≥1 #2 b2[m2M+2 Q7 ` M/QK K2 bm`2b UTQbbB#Hv /2}M2/ QM /Bz2`2Mi T`Q# #BHBiv bT +2b {(Ωn , Fn , Pn )}n≥1 V M/ H2i {Pn }n≥1 #2 i?2B` `2bT2+iBp2 /Bbi`B#miBQMbX h?2 b2[m2M+2 {Pn }n≥1 Bb b B/ iQ +QMp2`;2 r2 FHv iQ i?2 T`Q# #BHBiv P QM (M (E), M(E)) B7 7Q` HH #QmM/2/ M/ +QMiBMmQmb 7mM+iBQMb f : M (E) → R lim f (μ) Pn (dμ) = f (μ) P(dμ) , n↑∞
M (E)
M (E)
i? i Bb- 2[mBp H2MiHv f (μ) P(dμ) .
lim En [f (Nn )] =
n↑∞ w
M (E) w
h?Bb Bb /2MQi2/ #v Nn → P Q` Pn → PX
k3
*>Sh1_ RX :1L1_GAhA1a w
h?2Q`2K RX9X8 Nn → P B7 M/ QMHv B7 7Q` HH MQM@M2; iBp2 +QMiBMmQmb 7mM+iBQMb ϕ : E → R rBi? +QKT +i bmTTQ`i LNn (ϕ) → LP (ϕ) := e− E ϕ(x) μ(dx) P (dμ) . M (E)
M TT `2MiHv r2 F2` +QM+2Ti Q7 +QMp2`;2M+2 Bb r2 F +QMp2`;2M+2 BM i?2 b2Mb2 Q7 }/B U}MBi2@/BK2MbBQM HV /Bbi`B#miBQMbX +QMiBMmBiv b2i Q7 TQBMi T`Q+2bb N U`2bTX- Q7 T`Q# #BHBiv P QM (Mp (E), Mp (E))V Bb b2i A ∈ B(Rm ) bm+? i? i P (N (∂A) > 0) = 0 U`2bTX P ({μ ; μ(∂A) > 0}) = 0VX .2}MBiBQM RX9Xe h?2 b2[m2M+2 Q7 HQ+ HHv }MBi2 TQBMi T`Q+2bb2b {Nn }n≥1 Bb b B/ iQ +QMp2`;2 BM i?2 b2Mb2 Q7 }/B /Bbi`B#miBQMb B7 7Q` HH A1 - A2 - Ę- Ak i? i `2 +QMiBMmBiv b2ib Q7 N - i?2 DQBMi /Bbi`B#miBQM Q7 (Nn (A1 ), . . . , Nn (Ak )) +QMp2`;2b r2 FHv b n ↑ ∞ iQ i?2 /Bbi`B#miBQM Q7 (N (A1 ), . . . , N (Ak ))X 6Q` i?Bb Bi bm{+2b i? i limn↑∞ LNn (ϕ) = LN (ϕ) 7Q` HH ϕ Q7 i?2 7Q`K ϕ(x) = i=1 ti 1Ai (x) r?2`2 t1 , . . . , tk ∈ R+ X h?2`27Q`2,
k
h?2Q`2K RX9Xd PM Mp (E)- r2 F +QMp2`;2M+2 Bb 2[mBp H2Mi iQ r2 F +QMp2`;2M+2 BM i?2 b2Mb2 Q7 i?2 }/B /Bbi`B#miBQMbX 1t KTH2 RX9X3, SQBbbQM S`Q+2bb b GBKBi Q7 "BMQKB H SQBMi S`Q+2bb2bX AM i?2 1t KTH2 RXjXRy- H2i p = pn M/ + HH Nn i?2 +Q``2bTQM/BM; #BMQKB H TQBMi T`Q+2bb- Q7 G TH +2 i` Mb7Q`K n LNn (ϕ) = 1 − pn e−ϕ(x) Q(dx) . E
amTTQb2 MQr i? i npn → λ b n ↑ ∞X h FBM; HQ; `Bi?Kb log LNn (ϕ) = n log 1 + pn e−ϕ(x) Q(dx) → (e−ϕ(x) − 1) λQ(dx) . E
E
h?2`27Q`2 i?2 HBKBi BM /Bbi`B#miBQM Q7 i?2 b2[m2M+2 {Nn }n≥1 Bb SQBbbQM T`Q+2bb Q7 K2 M K2 bm`2 ν(·) = λQ(·)X
1t KTH2 RX9XN, h?BMMBM; M/ *QMi` +iBQMX G2i N #2 TQBMi T`Q+2bb QM Rm M/ H2i p ∈ (0, 1)X *QMbi`m+i TQBMi T`Q+2bb Np #v i?BMMBM; rBi? T`Q# #BHBiv p i?2 T`Q+2bb N +QMi` +i2/ #v p U7Q` r?B+? i?2 MmK#2` Q7 TQBMib BM C Bb N (p−1 C)VX lM/2` i?2 +QM/BiBQM i? i 7Q` Mv #QmM/2/ C ∈ B(Rm ) lim pN (p−1 C) = λ m (C) , p↓0
i?2 TQBMi T`Q+2bb Np +QMp2`;2b r2 FHv iQ M ?TT Q7 BMi2MbBiv λ b p → 0X
RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL
kN
S`QQ7X "v URX3V LNp (ϕ) = E exp
Rm
log 1 − p(1 − e−ϕ(x) ) N (p−1 dx)
.
LQr rBi? ϕ(x) = ki=1 ti 1Ai (x)- r?2`2 i?2 Ai Ƕb `2 #QmM/2/ b2ib M/ i?2 ti Ƕb `2 MQM@M2; iBp2 MmK#2`b- r2 ? p2 i? i k log 1 − p(1 − e−ϕ(x) ) = p(e−ϕ(x) − 1) + o(p) = p (e−ti − 1)1Ai (x) + o(p) i=1
M/ i?2`27Q`2- b p ↓ 0 Rm
k log 1 − p(1 − e−ϕ(x) N (p−1 dx) = (e−ti − 1)pN (p−1 Ai )
→
i=1 k
(e−ti − 1)λ m (Ai )
i=1
M/
E exp −
k
Np (Ai )
i=1
→ exp
k
(e
−ti
− 1)λ (Ai ) m
.
i=1
h?2`27Q`2- Np +QMp2`;2b r2 FHv iQ SQBbbQM T`Q+2bb Q7 BMi2MbBiv λX
h?2 7QHHQrBM; `2bmHi Bb +? ` +i2`Bx iBQM Q7 i?2 *Qt T`Q+2bb2b BM i2`Kb Q7 i?BMMBM;X h?2Q`2K RX9XRy URR V 6Q` TQBMi T`Q+2bb N QM E- i?2 7QHHQrBM; irQ bi i2K2Mib `2 2[mBp H2Mi, UBV N Bb *Qt T`Q+2bbX X UBBV 6Q` 2p2`v p ∈ (0, 1]- i?2`2 Bb TQBMi T`Q+2bb N bm+? i? i N = Nthin,p
h?2 T`QQ7 rBHH #2 # b2/ QM i?2 7QHHQrBM; H2KK , G2KK RX9XRR URk V G2i {Nn }n≥1 #2 b2[m2M+2 Q7 TQBMi T`Q+2bb2b QM E M/ H2i {pn }n≥1 #2 b2[m2M+2 Q7 `2 H MmK#2`b BM (0, 1] bm+? i? i limn↑+∞ pn = 0 mMB7Q`KHvX G2i Nn #2 7Q` 2 +? n ≥ 1 pn @i?BMMBM; Q7 Nn X h?2 irQ 7QHHQrBM; +QM/BiBQMb `2 2[mBp H2Mi, UBV h?2 b2[m2M+2 {Nn }n≥1 +QMp2`;2b BM /Bbi`B#miBQM iQ bQK2 TQBMi T`Q+2bb N X UBBV h?2`2 2tBbib ` M/QK K2 bm`2 ν QM E bm+? i? i i?2 b2[m2M+2 Q7 ` M/QK K2 bm`2b {pn Nn }n≥1 +QMp2`;2b BM /Bbi`B#miBQM iQ ` M/QK K2 bm`2 M X A7 i?2b2 +QM/BiBQMb `2 b iBb}2/- N Bb *Qt T`Q+2bb /B`2+i2/ #v M X RR Rk
(J2+F2- RNe3)X (E HH2M#2`;- RNd8)X
jy
*>Sh1_ RX :1L1_GAhA1a
S`QQ7X 6Q` i?2 2[mBp H2M+2- Bi bm{+2b iQ Q#b2`p2 i? i 7Q` 2 +? MQM@M2; iBp2 7mM+@ iBQM ϕ : E → R LNn (ϕ) = LNn − log 1 − pn − pn e−ϕ LNn pn (1 − e−ϕ ) = Lpn Nn (1 − e−ϕ ) . S bbBM; iQ i?2 HBKBi ;Bp2b LN (ϕ) = LM (1 − e−ϕ ) r?B+? Bb BM/22/ i?2 G TH +2 i` Mb7Q`K Q7 *Qt T`Q+2bb /B`2+i2/ #v M X
q2 + M MQr `2bmK2 i?2 T`QQ7 Q7 h?2Q`2K RX9XRy, S`QQ7X hQ T`Qp2 i? i UBV BKTHB2b UBBV i F2 7Q` N i?2 *Qt T`Q+2bb rBi? /B`2+iBM; K2 bm`2 p−1 M - r?2`2 M Bb i?2 /B`2+iBM; K2 bm`2 Q7 N X 6Q` i?2 T`QQ7 i? i UBBV BKTHB2b UBV- Q#b2`p2 i? i 7Q` 2 +? n ≥ 1- N ? b i?2 b K2 /Bbi`B#miBQM b i?2 1/n@ i?BMMBM; Q7 bQK2 TQBMi T`Q+2bb Nn M/ TTHv G2KK RX9XRRX *QMp2`;2M+2 BM o `B iBQM a2+iBQM XR Q7 i?2 TT2M/Bt `2pB2rb i?2 # bB+ i?2Q`v Q7 +QMp2`;2M+2 BM p `B iBQM M22/2/ BM i?Bb b2+iBQMX .2}MBiBQM RX9XRk G2i P1 M/ P2 #2 irQ T`Q# #BHBiv /Bbi`B#miBQMb (M (E), M(E))X h?2 /Bbi M+2 BM p `B iBQM #2ir22M P1 M/ P2 Bb i?2 [m MiBiv
QM
dV (P1 , P2 ) := sup (P1 (Γ) − P2 (Γ)) . Γ∈M(E)
1[mBp H2MiHv dV (P1 , P2 ) = sup (P r(N1 ∈ Γ) − P r(N2 ∈ Γ)) Γ∈M(E)
7Q` Mv T B` (N1 , N2 ) rBi? ;Bp2M K `;BM H /Bbi`B#miBQMb P1 M/ P2 X "v h?2Q`2K XRXkydV (P1 , P2 ) ≤ P (N1 = N2 ) URXRRV M/- KQM; HH T B`b (N1 , N2 ) rBi? ;Bp2M K `;BM H /Bbi`B#miBQMb P1 M/ P2 - i?2`2 2tBbib i H2 bi QM2 ii BMBM; 2[m HBiv BM URXRRVX AM Q`/2` iQ +QKTmi2 i?2 `B;?i@? M/ bB/2 Q7 URXRRV M/ i?2`27Q`2 Q#i BM #QmM/ 7Q` i?2 /Bbi M+2 BM p `B iBQM #2ir22M P1 M/ P2 - +H2p2` +?QB+2 Q7 N1 M/ N2 Bb M22/2/- b i?2 7QHHQrBM; 2t KTH2 b?QrbX 1t KTH2 RX9XRj, o `B iBQM .Bbi M+2 #2ir22M hrQ SQBbbQM S`Q+2bb2bX q2 b22F #QmM/ 7Q` i?2 /Bbi M+2 BM p `B iBQM #2ir22M irQ SQBbbQM /Bbi`B#miBQMb QM i?2 HBM2 rBi? i?2 `2bT2+iBp2 BMi2MbBiv 7mM+iBQMb λ1 (t) M/ λ2 (t)X 6Q` i?Bb- r2 +QMbi`m+i Ub22 1t2`+Bb2 RXdXRyV irQ SQBbbQM T`Q+2bb2b N1 M/ N2 QM i?2 HBM2 rBi?
RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL
jR
i?2 `2bT2+iBp2 BMi2MbBiv 7mM+iBQMb λ1 (t) M/ λ2 (t) 7`QK bi M/ `/ ?TT N QM R2 b 7QHHQrb Ni (a, b] = N ({(t, z) ; (t, z) ∈ (a, b] × {z ; 0 ≤ z ≤ λi (t)}}) (i = 1, 2) . *H2 `Hv- N1 = N2 B7 M/ QMHv B7 i?2`2 `2 TQBMib Q7 N #2ir22M i?2 +m`p2b z = λ1 (t) M/ z = λ2 (t)X h?2 T`Q# #BHBiv 7Q` i?Bb iQ ? TT2M Bb 1−exp − R |λ1 (t) − λ2 (t)| dt X h?2`27Q`2- #v h?2Q`2K XRXky dV (P1 , P2 ) ≤ 1 − exp − |λ1 (t) − λ2 (t)| dt . R
LQi2 i? i BM i?Bb 2t KTH2- +?QQbBM; N1 M/ N2 BM/2T2M/2Mi rQmH/ H2 / iQ i?2 i`BpB H #QmM/ 1X .2}MBiBQM RX9XR9 b2[m2M+2 Q7 ` M/QK K2 bm`2b {Nn }n≥1 rBi? `2bT2+iBp2 /Bb@ i`B#miBQMb {Pn }n≥1 Bb b B/ iQ +QMp2`;2 BM p `B iBQM iQ i?2 ` M/QK K2 bm`2 N rBi? i?2 T`Q# #BHBiv /Bbi`B#miBQM P B7 lim dV (Pn , P) = 0 .
n↑∞
h?2 M2ti `2bmHi Bb 2bT2+B HHv / Ti2/ iQ i?2 +QmTHBM; Q7 irQ biQ+? biB+ T`Q+2bb2bQM2 Q7 r?B+? Bb 2`;Q/B+X Ai rBHH #2 mb2/ BM i?2 H bi +? Ti2`X h?2Q`2K RX9XR8 URj V G2i (Ω, F, P ) #2 bQK2 T`Q# #BHBiv bT +2 2M/Qr2/ rBi? }Hi` iBQM {Ft }t≥0 M/ H2i X := {X(t)}t≥0 M/ Y := {Y (t)}t≥0 #2 irQ Ft @ / Ti2/ T`Q+2bb2bX bbmK2 i? i 7Q` HH s ≥ 0P (X ≡ Y QM [s, ∞) | Fs ) ≥ Z(s) − ε(s) ,
URXRkV
r?2`2 {ε(t)}t≥0 Bb `2 H@p Hm2/ T`Q+2bb bm+? i? i ε(s) i2M/b iQ 0 XbX b s → ∞- M/ r?2`2 Z := {Z(t)}t≥0 Bb `2 H@p Hm2/ 2`;Q/B+ T`Q+2bb bm+? i? i P (Z(0) > 0) > 0 .
URXRjV
h?2M {X(t)}t≥0 M/ {Y (t)}t≥0 +QmTH2 BM XbX }MBi2 iBK2X S`QQ7X UQ7 G2KK RX9XR8V ++Q`/BM; iQ URX9XR8V- i?2`2 2tBbib β > 0 bm+? i? i P (Z(s) > β) ≥ βX h?2 2p2Mi As := {X ≡ Y QM [s, ∞)} BM+`2 b2b b s → ∞ iQ i?2 2p2Mi A∞ := {X M/ Y +QmTH2} r?B+? Bb F∞ @K2 bm` #H2 bBM+2 #Qi? X M/ Y `2 Ft @ / Ti2/X 1[m iBQM URXRkV i?2`27Q`2 BKTHB2b 1 P (A∞ | Fs ) ≥ β1[β,∞) (Z(s))1[−∞, 1 β] (ε(s)) . 2 2 AMi2;` iBM; #2ir22M s M/ s + tRj
(J bbQmHBû- RNN8)X
jk
*>Sh1_ RX :1L1_GAhA1a 1 t
s+t s
1 1 P (A∞ | Fu ) du ≥ β1[−∞, 1 β] (sup ε(u)) 2 2 t u≥s
s+t
1[β,∞) (Z(u)) du .
URXR9V
s
aBM+2 {P (A∞ | Fu )}u≥0 Bb mMB7Q`KHv BMi2;` #H2 K `iBM; H2- Bi +QMp2`;2b HKQbi bm`2Hv iQ 1A∞ b u → ∞- M/ i?2`27Q`2 i?2 H27i@? M/ bB/2 Q7 URXR9V HbQ +QMp2`;2b XbX iQ 1A∞ X "v i?2 2`;Q/B+Biv Q7 Z- i?2 *2b¨`Q K2 M QM i?2 `B;?i@? M/ bB/2 Q7 URXR9V +QMp2`;2b XbX iQ P (Z(0) ≥ β) ≥ βX h?2`27Q`2 1 1A∞ ≥ β 2 1[−∞, 1 β] (sup ε(u)) . 2 2 u≥s b s → ∞ i?2 BM/B+ iQ` QM i?2 `B;?i@? M/ bB/2 Q7 i?2 #Qp2 2[m iBQM i2M/b iQ 1 XbX bQ i? i i?2 BM/B+ iQ` Q7 i?2 2p2Mi A∞ Bb XbX bi`B+iHv TQbBiBp2- M/ Bb i?2`27Q`2 XbX 2[m H iQ 1- i? i Bb- i?2 T`Q+2bb2b X M/ Y +QmTH2X
RX8 *Hmbi2` SQBMi S`Q+2bb2b _Qm;?Hv bT2 FBM;- +Hmbi2` TQBMi T`Q+2bb +QMbBbib Q7 TQBMi T`Q+2bb Ui?2 ;2`KV 2 +? TQBMi Q7 r?B+? Bb bm``QmM/2/ #v TQBMi T`Q+2bb Ui?2 +Hmbi2` i i?Bb TQBMiVX JQ`2 ;2M2` HHv- i?2 +Hmbi2`b K v #2 ` M/QK K2 bm`2bX 6Q`K HHv MQr- M/ KQ`2 ;2M2` HHv, G2i N0 #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT +2 E rBi? b2@ [m2M+2 Q7 TQBMib {X0,n }n∈N M/ HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 ν0 X G2i {(x, ω, C) ∈ E × Ω × B(E) → Zn (x, ω, C)}n≥1 #2 K TTBM;b bm+? i? i Zn (x, ω, ·) ∈ M (E) ((x, ω) ∈ E × Ω) M/ bm+? i? i 7Q` 2 +? C ∈ B(E) i?2 K TTBM; (x, ω) ∈ E × Ω → Zn (x, ω, C) Bb B(E) ⊗ F @K2 bm` #H2X q`Bi2 Zn (x)(ω, C) := Zn (x, ω, C) Ui?2`27Q`2 Zn (x) Bb ` M/QK K2 bm`2 QM (E, B(E)VVX amTTQb2 i? i 7Q` 2 +? x ∈ E- i?2 b2[m2M+2 Q7 ` M/QK K2 bm`2b {Zn (x)}n∈Z Bb BB/- BM/2T2M/2Mi Q7 N0 M/ bm+? i? i E [Z1 (x, C)] = KZ (x, C) 7Q` bQK2 K2 bm` #H2 F2`M2H KZ 7`QK (E, B(E)) iQ (E, B(E)) bm+? i? i 7Q` Mv +QKT +i b2i C ∈ B(E) KZ (x, C − x) ν0 (dx) < ∞ .
URXR8V
E
.2}MBiBQM RX8XR h?2 ` M/QK K2 bm`2 N QM E /2}M2/ #v Zn (X0,n , C − X0,n ) (C ∈ B(E)) N (C) :=
URXReV
n∈N
Bb + HH2/ +Hmbi2` ` M/QK K2 bm`2 rBi? ;2`K UTQBMi T`Q+2bbV N0 X h?2 ` M/QK K2 bm`2 Zn (X0,n , · − X0,n ) Bb i?2 +Hmbi2` i i?2 TQBMi X0,n X h?2 BMi2MbBiv K2 bm`2 ν Q7 N Bb
RX8X *Glah1_ SPALh S_P*1aa1a
jj
KZ (y, C − y) ν0 (dy) (C ∈ B(E)) .
ν(C) =
URXRdV
E
AM 7 +i- #v * KT#2HHǶb 7Q`KmH
E
Zn (X0,n , C − X0,n ) = E E
n∈N
n∈N
=E
E Zn (X0,n , C − X0,n ) | F
n∈N
=E
Zn (X0,n , C − X0,n ) | F
N0
N0
KZ (X0,n , C − X0,n )
n∈N
KZ (y, C − y)) ν0 (dy) .
= E
A7 i?Bb BMi2MbBiv K2 bm`2 Bb HQ+ HHv }MBi2- i?2 ` M/QK K2 bm`2 +QMbB/2`2/ Bb ` M/QK 2H2K2Mi Q7 M (E)X q?2M 7Q` HH x ∈ E M/ HH ω- Z1 (x, ω, ·) ∈ Mp (E)- URXReV /2}M2b +Hmbi2` TQBMi T`Q+2bbX h?Bb TQBMi T`Q+2bb Bb bBKTH2 B7- 7Q` BMbi M+2- ν0 Bb MQM@ iQKB+ K2 bm`2X 1t KTH2 RX8Xk, aT +2 >QKQ;2M2Qmb *Hmbi2`bX q?2M Z1 dz/Q2b MQi /2T2M/ QM xǴ- i? i Bb- r?2M Bi Bb ` M/QK K2 bm`2 QM (E, B(E)) UBM r?B+? + b2 KZ (x, ·) = νZ (·)- i?2 BMi2MbBiv K2 bm`2 Q7 Z1 V- r2 mb2 i?2 MQi iBQM N = N0 ∗ Z- r?2`2 Z bi M/b 7Q` i?2 dz;2M2`B+Ǵ +Hmbi2`- i? i Bb- Mv ` M/QK K2 bm`2 rBi? i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX AKTHB+Bi BM i?Bb MQi iBQM Bb i?2 bbmKTiBQM i? i i?2 K `Fb Zn Q7 N0 `2 BB/ M/ BM/2T2M/2Mi Q7 N0 X AM i?Bb + b2ν = ν0 ∗ νZ ,
URXR3V
r?2`2 νZ Bb i?2 +QKKQM BMi2MbBiv K2 bm`2 Q7 i?2 Zn Ƕb M/ ∗ /2MQi2b i?2 mbm H +QMpQHmiBQM T`Q/m+i Q7 irQ K2 bm`2bX
_2K `F RX8Xj LQi2 i? i BM i?2 + b2 Q7 TQBMi T`Q+2bb +Hmbi2`b- i?2 Zn Ƕb K v ? p2 TQBMi i 0 BM r?B+? + b2 bQK2 Q` HH TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb `2 T `i Q7 i?2 +Hmbi2` TQBMi T`Q+2bbX q?2M E = Rm - bm{+B2Mi +QM/BiBQM 7Q` i?2 +Hmbi2` TQBMi T`Q+2bb iQ #2 bBKTH2 Bb i? i Bib BMi2MbBiv K2 bm`2 #2 /Bzmb2X h?Bb Bb i?2 + b2 r?2M2p2` QM2 Q7 i?2 K2 bm`2b Q7 i?2 +QMpQHmiBQM URXR3V Bb KmHiBTH2 Q7 i?2 G2#2b;m2 K2 bm`2 M/ i?2 Qi?2` Bb }MBi2 K2 bm`2X 6Q` BMbi M+2- B7 i?2 BMi2MbBiv K2 bm`2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb Q7 i?2 7Q`K ν0 (dx) = λ0 m (dx)- i?2M
ν(C) = Rm
λ0 m (C − x)νZ (dx) =
λ0 m (C)νZ (dx) = λ0 νZ (E) m (C) . Rm
j9
*>Sh1_ RX :1L1_GAhA1a
1t KTH2 RX8X9, h?2 *Qt *Hmbi2` SQBMi S`Q+2bbX A7 7Q` HH x ∈ E- Z1 (x, ·) Bb SQBbbQM T`Q+2bb- i?2 +Hmbi2` TQBMi T`Q+2bb Bb + HH2/ *Qt +Hmbi2` TQBMi T`Q+2bbX
1t KTH2 RX8X8, h?2 L2vK Mėa+Qii *Hmbi2` SQBMi S`Q+2bbXR9 AM i?Bb KQ/2H- i?2 ;2M2`B+ +Hmbi2` Bb Q7 i?2 7Q`K Z := {U1 , U2 , . . . , UT } r?2`2 T Bb M BMi2;2`@p Hm2/ ` M/QK p `B #H2 M/ {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 ` M/QK p2+iQ`b Q7 Rm - BM/2T2M/2Mi Q7 T X
1t KTH2 RX8Xe, h?BMMBM;X A7 i?2 ;2M2`B+ +Hmbi2` 2Bi?2` ? b QMHv QM2 TQBMi i 0 UrBi? T`Q# #BHBiv pV Q` Bb 2KTiv UrBi? T`Q# #BHBiv 1 − pV- i?2 +Hmbi2` TQBMi T`Q+2bb N = N0 ∗ Z Bb BM 7 +i M BM/2T2M/2Mi p@i?BMMBM; Q7 N0 , 2 +? TQBMi Q7 N0 ? b #22M 2` b2/ UrBi? T`Q# #BHBiv 1 − pV Q` F2Ti UrBi? T`Q# #BHBiv pV BM/2T2M/2MiHv Q7 i?2 Qi?2` TQBMibX 1t KTH2 RX8Xd, .BbTH +BM;X A7 i?2 ;2M2`B+ +Hmbi2` Z ? b 2t +iHv QM2 TQBMi i HQ+ iBQM V - i?2 `2bmHiBM; +Hmbi2` TQBMi T`Q+2bb N = N0 ∗ Z Bb Q#i BM2/ #v BM/2@ T2M/2Mi ` M/QK /BbTH +2K2Mib- i?2 ;2M2`B+ /BbTH +2K2Mi #2BM; V X h?2 TQBMib Q7 N = N0 ∗ Z `2 {X0,n + Vn }n∈N - r?2`2 i?2 Vn Ƕb 7Q`K M BB/ b2[m2M+2 Q7 i?2 b K2 /Bbi`B#miBQM b i?2 ;2M2`B+ /BbTH +2K2Mi V X
1t KTH2 RX8X3, G TH +2 6mM+iBQM H Q7 *Hmbi2` SQBMi S`Q+2bbX *QM@ bB/2` i?2 TQBMi T`Q+2bb N /2}M2/ #v URXReV- bbmK2/ bBKTH2 M/ rBi? HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2X amTTQb2 KQ`2Qp2` i? i Zn (x, ·) = Zn (·) Ui?2 ;2M2` H + b2 r?2`2 i?2 /Bbi`B#miBQMb Q7 i?2 +Hmbi2`b /2T2M/ QM i?2B` TQbBiBQMb + M Q7 +Qm`b2 #2 i`2 i2/ bBKBH `HvVX .2MQi2 #v LN0 M/ LZ i?2 G TH +2 i` Mb7Q`Kb Q7 N0 M/ Mv Zn `2bT2+iBp2HvX h?2M LN (ϕ) := E exp − ϕ(x + X0,n ) Zn (dx) =E
E
n∈N
exp − ϕ(x + X0,n ) Zn (dx) E
n∈Z
N0 exp − ϕ(x + X0,n ) Zn (dx) | F =E E
n∈Z
E
N0 =E E exp − ϕ(x + X0,n ) Zn (dx) | F .
n∈N R9
(L2vK M M/ a+Qii- RN83)X
E
RX8X *Glah1_ SPALh S_P*1aa1a
j8
h?2`27Q`2- bBM+2
E exp − ϕ(x + X0,n ) Zn (dx) | F N0 = LZn (ϕ(· + X0,n )) = LZ (ϕ(· + X0,n )) , E
r2 Q#i BM
LN (ϕ) = E
LZ (ϕ(· + X0,n )
n∈N
= E exp
log LZ (ϕ(· + X0,n )
.
n∈N
6BM HHvURXRNV
LN (ϕ) = LN0 (−ψ) , r?2`2 ψ(x) := log LZ (ϕ(· + x)) .
Bb SQBbbQM AM i?2 bT2+B H + b2 r?2`2 E = Rm M/ r?2`2 i?2 ;2`K TQBMi T`Q+2bb T`Q+2bb Q7 HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 μ - LN0 (ϕ) = exp Rm e−ϕ(x) − 1 μ (dx) M/ i?2`27Q`2 LN (ϕ) = exp E e− Rm ϕ(y+x) Z(dy) − 1 μ (dx) . URXkyV Rm
1t KTH2 RX8XN, pQB/ M+2 S`Q# #BHBiv 6mM+iBQM Q7 SQBbbQM *Hmbi2` S`Q+2bbX *QMbB/2` i?2 bT +2@?QKQ;2M2Qmb +Hmbi2` TQBMi T`Q+2bb Q7 1t KTH2 RX8XkrBi? i?2 //BiBQM H bT2+B}+ iBQM i? i i?2 ;2`K N0 Bb SQBbbQM T`Q+2bbX q2 +QKTmi2 Bib pQB/ M+2 T`Q# #BHBiv 7mM+iBQMX 6Q` i?Bb- r2 }`bi Q#b2`p2 i? i P (N (C) = 0) = lim E e−tN (C) . t↑∞
LQr
−tN (C)
E e
=E e
−t
n
Zn (C−Xn )
=E
=E E =E
e
−tZn (C−Xn )
n
e
−tZn (C−Xn )
|F
n
−tZ1 (C−Xn )
E e
n
N0
=E
= E exp
−tZn (C−Xn )
E e
|F
N0
n
log E e
−tZ1 (C−Xn )
.
n
aBM+2 N0 Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν0 - i?2 H bi i2`K Q7 i?2 #Qp2 b2[m2M+2 Q7 2[m HBiB2b Bb −tZ1 (C−x) exp E e − 1 ν0 (dx) . () Rm
je
*>Sh1_ RX :1L1_GAhA1a
"mi
lim E e−tZ1 (C−x) − 1 = vZ (C − x) − 1 , t↑∞
r?2`2 vZ Bb i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM Q7 Z1 X h?2`27Q`2 i FBM; i?2 HBKBi b t ↑ ∞ BM UV vB2H/b #v /QKBM i2/ +QMp2`;2M+2, (vZ (C − x) − 1) ν0 (dx) . vN (C) = exp Rm
RXe h?2 aiB2HiD2bĜG2#2b;m2 * H+mHmb H `;2 MmK#2` Q7 biQ+? biB+ T`Q+2bb2b QM i?2 `2 H HBM2 ? p2 b KTH2 T i?b Q7 #QmM/2/ p `B iBQMX h?2 ivTB+ H + b2 Bb i? i Q7 TB2+2rBb2 +QMiBMmQmb T`Q+2bb r?Qb2 /Bb+QMiBMmBiB2b Q++m` i i?2 2p2Mi iBK2b Q7 TQBMi T`Q+2bb QM i?2 `2 H HBM2X hQ K @ MBTmH i2 BMi2;` Hb rBi? `2bT2+i iQ i?Bb ivT2 Q7 T`Q+2bb2b QM2 ? b i /BbTQbBiBQM bBKTH2 b2i Q7 `mH2b- bm+? b i?2 T`Q/m+i `mH2 M/ i?2 2tTQM2MiB H `mH2- r?B+? `2 [mBi2 +QMp2MB2Mi iQ 2tT`2bb i?2 2pQHmiBQM 2[m iBQMb Q++m``BM; BM i?2 bim/v Q7 /v@ M KB+ H bvbi2Kb- 2bT2+B HHv BM i?2 i?2Q`v Q7 biQ+? biB+ BMi2MbBiv U*? Ti2` 8V M/ BM i?2 i?2Q`v Q7 S HK T`Q# #BHBiv U*? Ti2` dVX h?2 S`Q/m+i _mH2 6B`bi `2+ HH i?2 /2}MBiBQM Q7 #QmM/2/ p `B iBQM 7mM+iBQM M/ i? i Q7 aiB2HiD2bĜ G2#2b;m2 BMi2;` H rBi? `2bT2+i iQ bm+? 7mM+iBQMbX .2}MBiBQM RXeXR G2i f : [0, ∞) → R #2 7mM+iBQM bm+? i? i 7Q` HH t ≥ 0Vf (t) := sup D
N
|f (ti ) − f (ti−1 )| < ∞,
i=1
r?2`2 D ` M;2b Qp2` HH i?2 bm#/BpBbBQMb Q7 [0, t], 0 = t0 < t1 < · · · < t N = t . h?2 7mM+iBQM f Bb i?2M b B/ iQ #2 Q7 #QmM/2/ p `B iBQM Qp2` }MBi2 BMi2`p HbX h?2 [m MiBiv Vf (t) Bb + HH2/ i?2 p `B iBQM Q7 f Qp2` i?2 BMi2`p H [0, t]X t 1t KTH2 RXeXk, hrQ 2t KTH2bX h?2 7Q`KmH f (t) = 0 g(s) ds- r?2`2 g : [0, ∞) → R Bb HQ+ HHv BMi2;` #H2 7mM+iBQM- /2}M2b 7mM+iBQM f Q7 #QmM/2/ p `B@ iBQM Qp2` }MBi2 BMi2`p HbX HbQ MQM@/2+`2 bBM; 7mM+iBQM a : [0, ∞) → R Bb Q7 #QmM/2/ p `B iBQM Qp2` }MBi2 BMi2`p Hb M/ Vf (t) = a(t) − a(0)X +H bbB+ H `2bmHi Q7 M HvbBbR8 b vb i? i Mv `B;?i@+QMiBMmQmb 7mM+iBQM f : [0, ∞) → R Q7 #QmM/2/ p `B iBQM Qp2` }MBi2 BMi2`p Hb + M #2 2tT`2bb2/ b i?2 /Bz2`2M+2 #2ir22M irQ MQM@/2+`2 bBM; `B;?i@+QMiBMmQmb 7mM+iBQMb, R8
a22 7Q` BMbi M+2 (h vHQ`- RN38)- *? Ti2` NX
RXeX h>1 ahA1GhC1aĜG1"1a:l1 *G*lGla
jd
f (t) = f (0) + a(t) − b(t). 6Q` BMbi M+2-
URXkRV
b(t) = −f (t) + f (0) + Vf (t).
a(t) = Vf (t);
h?Bb T `iB+mH ` /2+QKTQbBiBQM Bb `272``2/ iQ b i?2 + MQMB+ H /2+QKTQbBiBQM Q7 f X t 6Q` BMbi M+2- B7 f (t) = 0 g(s) ds r?2`2 g : [0, ∞) → R Bb HQ+ HHv BMi2;` #H2 t t 7mM+iBQM- i?2M f (t) = 0 |g(s)| ds − 0 2g − (s) dsX AM i?2 #b2M+2 Q7 2tTHB+Bi K2MiBQM Q7 i?2 +QMi` `v- i?2 #QmM/2/ p `B iBQM 7mM+iBQMb ?2M+27Q`i? +QMbB/2`2/ `2 bbmK2/ +Q`HQH- i? i Bb- `B;?i@+QMiBMmQmb rBi? H27i@? M/ HBKBib f (t−) = limh↓0 f (t − h)X G2i f : [0, ∞) → R #2 +Q`HQH #QmM/2/ p `B iBQM 7mM+iBQM rBi? + MQMB+ H /2+QKTQbBiBQM URXkRVX hQ a M/ b- r?B+? `2 +Q`HQH M/ MQM@/2+`2 bBM;- `2 bbQ+B i2/ i?2 K2 bm`2b μa M/ μb QM ((0, ∞), B((0, ∞)) `2bT2+iBp2Hv- /2}M2/ #v, μa ((0, t]) = a(t) = Vf (t);
μb ((0, t]) = b(t) .
LQi2 i? i μb ≤ 2μa X q2 b? HH mb2 i?2 MQi iBQM μa (dt) := |df (t)|X K2 bm` #H2 7mM+iBQM u : [0, ∞) → R bm+? i? i |u(s)| μa (ds) < ∞ (0,∞)
U M/ i?2`27Q`2 (0,∞) |u(s)| μb (ds) < ∞V Bb b B/ iQ #2 G2#2b;m2ĜaiB2HiD2b BMi2;` #H2 rBi? `2bT2+i iQ f - M/ Bib G2#2b;m2ĜaiB2HiD2b BMi2;` H Bb- #v /2}MBiBQM- i?2 [m MiBiv u(s) df (s) := u(s) μa (ds) − u(s) μb (ds). (0,∞)
(0,∞)
(0,∞)
qBi? i?Bb /2}MBiBQM- r2 + M i?2`27Q`2 r`Bi2 u(s) df (s) := u(s) da(s) − (0,∞)
(0,∞)
u(s) db(s). (0,∞)
G2i f1 , f2 : R+ → R #2 irQ +Q`HQH 7mM+iBQMb Q7 HQ+ HHv #QmM/2/ p `B iBQM rBi? + MQMB+ H `2T`2b2Mi iBQMb fi (t) = fi (0) + ai (t) − bi (t)- i = 1, 2X h?2 bvK#QHB+ 2tT`2bbBQM u(x, y) df1 (x) df2 (y) (0,∞)
()
(0,∞)
Bb iQ #2 BMi2`T`2i2/ #v r`BiBM; df1 (x) df2 (y) = μa1 (dx)μa2 (dy)+μb1 (dx)μb2 (dy)−μa1 (dx)μb2 (dy)−μa2 (dx)μb1 (dy). h?2 [m MiBiv UV Bb r2HH /2}M2/ B7 i?2 7Qm` +Q``2bTQM/BM; BMi2;` Hb `2 r2HH /2}M2/ M/ B7 KQ`2Qp2` i?2 //BiBQMb M/ bm#i` +iBQMb BMpQHp2/ /Q MQi H2 / iQ BM/2i2`KB@ M i2 7Q`KbX AM pB2r Q7 i?Bb /2+QKTQbBiBQM- i?2 +QM/BiBQMb Q7 TTHB+ iBQM Q7 6m#BMBǶb 7Q`KmH 7QHHQr 7`QK i?2 bi M/ `/ + b2 Q7 T`Q/m+i K2 bm`2bX h?2Q`2K RXeXj G2i f, g : R+ → R #2 irQ +Q`HQH 7mM+iBQMb Q7 HQ+ HHv #QmM/2/ p `B iBQMX h?2M- 7Q` HH t ≥ 0 f (t)g(t) = f (0)g(0) + f (s) dg(s) + g(s−) df (s) . URXkkV (0,t]
(0,t]
j3
*>Sh1_ RX :1L1_GAhA1a
S`QQ7X Ai Bb 2MQm;? iQ T`Qp2 i?Bb 7Q` MQM@M2; iBp2 7mM+iBQMb f M/ gX "v 6m#BMB df (x) dg(y) = df (x) dg(y) . (f (t) − f (0))(g(t) − g(0)) = (0,t]
(0,t]
(0,t]
(0,t]
G2i D := (0, t] × (0, t] ,
D1 := {(x, y) ∈ D ; x ≤ y} ,
M/ i?2`27Q`2
df (x) dg(y) = (0,t]
D2 := {(x, y) ∈ D ; x > y} ,
df (x) dg(y) .
df (x) dg(y) +
(0,t]
D1
D2
"v 6m#BMB df (x) dg(y) = df (x) dg(y) D1 (0,t] (0,y] (f (y) − f (0)) dg(y) = f (y) dg(y) − f (0)(g(t) − g(0)) . = (0,t]
(0,t]
aBKBH `Hv
df (x) dg(y) = D2
dg(y)
(0,t]
(0,t]
df (x)
(0,x)
(g(x−) − g(0)) df (x)
=
g(x−) df (x) − g(0)(f (t) − f (0)).
= (0,t]
*QK#BMBM; i?2 #Qp2 2[m HBiB2b H2 /b iQ i?2 MMQmM+2/ `2bmHiX
*Q`QHH `v RXeX9 G2i a : R+ → R #2 `B;?i@+QMiBMmQmb MQM@/2+`2 bBM; 7mM+iBQMX h?2M- 7Q` HH n ≥ 1 an (t) − an (0) ≤ an−1 (s−) da(s) ≤ an−1 (s) da(s) . n (0,t] (0,t] S`QQ7X q2 b? HH mb2 i?2 MQi iBQM h− iQ `2T`2b2Mi i?2 7mM+iBQM t → h(t−) r?2M h Bb +Q`HQH 7mM+iBQMX qBi? i?Bb MQi iBQM- URXkkV `2 /b d(f g) = f− dg + g df X TTHvBM; i?Bb `mH2 iQ f = an−1 M/ g = a- r2 Q#i BM n−1 d(an ) = an−1 ) − da + a d(a n−1 n−2 = a− da + aa− da + a2 d(an−2 ) = (an−1 + aan−2 + · · · + an−1 ) . − −
h?2 `2bmHi 7QHHQrb bBM+2 a− ≤ aX
RXeX h>1 ahA1GhC1aĜG1"1a:l1 *G*lGla
jN
h?2 1tTQM2MiB H _mH2 G2i a : R+ → R #2 `B;?i@+QMiBMmQmb 7mM+iBQM Q7 #QmM/2/ p `B iBQM bm+? i? i a(0) = 0- M/ H2i u : R+ → R #2 bm+? i? i 7Q` HH t ≥ 0 |u(s)| da(s) < ∞ . (0,t]
h?2Q`2K RXeX8 h?2 2[m iBQM
x(s−)u(s) da(s)
x(t) = x(0) + (0,t]
/KBib mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM ;Bp2M #v t x(t) = x(0) (1 + u(s)Δa(s)) × e o ac (ds) , 0 n0 X amTTQb2 BM //BiBQM i? i QM i?2 BMi2`p Hb [tn , tn+1 ) i? i HB2 BM R+
t
f (t) = f (tn ) +
f (s) ds ,
tn
7Q` bQK2 HQ+ HHv BMi2;` #H2 7mM+iBQM f Ui?2 /2`Bp iBp2VX G2i MQr G : R → R #2 /Bz2`2MiB #H2 7mM+iBQM rBi? /2`Bp iBp2 G X h?2M 7Q` HH [a, b) ⊂ R+ -
9k
*>Sh1_ RX :1L1_GAhA1a
G(f (b)) = G(f (a)) +
b
(G(f (tn )) − G(f (tn −)) +
f (s)G (f (s)) ds . URXk9V
a
tn ∈(a,b)
6Q` BMbi M+2- rBi? G(x) = eαx b eαf (b) = eαf (a) + f (s)αeαf (s) ds . eαf (tn ) − eαf (tn −) + a
tn ∈(a,b)
AM T `iB+mH `- B7 tn = Tn r?2`2 {Tn }n≥1 Bb i?2 b2[m2M+2 Q7 TQBMib Q7 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb N QM R+ b f (s)αeαf (s) ds , eα(f (Tn )−f (Tn −)) − 1 eαf (Tn −) + eαf (b) = eαf (a) + a
n≥1
i? i Bb- rBi? i?2 BMi2;` H MQi iBQM b α(f (s)−f (s−)) eαf (b) = eαf (a) + e − 1 eαf (s−) N (ds) + f (s)αeαf (s) ds . a
(a,b]
1t KTH2 RXeXN, h?2 HBF2HB?QQ/ ` iBQ 7Q`KmH - i F2 RX h?2 iBiH2 Q7 i?2 2t KTH2 `272`b iQ 7mM/ K2Mi H 7Q`KmH Q7 TQBMi T`Q+2bb i?2Q`v i? i r2 b? HH 2M+QmMi2` BM am#b2+iBQM 8X8 Uh?2Q`2K 8X8XRVX G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM i?2 HBM2- M/ H2i {λ(t)}t≥0 M/ {μ(t)}t≥0 #2 irQ MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 biQ+? biB+ T`Q+2bb2bX h?2 biQ+? biB+ T`Q+2bb {L(t)}t≥0 i? i Bb i?2 mMB[m2 bQHmiBQM Q7 i?2 2[m iBQM L(t) = L(0) + L(s−)(μ(s) − 1)dM (s) () r?2`2 M (t) := N (0, t] −
(0,t]
t
λ(s) ds- Bb t L(t) = L(0) μ(s)N ({s}) e− 0 (μ(s)−1)λ(s) ds , 0
()
0 u}- r?B+? Bb BM/22/ 2[m H iQ a(u)X *Q`QHH `v RXeXRk G2i a : R+ → R+ #2 MQM@/2+`2 bBM; +QMiBMmQmb 7mM+iBQM bm+? i? i a(0) = 0X h?2M- 7Q` HH K2 bm` #H2 7mM+iBQMb g : R+ → R+
∞
a(∞)
g((a(t)) da(t) = 0
g(t) dt.
URXkeV
0
S`QQ7X P#b2`p2 i? i bBM+2 a Bb +QMiBMmQmb- a(c(t)) = t r?2M t ∈ [0, a(+∞)) M/ TTHv 7Q`KmH URXk8V rBi? f = g ◦ aX
RXd
1t2`+Bb2b
1t2`+Bb2 RXdXRX LQi bBKTH2 TQBMi T`Q+2bb5 aBKTHB+Biv Q7 TQBMi T`Q+2bb N QM Rm Bb MQi ;m ` Mi22/ #v i?2 `2[mB`2K2Mi P (N ({a}) ≤ 1) = 1 7Q` HH a ∈ Rm X :Bp2 UbBKTH25V +QmMi2`@2t KTH2X 1t2`+Bb2 RXdXkX h?2 TQBMi b2[m2M+2 Q7 HQ+ HHv 7BMBi2 TQBMi T`Q+2bb X G2i μ #2 HQ+ HHv }MBi2 TQBMi K2 bm`2 QM (R, B(R))X a?Qr i? i UBV i?2`2 2tBbib MQM@/2+`2 bBM; b2[m2M+2 {tn }n∈Z BM R rBi? Ui?2 +QMp2MiBQMV t0 ≤ 0 < t1 M/ bm+? i? i 7Q` HH K2 bm` #H2 b2ib C εtn (C) μ(C) = n∈Z
99
*>Sh1_ RX :1L1_GAhA1a
Ur?2`2 εa Bb i?2 .B` + K2 bm`2 i a B7 a ∈ R- M/ i?2 MmHH K2 bm`2 B7 a = +∞ Q` −∞VUBBV i?2 K TTBM;b μ → tn = tn (μ) `2 K2 bm` #H2 7`QK (M (E), M(E)) iQ (R, B) M/ UBBBV B7- BM //BiBQM- μ Bb bBKTH2- i?2M |tn | < ∞ ⇒ tn < tn+1 X UAM Qi?2` rQ`/b{tn }n∈Z Bb bi`B+iHv BM+`2 bBM; QM RXV "X S`Qp2 G2KK RXRX8 r?2M E = Rm X 1t2`+Bb2 RXdXjX J2 bm` #BHBiv Q7 TQBMi T`Q+2bb BMi2;` Hb G2i N #2 ` M/QK K2 bm`2 QM (E, B(E)) M/ H2i ϕ: (E, E) → (R, B) #2 MQM@ M2; iBp2 K2 bm` #H2 7mM+iBQMX a?Qr i? i N (ϕ) := E ϕ(t) N (dt) Bb r2HH /2}M2/ M/ ` M/QK p `B #H2X 1t2`+Bb2 RXdX9X AMi2MbBiv J2 bm`2 Q7 J `F2/ SQBMi S`Q+2bb rBi? AM/2T2M/2Mi BB/ J `Fb S`Qp2 i? i i?2 BMi2MbBiv K2 bm`2 Q7 i?2 K `F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd Bb i?2 T`Q/m+i K2 bm`2 ν × QZ - r?2`2 ν Bb i?2 BMi2MbBiv K2 bm`2 Q7 i?2 # b2 TQBMi T`Q+2bbX 1t2`+Bb2 RXdX8X SQBMib QM i?2 ;`B/ AM 1t KTH2 RXRXRk- M/ BM i?2 + b2 i? i {Xn1 ,n2 }n1 ,n2 ∈Z Bb M BM/2T2M/2Mi b2[m2M+2+ M r2 b v i? i +QM/BiBQM limn1 ,n2 ↑∞ pn1 ,n2 = 0 BKTHB2b i? i i?2 +Q``2bTQM/BM; TQBMi T`Q+2bb Bb HQ+ HHv }MBi2\ 1t2`+Bb2 RXdXeX h?2 n@i? TQBMi Q7 SQBbbQM T`Q+2bb G2i N #2 SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv 7mM+iBQM λ : R → RX .2MQiBM; #v Tn i?2 n@i? TQBMi Q7 N bi`B+iHv iQ i?2 `B;?i Q7 i?2 Q`B;BM- T`Qp2 i? i Tn Bb M #bQHmi2Hv +QMiBMmQmb ` M/QK p `B #H2 M/ ;Bp2 Bib T`Q# #BHBiv /2MbBivX 1tT`2bb BM i2`Kb Q7 i?2 BMi2MbBiv 7mM+iBQM i?2 +mKmH iBp2 T`Q# #BHBiv /Bbi`B#miBQM FT1 Q7 T1 M/ Bib T`Q# #BHBiv /2MbBiv 7mM+iBQM fT1 X o2`B7v i? i λ : R → R Bb i?2 ? x `/ ` i2 7mM+iBQM Q7 T1 X 1t2`+Bb2 RXdXdX 6 +iQ`B H K2 bm`2b Q7 SQBbbQM T`Q+2bb2b S`Qp2 i? i i?2 n@i? 7 +iQ`B H KQK2Mi K2 bm`2 Q7 SQBbbQM T`Q+2bb QM Rm rBi? K2 M K2 bm`2 ν Bb ν ⊗n X 1t2`+Bb2 RXdX3X h?2 n@i? 7 +iQ`B H KQK2Mi K2 bm`2 Q7 i?BMM2/ TQBMi T`Q+2bbX G2i N #2 M n@i? Q`/2` TQBMi T`Q+2bb rBi? b2[m2M+2 Q7 TQBMib {Xn }n∈N X :Bp2 i?2 ! Q7 i?2 i?BMM2/ TQBMi T`Q+2bb Nthin,p(·) n@i? Q`/2` 7 +iQ`B H KQK2Mi K2 bm`2 M n Ua22 .2}MBiBQM RXjXkyVX .2/m+2 7`QK i?2 `2bmHi i? i i?BMM2/ /2i2`KBM Mi H TQBMi T`Q+2bb Bb HbQ /2i2`KBM Mi H TQBMi T`Q+2bbX 1t2`+Bb2 RXdXNX .2i2`KBM Mi H T`Q+2bb2b `2 bBKTH2 S`Qp2 i? i /2i2`KBM Mi H TQBMi T`Q+2bb Bb bBKTH2X
RXdX 1s1_*Aa1a
98
1t2`+Bb2 RXdXRyX SQBbbQM TQBMib mM/2` +m`p2 #2 M ?TT QM R2 rBi? BMi2MbBiv 1X G2i λ : R → R #2 MQM@M2; iBp2 HQ+ HHv G2i N BMi2;` #H2 7mM+iBQMX .2}M2 TQBMi T`Q+2bb N QM R #v N (C) = R
R
(dt × dz) . 1C (t) 1[0,λ(t)] (z) N
S`Qp2 i? i N Bb SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv 7mM+iBQM λ : R → RX 1t2`+Bb2 RXdXRRX JQK2Mi K2 bm`2b Q7 TQBMi T`Q+2bb X G2i N #2 b2+QM/@Q`/2` TQBMi T`Q+2bb QM i?2 HX+X/X#X EX a?Qr i? i i?2 7Q`KmH M2 (A × B) := E [N (A) N (B)] mMB[m2Hv /2}M2b K2 bm`2 M2 QM (E × E, B(E × E)) M/ i? i i?Bb K2 bm`2 Bb HQ+ HHv }MBi2X "X G2i N #2 SQBbbQM T`Q+2bb QM i?2 HX+X/X#X E rBi? BMi2MbBiv K2 bm`2 νX S`Qp2 i? i M2 (A1 × A2 ) = ν(A1 )ν(A2 ) − ν(A1 ∩ A2 ) M/ M2! (A1 × A2 ) = ν(A1 )ν(A2 ) . *X A7 N Bb M ?TT QM Rm Q7 BMi2MbBiv λ- T`Qp2 i? i i?2 n@i? 7 +iQ`B H KQK2Mi K2 bm`2 Mn! /KBib /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 n ;Bp2M #v ρ!n (x1 , . . . , xn ) = λn .
1t2`+Bb2 RXdXRkX _ M/QK TQBMib mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] *QMbi`m+i TQBMi T`Q+2bb N QM R BM i?2 7QHHQrBM; r vX 6B`bi /` r }MBi2 BMi2;2`@ p Hm2/ ` M/QK p `B #H2 T - M/ i?2M M BB/ b2[m2M+2 {Un }n≥1 mMB7Q`KHv /Bbi`B#mi2/ QM 0VX 6BM HHv- H2i N =
T[0, 1]- BM/2T2M/2Mi Q7 T X .2}M2 αk := P (T = k) Uk ≥ 0 ε r?2`2 ε Bb i?2 .B` + K2 bm`2 i a M/ r?2`2 U a k k=1 k=1 εUk Bb i?2 MmHH K2 bm`2 #v +QMp2MiBQMX q? i Bb i?2 G TH +2 i` Mb7Q`K Q7 N \ q? i #Qmi i?2 + b2 r?2`2 T Bb SQBbbQM p `B #H2 Q7 K2 M θ\ 1t2`+Bb2 RXdXRjX G TH +2 i` Mb7Q`K Q7 i?BMM2/ TQBMi T`Q+2bb S`Qp2 h?2Q`2K RXjXkRX 1t2`+Bb2 RXdXR9X h?2 G TH +2 7mM+iBQM H Q7 i?2 L2vK Mėa+Qii +Hmbi2` TQBMi T`Q+2bb *QKTmi2 i?2 G TH +2 7mM+iBQM H Q7 i?2 L2vK MĜa+Qii +Hmbi2` TQBMi T`Q+2bb Q7 1t@ KTH2 RX8X8 BM i2`Kb Q7 i?2 ;2M2` iBM; 7mM+iBQM Q7 T M/ Q7 i?2 +QKKQM T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 Un ǶbX
9e
*>Sh1_ RX :1L1_GAhA1a
1t2`+Bb2 RXdXR8X h?2 HBMF #2ir22M i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM M/ i?2 G TH +2 7mM+iBQM H G2i N #2 TQBMi T`Q+2bb QM Rm rBi? G TH +2 7mM+iBQM H LN X a?Qr i? i Bib pQB/@ M+2 T`Q# #BHBiv 7mM+iBQM v Bb ;Bp2M #v v(C) = lim LN (t1C ) . t↑+∞
1t2`+Bb2 RXdXReX +? ` +i2`Bx iBQM Q7 i?2 SQBbbQM T`Q+2bb 6Q` bBKTH2 TQBMi T`Q+2bb N QM Rm iQ #2 SQBbbQM T`Q+2bb Q7 σ@}MBi2 BMi2MbBiv K2 bm`2 ν Bi Bb M2+2bb `v M/ bm{+B2Mi i? i 7Q` HH "Q`2H b2i C ∈ Rm Q7 }MBi2 ν@K2 bm`2- N (C) #2 SQBbbQM p `B #H2 Q7 K2 M ν(C)X Uh?2 bm`T`BbBM; 7 +i Bb i? i QM2 /Q2b MQi M22/ i?2 BM/2T2M/2M+2 bbmKTiBQM Q7 .2}MBiBQM RXRXNXV 1t2`+Bb2 RXdXRdX T`QT2`iv Q7 i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM U V a?Qr i? i B7 v1 M/ v2 `2 i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQMb Q7 bQK2 TQBMi T`Q+2bb2b QM Rm - bQ Bb v1 × v2 X U#V G2i v #2 i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM Q7 bQK2 TQBMi T`Q+2bb QM Rm M/ H2i gT #2 i?2 ;2M2` iBM; 7mM+iBQM Q7 bQK2 ` M/QK p `B #H2 i FBM; Bib p Hm2b BM NX a?Qr i? i w(C) := gT (v(C)) Bb i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM Q7 bQK2 TQBMi T`Q+2bb QM Rm X 1t2`+Bb2 RXdXR3X *QKT `BbQM Q7 pQB/ M+2 7mM+iBQMb *QKT `2 i?2 pQB/ M+2 7mM+iBQMb Q7 M ?TT Q7 BMi2MbBiv λ M/ Q7 ?QKQ;2M2Qmb *Qt T`Q+2bb Q7 BMi2MbBiv Λ bm+? i? i E [Λ] = λX 1t2`+Bb2 RXdXRNX M 2tTQM2MiB H 7Q`KmH G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM (a, b]X a?Qr i? i 7Q` HH u ∈ R HH t ∈ (a, b]- r2 ? p2 i? i UT v ii2MiBQM iQ i?2 T `2Mi?2b2b M/ b[m `2 #` +F2ibV iuN ((a,b]) iu e = 1 + (e − 1) eiuN ((a,s)) N (ds) . (a,b]
1t2`+Bb2 RXdXkyX h?2 rQ`FHQ / T`Q+2bb G2i σ #2 TQbBiBp2 MmK#2` M/ H2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ X G2i {W (t)}t∈R+ #2 `B;?i@+QMiBMmQmb biQ+? biB+ T`Q+2bb rBi? H27i@? M/ HBKBib M/ /2+`2 bBM; +QMbi MiHv i ` i2 1 2t+2Ti BM irQ +B`+mKbi M+2b, UBV i DmKT Q7 N r?2`2 Bi DmKTb 7`QK W (t−) iQ W (t) = W (t−) + σ- M/ UBBV r?2M Bi `2 +?2b i?2 p Hm2 0- BM r?B+? + b2 Bi `2K BMb MmHH mMiBH i?2 M2ti 2p2Mi iBK2 Q7 N X a?Qr i? i 7Q` HH u ∈ R- HH t ∈ R+
eiuW (t) = eiuW (0) + (eiuσ − 1)
eiuW (s−) N (ds) − iu (0,t]
t
1{W (s)>0} . 0
RXdX 1s1_*Aa1a
9d
U*QKK2Mi, i?2 iBiH2 Q7 i?2 2t2`+Bb2 `272`b iQ b2`pB+2 bvbi2K BM r?B+? 2 +? +mb@ iQK2` 2Mi2`BM; i?2 bvbi2K `2[mB`2b b2`pB+2 iBK2 σ iQ mMB[m2 b2`p2` /2HBp2`BM; b2`pB+2 i mMBi iBK2 b HQM; i?2`2 Bb rQ`F iQ #2 /QM2- M/ r?2`2 i?2 +mbiQK2`bǶ ``Bp H iBK2b 7Q`K TQBMi T`Q+2bb N X h?2 [m MiBiv W (t) Bb i?2 iQi H `2K BMBM; rQ`F i iBK2 t- M/ W (0) Bb i?2 BMBiB H HQ /X Ua22 1t KTH2 8XRXjyXVV
*? Ti2` k SQBbbQM S`Q+2bb2b QM i?2 GBM2 SQBbbQM T`Q+2bb2b QM i?2 HBM2 #2HQM; Q7 +Qm`b2 iQ i?2 H `;2` + i2;Q`v Q7 bT iB H SQBb@ bQM T`Q+2bb2b i? i rBHH #2 i`2 i2/ BM i?2 M2ti +? Ti2`X >Qr2p2`- i?2 TQBMi Q7 pB2r /QTi2/ BM i?Bb +? Ti2` Bb ` /B+ HHv /Bz2`2MiX h?2 T`BM+BT H `2bmHi Bb i?2 bKQQi?BM; 7Q`KmH - r?B+? QT2Mb i?2 r v iQ i?2 biQ+? biB+ + H+mHmb Q7 TQBMi T`Q+2bb2b rBi? biQ+? biB+ BMi2MbBiv Q7 *? Ti2` 8X Ai rBHH #2 mb2/ BM a2+iBQM kX9 7Q` T`QQ7 Q7 q i M #2Ƕb i?2Q`2K Q7 +? ` +i2`Bx iBQM Q7 M ?TT BM i2`Kb Q7 K `iBM; H2bX a2+iBQM kX8 b?Qrb ?Qr +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J `FQp +? BM + M #2 /2b+`B#2/ BM i2`Kb Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQM Ub/2V /`Bp2M #v SQBbbQM T`Q+2bb2bX UJQ`2 2t KTH2b QM i?2 b/2 TQBMi Q7 pB2r BM TQBMi T`Q+2bb i?2Q`v rBHH #2 ;Bp2M i p `BQmb TH +2b BM i?2 b2[m2HXV MQi?2` `2K `F #H2 `2T`2b2Mi iBQM Q7 +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J `FQp +? BM BM i2`Kb Q7 iBK2@b+ H2/ ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Bb ;Bp2M BM a2+iBQM kXeX
kXR
*QmMiBM; S`Q+2bb M/ AMi2`p H a2[m2M+2
_2+ HH i? i bBKTH2 M/ HQ+ HHv }MBi2 biQ+? biB+ TQBMi T`Q+2bb QM i?2 TQbBiBp2 ? H7@ HBM2 Bb b2[m2M+2 {Tn }n≥1 Q7 TQbBiBp2 UTQbbB#Hv BM}MBi2V ` M/QK p `B #H2b bm+? i? i- HKQbi bm`2HvUBV 0 < Tn < Tn+1 r?2M2p2` Tn < ∞ UbBKTHB+BivV- M/ UBBV limn↑∞ Tn = +∞ UHQ+ H }MBi2M2bbVX h?Bb b2[m2M+2 Bb i?2 2p2Mi iBK2 b2[m2M+2X aQK2iBK2b- QM2 `272`b iQ Tn b i?2 n@i? dzTQBMiǴ Q7 i?2 TQBMi T`Q+2bbX h?2 BMi2;2`@p Hm2/ ` M/QK p `B #H2 N ((a, b]) = 1(a,b] (Tn ) n≥1
+QmMib i?2 2p2Mi iBK2b Q++m``BM; BM i?2 iBK2 BMi2`p H (a, b] ⊂ R+ X 6Q` ivTQ;` T?@ B+ H bBKTHB+Biv- r2 Q++ bBQM HHv /2MQi2 i?Bb p `B #H2 #v N (a, b]- i?mb QKBiiBM; i?2 2ti2`M H T `2Mi?2b2bX 6Q` t ≥ 0- H2i N (t) := N (0, t]. AM T `iB+mH `- N (0) = 0 M/ N (a, b] = N (b)−N (a)X aBM+2 i?2 BMi2`p H (0, t] Bb +HQb2/ QM i?2 `B;?i- i?2 i` D2+iQ`B2b UQ` b KTH2 T i?bV t → N (t, ω) `2 `B;?i@+QMiBMmQmb-
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_2
9N
8y
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
MQM@/2+`2 bBM;- ? p2 HBKBib QM i?2 H27i M/ DmKT QM2 mMBi mTr `/b i 2 +? 2p2Mi iBK2 Q7 i?2 TQBMi T`Q+2bbX h?2 biQ+? biB+ T`Q+2bb {N (t)}t≥0 Bb + HH2/ i?2 +QmMiBM; T`Q+2bb Q7 i?2 TQBMi T`Q+2bbX aBM+2 i?2 b2[m2M+2 Q7 2p2Mi iBK2b + M #2 `2+Qp2`2/ 7`QK N - i?2 H ii2` HbQ `2+2Bp2b i?2 TT2HH iBQM dzTQBMi T`Q+2bbǴX q2 `2T2 i .2}MBiBQM RXRXN BM i?2 T `iB+mH ` b2iiBM; Q7 TQBMi T`Q+2bb2b QM i?2 `2 H HBM2X .2}MBiBQM kXRXR SQBbbQM T`Q+2bb QM R+ rBi? HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 ν Bb TQBMi T`Q+2bb N QM R+ bm+? i? i (α) 7Q` HH k ∈ N+ - HH Kmim HHv /BbDQBMi BMi2`p Hb Ij := (aj , bj ] (1 ≤ j ≤ k)i?2 ` M/QK p `B #H2b N (Ij ) (1 ≤ j ≤ k) `2 BM/2T2M/2Mi- M/ (β) 7Q` Mv BMi2`p H (a, b] ⊂ R+ - N (a, b] Bb SQBbbQM ` M/QK p `B #H2 Q7 K2 M ν((a, b])X 1[mBp H2MiHv- 7Q` HH Kmim HHv /BbDQBMi BMi2`p Hb Ij = (aj , bj ] ⊂ R+ M/ HH `2 H MmK#2`b uj U1 ≤ j ≤ kV E e
i
k
j=1
uj N (Ij )
= exp
k
e
iuj
− 1 ν((aj , bj ])
.
j=1
"v h?2Q`2K jXRXj- r?B+? r2 MiB+BT i2- B7 ν Bb /Bzmb2- N Bb bBKTH2X .2}MBiBQM kXRXk A7 BM .2}MBiBQM kXRXR- ν(dt) = λ(t) dt 7Q` bQK2 HQ+ HHv BMi2;` #H2 K2 bm` #H2 7mM+iBQM t → λ(t)- N Bb + HH2/ SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t)X A7 KQ`2Qp2` λ(t) ≡ λ- N Bb + HH2/ ?QKQ;2M2Qmb SQBbbQM T`Q+2bb U?TTV rBi? BMi2MbBiv λX .2}MBiBQM kXRXj G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM (0, ∞) rBi? 2p2Mi iBK2b b2[m2M+2 {Tn }n≥1 - M/ H2i {Zn }n≥1 #2 M BB/ b2[m2M+2 Q7 `2 H@p Hm2/ K `FbBM/2T2M/2Mi Q7 N X h?2 biQ+? biB+ T`Q+2bb Y (t) :=
N (t)
Zn
(t ≥ 0)
n=1
Bb + HH2/ +QKTQmM/ U?QKQ;2M2QmbV SQBbbQM T`Q+2bbX 1t KTH2 kXRX9, h?2 _BbF JQ/2H- h F2 RX PM2 Q7 i?2 }`bi bB;MB}+ Mi mb2b Q7 SQBbbQM T`Q+2bb2b BM TTHB+ iBQMb Q++m``2/ BM i?2 BMbm` M+2 `BbF KQ/2H Q7 *` Kû` M/ GmM/#2`;X AM i?2 BMbm` M+2 #mbBM2bb- QM2 Q7 i?2 +2Mi` H T`Q#H2Kb Bb i?2 2p H@ m iBQM Q7 i?2 T`Q# #BHBiv Q7 `mBM Q7 ;Bp2M BMbm` M+2 +QKT MvX h?2 bi M/ `/ KQ/2H 72 im`2b i?2 +QKTQmM/ SQBbbQM T`Q+2bb Q7 .2}MBiBQM kXRXj- r?2`2 i?2 K `Fb ? p2 +QKKQM +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM G- bm+? i? i G(0) = 0- rBi? K2 M μ < ∞ M/ p `B M+2 σ 2 < ∞X h?2 `BbF T`Q+2bb Bb /2}M2/ QM [0, ∞) #v
kXRX *PlLhAL: S_P*1aa L. ALh1_oG a1Zl1L*1 X(t) = ct −
N (t)
8R
Zn ,
n=1
r?2`2 c Bb TQbBiBp2 `2 H +QMbi MiX h?2 BMi2`T`2i iBQM Bb i? i c Bb i?2 ;`Qbb `BbF T`2KBmK Ui? i Bb- i?2 ` i2 Q7 BM+QK2 T2` mMBi iBK2V- Tn Bb i?2 iBK2 Q7 Q++m``2M+2 Q7 +H BK M/ Zn Bb i?2 bBx2 Q7 i?2 +H BKX G2i F N := σ(N (s) ; s ≥ 0) #2 i?2 σ@}2H/ `2+Q`/BM; HH i?2 2p2Mib Q7 N X _2+ HH i? i i?2 ?TTb N1 , N2 , . . . `2 + HH2/ BM/2T2M/2Mi B7 i?2 σ@}2H/b F N1 , F N2 , . . . `2 BM/2T2M/2Mi- i? i Bb- B7 7Q` HH A1 ∈ F N1 , A2 ∈ F N2 , . . .- i?2 7 KBHv Q7 2p2Mib (A1 , A2 , . . .) Bb BM/2T2M/2MiX h?2Q`2K kXRX8 G2i Ni (i ≥ 1) #2 7 KBHv Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λi (i ≥ 1)X h?2MUBV irQ /BbiBM+i ?TTb Q7 i?Bb 7 KBHv ? p2 MQ TQBMib BM +QKKQM- M/
∞
UBBV B7 ∞ i=1 λi = λ < ∞- i?2 bmK N := i=1 Ni Bb M ?TT rBi? BMi2MbBiv λX S`QQ7X UBV Ai bm{+2b iQ T`Qp2 i?Bb 7Q` irQ ?TTb- M/ i?2 `2bmHi Bb i?2M /B`2+i +QMb2[m2M+2 Q7 h?2Q`2K RXkX9X 6Q` UBBV- r Bi 7Q` h?2Q`2K jXRXkX L2ti `2bmHi Bb + HH2/ i?2 +QKT2iBiBQM i?2Q`2K 7Q` ?TTbX
∞ h?2Q`2K kXRXe AM i?2 bBim iBQM Q7 h?2Q`2K i=1 λi = λ < ∞ ∞ kXRX8- r?2`2 /2MQi2 #v Z i?2 }`bi 2p2Mi iBK2 Q7 N := i=1 Ni M/ #v J i?2 BM/2t Q7 i?2 ?TT `2bTQMbB#H2 7Q` BiX AM T `iB+mH `- Z Bb i?2 }`bi 2p2Mi Q7 NJ X h?2M J M/ Z `2 BM/2T2M/2Mi- P (J = i) = λλi M/ Z Bb M 2tTQM2MiB H ` M/QK p `B #H2 Q7 K2 M λ−1 X X1
1
J=2
3
X2 =Z X3
S`QQ7X * HH X1 , X2 , . . . i?2 }`bi 2p2Mi iBK2b Q7 N1 , N2 , . . .X h?2b2 `2 BM/2T2M/2Mi 2tTQM2MiB H p `B #H2b rBi? `2bT2+iBp2 T ` K2i2`b λ1 , λ2 , . . .X aBM+2 i?2b2 ` M/QK p `B #H2b `2 #bQHmi2Hv +QMiBMmQmb M/ BM/2T2M/2Mi- i?2 BM/2t J Bb HKQbi bm`2Hv mM K#B;mQmbHv /2}M2/X q2 ? p2P (J = i, Z ≥ a) = P (Xi ≥ a, Xj ≥ Xi 7Q` HH j = i) ∞ = λi e−λi s P (Xj ≥ s 7Q` HH j = i) ds a ∞ ∞ λi = λi e−λi s Πj=i e−λj s ds = λi e−λs ds = e−λa . λ a a
8k
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
G2iiBM; a → ∞ vB2H/b P (J = i) = λλi X amKKBM; rBi? `2bT2+i iQ i ;Bp2b P (Z ≥ a) = e−λa X h?2`27Q`2 P (J = i, Z ≥ a) = P (J = i)P (Z ≥ a)- 7Q` HH i ∈ N+ M/ HH a ∈ R+ - r?B+? T`Qp2b BM/2T2M/2M+2X h?2 ai`QM; J `FQp S`QT2`iv 6Q` Mv ` M/QK p `B #H2 τ rBi? p Hm2b BM [0, +∞]- H2i N τ #2 i?2 `2bi`B+iBQM Q7 N iQ R+ ∩ (0, τ ], N τ (a, b] := N ((a, b] ∩ (0, τ ]) ((a, b] ⊂ R+ ) . h?2 TQBMi T`Q+2bb N τ Bb dzN #27Q`2 τ ǴX h?2 TQBMi T`Q+2bb Sτ N - + HH2/ dzN 7i2` τ ǴBb /2}M2/ #v Sτ N (a, b] := N (τ + a, τ + b] ((a, b] ⊂ R+ ) . AM T `iB+mH `- B7 τ = ∞- Sτ N Bb i?2 2KTiv TQBMi T`Q+2bbX _2+ HH i?2 /2}MBiBQM Q7 M Ft @biQTTBM; iBK2- r?2`2 {Ft }t≥0 Bb 7 KBHv Q7 MQM@ /2+`2 bBM; σ@}2H/b Ui? i Bb Fs ⊆ Ft r?2M2p2` 0 ≤ s ≤ tVX Ai Bb ` M/QK p `B #H2 τ rBi? p Hm2b BM [0, +∞] bm+? i? i 7Q` HH a ∈ R+ - {τ ≤ a} ∈ Fa Ub22 .2}MBiBQM XkXdVX h?2Q`2K kXRXd G2i A #2 M `#Bi` `v BM/2t M/ H2i Nα (α ∈ A) #2 M BM/2T2M/2Mi 7 KBHv Q7 ?TTb QM R+ rBi? `2bT2+iBp2 BMi2MbBiB2b λα (α ∈ A)X G2i G #2 M `#Bi` `v σ@}2H/ BM/2T2M/2Mi Q7 F N := ∨α∈A F Nα X 6Q` HH t ≥ 0- /2}M2 FtN := ∨α∈A FtNα X G2i τ #2 M FtN ∨ G@biQTTBM;@iBK2 bm+? i? i P (τ < ∞) > 0X h?2M- ;Bp2M {τ < ∞}(α) F N M/ G `2 BM/2T2M/2Mi Q7 F Sτ N - M/ τ
(β) i?2 7 KBHv Sτ Nα Uα ∈ AV Bb M BM/2T2M/2Mi 7 KBHv Q7 ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λα (α ∈ A)X S`QT2`iB2b (α) M/ (β) +QMbiBimi2 r? i Bb + HH2/ i?2 bi`QM; J `FQp T`QT2`iv Q7 ?TTbX S`QQ7X h?2 T`QQ7 Bb ;Bp2M 7Q` bBM;H2 TQBMi ?TT N X h?2 ;2M2` H + b2 Bb +QKTH2i2Hv bBKBH `- QM+2 Bi Bb Q#b2`p2/ i? i i?2 BM/2T2M/2M+2 bi i2K2Mib +QM+2`M QMHv }MBi2 +QHH2+iBQMb Q7 TQBMi T`Q+2bb2b Q7 i?2 7 KBHv Nα Uα ∈ AVX AM i?2 T`QQ7- i?2 mb2 Q7 h?2Q`2Kb kXRX3 M/ kXRXN #2HQr Bb BKTHB+BiX Uh?2b2 `2bmHib HHQr mb iQ `2bi`B+i Qm` ii2MiBQM iQ }MBi2@/BK2MbBQM H /Bbi`B#miBQMb BM Q`/2` iQ T`Qp2 2Bi?2` B/2MiBiv Q7 i?2 /Bbi`B#miBQMb Q7 biQ+? biB+ T`Q+2bb2b Q` i?2B` BM/2T2M/2M+2XV h?2Q`2K kXRX3 G2i E #2 SQHBb? bT +2 M/ H2i E := B(E) #2 Bib "Q`2H σ@}2H/X G2i Q = {Q(t1 ,...,tk ) ; k ≥ 1, t1 , . . . , tk ∈ R} #2 7 KBHv Q7 T`Q# #BHBiv /Bbi`B#miBQMb QM (E k , E k ) b iBb7vBM; i?2 7QHHQrBM; +QKT iB#BHBiv +QM/BiBQMb C1 X 6Q` HH (t1 , . . . , tk ) ∈ Rk - M/ Mv T2`Kmi iBQM σ QM Rk Qσ(t1 ,...,tk ) = Qt1 ,...,tk ◦ σ −1 .
UkXRV
C2 X 6Q` HH (t1 , . . . , tk , tk+1 ) ∈ Rk+1 M/ HH A ∈ E k Q(t1 ,...,tk ) (A) = Q(t1 ,...,tk+1 ) (A × E).
UkXkV
kXRX *PlLhAL: S_P*1aa L. ALh1_oG a1Zl1L*1
8j
h?2M i?2`2 2tBbib mMB[m2 T`Q# #BHBiv K2 bm`2 P QM i?2 + MQMB+ H K2 bm` #H2 bT +2 (E R , E R ) bm+? i? i i?2 +QQ`/BM i2 T`Q+2bb {πt }t∈R /KBib i?2 }MBi2@/BK2MbBQM H /Bbi`B#miBQM QX h?2Q`2K kXRXN hrQ biQ+? biB+ T`Q+2bb2b {X(t)}t∈R M/ {Y (t)}t∈R - rBi? p Hm2b BM (E, E)- `2 BM/2T2M/2Mi B7 7Q` HH t1 , . . . , tk ∈ R M/ HH s1 , . . . , s ∈ R- i?2 p2+iQ`b (X(t1 ), . . . , X(tk )) M/ (Y (s1 ), . . . , Y (s )) `2 BM/2T2M/2MiX _2im`MBM; iQ i?2 T`QQ7 Q7 h?2Q`2K kXRXd- r2 }`bi b?Qr i? i Bi bm{+2b iQ T`Qp2 i? i 7Q` HH BMi2;2`b k, - HH u1 , . . . , uk - v1 , . . . , v - w ∈ R- HH Kmim HHv /BbDQBMi BMi2`p Hb (aj , bj ] U1 ≤ j ≤ kV- HH Kmim HHv /BbDQBMi BMi2`p Hb [cj , dj ] U1 ≤ j ≤ V M/ Mv `2 H@p Hm2/ ` M/QK p `B #H2 Z i? i Bb G@K2 bm` #H2- Bi ?QH/b i? i k τ E ei( j=1 uj (Sτ N )(aj ,bj ]+ j=1 vj N (cj ,dj ]+wZ ) 1{τ 1 aJPPh>AL: 6P_JlG
88
h?2 AMi2`@2p2Mi a2[m2M+2 h?2 b2[m2M+2 {Sn }n≥1 /2}M2/ #v T0 = 0 M/ Sn := Tn − Tn−1 U= ∞ B7 Tn−1 = ∞V Bb + HH2/ i?2 BMi2`@2p2Mi b2[m2M+2 Q`- BM i?2 TT`QT`B i2 +QMi2tii?2 BMi2`@ ``Bp H b2[m2M+2X h?2 bi`QM; J `FQp T`QT2`iv vB2H/b i?2 T`Q# #BHBbiB+ /2b+`BTiBQM Q7 M ?TT BM i2`Kb Bib BMi2`@2p2Mi iBK2bX h?2Q`2K kXRXRR h?2 BMi2`@2p2Mi b2[m2M+2 {Sn }n≥1 Q7 M ?TT QM i?2 TQbBiBp2 ? H7@HBM2 rBi? i?2 BMi2MbBiv λ > 0 Bb BB/- rBi? 2tTQM2MiB H /Bbi`B#miBQM Q7 T ` K2i2` λ- i? i Bb P (Sn ≤ t) = 1 − e−λt M/ BM T `iB+mH ` E[Sn ] = λ−1 X S`QQ7X TTHv h?2Q`2K kXRXd rBi? G = (Ω, ∅) M/ τ = Tn - iQ Q#i BM i? i i?2 }`bi 2p2Mi iBK2 Q7 STn N - M K2Hv Sn+1 - Bb BM/2T2M/2Mi Q7 N Tn U M/ BM T `iB+mH ` Q7 S1 , . . . , Sn V M/ Bb M 2tTQM2MiB H ` M/QK p `B #H2 rBi? T ` K2i2` λX h?Bb #2BM; i`m2 7Q` HH n ≥ 1- i?2 T`QQ7 Bb /QM2X
kXk
h?2 aKQQi?BM; 6Q`KmH
_2+ HH bQK2 MQi iBQM M/ i2`KBMQHQ;vX ?BbiQ`v QM (Ω, F) Bb- #v /2}MBiBQM- MQM@ /2+`2 bBM; 7 KBHv {Ft }t∈R Q7 bm#@σ@}2H/b Q7 FX G2i F∞ := ∨t∈R Ft #2 i?2 bK HH2bi σ@}2H/ +QMi BMBM; 7Q` HH t ∈ R HH b2ib Q7 Ft X biQ+? biB+ T`Q+2bb {X(t)}t∈R Bb b B/ iQ #2 Ft @ / Ti2/ B7 i?2 ` M/QK p `B #H2 X(t) Bb Ft @K2 bm` #H2 7Q` HH t ∈ RX 6Q` ;Bp2M TQBMi T`Q+2bb N QM R- M/ t ∈ R- /2}M2 i?2 σ@}2H/ FtN := σ(N (a, b] ; (a, b] ⊆ (−∞, t]) `2+Q`/BM; i?2 2p2Mib Q7 N mT iQ iBK2 tX Mv ?BbiQ`v {Ft }t∈R bm+? i? i FtN ⊆ Ft Ut ∈ RV Bb + HH2/ ?BbiQ`v Q7 N X h?2 ?BbiQ`v {FtN }t∈R Bb + HH2/ i?2 BMi2`M H ?BbiQ`v Q7 N X AM i?2 #Qp2 /2}MBiBQMb- R Bb i?2 BM/2t b2i- #mi Q7 +Qm`b2 bBKBH ` /2}MBiBQMb ?QH/ 7Q` T`Q+2bb2b M/ ?BbiQ`B2b BM/2t2/ #v Mv bm#b2i Q7 R UBM i?Bb b2+iBQM, R+ VX h?2Q`2K kXkXR G2i {Ft }t≥0 #2 ?BbiQ`v Q7 N - SQBbbQM T`Q+2bb QM R+ rBi? BMi2MbBiv K2 bm`2 νX amTTQb2 KQ`2Qp2` i? i 7Q` HH a > 0 M/ HH c, d (a ≤ c ≤ d)N ((c, d]) Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2 H27i@+QMiBMmQmb +QKTH2t biQ+? biB+ T`Q+2bb / Ti2/ iQ {Ft }t≥0 X amTTQb2 i? i i H2 bi QM2 Q7 i?2 irQ +QM/BiBQMb #2HQr Bb b iBb}2/, UBV {Z(t)}t≥0 Bb `2 H MQM@M2; iBp2X UBBV {Z(t)}t≥0 Bb +QKTH2t@p Hm2/ M/ E h?2M-
E (0,∞)
(0,∞)
Z(s)N (ds) = E 0
|Z(t)| ν(dt) < ∞X
∞
Z(s) ν(ds) .
UkXeV
8e
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
S`QQ7X X q2 }`bi i`2 i i?2 + b2 r?2`2 {Z(t)}t≥0 Bb `2 H- MQM@M2; iBp2 M/ #QmM/2/ M/ T`Qp2 i? i
T
E Z(s)N (ds) = E Z(s) ν(ds) , () 0
(0,T ]
r?2`2 T < ∞X G2i 7Q` HH n ≥ 1- HH ω ∈ Ω M/ HH t ≥ 0Zn (t, ω) :=
n −1 2
Z kT 2−n , ω 1(kT 2−n ,(k+1)T 2−n ] (t).
UkXdV
k=0
"v i?2 H27i@+QMiBMmBiv ?vTQi?2bBb- 7Q` HH ω ∈ Ω M/ HH t ∈ [0, T ]lim Zn (t, ω) = Z(t, ω).
n↑∞
q2 }`bi +?2+F i? i UV Bb i`m2 r?2M {Z(t)}t≥0 Bb `2TH +2/ #v Bib TT`QtBK iBQM {Zn (t)}t≥0 X AM/22/- i?2 H27i@? M/ bB/2 Q7 i?Bb 2[m HBiv Bb i?2M 2n −1
−n −n −n E N ((kT 2 , (k + 1)T 2 ]) Zn (t)N (dt) = E Z kT 2 (0,T ]
=
k=0 n −1 2
E Z kT 2−n N ((kT 2−n , (k + 1)T 2−n ]) .
k=0
"mi Z (kT 2−n ) Bb BM/2T2M/2Mi Q7 N ((kT 2−n , (k + 1)T 2−n ]) bBM+2 Bi Bb FkT 2−n @ K2 bm` #H2- bQ i? i i?2 H bi i2`K BM i?2 #Qp2 +? BM Q7 2[m HBiB2b Bb 2[m H iQ n −1 2
E Z kT 2−n E N ((kT 2−n , (k + 1)T 2−n ])
k=0
=
n −1 2
E Z kT 2−n ν((kT 2−n , (k + 1)kT 2−n ])
k=0 2n −1
=E
k=0 T
=E
Z kT 2
−n
ν((kT 2
−n
−n
, (k + 1)kT 2
])
Zn (s) ν(ds) .
0
h?2`27Q`2-
Zn (s)N (ds) = E
E (0,T ]
T 0
Zn (s) ν(ds) .
UkX3V
T .2MQiBM; #v K i?2 mTT2` #QmM/ Q7 Z(t, ω)- r2 ? p2 i? i 0 Zn (t) ν(dt) ≤ Kν((0, T ]) < ∞ M/ E (0,T ] Zn (t)N (dt) ≤ KE [N (T )] = Kν((0, T ]) < ∞X h?2`27Q`2- #v /QKBM i2/ +QMp2`;2M+2- H2iiBM; n ↑ ∞ BM #Qi? bB/2b Q7 UkX3V- r2 Q#i BM UVX
kXjX SPAaaPL J_hAL:G1a L. ahP*>ahA* ALh1:_Ga "X q2 MQr i`2 i i?2 MQM@M2; iBp2 M/ ⎧ ⎪ ⎨1 gK (x) = 0 ⎪ ⎩ −x + K + 1
8d
mM#QmM/2/ + b2X .2}MBM; B7 x ≤ K, B7 x ≥ K + 1, B7 K ≤ x ≤ K + 1,
i?2 T`Q+2bb {gK (Z(t))}t≥0 b iBb}2b i?2 +QM/BiBQMb 7Q` UV M/ i?2`27Q`2
gK (Z(s))N (ds) = E
E
T
gK (Z(s)) ν(ds) ,
0
(0,T ]
r?B+? ;Bp2b UV b K → ∞- #v KQMQiQM2 +QMp2`;2M+2X Ai i?2M bm{+2b iQ H2i T ↑ ∞ M/ iQ BMpQF2 i?2 KQMQiQM2 +QMp2`;2M+2 i?2Q`2K ; BM iQ Q#i BM UkXeV BM i?2 MQM@ M2; iBp2 mM#QmM/2/ + b2X LQi2 i? i i?2 [m MiBiB2b BM UkXeV + M MQr p2`v r2HH #2 BM}MBi2X *X h?2 `2 H@p Hm2/ + b2 7QHHQrb 2 bBHv- #v }`bi +QMbB/2`BM; b2T ` i2Hv i?2 TQbB@ iBp2 M/ M2; iBp2 T `ib Q7 i?2 BMi2;` M/X h?2 +QKTH2t + b2 Bb /B`2+i +QMb2[m2M+2 Q7 i?2 `2 H + b2 r?2M QM2 +QMbB/2`b b2T ` i2Hv i?2 `2 H M/ BK ;BM `v T `ibX
kXj
SQBbbQM J `iBM; H2b M/ aiQ+? biB+ AMi2;` Hb
G2i N #2 SQBbbQM T`Q+2bb QM R+ rBi? i?2 BMi2MbBiv K2 bm`2 νX h?2M M (t) := N (0, t] − ν((0, t]) (t ≥ 0) /2}M2b M FtN @K `iBM; H2 {M (t)}t≥0 XR AM/22/ M (t) Bb BMi2;` #H2 7Q` HH t ≥ 0 M/ #v i?2 BM/2T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b- 7Q` HH (a, b] ⊂ R+ E N (a, b] | FaN = E [N (a, b]] = ν((a, b]) . G `;2 .2pB iBQMb aBM+2 i?2 2tT2+i iBQM Q7 i?2 SQBbbQM K `iBM; H2 Bb MmHH- QM2 K v #2 BMi2`2bi2/ BM i?2 H `;2 /2pB iBQMb 7`QK Bib K2 M 0X h?2 M2ti i?2Q`2K ;Bp2b M 2biBK i2 i? i rBHH #2 mb2/ H i2` QMX h?2Q`2K kXjXR G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv 1X h?2M- 7Q` Mv T > 0 M/ ε > 0 bm+? i? i Tε ≤ 1
P
sup |N (t) − t| ≥ ε
≤ 2e−T h( T ) , ε
t∈[0,T ]
r?2`2 h(x) := (1 + x) log(1 + x) − xX R
a2+iBQM Xj ;Bp2b i?2 /2}MBiBQM M/ # bB+ T`QT2`iB2b Q7 K `iBM; H2bX
83
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
S`QQ7X q2 b? HH mb2 EQHKQ;Q`QpǶb bm#K `iBM; H2 BM2[m HBiv Uh?2Q`2K XjX9V r?B+? i2HHb mb i? i B7 {S(t)}t∈[0,T ] Bb bm#K `iBM; H2, 1 sup |S(t)| ≥ a ≤ E [|S(T )|] . a t∈[0,T ] 6Q` θ > 0 P
sup |N (t) − t| ≥ ε t∈[0,T ]
≤P
sup (N (t) − t) ≥ ε
=P
t∈[0,T ]
sup e
sup (t − N (t)) ≥ ε
+P
t∈[0,T ]
θ(N (t)−t)
≥e
θε
+P
t∈[0,T ]
sup e
θ(t−N (t))
≥e
θε
=A+B.
t∈[0,T ]
aBM+2 {N (t) − t}t∈[0,T ] M/ {t − N (t)}t∈[0,T ] `2 K `iBM; H2b M/ bBM+2 θ > 0{eθ(N (t)−t) }t∈[0,T ] M/ {eθ(t−N (t) }t∈[0,T ] `2 bm#K `iBM; H2b U b +QMp2t 7mM+iBQMb Q7 K `iBM; H2bV iQ r?B+? r2 TTHv EQHKQ;Q`QpǶb BM2[m HBiv iQ Q#i BM A ≤ e−θε E eθ(N (T )−T ) M/ B ≤ e−θε E eθ(T −N (T )) . h FBM; BMiQ ++QmMi i?2 7 +i i? i N (T ) Bb SQBbbQM p `B #H2 rBi? K2 M T θ θ A ≤ e−θ(ε+T ) E eθN (T ) = e−θ(ε+T ) eT (e −1) = e−θ(ε+T )+T (e −1) . h?Bb [m MiBiv Bb KBMBKBx2/ KQM; HH TQbBiBp2 p Hm2b Q7 θ #v θ = log 1 + Tε X ε ε h?2`27Q`2 A ≤ e−T h( T ) M/- bBKBH `Hv- B ≤ e−T h(− T ) X h?2 +QM+HmbBQM 7QHHQrb #v Q#b2`pBM; i? i h(−x) ≥ h(x) r?2M x ∈ [0, 1]X
aiQ+? biB+ BMi2;` Hb rBi? `2bT2+i iQ i?2 SQBbbQM K `iBM; H2 aiQ+? biB+ BMi2;` Hb rBi? `2bT2+i iQ SQBbbQM K `iBM; H2 vB2H/ K `iBM; H2bX JQ`2 T`2+Bb2Hv- r2 ? p2 i?2 7QHHQrBM; `2bmHi- r?Qb2 T`QQ7 Bb H27i b 1t2`+Bb2 kXdXNX h?2Q`2K kXjXk G2i {Ft }t≥0 #2 ?BbiQ`v Q7 N - M ?TT QM R+ Q7 BMi2MbBiv K2 bm`2 νX amTTQb2 KQ`2Qp2` i? i 7Q` HH a > 0 M/ HH c, d (a ≤ c ≤ d)- N (c, d] Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2 +QKTH2t biQ+? biB+ T`Q+2bb / Ti2/ iQ {Ft }t≥0 M/ rBi? H27i@+QMiBMmQmb i` D2+iQ`B2bX X A7
E
t
|Z(s)| ν(ds) < ∞
(t ≥ 0) ,
()
0
i?2M
(0,t]
t
Z(s) N (ds) −
Z(s) dM (s) :=
Y (t) :=
(0,t]
Z(s) ν(ds)
()
0
Bb r2HH@/2}M2/ [m MiBiv UMQi ∞ − ∞ 7Q`KV M/ i?2 biQ+? biB+ T`Q+2bb {Y (t)}t≥0 Bb M Ft @K `iBM; H2X
kX9X qhL"1Ƕa *>_*h1_AwhAPL
8N
"X G2i i?BM;b #2 b #Qp2- 2t+2Ti 7Q` +QM/BiBQM UV- MQr `2TH +2/ #v i?2 HQ+ H BMi2;` #BHBiv Q7 {Z(t)}t≥0 - i? i Bb- HKQbi@bm`2Hv ∞ |Z(t)| ν(dt) < ∞ 7Q` HH t ≥ 0 . 0
h?2M Y (t) ;Bp2M #v UV Bb r2HH@/2}M2/ [m MiBiv UMQi ∞ − ∞ 7Q`KV M/ i?2 biQ+? biB+ T`Q+2bb {Y (t)}t≥0 Bb HQ+ H Ft @K `iBM; H2X
kX9
q i M #2Ƕb *? ` +i2`Bx iBQM
h?2Q`2K kX9XR Uk V G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ - M/ H2i {Ft }t≥0 #2 ?BbiQ`v Q7 N X G2i ν #2 /Bzmb2 HQ+ HHv }MBi2 K2 bm`2 QM R+ X amTTQb2 i? i
T
E Z(t)N (dt) = E Z(t) ν(dt) UkXNV 0
(0,T ]
?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH MQM@M2; iBp2 `2 H@p Hm2/ H27i@+QMiBMmQmb biQ+? biB+ T`Q+2bb2b {Z(t)}t≥0 / Ti2/ iQ {Ft }t≥0 X h?2M- N Bb SQBbbQM T`Q@ +2bb rBi? i?2 BMi2MbBiv K2 bm`2 ν- M/ 7Q` Mv BMi2`p H (a, b] ∈ R+ - N (a, b] Bb BM/2T2M/2Mi Q7 Fa X S`QQ7X h?2 T`QQ7 ;Bp2M #2HQr Bb ivTB+ H Q7 i?2 biQ+? biB+ + H+mHmb Q7 ?TTbXj 6B`biMQi2 i? i 7Q`KmH UkXNV ?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH +QKTH2t@p Hm2/ biQ+? b@ iB+ T`Q+2bb2b {Z(t)}t≥0 rBi? H27i@+QMiBMmQmb i` D2+iQ`B2b- / Ti2/ iQ {Ft }t≥0 M/ T bm+? i? i E 0 |Z(t)| ν(dt) < ∞ Ub K2 `;mK2Mib b BM T `i * Q7 i?2 T`QQ7 Q7 h?2Q`2K kXkXRVX q2 b?Qr i? i Bi Bb 2MQm;? iQ T`Qp2 i? i iu E 1A eiuN (a,b] = P (A) e(e −1)ν((a,b]) , () r?2`2 (a, b] ⊆ R+ M/ u ∈ R `2 `#Bi` `v- M/ A Bb M `#Bi` `v 2p2Mi BM Fa X AM 7 +i- rBi? A = Ω BM UVE[eiuN (a,b] ] = e(e
iu −1)ν((a,b])
,
r?B+? b?Qrb i? i N (a, b] Bb SQBbbQM ` M/QK p `B #H2 Q7 K2 M ν((a, b])X 1[m HBiv UV i?2M `2 /b E[1A eiuN (a,b] ] = P (A)E[eiuN (a,b] ] , 7`QK r?B+? Bi 7QHHQrb i? i N (a, b] M/ A `2 BM/2T2M/2MiX aBM+2 A Bb M `#Bi` `v 2p2Mi BM Fa - r2 ? p2 T`Qp2/ i? i N (a, b] Bb BM/2T2M/2Mi Q7 Fa M/- bBM+2 Fa ⊇ FaN i? i N ? b BM/2T2M/2Mi BM+`2K2MibX 6Q` i?2 T`QQ7 Q7 UV- +QMbB/2` i?2 T`Q+2bb k
(q i M #2- RNe9)X h?Bb T`QQ7 U("`ûK m/- RNd8 #)V BKBi i2b i?2 T`QQ7 Q7 GûpvǶb +? ` +i2`Bx iBQM Q7 i?2 "`QrMB M KQiBQM ;Bp2M BM (EmMBi M/ q i M #2- RNed)- h?KX kXj- TX kRdX h?2 BMi2`2bi Q7 i?Bb T`QQ7 HB2b BM i?2 2 bv 2ti2MbBQMb Q7 i?2 i?2Q`2K i? i Bi HHQrb- BM+Hm/BM; i?2 T`2b2Mi QM2X a22 7Q` BMbi M+2 1t2`+Bb2 kXdXRk M/ h?2Q`2K 8XdXRX j
ey
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1 X(t) := 1A eiuN (a,t]
(t ≥ a) ,
M/ Q#b2`p2 i? i Bi Bb TB2+2rBb2 +QMbi Mi M/ i? i HH Bib iBK2b Q7 /Bb+QMiBMmBiv `2 HQ+ i2/ i i?2 2p2Mi iBK2b Q7 N X h?2`27Q`2 X(b) = X(a) + {X(Tn ) − X(Tn− )}1(a,b] (Tn ). n≥1
"mi X(0) = 1A M/- 7Q` Mv Tn ∈ (a, b] Ur i+? i?2 T `2Mi?2b2b M/ i?2 b[m `2 #` +F2ibVX(Tn ) = eiuN (a,Tn ] = eiu(N (a,Tn )+1) = eiuN (a,Tn ) eiu = X(Tn− ) eiu . h?2`27Q`2X(b) = 1A +
X(Tn− )(eiu − 1)1(a,b] (Tn )
n≥1
Q`- BM i?2 BMi2;` H MQi iBQM X(s−)(eiu − 1) N (ds).
X(b) = 1A +
UkXRyV
(a,b]
aBM+2 Z(t) = X(t−)(eiu − 1) /2}M2b #QmM/2/ H27i@+QMiBMmQmb +QKTH2t@p Hm2/ biQ+? biB+ T`Q+2bb / Ti2/ iQ {Ft }t≥0 - 2[m HBiv UkXNV ?QH/b i`m2 #v ?vTQi?2bBbX h?2`27Q`2
b
iu iu E X(s−)(e − 1)N (ds) = E X(s−)(e − 1) ν(ds) . a
(a,b]
LQr- 7Q` 2 +? ω ∈ Ω- X(t−, ω) = X(t, ω) 2t+2Ti QM +QmMi #H2 b2i- M/ i?2`27Q`2 QM2 K v `2TH +2 X(t−) #v X(t) BM i?2 `B;?i@? M/ bB/2 Q7 i?2 #Qp2 2[m HBiv U`2+ HH i? i ν Bb bbmK2/ /Bzmb2VX h FBM; 2tT2+i iBQMb BM UkXRyV i?2`27Q`2 vB2H/b
b
E[X(s)](eiu − 1) ν(ds) .
E[X(b)] = P (A) + a
h?Bb #2BM; i`m2 7Q` HH b ≥ a- UV 7QHHQrb Uh?2Q`2K RXeX8VX
KQbi mb27mH ;2M2` HBx iBQM Q7 q i M #2Ƕb i?2Q`2K rBHH #2 ;Bp2M BM a2+iBQM 8Xd- h?2Q`2K 8XdXRX h?2 7QHHQrBM; +Q`QHH `v Q7 h?2Q`2K kX9XR Bb i?2 Q`B;BM H p2`bBQM Q7 q i M #2Ƕb `2bmHi- r?B+? r b ;Bp2M BM i2`Kb Q7 K `iBM; H2bX *Q`QHH `v kX9Xk G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ bm+? i? i i?2 T`Q+2bb M (t) := N ((0, t]) − ν((0, t]) (t ≥ 0) Bb M Ft @K `iBM; H2- r?2`2 ν Bb HQ+ HHv }MBi2 /Bzmb2 K2 bm`2 QM R+ X h?2M N Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? i?2 BMi2MbBiv K2 bm`2 ν- M/ 7Q` Mv BMi2`p H (a, b] ∈ R+ - N (a, b] Bb BM/2T2M/2Mi Q7 Fa X
kX9X qhL"1Ƕa *>_*h1_AwhAPL
eR
S`QQ7X "v h?2Q`2K kX9XR- Bi bm{+2b iQ b?Qr i? i 7Q` HH T > 0 M/ HH MQM@M2; iBp2 #QmM/2/ `2 H@p Hm2/ H27i@+QMiBMmQmb biQ+? biB+ T`Q+2bb2b {Z(t)}t≥0 / Ti2/ iQ {Ft }t≥0
T
Z(t) N (dt) = E
E
Z(t) ν(dt)t .
()
0
(0,T ]
1[m HBiv UV Bb i`m2 7Q` Z(t, ω) := 1A (ω) 1(a,b] (t) 7Q` Mv BMi2`p H (a, b] ⊂ R+ M/ Mv A ∈ Fa - bBM+2 BM i?Bb + b2- UV `2 /b E [1A N (a, b]] = E [1A ν((a, b])] , i? i Bb- bBM+2 A Bb `#Bi` `v BM Fa E [N (a, b] | Fa ] = ν((a, b]), r?B+? Bb i?2 K `iBM; H2 ?vTQi?2bBbX h?2 T`QQ7 i?2M 7QHHQrb i?2 b K2 HBM2b b i? i Q7 h?2Q`2K kX9XR pB i?2 TT`QtBK iBQM UkXdVX *Q`QHH `v kX9Xj G2i N #2 TQBMi T`Q+2bb QM R+ rBi? HQ+ HHv }MBi2 /Bzmb2 BMi2MbBiv K2 bm`2 ν bm+? i? i 7Q` HH #QmM/2/ /BbDQBMi b2ib A M/ B BM B(R+ )- N (A) M/ N (B) `2 BM/2T2M/2Mi p `B #H2bX h?2M N Bb SQBbbQM T`Q+2bbX S`QQ7X h?2 bbmKTiBQMb BKTHv i? i 7Q` HH s, t U0 ≤ s ≤ tV i?2 ` M/QK p `B #H2 N ((s, t]) Bb BM/2T2M/2Mi Q7 FsN - M/ i?2`27Q`2 E N ((s, t]) | FsN = E [N ((s, t])] = ν((s, t]) = E ν((s, t]) | FsN , M/ i?2`27Q`2 Bb M
M (t) := N ((0, t]) − ν((0, t])
(t ≥ 0)
FtN @K `iBM; H2X
ai`QM; J `FQp S`QT2`iv Q7 i?2 ?TT pB q i M #2Ƕb i?2Q`2K h?2 bi`QM; J `FQp T`QT2`iv Q7 ?TTb + M #2 Q#i BM2/ b +QMb2[m2M+2 Q7 q i M@ #2Ƕb i?2Q`2KX 6Q` BMbi M+2- H2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ M/ H2i τ #2 }MBi2 FtN @biQTTBM; iBK2X .2}M2 i?2 TQBMi T`Q+2bb N τ QM R+ #v N τ (0, t] := N (τ, τ + t) (t ≥ 0) . h?2M N τ Bb M ?TT BM/2T2M/2Mi Q7 FτN X S`QQ7X G2i (a, b] ⊂ R+ M/ A ∈ FτN+a X h?2 T`Q+2bb Z(t) = 1A 1(τ +a,τ +b] (t) Bb H27i@+QMiBMmQmb M/ / Ti2/ iQ {FtN }t≥0 M/ i?2`27Q`2
ek
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
Z(t) N (dt) = E
E R+
Z(t) λ dt ,
R+
r?B+? `2 /b BM i?Bb T `iB+mH ` + b2 E [1A N τ (a, b]] = E [1A λ(b − a)] . aBM+2 i?Bb ?QH/b 7Q` HH (a, b] ⊂ R+ M/ HH A ∈ FτN+a - {N τ (0, t] − λt}t≥0 Bb M FτN+t @K `iBM; H2X h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK *Q`QHH `v kX9XkX
kX8 ?K+b M/ aiQ+? biB+ .Bz2`2MiB H 1[m iBQMb h?Bb bm#b2+iBQM BMi`Q/m+2b i?2 MQiBQM Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQMb /`Bp2M #v SQBbbQM T`Q+2bb2b #v K2 Mb Q7 i?2 +H bbB+ H 2t KTH2 Q7 +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J `FQp +? BMb U?K+VX9 Ai +QM+2`Mb i?2 T`Q#H2K Q7 `2 HBx iBQM Q7 M BM}MBi2bBK H ;2M2` iQ`- i? i Bb- i?2 T`Q#H2K Q7 bbQ+B iBM; iQ ;2M2` iQ` A +QMiBMmQmb@iBK2 ?K+ /KBiiBM; Bi b M BM}MBi2bBK H ;2M2` iQ`X JQ`2 T`2+Bb2HvH2i E #2 +QmMi #H2 bT +2 M/ H2i A = {qij }i,j∈E #2 K i`Bt Q7 `2 H MmK#2`b BM/2t2/ #v E- bm+? i? i 7Q` HH i, j ∈ E bm+? i? i i = jqi ∈ [0, ∞],
qij ∈ [0, ∞) .
A Bb i?2M + HH2/ ;2M2` iQ` UMQi v2i M BM}MBi2bBK H ;2M2` iQ`VX bbmK2 KQ`2Qp2` i? i i?Bb ;2M2` iQ` Bb bi #H2 M/ +QMb2`p iBp2- i? i Bb qi < ∞, qi = qik , k∈E k=i
r?2`2 qi = −qii X LQi2 HbQ i? i r?2M bT2 FBM; #Qmi ;2M2` iQ` A- QM2 /Q2b MQi `272` iQ ?QKQ;2M2Qmb +QMiBMmQmb@iBK2 J `FQp +? BMX >Qr2p2`- r2 + M +QMbi`m+i +QMiBMmQmb@iBK2 ?K+ /KBiiBM; i?2 ;Bp2M ;2M2` iQ` b M BM}MBi2bBK H ;2M2` iQ`X hQ /Q i?Bb- bi `i rBi? 7 KBHv Ni,j Ui, j ∈ E, i = j) Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b qij Ui, j ∈ E, i = j) M/ BM/2T2M/2Mi Q7 i?2 BMBiB H bi i2 X(0) ∈ EX h?2 T`Q+2bb {X(t)}t≥0 Bb +QMbi`m+i2/ b DmKT T`Q+2bb, X(t) = Xn 7Q` t ∈ [τn , τn+1 ) , r?2`2 i?2 b2[m2M+2 {τn , Xn }n≥0 Bb /2}M2/ `2+m`bBp2Hv b 7QHHQrbX 6B`bi- τ0 = 0X0 = X(0)X "27Q`2 T`Q+22/BM; iQ i?2 ;2M2` H bi2T- `2+ HH i? i Sa N Ua ∈ R+ V Bb #v /2}MBiBQM i?2 TQBMi T`Q+2bb N QM i?2 HBM2 b?B7i2/ #v a- i? i Bb- 7Q` Mv "Q`2H b2i C Q7 i?2 HBM2(Sa N )(C) := N (C + a) . A7 τn < ∞ M/ Xn = X(τn ) = i ∈ E- i?2M τn+1 − τn Bb i?2 }`bi 2p2Mi Q7 i?2 7 KBHv Q7 ?TTb Sτn Nij Uj ∈ E , j = i)X A7 τn+1 − τn < ∞- Xn+1 Bb i?2 BM/2t k = i bm+? 9
RkX
JQ`2 QM i?2 i?2K2 Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQMb /`Bp2M #v TQBMi T`Q+2bb2b BM *? Ti2`
kX8X >J*a L. ahP*>ahA* .A661_1LhAG 1ZlhAPLa
ej
i? i Sτn Nik Bb i?2 }`bi KQM; i?2 ?TTb Sτn Nij Uj ∈ E , j = iV i? i T`Q/m+2b M 2p2MiX Ai K v Q++m` i? i τn+1 − τn = ∞X h?Bb Bb i?2 + b2 B7 M/ QMHv B7 qi = 0 UMQ 2p2Mib5V- M/ i?2M i?2 +QMbi`m+iBQM 2M/b #v H2iiBM; Xn+m = i M/ τn+m = ∞ 7Q` HH m ≥ 1- M/ X(t) = Xn 7Q` HH t ≥ τn X h?2 T`Q+2bb +QMbi`m+i2/ BM i?Bb r v Bb /2}M2/ QM [0, τ∞ )- r?2`2 τ∞ = limn↑∞ τn X A7 τ∞ = ∞ S@ Xb- i?2M i?2 T`Q+2bb Bb 7mHHv /2}M2/ QM R+ M/ Bb i?2`27Q`2 `2;mH ` DmKT T`Q+2bb QM EX _2K `F kX8XR h?2 b K2 ivT2 Q7 +QMbi`m+iBQM rBHH #2 mb2/ 7Q` KQ`2 ;2M2` H ivT2 Q7 biQ+? biB+ bvbi2Kb- i?2 SQBbbQM bvbi2Kb Q7 a2+iBQM RRXkX h?2Q`2K kX8Xk X 6Q` HH n ≥ 0, ;Bp2M Xn - i?2 ` M/QK p `B #H2b Xn+1 M/ τn+1 − τn `2 BM/2T2M/2Mi Q7 X0 , . . . , Xn−1 M/ τ1 , . . . , τn - M/ 7Q` HH i, j ∈ Ei = j- HH a ≥ 0P (Xn+1 = j, τn+1 − τn > a | Xn = i) = e−qi a pij , r?2`2 pij =
qij qi
UkXRRV
B7 qi > 0- pij = 0 B7 qi = 0X UAM T `iB+mH `- {Xn }n≥0 Bb M ?K+XV
"X A7 P (τ∞ = ∞) = 1- i?2 T`Q+2bb {X(t)}t≥0 +QMbi`m+i2/ b #Qp2 Bb `2;mH ` DmKT ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` AX N S`QQ7X X G2i Gt := σ(X0 ) ∨ ∨i,j∈E Ft ij M/ Q#b2`p2 i? i i?2 τn Ƕb 7Q`K b2[m2M+2 Q7 Gt @biQTTBM; iBK2bX h?2 MMQmM+2/ `2bmHi i?2M 7QHHQrb 7`QK i?2 bi`QM; J `FQp T`QT2`iv 7Q` ?TTb Uh?2Q`2K kXRXdV M/ i?2 +QKT2iBiBQM i?2Q`2K Uh?2Q`2K kXRXeVX "X "v +QMbi`m+iBQM- 7Q` ;Bp2M iBK2 t- i?2 T`Q+2bb 7i2` iBK2 t /2T2M/b QMHv mTQM X(t) M/ i?2 ?TTb St Nij Ui, j ∈ E , i = j)X h?2 ?QKQ;2M2Qmb J `FQp T`QT@ 2`iv 7QHHQrb BKK2/B i2Hv 7`QK i?Bb Q#b2`p iBQMX Ai `2K BMb iQ p2`B7v i? i A Bb BM/22/ i?2 BM}MBi2bBK H ;2M2` iQ` Q7 i?2 ?K+X q2 }`bi +?2+F i? i- 7Q` i = j1 lim Pi (X(t) = j) = qij . t↓0 t 6Q` i?Bb- Q#b2`p2 i? i r?2M X(t) = X(0)- M2+2bb `BHv τ1 < t- M/ r`Bi2 UBMi`Q/m+BM; i?2 MQi iBQM Pi (·) := P (· | X(0) = i)VPi (X(t) = j) = Pi (τ2 ≤ t, X(t) = j) + Pi (τ2 > t, X(t) = j) = Pi (τ2 ≤ t, X(t) = j) + Pi (τ2 > t, X1 = j, τ1 < t) = Pi (τ2 ≤ t, X(t) = j) + Pi (X1 = j, τ1 < t) −Pi (τ2 ≤ t, X1 = j, τ1 < t) . "v h?2Q`2K kXRXe- Pi (X1 = j, τ1 < t) = (1 − e−qi t ) qiji - M/ i?2`27Q`2 q
1 lim Pi (X1 = j, τ1 < t) = qij . t↓0 t
e9
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
Ai i?2`27Q`2 `2K BMb iQ b?Qr i? i Pi (τ2 ≤ t, X(t) = j) M/ Pi (τ2 ≤ t, X1 = j, τ1 ≤ t) `2 o(t) UQ#pBQmb B7 qi = 0- M/ i?2`27Q`2 r2 bmTTQb2 qi > 0VX "Qi? i2`Kb `2 #QmM/2/ #v Pi (τ2 ≤ t)- M/ Pi (τ1 ≤ t, X1 = k, τ2 − τ1 ≤ t) Pi (τ2 ≤ t) ≤ Pi (τ1 ≤ t, τ2 − τ1 ≤ t) = k∈E k=j
qik qik (1 − e−qk t ). (1 − e−qi t ) (1 − e−qk t ) = (1 − e−qi t ) = q q i i k∈E k∈E k=j
"mi (1 − e−qi t ) Bb O(t) UB/2MiB+ HHv MmHH B7 qi = 0V M/ limt↓0 #v /QKBM i2/ +QMp2`;2M+2X h?2`27Q`2 Pi (τ2 ≤ t) Bb o(t)X q2 MQr +?2+F i? i limt↓0
1−pii (t) t
k=i
k∈E k=i
qik (1 − e−qk t ) qi
= 0-
= qi X 6`QK
1 − pii (t) = 1 − Pi (X(t) = i) = 1 − Pi (X(t) = i, τ1 > t) − Pi (X(t) = i, τ1 ≤ t) = 1 − Pi (τ1 > t) − Pi (X(t) = i, τ1 ≤ t, τ2 ≤ t) = 1 − e−qi t − Pi (X(t) = i, τ1 ≤ t, τ2 ≤ t), M/ i?2 MMQmM+2/ `2bmHi 7QHHQrb 7`QK i?2 7 +i UT`Qp2/ #Qp2V i? i Pi (τ2 ≤ t) Bb o(t)X _2K `F kX8Xj G2i Zi (t) := 1{X(t)=i} X h?2 +QMbi`m+iBQM Q7 i?2 bi i2 T`Q+2bb {X(t)}t≥0 QM [0, τ∞ ) Bb bmKK `Bx2/ #v i?2 7QHHQrBM; 2[m iBQMb, 7Q` HH i ∈ E Zi (t) = Zi (0) + Zj (s−) Nji (ds) − Zi (s−) Nij (ds) . UkXRkV j∈E;j=i
(0,t]
j∈E;j=i
(0,t]
a2+iBQM RRXk rBHH 72 im`2 M 2ti2MbBQM Q7 `2T`2b2Mi iBQM UkXRkV T `iB+mH `Hv / Ti2/ iQ [m2m2BM; +QMi2tiX 1[m iBQMb UkXRkV +QMbiBimi2 bvbi2K Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQMb /`Bp2M #v i?2 SQBbbQM T`Q+2bb2b Nij Ui, j ∈ E , i = jV 7Q` i?2 T`Q+2bb2b {Zi (t)}t≥0 Ui ∈ EVX Ai Bb bQK2iBK2b +QMp2MB2Mi iQ bi i2 +QM+HmbBQM " Q7 h?2Q`2K kX8Xk b 7QHHQrb, h?2Q`2K kX8X9 G2i {X(t)}t≥0 #2 M E@p Hm2/ `2;mH ` DmKT T`Q+2bb b iBb7vBM; UkXRkV r?2`2 Zi (t) = 1{X(t)=i} M/ r?2`2 Ni,j (i, j ∈ E , i = j) Bb 7 KBHv Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b qij (i, j ∈ E , i = j)- M/ BM/2T2M/2Mi Q7 i?2 BMBiB H bi i2 X(0)X h?2M {X(t)}t≥0 Bb `2;mH ` DmKT ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` AX _2K `F kX8X8 LQi2 i? i UkXRkV Bb 2[mBp H2Mi iQ i?2 `2[mB`2K2Mi i? i f (X(t)) − f (X(0)) = {f (j) − f (i)} 1{X(s−)=i} dNij (s) i,j∈E i=j
UkXRjV
(0,t]
7Q` HH MQM@M2; iBp2 7mM+iBQMb f : E → RX q2 b? HH MQr 2tTHQBi i?Bb + MQMB+ H `2T`2b2Mi iBQMX
kX8X >J*a L. ahP*>ahA* .A661_1LhAG 1ZlhAPLa
e8
;;`2; iBQM Q7 ai i2b *QMbB/2` `2;mH ` DmKT ?K+ {X(t)}t≥0 rBi? bi i2 bT +2 E M/ BM}MBi2bBK H ;2M@ ˜ 2` iQ` AX G2i E˜ = {α, β, . . .} #2 T `iBiBQM Q7 E- M/ /2}M2 i?2 T`Q+2bb {X(t)} t≥0 i FBM; Bib p Hm2b BM E˜ #v ˜ X(t) = α Bz X(t) ∈ α .
UkXR9V
˜ h?2 ?K+ {X(t)} t≥0 Bb i?2 ;;`2; i2/ +? BM Q7 {X(t)}t≥0 UrBi? `2bT2+i iQ i?2 ˜ T `iBiBQM EVX ˜ i ∈ α- β ∈ E˜ rBi? α = βh?2Q`2K kX8Xe amTTQb2 i? i 7Q` HH α ∈ E qij = q˜αβ .
UkXR8V
j∈β
Uh?Bb 2[m HBiv MQi QMHv /2}M2b i?2 [m MiBiv QM i?2 `B;?i@? M/ bB/2 #mi HbQ b vb ˜ BKTHB+BiHv i? i i?2 H27i@? M/ bB/2 Bb BM/2T2M/2Mi Q7 i ∈ αVX h?2M {X(t)} t≥0 Bb ˜ rBi? Qz@ ˜ `2;mH ` DmKT ?K+ rBi? bi i2 bT +2 E M/ BM}MBi2bBK H ;2M2` iQ` A/B ;QM H i2`Kb ;Bp2M #v UkXR8VX S`QQ7X h?Bb bi i2K2Mi +QM+2`Mb i?2 /Bbi`B#miBQM Q7 {X(t)}t≥0 - M/ i?2`27Q`2 r2 K v bmTTQb2 i? i i?Bb T`Q+2bb Bb Q7 i?2 7Q`K UkXRjVX h?2M- 7Q` f : E˜ → R M/ s ≤ t ˜ ˜ ˜ ˜ f (X(t)) = f (X(0)) + {f (X(u)) − f (X(u−))}1 {X(u−)=i} dNij (u) (0,t]
i,j∈E i=j
˜ = f (X(0)) +
{f (β) − f (α)}
˜ α,β∈E α=β
1{X(u−)=i} (0,t]
i∈α
(0,t]
j∈β
dNij (u)
j∈β
˜ α = β- i?2 TQBMi T`Q+2bb N ˜αβ #v .2}M2 7Q` HH α, β ∈ E, ˜ 1{X(s−)=i} dNij (s) + Nαβ (0, t] = (0,t] i∈α
ˆ 1{X(s−) ˜ =α} dNα,β (s) ,
ˆαβ } α,β∈E˜ 7Q`K M BM/2T2M/2Mi 7 KBHv Q7 r?2`2 i?2 dz/mKKvǴ TQBMi T`Q+2bb2b {N α=β
?TTb rBi? BMi2MbBiB2b {˜ qαβ } α,β∈E˜ - `2bT2+iBp2Hv- M/ `2 BM/2T2M/2Mi Q7 X(0) M/ {Nij } i,j∈E X h?2M
α=β
i=j
˜ ˜ f (X(t)) = f (X(0)) +
˜ α,β∈E α=β
(f (β) − f (α)) (0,t]
˜αβ (u). 1{X(u−)=α} dN ˜
UkXReV
˜α,β Uα, β ∈ E˜ , α = βV Bb 7 KBHv AM pB2r Q7 _2K `F kX8X8- Bi bm{+2b iQ T`Qp2 i? i N Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b q˜αβ Uα, β ∈ E˜ - α = βVX 6Q` i?Bb-
.
ee
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
r2 TTHv q i M #2Ƕb i?2Q`2K Uh?2Q`2K kX9XRVX G2i {Z(t)}t≥0 #2 H27i@+QMiBMmQmb Ft @ / Ti2/ biQ+? biB+ T`Q+2bb- r?2`2 ˜ ˜uv N N . Ft = σ(X(0)) ∨ Ft αβ ∨ ∨u,v∈E;(u,v) ˜ =(α,β) F∞ q2 Q#i BM
E
˜αβ (t) = Z(t)dN E
(0,T ]
i∈α j∈β
Z(t)1{X(t−)=i} dNij (t) (0,T ]
+E (0,T ]
ˆαβ , Z(t)1{X(t−) d N ˜ =α}
M/ i?Bb [m MiBiv Bb 2[m H- #v i?2 bKQQi?BM; 7Q`KmH Uh?2Q`2K kXkXRV- iQ T
T
E Z(t)1{X(t−)=i} qij dt + E Z(t)1{X(t) ˜αβ dt ˜ =α} q 0
i∈α j∈β
T
Z(t)
=E
0
=E
0
qij
+ q˜αβ 1{X(t−) 1{X(t−)=α} ˜ ˜ =α} dt
j∈β
T
Z(t)˜ qαβ dt .
0
˜αβ Bb M ?TT rBi? BMi2MbBiv q˜αβ BM/2T2M/2Mi Q7 h?2`27Q`2- #v h?2Q`2K kX9XR- N ˜ Nuv X σ(X(0)) ∨ ∨u,v∈E;(u,v) ˜ =(α,β) F∞
kXe ?K+b M/ hBK2@a+ H2/ ?TTb h?2 K BM `2bmHi Q7 i?Bb b2+iBQM8 2tTHQBib `2T`2b2Mi iBQM Q7 +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J `FQp +? BM /Bz2`2Mi 7`QK i?2 QM2 +QMbB/2`2/ BM i?2 T`2pBQmb b2+iBQMBM i2`Kb Q7 iBK2@b+ H2/ ?TTX aT2+B}+ HHv- r2 `2 ;QBM; iQ +QMbi`m+i ?QKQ;2M2Qmb +QMiBMmQmb@iBK2 J `FQp +? BM {X(t)}t≥0 rBi? p Hm2b BM E = Zd 7Q` r?B+? HH TQbbB#H2 i` MbBiBQMb `2 Q7 i?2 ivT2 x → x + ei r?2`2 ei ∈ E U1 ≤ i ≤ kVX Aib BM}MBi2bBK H ;2M2` iQ` Q Bb ;Bp2M M/ bbmK2/ bi #H2- +QMb2`p iBp2 M/ `2;mH `- M/ Bib MQM@/B ;QM H 2H2K2Mib i? i `2 TQbBiBp2 `2 /2MQi2/ #v qx,x+ei X h?2 +QMbi`m+iBQM Bb b 7QHHQrb X(t) := X(0) +
k i=1
t
e i Ni
qX(s),X(s)+ei ds
,
UkXRdV
0
r?2`2 Ni U1 ≤ i ≤ kV `2 BM/2T2M/2Mi bi M/ `/ ?TTb BM/2T2M/2Mi Q7 i?2 BMBiB H bi i2 X(0) ∈ EX Ai 7QHHQrb 7`QK i?2 J `FQp T`QT2`iv Q7 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b M/ i?2 +QKT2iBiBQM i?2Q`2K i? i i?2 biQ+? biB+ T`Q+2bb bQ /2}M2/ Bb BM/22/ M ?K+ rBi? i?2 ;Bp2M BM}MBi2bBK H ;2M2` iQ` U1t2`+Bb2 kXdXReVX 8
(Em`ix- RNdR- RN3y)X
kXeX >J*a L. hAJ1@a*G1. >SSa
ed
1t KTH2 kXeXR, *H bbB+ H JQ/2H Q7 1TB/2Kv- i F2 RX h?2 #Qp2 +QM@ bi`m+iBQM rBHH #2 TTHB2/ iQ bBKTH2 KQ/2H Q7 2TB/2Kv BM TQTmH iBQM Q7 N BM/BpB/m Hb r?Q #2HQM; iQ QM2 Q7 irQ +H bb2b, i?2 BM72+i2/ M/ i?2 bmb+2TiB#H2b UMQi BM72+i2/- 2Bi?2` #2+ mb2 i?2v ? p2 M2p2` #22M BM72+i2/ Q` #2+ mb2 i?2v ? p2 `2@ +Qp2`2/- rBi? ?Qr2p2` TQbbB#H2 `2H Tb2VX h?2 bi i2 Q7 i?2 2TB/2Kv i ;Bp2M iBK2 Bb ;Bp2M #v i?2 MmK#2` i U0 ≤ i ≤ N V Q7 BM72+i2/ BM/BpB/m Hb U M/ i?2`27Q`2 i?2 MmK#2` Q7 bmb+2TiB#H2b i i? i iBK2 Bb N − iVX G2i XN (t) #2 i?2 MmK#2` Q7 BM72+i2/ BM/BpB/m Hb i iBK2 tX Mv ;Bp2M BM72+i2/ BM/BpB/m H K22ib MQi?2` K2K@ #2` Q7 i?2 TQTmH iBQM i ` i2 βX h?2 BM/BpB/m H K2i #v i?Bb BM72+i2/ BM/BpB/m H Bb +?Qb2M i ` M/QK BM i?2 TQTmH iBQM- M/ i?2`27Q`2 i?2 ` i2 i r?B+? bQK2 ;Bp2M BM72+i2/ BM/BpB/m H K22ib bmb+2TiB#H2 Bb β NN−i X h?2 ` i2 i r?B+? i?2 BM72+i2/ TQTmH iBQM BM+`2 b2b #v QM2 mMBi Bb i?2`27Q`2 iβ NN−i X LQr- i?2 ` i2 i r?B+? bQK2 ;Bp2M BM72+i2/ BM/BpB/m H Bb +m`2/ U#2+QK2b bmb+2TiB#H2V Bb δX h?2`27Q`2 i?2 ` i2 i r?B+? i?2 BM72+i2/ TQTmH iBQM /2+`2 b2b #v QM2 mMBi Bb δiX HH i?Bb DmbiB}2b i?2 KQ/2HBx iBQM Q7 i?2 biQ+? biB+ T`Q+2bb {XN (t)}t≥0 b M ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` ;Bp2M #v β qi,i+1 = i(N − i), qi,i−1 = iδ . N
h?Bb T `iB+mH ` 2t KTH2 }ib i?2 7QHHQrBM; ;2M2` H KQ/2H BMi`Q/m+2/ i i?2 #2;BMMBM; Q7 i?Bb b2+iBQM- M/ XN (t) = XN (0) +
k
t
e i Ni
N λi (XN (s)/N ) ds
,
UkXR3V
0
i=1
r?2`2 ei ∈ Zd U1 ≤ i ≤ kV M/ λi : Rd → R+ U1 ≤ i ≤ kVX h?2 7QHHQrBM; ?vTQi?2b2b `2 K /2, >R λ := maxi∈{1,...,k} supx∈Rd λi (x) < ∞X
>k h?2 7mM+iBQM x → F (x) := ki=1 ei λi (x) Bb GBTb+?Bix- i? i Bb- i?2`2 2tBbib }MBi2 +QMbi Mi M bm+? i? i |F (x) − F (y)| ≤ M |x − y|
(x, y ∈ Rd ) .
>j 6BM HHv r2 bmTTQb2 i? i y(0) := lim
N ↑∞
XN (0) 2tBbib S@ XbX N
h?2Q`2K kXeXk bbmK2 i?2 ?vTQi?2b2b >R- >k M/ >j p2`B}2/X G2i y : R+ → Rd #2 i?2 bQHmiBQM Q7 i?2 BMi2;` H 2[m iBQM t y(t) = y(0) + F (y(s)) ds . UkXRNV 0
h?2M 7Q` Mv }t2/ ε > 0 M/ T > 0 M/ 7Q` bm{+B2MiHv H `;2 N -
e3
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1 P
& & &1 & εe−M T & & sup & XN (t) − y(t)& ≥ ε ≤ 2k exp −N T λh , N 2kT λe
t∈[0,T ]
UkXkyV
max
r?2`2 emax := maxi∈{1,...,k} |ei |X JQ`2Qp2`- P @ XbX & & & &1 lim sup && XN (t) − y(t)&& = 0 . N ↑∞ t∈[0,T ] N
UkXkRV
_2K `F kXeXj lM/2` i?2 GBTb+?Bix ?vTQi?2bBb >k- i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 bQHmiBQM Q7 i?2 BMi2;` H 2[m iBQM UkXRNV Bb ;m ` Mi22/ #v i?2 7mM/ K2Mi H `2bmHi Q7 i?2 i?2Q`v Q7 /Bz2`2MiB H 2[m iBQMbX 1t KTH2 kXeX9, *H bbB+ H JQ/2H Q7 1TB/2Kv- i F2 kX AM i?2 KQ/2H Q7 1t KTH2 kXeXR i?2`2 `2 QMHv irQ DmKT K ;MBim/2b, e1 = +1 M/ e2 = −1- M/ λ1 (x) = βx(1 − x)- λ2 (x) = δx- M/ i?2`27Q`2 F (x) = βx(1 − x) − δx . "Qi? ?vTQi?2b2b >R M/ >k `2 b iBb}2/X "v M TT`QT`B i2 b+ HBM;- r2 K v bbmK2 i? i δ = 1X bbmKBM; ?vTQi?2bBb >j- i?2 bQHmiBQM Q7 UkXRNV Bb ;Bp2M 7Q` β = 1 #v (β − 1)y(0)e(β−1)t y(t) = . (β − 1) − βy(0)(1 − e(β−1)t ) AM T `iB+mH `- y(t) →
β−1 β
B7 β > 1- r?2`2 b y(t) → 0 B7 β < 1X 1 X (t)N N
S`QQ7X AMi`Q/m+BM; i?2 MQi iBQM YN (t) := YN (t) = YN (0) +
k
UkXR3V #2+QK2b
t
e i Ni
N λi (YN (s)) ds 0
i=1
Q`- rBi? Mi (t) := Ni (t) − t Ut ≥ 0VYN (t) = YN (0) +
k
t
e i Mi
t nλi (YN (s)) ds + F (YN (s)) ds .
0
i=1
0
am#i` +iBM; UkXRNV M/ i FBM; #bQHmi2 p Hm2b, t |F (YN (s)) − F (y(s))| ds + · · · |YN (t) − y(t)| ≤ |YN (0) − y(0)| + 0
& t & k & |ei | && Mi N λi (YN (s)) ds && . ··· + N & 0
i=1
"v ?vTQi?2b2b >j M/ >k- 7Q` bm{+B2MiHv H `;2 N
t
|YN (t) − y(t)| ≤ ε + M
|YN (s) − y(s)| ds + 0
k 1 αN,i (t) , N i=1
UkXkkV
kXeX >J*a L. hAJ1@a*G1. >SSa
eN
& t & & & & λi (YN (s)) ds && . αN,i (t) = |ei | &Mi N
r?2`2
0
"v i?2 #QmM/ Q7 h?2Q`2K kXjXR k k 1 Nε P sup αN,i (t) ≥ αN,i (t) ≥ ε ≤ P sup k t∈[0,T ] i=1 N t∈[0,T ] i=1 k Nε P sup emax |M (t)| ≥ ≤ k t∈[0,T ] i=1 ε . ≤ 2k exp −N T λh kT λemax *QK#BMBM; i?2 H bi irQ BM2[m HBiB2b vB2H/b t P sup |YN (t) − y(t)| − M |YN (s) − y(s)| ds ≥ 2ε t∈[0,T ]
0
≤ 2k exp −N T λh
ε kT λemax
UkXkjV
.
G2i ZN (t) := |YN (t) − y(t)|X "v :`QMr HHǶb H2KK Uh?2Q`2K RXeXeV- B7 7Q` HH t ∈ [0, T ] t ZN (s) ds ZN (t) ≤ 2ε + M 0
i?2M 7Q` HH t ∈ [0, T ]- ZN (t) ≤ 2εeM t ≤ 2εeM T - M/ BM T `iB+mH ` supt∈[0,T ] ZN (t) ≤ 2εeM T X h?2`27Q`2 t MT P sup ZN (t) − M ZN (s) ds ≤ 2ε ≤ P sup ZN (t) ≤ 2εe t∈[0,T ]
Q`- 2[mBp H2MiHv P
t∈[0,T ]
0
sup ZN (t) > 2εeM T
≤P
t∈[0,T ]
sup ZN (t) > 2εe
MT
P
sup ZN (t) > ε
≤ 2k exp −N T λh
≤ 2k exp −N T λh
t∈[0,T ]
AM T `iB+mH `
N ≥1
P
ZN (s) ds
> 2ε
.
0
t∈[0,T ]
Q`- Hi2`M iBp2Hv
t
ZN (t) − M
sup t∈[0,T ]
h?2`27Q`2- #v UkXkjV P
ε 2kT λemax
εeM T 2kT λemax
.
sup ZN (t) > ε
0) ,
t∈[0,T ]
r?B+?- #v i?2 "Q`2HĜ* Mi2HHB H2KK - BKTHB2b i?2 MMQmM+2/ +QMp2`;2M+2X
dy
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
kXd 1t2`+Bb2b 1t2`+Bb2 kXdXRX SQBbbQM BM #Qt2b G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λX G2i I1 - Ę- IK #2 /BbDQBMi BMi2`p Hb Q7 R M/ /2MQi2 #v I i?2B` mMBQMX G2i n #2 M BMi2;2`X q? i Bb i?2 +QM/BiBQM H /Bbi`B#miBQM Q7 i?2 p2+iQ` (N (I1 ), . . . , N (IK )) ;Bp2M i? i N (I) = n\ 1t2`+Bb2 kXdXkX h?2 `BbF T`Q+2bb *QKTmi2 i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 ` M/QK p `B #H2 X(t) Q7 1t KTH2 kXRX9X 1t2`+Bb2 kXdXjX " +Fr `/ M/ 7Q`r `/ `2+m``2M+2 iBK2b G2i {Tn }n≥1 #2 i?2 b2[m2M+2 Q7 2p2Mi iBK2b Q7 M ?TT QM R rBi? i?2 BMi2MbBiv λ > 0 M/ H2i T0 ≡ 0X h?2 7Q`r `/ `2+m``2M+2 T`Q+2bb {A(t)}t≥0 M/ i?2 # +Fr `/ `2+m`@ `2M+2 T`Q+2bb {B(t)}t≥0 `2 /2}M2/ b 7QHHQrbX "Qi? T`Q+2bb2b `2 `B;?i@+QMiBMmQmb rBi? H27i@? M/ HBKBibX 6Q` n ≥ 0- i?2v ? p2 HBM2 ` i` D2+iQ`B2b BM (Tn , Tn+1 ) rBi? `2bT2+iBp2 bHQT2b −1 M/ +1- M/ i `2M2r H TQBMi Tn A(Tn ) = Tn+1 − Tn , A(Tn+1 −) = 0 , B(Tn ) = 0 , B(Tn+1 −) = Tn+1 − Tn . 6Q` 0 ≤ t < T0 - A(t) = T0 − t M/ B(t) = tX 6Q` }t2/ t ∈ R- r? i Bb i?2 /Bbi`B#miBQM Q7 i?2 p2+iQ` (B(t), F (t))\ q? i B7 t ↑ ∞\ *QKTmi2 limt↑∞ E[B(t) + F (t)]X 1t2`+Bb2 kXdX9X 6HBT@7HQT G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X .2}M2 i?2 Ui2H2;` T? Q` ~BT@~QTV T`Q+2bb {X (t)}t≥0 rBi? bi i2 bT +2 E = {+1, −1} #v X (t) := X(0) (−1)N (t) Ui?Bb biQ+? biB+ T`Q+2bb Hi2`M i2b #2ir22M −1 M/ +1 i 2 +? 2p2Mi Q7 N V- r?2`2 X (0) Bb M E@p Hm2/ ` M/QK p `B #H2 BM/2T2M/2Mi Q7 i?2 +QmMiBM; T`Q+2bb N X h?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 X(0) Bb `#Bi` `vX RX *QKTmi2 P (X (t + s) = j|X (s) = i) 7Q` HH t, s ≥ 0 M/ HH i, j ∈ EX kX :Bp2 7Q` HH i ∈ E i?2 HBKBi Q7 P (X (t) = i) b t i2M/b iQ ∞X jX *QKTmi2 P (X (t) = i) 7Q` HH i ∈ E M/ HH t ≥ 0 r?2M P (X (0) = i) = 7Q` HH i ∈ EX
1 2
1t2`+Bb2 kXdX8X 1`;Q/B+ BMi2`T`2i iBQM Q7 i?2 BMi2MbBiv Q7 M ?TT G2i N #2 M ?TT QM R rBi? TQbBiBp2 }MBi2 BMi2MbBiv λX a?Qr i? i lim
t→∞
N (t) 1 = - S@ XbX t E[S1 ]
G2i MQr N #2 ?QKQ;2M2Qmb *Qt QM R rBi? U` M/QKV BMi2MbBiv ΛX *QK@ T`Q+2bb Tmi2 limt→∞ Nt(t) M/ limt→∞ E N t(t) X
kXdX 1s1_*Aa1a
dR
1t2`+Bb2 kXdXeX Jmim HHv bBM;mH ` ?TTb G2i N #2 TQBMi T`Q+2bb QM R /2}M2/ QM K2 bm` #H2 bT +2 (Ω, F)X G2i P1 M/ P2 #2 irQ T`Q# #BHBiv K2 bm`2b QM (Ω, F) i? i K F2 Q7 N ?TTb Q7 BMi2MbBiB2b λ1 > 0 M/ λ2 > 0 `2bT2+iBp2HvX a?Qr i? i B7 λ1 = λ2 - P1 M/ P2 `2 Kmim HHv bBM;mH `¯ = 1X i? i Bb iQ b v- i?2`2 2tBbib b2i A ∈ F bm+? i? i P1 (A) = 1 M/ P2 (A) 1t2`+Bb2 kXdXdX h?2 r `2?Qmb2 G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X G2i T #2 }t2/ }MBi2 TQbBiBp2 iBK2 M/ H2i ST #2 i?2 +QHH2+iBQM Q7 FtN @biQTTBM; iBK2b bK HH2` i? M Q` 2[m H iQ T X q2 b22F iQ }M/- B7 Bi 2tBbib- biQTTBM; iBK2 τ BM ST KBMBKBxBM; E [N (τ )(T − τ )]X U TQbbB#H2 BMi2`T`2i iBQM Bb i?2 7QHHQrBM;X N +QmMib i?2 Bi2Kb Q7 bT2+B}+ FBM/ 2Mi2`BM; r `2?Qmb2X h?2 biQ` ;2 722 Bb Q7 QM2 mMBi T2` Bi2K M/ T2` mMBi Q7 bQDQm`M iBK2 BM i?2 r `2?Qmb2X h?2`2 Bb MQ biQ` ;2 722 7i2` iBK2 T bBM+2 HH i?2 Bi2Kb MQi /BbT i+?2/ `2 i F2M 2Hb2r?2`2 Mvr vX i M BMi2`K2/B `v ` M/QK iBK2 τ ∈ [0, T ]- i?2 N (τ ) Bi2Kb T`2b2Mi `2 i F2M r v 7`QK i?2 biQ` ;2 ?Qmb2- bQ i? i MQ KQ`2 biQ` ;2 +Qbi Bb BM+m``2/ #v i?2b2 Bi2KbX h?Bb b p2b N (τ )(T − τ ) mMBib Q7 biQ` ;2 +QbiX h?2 BMi2`K2/B `v /BbT i+?BM; iBK2 Q7 +Qm`b2 + MMQi MiB+BT i2 i?2 7mim`2 iBK2 Q7 ``Bp Hb- i?Bb Bb r?v Bi Kmbi #2 M FtN @biQTTBM; iBK2XV 6BM/ i?2 QTiBK H biQTTBM; iBK2 τ ∗ X 1t2`+Bb2 kXdX3X aQK2 K `iBM; H2b G2i N #2 SQBbbQM T`Q+2bb QM R+ rBi? i?2 HQ+ HHv BMi2;` #H2 BMi2MbBiv 7mM+iBQM λ : R+ → R+ X 6Q` Mv t ≥ 0- /2}M2 N (t) := N ((0, t])X G2i M (t) := N (t) − λtX S`Qp2 i? i {M (t)2 − λt}t≥0 Bb M FtN @K `iBM; H2X 1t2`+Bb2 kXdXNX aiQ+? biB+ BMi2;` Hb M/ K `iBM; H2b G2i {Ft }t≥0 #2 ?BbiQ`v Q7 N - M ?TT QM R+ Q7 BMi2MbBiv λX amTTQb2 KQ`2Qp2` i? i 7Q` HH a > 0 M/ HH c, d Ua ≤ c ≤ dV- N (c, d] Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2 H27i@+QMiBMmQmb +QKTH2t biQ+? biB+ T`Q+2bb / Ti2/ iQ {Ft }t≥0 X X amTTQb2 i? i 7Q` HH t ≥ 0
t
E
|Z(s)| ds < ∞ .
()
0
a?Qr i? i
(0,t]
Z(s) N (ds) −
Z(s) dM (s) :=
Y (t) :=
(0,t]
t
Z(s) λ ds
()
0
Bb r2HH@/2}M2/ [m MiBiv UMQi ∞ − ∞ 7Q`KV M/ i? i i?2 biQ+? biB+ T`Q+2bb {Y (t)}t≥0 Bb M Ft @K `iBM; H2X "X G2i i?BM;b #2 b #Qp2- 2t+2Ti 7Q` +QM/BiBQM UV- MQr `2TH +2/ #v i?2 HQ+ H BMi2;` #BHBiv Q7 {Z(t)}t≥0 - i? i Bb,
∞ 0
|Z(t)| dt < ∞ 7Q` HH t ≥ 0 .
dk
*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1
a?Qr i? i Y (t) ;Bp2M #v UV Bb r2HH@/2}M2/ [m MiBiv UMQi ∞ − ∞ 7Q`KV M/ i? i i?2 biQ+? biB+ T`Q+2bb {Y (t)}t≥0 Bb HQ+ H Ft @K `iBM; H2X 1t2`+Bb2 kXdXRyX AMi2;` Hb rX`XiX K `iBM; H2 Q7 #QmM/2/ p `B iBQM G2i {M (t)}t∈R+ #2 +Q`HQH U+QMiBMmQmb QM i?2 `B;?i M/ rBi? HBKBib QM i?2 H27iV F t @K `iBM; H2 Q7 #QmM/2/ p `B iBQM- M/ rBi? HQ+ HHv BMi2;` #H2 p `B iBQM Ui? i Bb |dM (s)| < ∞ 7Q` HH t ≥ 0VX G2i {Z(t)}t∈R+ #2 H27i@+QMiBMmQmb Ft @ / Ti2/ (0,t] `2 H biQ+? biB+ T`Q+2bb2b bm+? i? i
E |Z(s)||dM (s)| < ∞ (t ≥ 0) . (0,t]
S`Qp2 i? i i?2 biQ+? biB+ T`Q+2bb {m(t)}t∈R+ /2}M2/ #v m(t) := Bb M Ft @K `iBM; H2X
(0,t]
Z(s) dM (s)
1t2`+Bb2 kXdXRRX Hi2`M iBM; ?TTb G2i N1 M/ N2 #2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b QM R+ rBi? i?2 b K2 TQbBiBp2 BMi2MbBiv λX G2i 7Q` HH t ≥ 0- Ft := FtN1 ∨ FtN2 - M/ H2i {τn }n≥0 #2 MQM@/2+`2 bBM; b2[m2M+2 Q7 Ft @biQTTBM; iBK2b- r?2`2 τ0 ≡ 0 M/ limn↑∞ τn = +∞X 2 QM R+ #v 1 M/ N .2}M2 i?2 TQBMi T`Q+2bb2b N 1 ((τ2n , t]) = N1 ((τ2n , t]) 7Q` HH t ∈ (τ2n , τ2n+1 ] (n ≥ 0) , N 1 ((τ2n+1 , t]) = N2 ((τ2n+1 , t]) 7Q` HH t ∈ (τ2n+1 , τ2n+2 ] (n ≥ 0) , N M/ 2 ((τ2n , t]) = N2 ((τ2n , t]) 7Q` HH t ∈ (τ2n , τ2n+1 ] (n ≥ 0) , N 2 ((τ2n+1 , t]) = N1 ((τ2n+1 , t]) 7Q` HH t ∈ (τ2n+1 , τ2n+2 ] (n ≥ 0) . N 2 `2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b QM R+ rBi? 1 M/ N S`Qp2 i? i N BMi2MbBiv λX 1t2`+Bb2 kXdXRkX q i M #2ǰb i?2Q`2K 7Q` *Qt T`Q+2bb2b G2i N #2 bBKTH2 TQBMi T`Q+2bb QM R+ M/ H2i {λ(t)}t≥0 #2 HQ+ HHv BMi2;` #H2 MQM@M2; iBp2 biQ+? biB+ T`Q+2bb bm+? i? i
T
E Z(t)N (dt) = E Z(t)λ(t) dt 0
(0,T ]
?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH MQM@M2; iBp2 `2 H@p Hm2/ H27i@+QMiBMmQmb biQ+? biB+ T`Q+2bb2b {Z(t)}t≥0 / Ti2/ iQ {Ft }t≥0 - r?2`2 Ft := FtN ∨ F λ
(t ≥ 0).
S`Qp2 i? i N Bb /Qm#Hv biQ+? biB+ SQBbbQM T`Q+2bb UQ` *Qt T`Q+2bbV rBi? `2bT2+i iQ F λ rBi? i?2 +QM/BiBQM H BMi2MbBiv {λ(t)}t≥0 X
kXdX 1s1_*Aa1a
dj
1t2`+Bb2 kXdXRjX _2;mH `Biv bbmK2 i?2 K i`Bt P /2}M2/ BM bi i2K2Mi Q7 h?2Q`2K kX8Xk Bb B``2/m+B#H2 M/ `2+m``2MiX a?Qr i? i τ∞ = ∞- HKQbi bm`2HvX 1t2`+Bb2 kXdXR9X EQHKQ;Q`Qpǰb 6Q`r `/ 1[m iBQMb G2i {X(t)}t≥0 #2 `2;mH ` DmKT ?K+ rBi? i` MbBiBQM b2KB@;`QmT {P(t)}t≥0 M/ BM}MBi2bBK H ;2M2` iQ` AX S`Qp2 i? i t P(t) = I + P(s)Ads (t ≥ 0) . 0
1t2`+Bb2 kXdXR8X AM7BMBi2bBK H :2M2` iQ` M/ J `iBM; H2b ij i?2 TQBMi T`Q+2bb +QmMi@ _272` iQ h?2Q`2K kX8X9X AM i?2 `2;mH ` + b2- /2MQi2 #v N BM; i?2 i` MbBiBQMb Q7 i?2 ?K+ {X(t)}t≥0 7`QK bi i2 i iQ bi i2 j = iX S`Qp2 i? i t ij (t) := N ij (t) − M 1{X(s−)=i} qij ds 0
/2}M2b M
FtX @K `iBM; H2
{Mij (t)}t≥0 X a?Qr i? i
Z(t) = Z(0) +
t
(t) , AZ(s) ds + M
UkXk9V
0
(t) = {M i (t)}i∈E M/ r?2`2 {Mi (t)}t≥0 Ui ∈ EV Bb M r?2`2 Z(t) = {Zi (t)}i∈E - M X Ft @K `iBM; H2X 1t2`+Bb2 kXdXReX Em`ixǰ `2T`2b2Mi iBQM S`Qp2 i?2 bi i2K2Mi 7QHHQrBM; 1[MX UkXRdVX
*? Ti2` j aT iB H SQBbbQM S`Q+2bb2b aT iB H SQBbbQM T`Q+2bb KQ/2Hb `2 TQTmH ` T `iHv #2+ mb2 Q7 i?2B` K i?2K iB+ H i` +i #BHBiv # b2/ QM bBKTH2 M/ TQr2`7mH + H+mHmb r?Qb2 K BM BM;`2/B2Mib- #2@ bB/2b * KT#2HHǶb 7Q`KmH - `2 i?2 +Qp `B M+2 7Q`KmH M/ i?2 2tTQM2MiB H 7Q`KmH U HbQ FMQrM b * KT#2HHǶb b2+QM/ 7Q`KmH VX h?2b2 7Q`KmH b 2ti2M/ BKK2/B i2Hv iQ K `F2/ SQBbbQM T`Q+2bb2b rBi? BM/2T2M/2Mi UQ` HQ+ iBQM /2T2M/2MiV BB/ K `Fb b +QMb2[m2M+2 Q7 i?2 7 +i i? i bm+? K `F2/ SQBbbQM T`Q+2bb2b + M #2 /2b+`B#2/ b mMK `F2/ SQBbbQM T`Q+2bb2b BM ?B;?2`@/BK2MbBQM H bT +2X h?Bb bHB;?i KQ/B@ }+ iBQM Q7 TQBMi Q7 pB2r rBHH #2 TTHB2/ iQ i?2 bim/v Q7 i?2 TQBMi T`Q+2bb2b Q#@ i BM2/ #v K2 Mb Q7 2H2K2Mi `v i` Mb7Q`K iBQMb Q7 # bB+ SQBbbQM T`Q+2bb- bm+? b i?BMMBM;- i` MbH iBQM Q` +Hmbi2`BM;X HH i?2 `2bmHib +QM+2`MBM; SQBbbQM T`Q+2bb2b }M/ bi` B;?i7Q`r `/ 2ti2MbBQMb +QM+2`MBM; *Qt T`Q+2bb2bX h?2 + H+mHmb Q7 bT iB H K `F2/ SQBbbQM Q` *Qt T`Q+2bb2b rBHH #2 TTHB2/ iQ T `iB+mH ` ivT2 Q7 biQ+? biB+ ;2QK2i`v U72 im`BM; i?2 "QQH2 M KQ/2HV M/ iQ i?2 2t +i b KTHBM; Q7 +Hmbi2` TQBMi T`Q+2bb2bX
jXR
a KTHBM; SQBbbQM S`Q+2bb
h?2 M2ti `2bmHi + M #2 +QMbB/2`2/ T`QQ7 Q7 2tBbi2M+2 Q7 i?2 SQBbbQM T`Q+2bb Q7 .2}MBiBQM RXRXNX >Qr2p2`- Bib K BM BMi2`2bi HB2b BM i?2 7 +i i? i Bi + M #2 mb2/ 7Q` b KTHBM; Ui?2 /Bbi`B#miBQM Q7V bm+? TQBMi T`Q+2bb Ub22 .2}MBiBQM RXjXkVX G2i E #2 M HX+X/X#X bT +2 rBi? "Q`2H σ@}2H/ B(E)X h?2Q`2K jXRXR G2i {Zn }n≥1 #2 M BB/ E@p Hm2/ ` M/QK b2[m2M+2 rBi? +QKKQM /Bbi`B#miBQM QX G2i T #2 SQBbbQM ` M/QK p `B #H2 rBi? K2 M θ M/ BM/2T2M/2Mi Q7 {Zn }n≥1 X h?2 TQBMi T`Q+2bb N QM E /2}M2/ #v N (C) :=
T
1C (Zn ) (C ∈ B(E))
n=1
Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν(·) = θ × Q(·)X S`QQ7X Ai bm{+2b iQ b?Qr i? i 7Q` Mv }MBi2 7 KBHv C1 , . . . , CK Q7 T B`rBb2 /BbDQBMi K2 bm` #H2 b2ib Q7 E rBi? }MBi2 ν@K2 bm`2 M/ 7Q` HH MQM@M2; iBp2 `2 H MmK#2`b t 1 , . . . , tK -
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_3
d8
de
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a E[e−
K
j=1 tj N (Cj )
−tj ] = ΠK − 1) . j=1 exp ν(Cj )(e
q2 ? p2 K
tj N (Cj ) =
j=1
K j=1
tj
T
1Cj (Zn )
=
n=1
K T n=1
tj 1Cj (Zn )
=
T
Yn ,
n=1
j=1
r?2`2 Yn = K j=1 tj 1Cj (Zn )X "v +H bbB+ H `2bmHi QM i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 ` M/QK bmKbT E[e− n=1 Yn ] = gT (E[e−Y1 ]) , r?2`2 gT Bb i?2 ;2M2` iBM; 7mM+iBQM Q7 i?2 ` M/QK p `B #H2 T X aBM+2 T Bb SQBbbQM ` M/QK p `B #H2 rBi? K2 M θgT (z) = exp {θ(z − 1)} . h?2 ` M/QK p `B #H2 Y1 i F2b i?2 p Hm2b
t1 , . . . , tK M/ 0 rBi? i?2 `2bT2+iBp2 T`Q#@ #BHBiB2b Q(C1 )- Ę- Q(CK ) M/ 1 − K j=1 Q(Cj )X h?2`27Q`2 E[e−Y1 ] =
K
e−tj Q(Cj ) + 1 −
j=1
K
Q(Cj ) = 1 +
j=1
K
e−tj − 1 Q(Cj ) ,
j=1
7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX
h?2Q`2K jXR Bb bT2+B H + b2 Q7 r? i Bb iQ #2 /QM2- i? i Bb- iQ +QMbi`m+i SQBbbQM T`Q+2bb QM E rBi? M BMi2MbBiv K2 bm`2 ν i? i Bb σ@}MBi2 UMQi Dmbi }MBi2VX am+? K2 bm`2 + M #2 /2+QKTQb2/ b ν(·) =
∞
θj × Qj (·) ,
j=1
r?2`2 i?2 θj Ƕb `2 TQbBiBp2 `2 H MmK#2`b M/ i?2 Qj Ƕb `2 T`Q# #BHBiv /Bbi`B#miBQMb QM EX PM2 + M +QMbi`m+i BM/2T2M/2Mi SQBbbQM T`Q+2bb2b Nj QM E rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b θj Qj (·)X h?2 MMQmM+2/ `2bmHi i?2M 7QHHQrb 7`QK i?2 7QHHQrBM;,
∞ h?2Q`2K jXRXk G2i ν #2 σ@}MBi2 K2 bm`2 QM E Q7 i?2 7Q`K ν = i=1 νi r?2`2 i?2 νi Ƕb (i ≥ 1) `2 σ@}MBi2 K2 bm`2b QM EX G2i Ni (i ≥ 1) #2 7 KBHv Q7 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM E rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b νi (i ≥ 1)X h?2M i?2 TQBMi T`Q+2bb ∞ N := Nj j=1
Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 νX S`QQ7X 6Q` Kmim HHv /BbDQBMi K2 bm` #H2 b2ib C1 - Ę- CK Q7 }MBi2 ν@K2 bm`2b- M/ MQM@M2; iBp2 `2 Hb t1 - Ę- tK -
jXRX aJSGAL: SPAaaPL S_P*1aa
dd
K K ∞ E e− =1 t N (C ) = E e− =1 t ( j=1 Nj (C )) K n = E e− limn↑∞ =1 t ( j=1 Nj (C )) K n = lim E e− =1 t ( j=1 Nj (C )) , n↑∞
#v /QKBM i2/ +QMp2`;2M+2X "mi n K K n E e− =1 t ( j=1 Nj (C )) = E e− j=1 ( =1 t Nj (C )) =
n
n K K E e− =1 t Nj (C ) = e−t Nj (C )
j=1
=
j=1 =1
n K
exp (e−t − 1)νj (C ) =
j=1 =1
= exp
K
e
=1
−t
−1
n
n
exp
j=1
e−t − 1 νj (C )
=1
νj (C ))
K
.
j=1
G2iiBM; n ↑ ∞- r2 Q#i BM- #v /QKBM i2/ +QMp2`;2M+2 K K − =1 t N (C ) −t E e e − 1 ν(C ) . = exp =1
h?2`27Q`2 N (C1 )- Ę- N (CK ) `2 BM/2T2M/2Mi SQBbbQM ` M/QK p `B #H2b rBi? `2@ bT2+iBp2 K2 Mb ν(C1 )- Ę- ν(CK )X h?2Q`2K jXRXj G2i N #2 SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 νX U V A7 ν Bb HQ+ HHv }MBi2- i?2M N Bb XbX HQ+ HHv }MBi2X U#V A7 ν Bb HQ+ HHv }MBi2 M/ MQM@ iQKB+- i?2M N Bb XbX bBKTH2X S`QQ7X U V A7 C Bb #QmM/2/ K2 bm` #H2 b2i- Bi Bb Q7 }MBi2 ν@K2 bm`2- M/ i?2`27Q`2 E[N (C)] = ν(C) < ∞- r?B+? BKTHB2b i? i N (C) < ∞- P @ HKQbi bm`2HvX U#V Ai bm{+2b iQ b?Qr i?Bb 7Q` }MBi2 BMi2MbBiv K2 bm`2 ν(·) = θ(·) Q r?2`2 θ Bb TQbBiBp2 `2 H MmK#2` M/ Q Bb MQM@ iQKB+ T`Q# #BHBiv K2 bm`2 QM Rm M/ i?2M mb2 i?2 +QMbi`m+iBQM Q7 h?2Q`2K jXRXRX AM im`M- Bi bm{+2b iQ b?Qr i? i 7Q` HH n ≥ 1- P (Zi = Zj 7Q` bQK2 T B` (i, j) (1 ≤ i < j ≤ n) | N (Rm ) = n) = 0X h?Bb Bb i?2 + b2 #2+ mb2 7Q` BB/ p2+iQ`b Z1 , . . . , Zn rBi? MQM@ iQKB+ T`Q# #BHBiv /Bbi`B#miBQM- P (Zi = Zj 7Q` bQK2 T B` (i, j) (1 ≤ i < j ≤ n)) = 0X 1t KTH2 jXRX9, h?2 bi M/ `/ b KTHBM; K2i?Q/ 7Q` M ?TTX h?2 +QM@ bi`m+iBQM Q7 h?2Q`2K jXRXR ;Bp2b K2i?Q/ 7Q` b KTHBM; SQBbbQM T`Q+2bb Q7 BMi2M@ bBiv K2 bm`2 ν BM rBM/Qr W ⊂ Rm Q7 }MBi2 ν@K2 bm`2X 6B`bi b KTH2 SQBbbQM p `B #H2 T Q7 K2 M θ := ν(W )X A7 T = k- b KTH2 k BB/ p2+iQ`b Z1 , . . . , Zk rBi? i?2
d3
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
+QKKQM /Bbi`B#miBQM P (Z1 ∈ C) = ν(C)/ν(W ) UC ⊂ W VX h?2 /2bB`2/ b KTH2 Bb i?2M i?2 b2i Q7 TQBMib {Z1 , . . . , Zk }X 6Q` i?2 b KTH2b Zn - mb2 QM2 Q7 i?2 K Mv K2i?@ Q/b p BH #H2 UBMp2`b2 7mM+iBQM- ++2Ti iBQM@`2D2+iBQM- 2i+XVX h?2 b KTHBM; Q7 T K v i #2 /QM2 #v i?2 +H bbB+ H BMp2`b2 7mM+iBQM K2i?Q/ b 7QHHQrbX G2iiBM; pi := e−θ θi! b KTH2 ` M/QK p `B #H2 /Bbi`B#mi2/ QM [0, 1] M/ b2i T = k B7 U 7 HHb
U mMB7Q`KHv
k BM i?2 BMi2`p H Ii := [ k−1 p , p ]X i=0 i i=0 i +`m/2 p2`bBQM Q7 i?2 b KTHBM; H;Q`Bi?K +QMbBbib BM 2t KBMBM; i?2 BMi2`p Hb Ii b2[m2MiB HHv mMiBH QM2 Bb 7QmM/ i? i +QMi BMb U X h?Bb rQmH/ `2[mB`2 QM i?2 p2` ;2 1 + E[T ] = 1 + θ i`B HbX A7 θ Bb p2`v H `;2 KQ`2 2+QMQKB+ H T`Q+2/m`2 Bb p BH #H2XR Ai i F2b BMiQ ++QmMi i?2 7 +i i? i i?2 T`Q# #BHBiv K bb Q7 SQBbbQM p `B #H2 Bb K tBK H i p Hm2 i0 M2 ` i?2 p2` ;2 p Hm2 M/ /2+`2 b2b b QM2 ;2ib 7 `i?2` r v 7`QK i?Bb p Hm2X h?2 2tTHQ` iBQM bi `ib rBi? i?2 p Hm2 i0 - M/ i?2M T`Q+22/b iQ i0 − 1- i0 + 1- i0 − 2- i0 + 2- 2i+X h?2 p2` ;2 MmK#2` Q7 i`B Hb Bb i?2M `Qm;?Hv 2[m H iQ
√ |T − θ| √ . 1 + E [|T − θ|] = 1 + θE θ Bb TT`QtBK i2Hv /Bbi`B#mi2/ b bi M/ `/ "v i?2 +2Mi` H HBKBi i?2Q`2K- Z := T√−θ θ : mbbB M p `B #H2X h?2`27Q`2 i?2 p2` ;2 MmK#2` Q7 i`B Hb 7Q` H `;2 θ Bb TT`QtB@ K i2Hv √ √ 1 + θE [|N (0, 1)|] 1 + 0.82 θ .
MQi?2` r v Q7 bi iBM; h?2Q`2K jXRXR Bb b 7QHHQrbX G2i ν #2 }MBi2 K2 bm`2 QM E M/ H2i T #2 SQBbbQM p `B #H2 rBi? K2 M ν(E)X 6Q` 2 +? n ≥ 0- /2MQi2 #v Pn i?2 T`Q# #BHBiv QM (Mp (E), Mp (E)) bm+? i? i i?2 + MQMB+ H TQBMi T`Q+2bb ? b 2t +iHv X n TQBMib- TH +2/ BM/2T2M/2MiHv QM E ++Q`/BM; iQ i?2 T`Q# #BHBiv /Bbi`B#miBQM ν(dx) ν(W ) q? i h?2Q`2K jXR b?Qrb Bb i? i i?2 T`Q# #BHBiv P QM (Mp (E), Mp (E)) K FBM; Q7 i?2 + MQMB+ H TQBMi T`Q+2bb SQBbbQM T`Q+2bb rBi? U`2+ HH, }MBi2V BMi2MbBiv K2 bm`2 ν + M #2 `2T`2b2Mi2/ b ν(E)n P = Pn . e−ν(E) UjXRV n! n=0 1t KTH2 jXRX8, _ `2 S2`im`# iBQM a2MbBiBpBiv M HvbBbXk G2i N #2 bT iB H SQBbbQM T`Q+2bb QM 1 rBi? K2 M K2 bm`2 ν = αQ- r?2`2 α > 0 M/ Q(E) < ∞X .2MQi2 #v Pα #2 +Q``2bTQM/BM; T`Q# #BHBivX G2i f (N ) #2 bQK2 #QmM/2/ [f (N )] K2 bm` #H2 7mM+iBQM H Q7 N X h?2 ;Q H Bb iQ }M/ M 2tT`2bbBQM 7Q` dEαdα X 6Q` i?Bbr2 mb2 i?2 `2T`2b2Mi iBQM UjXRV- r?B+? ;Bp2b Eα [f (N )] = P (Tα = n)En [f (N )] , UjXkV n=0
r?2`2 Tα Bb SQBbbQM p `B #H2 rBi? K2 M α M/ r?2`2 En /2MQi2b 2tT2+i iBQM rBi? `2bT2+i iQ i?2 T`Q# #BHBiv Pn mM/2` r?B+? N ? b 2t +iHv n TQBMib BM/2T2M/2MiHv R k
(_Qbb- RNNk)- TTX 9NĜ8yX ("`ûK m/ M/ ox[m2x@# /- RNNk)X
jXRX aJSGAL: SPAaaPL S_P*1aa
dN
M/ B/2MiB+ HHv /Bbi`B#mi2/ ++Q`/BM; iQ i?2 /Bbi`B#miBQM +QKTmi iBQM ;Bp2b
Q(dx) X Q(E)
dP (Tα = n) = −P (Tα = n) + P (Tα = n − 1) dα
bi` B;?i7Q`r `/
(n ≥ 1) ,
M/ dP (Tα = 0) = −P (Tα = 0) dα
(n ≥ 1) ,
M/ i?2`27Q`2- mbBM; `2T`2b2Mi iBQM UjXRVdEα [f (N )] = − P (Tα = 0)E0 (f (N )) dα + P (Tα = 0)E1 (f (N )) − P (Tα = 1)E1 (f (N )) + P (Tα = 1)E2 (f (N )) − P (Tα = 2)E2 (f (N )) + ··· _2 `` M;BM; i?2 i2`Kb- r2 Q#i BM dEα [f (N )] =P (Tα = 0) (E1 [f (N )] − E0 [f (N )]) dα + P (Tα = 1) (E2 [f (N )] E1 [f (N )]) + ··· LQr- 7Q` HH n ≥ 1En+1 [f (N )] =
1 Q(E)
En [f (N + εx )] Q(dx) E
M/ i?2`27Q`2- }M HHv, dEα [f (N )] 1 = dα Q(E)
(Eα [f (N + εx )] − Eα [f (N )]) Q(dx) . E
6`QK i?2 bBKmH iBQM TQBMi Q7 pB2r-j i?Bb /2`Bp iBp2 + M #2 TT`QtBK i2/ #v i?2 2KTB`B+ H K2 M n 1 (f (Nk + εZk )) − (f (Nk )) n k=1 7Q` bm{+B2MiHv H `;2 n- r?2`2 {Nk }1≤k≤n Bb M BB/ b2[m2M+2 Q7 +QTB2b Q7 N mM/2` T`Q# #BHBiv Pα - {Zk }1≤k≤n Bb M BB/ b2[m2M+2 Q7 E@p Hm2/ ` M/QK 2H2K2Mib rBi? i?2 /Bbi`B#miBQM Q(dx) M/ BM/2T2M/2Mi Q7 i?2 b2[m2M+2 {Nk }1≤k≤n X Q(E) j a22 (bKmbb2M M/ :HvMM- kyyd)- *? Ti2` oAA- 7Q` i?2 mb2 Q7 /2`Bp iBp2b BM bBKmH iBQM M/ biQ+? biB+ QTiBKBx iBQMX
3y
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
jXk h?2 *Qp `B M+2 M/ 1tTQM2MiB H 6Q`KmH b G2i N #2 SQBbbQM T`Q+2bb QM i?2 HX+X/X#X bT +2 E- rBi? BMi2MbBiv K2 bm`2 νX _2+ HH * KT#2HHǶb i?2Q`2K Uh?2Q`2K RXkXRVX G2i ϕ : E → R #2 ν@BMi2;` #H2 K2 bm` #H2 7mM+iBQMX h?2M N (ϕ) Bb r2HH@/2}M2/ BMi2;` #H2 ` M/QK p `B #H2 M/
E ϕ(x) N (dx) = ϕ(x) ν(dx) . UjXjV E
E
h?2Q`2K jXkXR G2i N #2 b #Qp2X G2i ϕ, ψ : E → C #2 irQ ν@BMi2;` #H2 K2 @ bm` #H2 7mM+iBQMb bm+? i? i KQ`2Qp2` |ϕ|2 M/ |ψ|2 `2 ν@BMi2;` #H2X h?2M N (ϕ) M/ N (ψ) `2 r2HHĜ/2}M2/ b[m `2@BMi2;` #H2 ` M/QK p `B #H2b M/ r2 ? p2 i?2 +Qp `B M+2 7Q`KmH 7Q` SQBbbQM T`Q+2bb2b, cov ϕ(x) N (dx), ψ(x) N (dx) = ϕ(x)ψ(x)∗ ν(dx) . UjX9V E
E
E
S`QQ7X Ai Bb 2MQm;? iQ +QMbB/2` i?2 + b2 Q7 `2 H 7mM+iBQMbX 6B`bi bmTTQb2 i? i ϕ M/ ψ `2 bBKTH2 MQM@M2; iBp2 "Q`2H 7mM+iBQMb- b vϕ :=
K
ah 1Ch M/
ψ :=
h=1
K
b h 1 Ch ,
h=1
r?2`2 C1 , . . . , CK `2 /BbDQBMi K2 bm` #H2 bm#b2ib Q7 E UQM2 K v Hr vb bbmK2 i? i
K i?2 irQ 7mM+iBQMb ? p2 i?2 Ci Ƕb BM +QKKQMVX AM T `iB+mH `- ϕ(x)ψ(x) = h=1 ah bh 1Ch (x)X lbBM; i?2 7 +ib i? i N (Ci ) M/ N (Cj ) `2 BM/2T2M/2Mi B7 i = j M/ i? i SQBbbQM ` M/QK p `B #H2 rBi? K2 M θ ? b p `B M+2 θE[N (ϕ)N (ψ)] =
K
ah bl E[N (Ch )N (Cl )]
h,l=1
=
K
ah bl E[N (Ch )N (Cl )] +
h,l=1 h=l
=
K
K
al bl E[N (Cl )2 ]
l=1
ah bl E[N (Ch )]E[N (Cl )] +
h,l=1 h=l
K
al bl E[N (Cl )2 ] ,
l=1
M/ i?2`27Q`2 E[N (ϕ)N (ψ)] =
K
ah bl ν(Ch )ν(Cl ) +
h,l=1 h=l
=
k h,l=1
k
al bl [ν(Cl ) + ν(Cl )2 ]
l=1
ah bl ν(Ch )ν(Cl ) +
k l=1
al bl ν(Cl ) = ν(ϕ)ν(ψ) + ν(ϕψ) .
jXkX h>1 *Po_AL*1 L. 1sSPL1LhAG 6P_JlGa
3R
G2i MQr ϕ- ψ #2 MQM@M2; iBp2 M/ H2i {ϕn }n≥1 - {ψn }n≥1 #2 MQM@/2+`2 bBM; b2@ [m2M+2b Q7 bBKTH2 MQM@M2; iBp2 7mM+iBQMb- rBi? `2bT2+iBp2 HBKBib ϕ M/ ψX G2iiBM; n ;Q iQ ∞ BM i?2 2[m HBiv E[N (ϕn )N (ψn )] = ν(ϕn ψn ) + ν(ϕn )ν(ψn ) vB2H/b i?2 MMQmM+2/ `2bmHib- #v KQMQiQM2 +QMp2`;2M+2X G2i ϕ+ := ϕ 1{ϕ≥0} M/ ϕ− := −ϕ 1{ϕSh1_ jX aShAG SPAaaPL S_P*1aa1a
E[e
−N (ϕ)
] = E e
−
K
h=1
ah N (Ch ))
=E
K
e
−ah N (Ch )
h=1
=
K
K E e−ah N (Ch ) = exp (e−ah − 1)ν(Ch )
h=1
= exp
h=1 K
(e
−ah
− 1)ν(Ch )
= exp ν(e−ϕ − 1) .
h=1
h?2 7Q`KmH Bb i?2`27Q`2 i`m2 7Q` MQM@M2; iBp2 bBKTH2 7mM+iBQMbX h F2 MQr MQM@ /2+`2 bBM; b2[m2M+2 {ϕn }n≥1 Q7 bm+? 7mM+iBQMb +QMp2`;BM; iQ ϕX 6Q` HH n ≥ 1 E[e−N (ϕn ) ] = exp ν(e−ϕn − 1) . "v KQMQiQM2 +QMp2`;2M+2- i?2 HBKBi b n i2M/b iQ ∞ Q7 N (ϕn ) Bb N (ϕ)X *QM@ b2[m2MiHv- #v /QKBM i2/ +QMp2`;2M+2- i?2 HBKBi Q7 i?2 H27i@? M/ bB/2 BM i?2 H bi /BbTH v Bb E[e−N (ϕ) ]X h?2 7mM+iBQM gn = −(e−ϕn − 1) Bb MQM@M2; iBp2 7mM+@ iBQM BM+`2 bBM; iQ g = −(e−ϕ − 1)- M/ i?2`27Q`2- #v KQMQiQM2 +QMp2`;2M+2ν(e−ϕn − 1) = −ν(gn ) +QMp2`;2b iQ ν(e−ϕ − 1) = −ν(g)- r?B+? BM im`M BKTHB2b i? i i?2 `B;?i@? M/ bB/2 BM i?2 H bi /BbTH v i2M/b iQ exp {ν(e−ϕ − 1)} b n i2M/b iQ ∞X 1t KTH2 jXkX9, h?2 G TH +2 6mM+iBQM H Q7 SQBbbQM S`Q+2bbX 6`QK h?2Q`2K jXkXk- r2 Q#i BM 7Q` i?2 G TH +2 7mM+iBQM H Q7 SQBbbQM T`Q+2bb N QM E rBi? BMi2MbBiv K2 bm`2 ν, LN (ϕ) = exp ν e−ϕ − 1 .
h?2 * KT#2HH 7Q`KmH BKTHB2b i? i 7Q` SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2 bm`2 ν- B7 ν(ϕ) < ∞- i?2M N (ϕ) < ∞X AM 7 +ih?2Q`2K jXkX8 P (N (ϕ) < ∞) = 1 ↔ ν(ϕ ∧ 1) < ∞ . S`QQ7X h?2 T`QQ7 mb2b i?2 7QHHQrBM; 7 +ib, 1 − e−a ≤ a ∧ 1 7Q` HH a ≥ 0 M/ P (N (ϕ) < ∞) = lim e−tN (ϕ) = lim exp −ν(1 − e−tϕ . t↓0
t↓0
h?2`27Q`2- BM i?2 + b2 ν(ϕ ∧ 1) < ∞ P (N (ϕ) < ∞) = lim exp −ν(1 − e−tϕ ≥ lim exp (−ν(tϕ ∧ 1)) = 1 t↓0
t↓0
U#v /QKBM i2/ +QMp2`;2M+2 bBM+2 limt↓0 ν(tϕ∧1) = 0 M/ ν(tϕ∧1) ≤ ν(ϕ∧1) < ∞ 7Q` bm{+B2MiHv bK HH tVX
jXkX h>1 *Po_AL*1 L. 1sSPL1LhAG 6P_JlGa
3j
A7 ν(ϕ ∧ 1) = ∞ E e−N (ϕ) = exp −ν(1 − e−ϕ ) ≥ exp (−ν(ϕ ∧ 1)) = 0 r?B+? BKTHB2b i? i P (N (ϕ) = ∞) = 1X
1t KTH2 jXkXe, h?2 J tBKmK 6Q`KmH X G2i N #2 bBKTH2 SQBbbQM T`Q+2bb QM E rBi? BMi2MbBiv K2 bm`2 ν M/ H2i ϕ : E → RX h?2M P sup ϕ(Xn ) ≤ a = exp − 1{ϕ(x)>a} ν(dx) . n∈N
E
/B`2+i T`QQ7 # b2/ QM i?2 +QMbi`m+iBQM Q7 i?2 SQBbbQM T`Q+2bb BM a2+iBQM RXR Bb TQbbB#H2 U1t2`+Bb2 jX3XjVX q2 i F2 MQi?2` T i? M/ }`bi T`Qp2 i? i lim E e−θ n∈N 1{ϕ(Xn )>a} = P sup ϕ(Xn ) ≤ a . () θ↑∞
n∈N
AM/22/- i?2 bmK n∈N 1{ϕ(Xn )>a} Bb bi`B+iHv TQbBiBp2- 2t+2Ti r?2M supn∈N ϕ(Xn ) ≤ a- BM r?B+? + b2 Bi Bb MmHHX h?2`27Q`2 lim e−θ
n∈N
1{ϕ(Xn )>a}
θ↑∞
= 1{supn∈N ϕ(Xn )≤a} .
h FBM; 2tT2+i iBQMb vB2H/b UV- #v /QKBM i2/ +QMp2`;2M+2X LQr- #v h?2Q`2K jXkXk −θ1 e {ϕ(x)>a} − 1 ν(dx) E e−θ n∈N 1{ϕ(Xn )>a} = exp E −θ = exp e − 1 1{ϕ(x)>a} ν(dx) E
M/ i?2 HBKBi Q7 i?2 H ii2` [m MiBiv b θ ↑ ∞ Bb exp − E 1{ϕ(x)>a} ν(dx) X h?2Q`2K jXkXd G2i Ni (i ∈ J) #2 }MBi2 +QHH2+iBQM Q7 bBKTH2 TQBMi T`Q+2bb2b QM EX A7 7Q` Mv +QHH2+iBQM ϕi : E → R+ (i ∈ J) Q7 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb − Ni (ϕi ) −ϕi (x) i∈J E e = e exp − 1 νi (dx) , UjX8V E
i∈J
r?2`2 νi (i ∈ J) Bb +QHH2+iBQM Q7 σ@}MBi2 K2 bm`2b QM E- i?2M Ni Ui ∈ JV Bb 7 KBHv Q7 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b νi (i ∈ J)X S`QQ7X G2iiBM; HH i?2 ϕi 2t+2Ti ϕ1 #2 B/2MiB+ HHv MmHH- r2 ? p2 −N1 (ϕ1 ) −ϕ1 (x) E e − 1 ν1 (dx) , = exp e E
r?B+? T`Qp2b i? i N1 Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν1 X aBKBH `Hv- 7Q` Mv i ∈ J- Ni Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 νi X AM/2T2M/2M+2 i?2M 7QHHQrb 7`QK h?2Q`2K RXjXRdX
39
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
jXj J `F2/ aT iB H SQBbbQM S`Q+2bb2b h?2 2ti2MbBQM Q7 i?2 +Qp `B M+2 M/ 2tTQM2MiB H 7Q`KmH b iQ K `F2/ SQBbbQM T`Q@ +2bb2b /2T2M/ QM i?2 7mM/ K2Mi H `2bmHi #2HQr Uh?2Q`2K jXjXRVX G2i UαV N #2 bBKTH2 M/ HQ+ HHv }MBi2 T`Q+2bb QM E rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ H2i UβV {Zn }n∈N #2 b2[m2M+2 Q7 ` M/QK 2H2K2Mib i FBM; i?2B` p Hm2b BM i?2 K2 @ bm` #H2 bT +2 (K, K)X h?2 b2[m2M+2 {Xn , Zn }n∈N Bb K `F2/ TQBMi T`Q+2bb- rBi? i?2 BMi2`T`2i iBQM i? i Zn Bb i?2 K `F bbQ+B i2/ rBi? i?2 TQBMi Xn X h?2 TQBMi T`Q+2bb N Bb + HH2/ i?2 # b2 TQBMi T`Q+2bb Q7 i?2 K `F2/ TQBMi T`Q+2bb M/ {Zn }n∈N Bb i?2 bbQ+B i2/ b2[m2M+2 Q7 K `FbX AM Qi?2` rQ`/b- N Bb bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM E rBi? K `Fb {Zn }n∈N BM KX A7 KQ`2Qp2` URV N Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 νUkV {Zn }n∈N Bb M BB/ b2[m2M+2- M/ UjV {Zn }n∈N M/ N `2 BM/2T2M/2Mii?2 +Q``2bTQM/BM; K `F2/ TQBMi T`Q+2bb Bb + HH2/ SQBbbQM T`Q+2bb2b QM E rBi? BM/2T2M/2Mi BB/ K `FbX h?Bb KQ/2H + M #2 bHB;?iHv ;2M2` HBx2/ #v HHQrBM; i?2 K `F /Bbi`B#miBQM iQ /2T2M/ QM i?2 HQ+ iBQM Q7 i?2 K `F2/ TQBMiX JQ`2 T`2+Bb2Hvr2 `2TH +2 UkV M/ UjV #v UkǶV {Zn }n∈N Bb- +QM/BiBQM HHv QM N - M BM/2T2M/2Mi b2[m2M+2UjǶV ;Bp2M Xn - i?2 ` M/QK 2H2K2Mi Zn Bb BM/2T2M/2Mi Q7 Xk Uk ∈ N, k = nV M/ U9ǶV 7Q` HH n ∈ N M/ HH L ∈ KP (Zn ∈ L | Xn = x) = Q(x, L) , r?2`2 Q(·, ·) Bb biQ+? biB+ F2`M2H 7`QK (E, B(E)) iQ (K, K)- i? i Bb- 7mM+iBQM 7`QK E×K iQ [0, 1] bm+? i? i 7Q` HH L ∈ K i?2 K TTBM; x → Q(x, L) Bb K2 bm` #H2 M/ 7Q` HH x ∈ E- Q(x, ·) Bb T`Q# #BHBiv K2 bm`2 QM (K, K)X h?2Q`2K jXjXR G2i {Xn , Zn }n∈N #2 b BM UαV M/ UβV #Qp2 M/ /2}M2 i?2 TQBMi QM E × K #v T`Q+2bb N (A) := N
1A (Xn , Zn )
(A ∈ B(E) ⊗ K) .
UjXeV
n∈N
Bb bBKTH2 SQBbbQM A7 +QM/BiBQMb URV- UkǶV- UjǶV- M/ U9ǶV #Qp2 `2 b iBb}2/- i?2M N T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν ;Bp2M #v ν(C × L) = Q(x, L) ν(dx) (C ∈ B(E) , L ∈ K) . C
jXjX J_E1. aShAG SPAaaPL S_P*1aa1a
38
S`QQ7X AM pB2r Q7 h?2Q`2K RXjXR9- Bi bm{+2b iQ b?Qr i? i i?2 G TH +2 i` Mb7Q`K ? b i?2 TT`QT`B i2 7Q`K- i? i BbQ7 N − 1 ν(dt × dz) E e−N (ϕ) = exp E K e−ϕ(t,z) 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM ϕ : E × K → RX "v /QKBM i2/ +QMp2`@ ;2M+2 n ,Zn ) n ,Zn ) E e−N (ϕ) = lim E e− n≤k ϕ(X = E e− n∈N ϕ(X . k↑∞
6Q` i?2 iBK2 #2BM;- }t i?2 TQbBiBp2 BMi2;2` kX h?2M- i FBM; BMiQ ++QmMi bbmKT@ iBQMb UkǶV M/ UjǶV n ,Zn ) n ,Zn ) n ,Zn ) E e− n≤k ϕ(X =E E =E e−ϕ(X e−ϕ(X | Xj ; j ≤ k =E
n≤k
n≤k
e
−ϕ(X n ,z)
Q(Xn , dz) = E exp
K
n≤k
n≤k
log
e
−ϕ(X n ,z)
Q(Xn , dz)
K
= E e− n≤k ψ(Xn ) ,
r?2`2 ψ(x) := − log K e−ϕ(x,z) Q(x, dz)- MQM@M2; iBp2 7mM+iBQMX G2iiBM; k ↑ ∞r2 Q#i BM #v /QKBM i2/ +QMp2`;2M+2 i? i E e−N (ϕ) = E e− n∈N ψ(Xn ) = E e−N (ψ) −ψ(x) = exp e − 1 ν(dx) E
= exp e−ϕ(x,z) Q(x, dz) − 1 ν(dx) E K
−ϕ(x,z) = exp e − 1 Q(x, dz) ν(dx) E K −ϕ(x,z) = exp − 1 ν(dx × dz) , e r?2`2 i?2 7 +i i? i
K
E
K
Q(x, dz) = 1 r b i F2M BMiQ ++QmMiX
1t KTH2 jXjXk, h?2 Jf:Af∞ JQ/2H- h F2 RX h?Bb KQ/2H Bb Q7 BMi2`2bi BM [m2m2BM; i?2Q`v M/ BM i?2 i` {+ M HvbBb Q7 +QKKmMB+ iBQMb M2irQ`FbX q2 /QTi i?2 [m2m2BM; BMi2`T`2i iBQMX G2i N #2 ?TT QM R rBi? BMi2MbBiv λ M/ H2i {σn }n∈Z #2 b2[m2M+2 Q7 ` M/QK p `B #H2b i FBM; i?2B` p Hm2b BM R+ rBi? i?2 T`Q# #BHBiv /Bbi`B#miBQM QX bbmK2 KQ`2Qp2` i? i {σn }n∈Z M/ N `2 BM/2T2M/2MiX h?2 n@i? 2p2Mi iBK2 Tn Q7 N Bb BMi2`T`2i2/ b i?2 ``Bp H iBK2 Q7 i?2 n@i? +mbiQK2` M/ σn QM R × R+ /2}M2/ #v Bb ?2` b2`pB+2 `2[m2bi UBM iBK2 mMBibVX h?2 TQBMi T`Q+2bb N (C) := 1C (Tn , σn ) (C ∈ B(R) ⊗ B(R+ )) N n∈Z
Bb- ++Q`/BM; iQ h?2Q`2K jXjXR- bBKTH2 SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2
3e
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a ν(dt × dz) = λdt × Q(dz) .
AM i?2 Jf:Af∞ KQ/2H-9 +mbiQK2` ``BpBM; i iBK2 Tn Bb BKK2/B i2Hv b2`p2/ M/ i?2`27Q`2 /2T `ib 7`QK i?2 dzbvbi2KǴ i iBK2 Tn + σn X h?2 MmK#2` X(t) Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t Bb i?2`27Q`2 ;Bp2M #v i?2 7Q`KmH X(t) = 1(−∞,t] (Tn )1(t,∞) (Tn + σn ) . n∈Z
Uh?2 n@i? +mbiQK2` Bb BM i?2 bvbi2K i iBK2 t B7 M/ QMHv B7 b?2 ``Bp2/ i iBK2 Tn ≤ t M/ /2T `i2/ i iBK2 Tn + σn > tXV bbmK2 i? i i?2 b2`pB+2 iBK2b ? p2 }MBi2 2tT2+i iBQM, E [σ1 ] < ∞X h?2M- 7Q` HH t ∈ R- X(t) Bb SQBbbQM ` M/QK p `B #H2 rBi? K2 M λE [σ1 ]X S`QQ7X LQiB+2 i? i
˜ (C(t)) , X(t) = N
r?2`2 C(t) := {(s, σ); s ≤ t, s + σ > t} ⊂ R × R+ Ui?2 b? /2/ `2;BQM BM 6B;m`2 jXjXRVX
C(t)
t
6B;m`2 jXjXR AM T `iB+mH `- X(t) Bb SQBbbQM ` M/QK p `B #H2 rBi? K2 M 1{s+σ>t} 1{s≤t} ν˜(ds × dσ) = 1{s+σ>t} 1{s≤t} λ ds × Q(dσ) ν˜(C(t)) = R R+ R R+ t 1{s+σ>t} Q(dσ) 1{s≤t} λ ds = λ Q((t − s, +∞)) ds = −∞ R R+ ∞ ∞ Q((s, +∞)) ds = λ P (σ1 > s)ds = λE[σ1 ] . =λ 0
0
Ai + M #2 b?QrM i? i i?2 /2T `im`2 T`Q+2bb D Q7 /2T `im`2 iBK2b /2}M2/ #v D(C) := 1C (Tn + σn ) (C ∈ B(R)) n∈Z 9
dz∞Ǵ `2T`2b2Mib i?2 MmK#2` Q7 b2`p2`bX h?Bb KQ/2H Bb bQK2iBK2b + HH2/ dz[m2m2BM;Ǵ bvbi2K Hi?Qm;? BM `2 HBiv i?2`2 Bb MQ [m2m2BM;- bBM+2 +mbiQK2`b `2 b2`p2/ BKK2/B i2Hv mTQM ``Bp H M/ rBi?Qmi BMi2``mTiBQMX Ai Bb BM 7 +i dzTm`2 /2H vǴ bvbi2KX
jXjX J_E1. aShAG SPAaaPL S_P*1aa1a
3d
Bb M ?TT rBi? BMi2MbBiv λ U1t2`+Bb2 jX3XNVX 6Q`KmH b bm+? b * KT#2HHǶb }`bi 7Q`KmH M/ i?2 SQBbbQM 2tTQM2MiB H 7Q`KmH + M #2 bi` B;?i7Q`r `/Hv 2ti2M/2/ iQ +Qp2` i?2 + b2 Q7 K `F2/ TQBMi T`Q+2bb2bX AM i?2 bBim iBQM T`2p BHBM; BM h?2Q`2K jXjXR +QMbB/2`- 7Q` 7mM+iBQM ϕ˜ : E × K → RbmKb Q7 i?2 ivT2 ˜ (ϕ) N ˜ := ϕ(X ˜ n , Zn ) . UjXdV n∈N
LQi2 i? i- /2MQiBM; #v Z1 (x) Mv ` M/QK 2H2K2Mi Q7 K rBi? i?2 /Bbi`B#miBQM Q(x, dz)
ν˜(ϕ) ˜ =
ϕ(x, ˜ z)Q(x, dz) ν(dx) = E
K
E [ϕ(x, ˜ Z1 (x))] ν(dx) , E
r?2M2p2` i?2 [m MiBiB2b BMpQHp2/ `2 r2HH /2}M2/X lbBM; i?Bb Q#b2`p iBQM- i?2 7Q`@ KmH b Q#i BM2/ BM i?2 T`2pBQmb bm#b2+iBQM + M #2 TTHB2/ BM i2`Kb Q7 K `F2/ TQBMi T`Q+2bb2b- M/ BM T `iB+mH ` i?2 +Q`QHH `B2b #2HQr /Q MQi `2[mB`2 T`QQ7b- bBM+2 i?2v #2+QK2 `27Q`KmH iBQMb Q7 h?2Q`2K jXkXR- h?2Q`2K jXkXk M/ 1t2`+Bb2 jX3X8X G2i 0 < p < ∞X _2+ HH i? i K2 bm` #H2 7mM+iBQM ϕ˜ : E × K → R U`2bTX → CV Bb b B/ iQ #2 BM LpR (˜ ν ) U`2bTX LpC (˜ ν )V B7 |ϕ(x, ˜ z)|p ν(dx) Q(x, dz) < ∞ . E
K
*Q`QHH `v jXjXj amTTQb2 i? i ϕ˜ ∈ L1C (˜ ν )X h?2M i?2 bmK UjXdV Bb r2HH /2}M2/- M/ KQ`2Qp2` E ϕ(X ˜ n , Zn ) = E [ϕ(x, ˜ Z1 (x))] ν(dx) . E
n∈N
ν ) ∩ L2C (˜ ν )X h?2M G2i ϕ, ˜ ψ˜ : R × E → C #2 irQ K2 bm` #H2 7mM+iBQMb BM L1C (˜ cov
n∈N
ϕ(X ˜ n , Zn ),
˜ n , Zn ) ψ(X
˜ Z1 (x))∗ ν(dx) . E ϕ(x, ˜ Z1 (x))ψ(x,
= E
n∈N
*Q`QHH `v jXjX9 G2i ϕ˜ #2 MQM@M2; iBp2 7mM+iBQM 7`QK E × K iQ RX h?2M ˜ n ,Zn ) E e− n∈N ϕ(X = exp
˜ 1 (x)) E e−ϕ(x,Z − 1 ν(dx) E
.
33
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
jX9 PT2` iBQMb QM SQBbbQM S`Q+2bb2b h?2 QT2` iBQMb +QMbB/2`2/ `2 +QHQm`BM;- i?BMMBM;- i` MbTQ`iBM;- i` MbH iBM; M/ }Hi2`BM;X *QHQm`BM; h?2Q`2K jX9XR *QMbB/2` i?2 bBim iBQM /2TB+i2/ BM h?2Q`2K jXjXRX G2i I #2 M `#Bi` `v BM/2t b2i M/ H2i {Li }i∈I #2 7 KBHv Q7 /BbDQBMi K2 bm` #H2 b2ib Q7 KX .2}M2 7Q` 2 +? i ∈ I i?2 bBKTH2 TQBMi T`Q+2bb Ni QM Rm #v Ni (C) = 1C (Xn )1Li (Zn ) . n∈N
Uh?2 TQBMib Q7 Ni `2+2Bp2 i?2 dz+QHQm`Ǵ iXV h?2M i?2 7 KBHv Ni (i ∈ I) Bb M BM/2@ T2M/2Mi 7 KBHv Q7 SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b νi (i ∈ I) r?2`2 νi (dx) = Q(x, Li ) ν(dx) (i ∈ I) . S`QQ7X ++Q`/BM; iQ i?2 /2}MBiBQM Q7 BM/2T2M/2M+2- Bi bm{+2b iQ +QMbB/2` }MBi2 ˜ QM Rm × K b BM UjXeVX h?2M N ˜ BM/2t b2i IX .2}M2 i?2 bBKTH2 TQBMi T`Q+2bb N Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν˜(C × L) = C Q(x, L)ν(dx)X qBi?
˜ (ϕ)X ϕ(x, ˜ z) := i∈I ϕi (x)1Li (z)- r2 ? p2 i∈I Ni (ϕi ) = N ˜ h?2`27Q`2 − Ni (ϕi ) ˜ ˜ ˜ = E e−N (ϕ) − 1 ν˜(dx × dz) = exp e−ϕ(x,z) E e i∈I Rm K −ϕ(x,z) ˜ = exp − 1 Q(x, dz)ν(dx) e m R K − ϕi (x)1 (z) Li = exp e i∈I − 1 Q(x, dz)ν(dx) m R K = exp e−ϕi (x) − 1 1Li (z)Q(x, dz)ν(dx)
Rm
e
= exp =
K i∈I
−ϕi (x)
Rm i∈I
Rm
e−ϕi (x) − 1 Q(x, Li )ν(dx)
h?2`27Q`2
−
E e
i∈I
Ni (ϕi )
=
− 1 Q(x, Li )ν(dx)
exp
i∈I
e
exp
i∈I
M/ i?2 `2bmHi 7QHHQrb 7`QK h?2Q`2K jXkXdX
Rm
−(ϕi )
.
− 1 νi (dx)
_2K `F jX9Xk h?2 #Qp2 i?2Q`2K +QM+2`Mb BM T `iB+mH ` i?BMMBM; Ui?2 QT2` iBQM Q7 ` M/QKHv 2` bBM; TQBMibVX 6Q` BMbi M+2 i?2 TQBMi T`Q+2bb N1 Bb Q#i BM2/ #v i?BMMBM; Q7 N - 2 +? TQBMi x Q7 r?B+? #2BM; b p2/ rBi? T`Q# #BHBiv Q(x, L1 )X
jX9X PS1_hAPLa PL SPAaaPL S_P*1aa1a
3N
h` MbTQ`i iBQM h?2Q`2K jX9Xj G2i N #2 TQBMi T`Q+2bb QM bQK2 K2 bm` #H2 bT +2 (E, E) M/ H2i T #2 K2 bm` #H2 K TTBM; 7`QK (E, E) iQ MQi?2` K2 bm` #H2 bT +2 (E , E )X amTTQb2 i? i N Bb SQBbbQM T`Q+2bb QM E rBi? i?2 HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 ν M/ i? i i?2 K2 bm`2 ν r?B+? Bb i?2 BK ;2 Q7 ν #v T - i? i Bbν (C ) = ν(T −1 (C )) (C ∈ E ) , Bb σ@}MBi2X h?2M- i?2 TQBMi T`Q+2bb N QM E /2}M2/ #v N (C ) = N (T −1 (C ))
(C ∈ E )
Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν X S`QQ7X
E e− E ϕ(x) N (dx) = e− n ϕ(T (Xn )) = e− E ϕ(T (x)) N (dx) −ϕ(T (x)) e − 1 ν(dx) = exp E −ϕ(y) = exp e − 1 ν (dx) . E
h?Bb i?2Q`2K +QK#BM2/ rBi? h?2Q`2K jXjXR Ur?B+? T2`KBib mb iQ pB2r K `F2/ SQBbbQM T`Q+2bb b SQBbbQM T`Q+2bb BM H `;2` bT +2V +Qp2`b K Mv bBim iBQMbX 1t KTH2 jX9X9, h?2 SQBMi S`Q+2bb Q7 J `FbX *QMbB/2` i?2 bBim iBQM /2@ TB+i2/ BM h?2Q`2K jXjXR- r?B+? r2 K v BMi2`T`2i b 7QHHQrbX TQBMi T`Q+2bb N ∗ QM K Bb 7Q`K2/ #v bbQ+B iBM; iQ TQBMi Xn ∈ Rm TQBMi Zn ∈ KX 6Q`K HHv 1L (Zn ) , N ∗ (L) = n∈N
r?2`2 L ∈ B(Rm )X q2 i?2M b v i? i N ∗ Bb Q#i BM2/ #v i` MbTQ`iBM; N pB i?2 biQ+? biB+ F2`M2H Q(x, ·)X bbmK2 i? i K Bb M HX+X/X#X bT +2 M/ i? i 7Q` HH +QKT +i bm#b2ib L ∈ K Rm
Q(x, L) ν(dx) < ∞ .
lM/2` i?2 #Qp2 +QM/BiBQMb- N ∗ Bb SQBbbQM T`Q+2bb QM K rBi? BMi2MbBiv K2 bm`2 ν ∗ ;Bp2M #v ∗ Q(x, L) ν(dx) . ν (L) = Rm
S`QQ7X G2i ϕ∗ : K → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX q2 ? p2
Ny
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a ∗ ∗ ∗ E e−N (ϕ ) = E e− n∈N ϕ (Zn ) −ϕ∗ (z) e − 1 ν(dx)Q(x, dz) = exp m R K −ϕ∗ (z) = exp e −1 ν(dx)Q(x, dz) . Rm
K
1t KTH2 jX9X8, h` MbH iBQMX G2i N #2 SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 ν M/ H2i {Vn }n∈N #2 M BB/ b2[m2M+2 ` M/QK p2+iQ`b Q7 Rm rBi? +QKKQM /Bbi`B#miBQM QX 6Q`K i?2 TQBMi T`Q+2bb N ∗ QM Rm #v i` MbH iBM; 2 +? TQBMi Xn Q7 N #v Vn X 6Q`K HHv 1C (Xn + Vn ) (C ∈ B(Rm )) . N ∗ (C) := n∈N
q2 `2 BM i?2 bBim iBQM Q7 h?2Q`2K jX9Xj rBi? T (Xn , Zn ) = Xn + Vn X AM T `iB+mH ` Q(x, A) = Q(A − x)X Ai 7QHHQrb i? i N ∗ Bb SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 ν ∗ (L) = Q(L − x) ν(dx) , Rm
i?2 +QMpQHmiBQM Q7 ν M/ QX
SQBbbQM a?Qi LQBb2 U6BHi2`BM;V G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ rBi? K `Fb {Zn }n∈N BM i?2 K2 bm` #H2 bT +2 (K, K)X G2i h : Rm × K → C #2 K2 bm` #H2 7mM+iBQMX h?2 +QKTH2t@p Hm2/ bT iB H biQ+? biB+ T`Q+2bb {X(y)}y∈Rm ;Bp2M #v X(y) := h(y − Xn , Zn ) , UjX3V n∈N
r?2`2 i?2 `B;?i@? M/ bB/2 Bb bbmK2/ r2HH /2}M2/ U7Q` BMbi M+2- r?2M h i F2b `2 H MQM@M2; iBp2 p Hm2bV- Bb + HH2/ bT iB H b?Qi MQBb2 rBi? ` M/QK BKTmHb2 `2bTQMb2X A7 N Bb bBKTH2 M/ HQ+ HHv }MBi2 SQBbbQM T`Q+2bb QM Rm rBi? BM/2T2M/2Mi BB/ K `Fb {Zn }n∈N - {X(y)}y∈Rm Bb + HH2/ SQBbbQM bT iB H b?Qi MQBb2 rBi? ` M/QK BKTmHb2 `2bTQMb2 M/ BM/2T2M/2Mi BB/ K `FbX h?2 7QHHQrBM; `2bmHi Bb /B`2+i TTHB+ iBQM Q7 h?2Q`2Kb jXkXR M/ jXjXRX h?2Q`2K jX9Xe *QMbB/2` i?2 #Qp2 SQBbbQM bT iB H b?Qi MQBb2 rBi? ` M/QK BK@ TmHb2 `2bTQMb2 M/ BM/2T2M/2Mi BB/ K `FbX amTTQb2 i? i 7Q` HH y ∈ Rm E [|h(y − x, Z1 )|] ν(dx) < ∞ Rm
M/
jX8X *>L:1 P6 S_P""AGAhu Rm
NR
E |h(y − x, Z1 )|2 ν(dx) < ∞ .
h?2M i?2 +QKTH2t@p Hm2/ bT iB H biQ+? biB+ T`Q+2bb {X(y)}y∈Rm ;Bp2M #v UjX3V Bb r2HH /2}M2/- M/ 7Q` HH y, ξ ∈ Rm E [X(y)] = E [h(y − x, Z1 )] ν(dx) Rm
M/
cov(X(y + ξ), X(y)) = Rm
E [h(y − x, Z1 )h∗ (y + ξ − x, Z1 )] ν(dt) .
AM i?2 + b2 r?2`2 i?2 # b2 TQBMi T`Q+2bb N Bb M ?TT rBi? BMi2MbBiv λ E [X(y)] = λ E [h(x, Z1 )] dx Rm
M/
E [h(x, Z1 )h∗ (ξ + x, Z1 )] dx .
cov(X(y + ξ), X(y)) = λ Rm
P#b2`p2 i? i i?2b2 [m MiBiB2b /Q MQi /2T2M/ QM y ∈ Rm X h?2 T`Q+2bb {X(y)}y∈Rm Bb 7Q` i? i `2 bQM + HH2/ rB/2@b2Mb2 bi iBQM `v T`Q+2bbX
jX8
*? M;2 Q7 S`Q# #BHBiv
G2i (Ω, F, P ) #2 T`Q# #BHBiv bT +2 QM r?B+? Bb ;Bp2M SQBbbQM T`Q+2bb N QM E rBi? HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 νX q2 b? HH `2TH +2 i?2 T`Q# #BHBiv P #v MQi?2` T`Q# #BHBiv P BM bm+? r v i? i rBi? `2bT2+i iQ i?Bb M2r T`Q# #BHBiv- i?2 b K2 TQBMi T`Q+2bb N Bb SQBbbQM T`Q+2bb- #mi rBi? i?2 BMi2MbBiv K2 bm`2 ν ;Bp2M #v ν(C) = μ(x) ν(dx), UjXNV E
7Q` bQK2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM μ : E → RX h?2 + b2 Q7 }MBi2 BMi2MbBiv K2 bm`2b h?Bb T`Q;` K Bb }`bi + ``B2/ Qmi mM/2` i?2 7QHHQrBM; ?vTQi?2b2b, H1 , ν Bb }MBi2 K2 bm`2- M/ H2 , μ Bb ν@BMi2;` #H2 UQ`- 2[mBp H2MiHv- ν Bb }MBi2VX h?2 +? M;2 Q7 T`Q# #BHBiv P → P rBHH #2 M #bQHmi2Hv +QMiBMmQmb QM2- i? i Bb7Q` HH A ∈ F UjXRyV P (A) = E[L 1A ], r?2`2 L Bb MQM@M2; iBp2 ` M/QK p `B #H2 bm+? i? i E[L] = 1 ,
UjXRRV
+ HH2/ i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 Q7 P rBi? `2bT2+i iQ P - HbQ /2MQi2/ #v
dP X dP
Nk
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
G2KK jX8XR lM/2` ?vTQi?2b2b H1 M/ H2 - i?2 ` M/QK p `B #H2 μ(Xn ) exp − (μ(x) − 1)ν(dx) L :=
UjXRkV
E
n∈N
b iBb}2b i?2 `2[mB`2K2Mi UjXRRVX S`QQ7X G2i g(x) = log(μ(x)) M/ /2+QKTQb2 i?Bb 7mM+iBQM BMiQ Bib TQbBiBp2 M/ M2; iBp2 T `i- g = g+ − g− X "v h?2Q`2K jXkXk r2 ? p2 i? i −N (g− ) −g− (x) E e − 1 ν(dx) = exp e E
M/
N (g+ )
E e
= exp
e
g+ (x)
− 1 ν(dx)
.
E
G2i B1 = {x ∈ E ; g(x) > 0}X "v h?2Q`2K jX9XR- i?2 `2bi`B+iBQMb Q7 N iQ B1 M/ B2 = B¯1 `2 BM/2T2M/2Mi- M/ i?2`27Q`2 i?2 p `B #H2b e−N (g− ) M/ eN (g+ ) `2 BM/2T2M/2MiX AM T `iB+mH `- 7`QK i?2 irQ H bi /BbTH vb E μ(Xn ) n∈N
= E eN (log(μ)) = E eN (g) = E eN (g+ )−N (g− ) = E e−N (g− ) eN (g+ ) = E e−N (g− ) E eN (g+ ) g+ (x) −g− (x) − 1 ν(dx) exp − 1 ν(dx) e e = exp E E g(x) g(x) = exp e e − 1 1{g(x)>0} ν(dx) exp − 1 1{g(x)≤0} ν(dx) E E g(x) g(x) = exp e e − 1 1{g(x)>0} ν(dx) + − 1 1{g(x)≤0} ν(dx) E E g(x) = exp e − 1 ν(dx) = exp (μ(x) − 1) ν(dx) . E
Rm
"v bbmKTiBQMb H1 M/ H2 i?2 H bi [m MiBiv Bb }MBi2 M/ i?2`27Q`2- #v /BpB/BM; i?2 }`bi M/ H bi i2`Kb Q7 i?2 #Qp2 +? BM Q7 2[m HBiB2b #v Bi- r2 Q#i BM UjXRRVX h?2Q`2K jX8Xk lM/2` i?2 bbmKTiBQMb H1 M/ H2 - B7 r2 /2}M2 T`Q# #BHBiv P #v UjXRyV M/ UjXRkV- N Bb mM/2` T`Q# #BHBiv P SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν ;Bp2M #v UjXNVX
S`QQ7X .2MQi2 2tT2+i iBQM rBi? `2bT2+i iQ P #v EX Ai bm{+2b iQ b?Qr i? i i?2 G TH +2 i` Mb7Q`K Q7 N mM/2` T`Q# #BHBiv P Bb i? i Q7 SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 ν- i? i Bb- 7Q` Mv #QmM/2/ MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM ϕ : Rm → R-
jX8X *>L:1 P6 S_P""AGAhu E e−N (ϕ) = exp
Nj
e−ϕ(x) − 1 ν(dx) .
E
"mi E e−N (ϕ) = E L e−N (ϕ) = E eN (log(μ))− E (μ(x)−1)ν(dx) e−N (ϕ) = E eN (−ϕ+log(μ)) e− E (μ(x)−1)ν(dx) −ϕ(x)+log(μ(x)) = exp − 1 ν(dx) exp − (μ(x) − 1)ν(dx) e E E −ϕ(x) = exp e μ(x) − 1 ν(dx) exp − (μ(x) − 1)ν(dx) E E −ϕ(x) −ϕ(x) = exp e e − 1 μ(x)ν(dx) = exp − 1 ν(dx) . E
E
1t KTH2 jX8Xj, .2`Bp iBp2 Q7 i?2 1tT2+i iBQM Q7 SQBbbQM 6mM+@ iBQM HX8 h?2 b2iiBM; Bb i? i Q7 1t KTH2 jXRX8- rBi? i?2 b K2 Q#D2+iBp2X h?Bb iBK2 r2 mb2 MQi?2` K2i?Q/X h?2 T`Q# #BHBiv Pα Bb Q#i BM2/ 7`QK i?2 T`Q# #BHBiv P1 K FBM; Q7 N SQBbbQM T`Q+2bb QM E rBi? K2 M K2 bm`2 Q pB i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 dPα = αN (E) exp (−(α − 1)Q(E)) . dP1 h?2`27Q`2
dPα Eα [f (N )] = E1 f (N ) dP1 = E1 f (N ) αN (E) exp (−(α − 1)Q(E)) . .Bz2`2MiB iBM; mM/2` i?2 2tT2+i iBQM bB;M vB2H/b
dPα 1 dEα [f (N )] = E1 f (N ) N (E) − Q(E) dα α dP 1 1 = Eα f (N ) N (E) − Q(E) . α U ;2M2` HBx iBQM Q7 i?Bb 2t KTH2 Bb ;Bp2M BM 1t KTH2 8X8X9XV h?2 + b2 Q7 HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2b aQ 7 ` r2 ? p2 +QMbB/2`2/ i?2 + b2 r?2`2 P M/ P ? p2 }MBi2 BMi2MbBiv K2 bm`2bX q2 MQr 2t KBM2 i?2 + b2 r?2`2 i?2b2 BMi2MbBiv K2 bm`2b `2 QMHv HQ+ HHv }MBi2X G2i N #2 i?2 +QQ`/BM i2 T`Q+2bb QM (Mp (E), Mp (E)) M/ H2i P #2 i?2 T`Q#@ #BHBiv K2 bm`2 i? i K F2b Q7 Bi SQBbbQM T`Q+2bb N QM E rBi? HQ+ HHv }MBi2 8
("`ûK m/ M/ ox[m2x@# /- RNNk)X
N9
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
BMi2MbBiv K2 bm`2 νX G2i μ : E → R #2 bQK2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM HQ+ HHv BMi2;` #H2 rBi? `2bT2+i iQ ν ;m ` Mi22BM; i? i i?2 K2 bm`2 ν ;Bp2M #v ν(C) := μ(x) ν(dx) (C ∈ B(E)) UjXRjV E
Bb HbQ HQ+ HHv }MBi2X G2i {Bn }n≥0 #2 M BM+`2 bBM; b2[m2M+2 Q7 +QKT +i b2ib r?Qb2 mMBQM Bb E M/ bm+? i? i ν(Bn ) M/ ν(Bn ) `2 }MBi2X h?2 +QM/BiBQMb QM ν M/ μ ;m ` Mi22 i?2 2tBbi2M+2 Q7 bm+? b2[m2M+2X .2MQi2 #v Fn i?2 σ@}2H/ ;2M2` i2/ #v i?2 `2bi`B+iBQM Q7 N iQ Bn X G2i An := Bn − Bn−1 Un ≥ 1VX G2i PAn #2 i?2 `2bi`B+iBQM Q7 P iQ i?2 σ@}2H/ Gn ;2M2` i2/ #v i?2 `2bi`B+iBQM Q7 N iQ An X "v h?2Q`2K jX8Xk- i?2 T`Q# #BHBiv PAn QM (Ω, Gn ) /2}M2/ #v dPAn = μ(Xi )1An (Xi ) exp (μ(x) − 1) ν(dx) () dPAn An i Bb bm+? i? i i?2 `2bi`B+iBQM Q7 N iQ An Bb SQBbbQM T`Q+2bb r?Qb2 BMi2MbBiv K2 bm`2 Bb i?2 `2bi`B+iBQM Q7 ν iQ An X .2MQiBM; #v Zn i?2 [m MiBiv UV- i?2 b2[m2M+2 {Zn }n≥1 Bb- BM pB2r Q7 i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b- BB/X :HmBM; iQ;2i?2` i?2 `2bi`B+iBQMb Q7 N iQ i?2 An Ƕb- r2 Q#i BM T`Q# #BHBiv K2 bm`2 P QM (Ω, F) bm+? i? i n n d PA k dP Ln := | Fn = = Zk . dP dPAk k=1 k=1 h?2 ;2M2` H `2bmHib Q7 a2+iBQM X8 i2HH mb i? i i?2`2 `2 QMHv irQ TQbbB#BHBiB2b, P P Q` P ⊥P - M/ i? i P P ⇐⇒
∞
1
E[Zn2 ] > 0 .
n=1
LQr-
1 (μ(x) − 1) ν(dx) (μ(Xi ) )1An (Xi ) exp Zn = An 2 i 1 1 2 2 (μ(Xi ) )1An (Xi ) exp (μ(x) − 1) ν(dx) = 1 2
1 2
An
1 1 (μ(x) − 1) ν(dx) exp{ −(μ(x) 2 − 1) ν(dx)} exp 2 An An 1 1 n exp =L (μ(x) − 1) − (μ(x) 2 − 1) ν(dx) , An 2
i
r?2`2 n := L
i
1 2
1
(μ(x) 2 − 1) ν(dx)
(μ(Xi ) )1An (Xi ) exp An
n = 1 U TTHv G2KK jX8XR rBi? μ(x) 12 BMbi2 / Q7 μ(x)VX h?2`27Q`2 Bb bm+? i? i E L
jX8X *>L:1 P6 S_P""AGAhu
N8
1 1 (μ(x) − 1) − (μ(x) 2 − 1) ν(dx) 2 An 1 = exp (μ(x) 2 − 1)2 ν(dx)
1
E[Zn2 ] = exp
An
M/ BM T `iB+mH ` ∞
1
E[Zn2 ] = exp
n=1
1 1 − (μ(x) 2 − 1)2 ν(dx) 2 E
.
h?2`27Q`2, h?2Q`2K jX8X9 ' P P ⇐⇒ ( μ(x) − 1)2 ν(dx) < ∞ . E
h?2 JBt2/ SQBbbQM * b2 G2i N #2 SQBbbQM T`Q+2bb QM E Q7 }MBi2 BMi2MbBiv K2 bm`2 ν M/ H2i Λ #2 MQM@M2; iBp2 ` M/QK p `B #H2 BM/2T2M/2Mi Q7 N X G2i L := ΛN (E) exp{−(Λ − 1)N (E)} . h?2 `;mK2Mib Q7 i?2 T`QQ7 Q7 G2KK jX8XR M/ h?2Q`2K jX8Xk `2 BKK2/B i2Hv / Ti #H2 iQ b?Qr i? i EP [L] = 1 M/ i? i mM/2` i?2 T`Q# #BHBiv K2 bm`2 P /2}M2/ #v ddPP = L- N Bb *Qt T`Q+2bb U?2`2 KBt2/ SQBbbQM T`Q+2bbV rBi? σ(Λ)@ +QM/BiBQM H BMi2MbBiv K2 bm`2 Λν(dx)X h?2Q`2K jX8X8 lM/2` i?2 #Qp2 +QM/BiBQMb- 7Q` Mv MQM@M2; iBp2 7mM+iBQM g : R+ → R g(λ)λN (E) e−λN (E) F (dλ) N . UjXR9V EP g(Λ) | F = N (E) −λN (E) λ e F (dλ) S`QQ7X h?2 T`QQ7 Bb # b2/ QM i?2 7QHHQrBM; 7mM/ K2Mi H H2KK , G2KK jX8Xe G2i P M/ Q #2 irQ T`Q# #BHBiv K2 bm`2b QM i?2 K2 bm` #H2 bT +2 dP (Ω, F) bm+? i? i P Q M/ H2i L := dQ X G2i Z #2 MQM@M2; iBp2 ` M/QK p `B #H2X 6Q` Mv bm#@σ@}2H/ G Q7 FEQ [L | G] EP [Z | G] = EQ [ZL | G]
Q@ XbX
UjXR8V
Q`- 2[mBp H2MiHvEP [Z | G] =
EQ [ZL | G] EQ [L | G]
P @ XbX
UjXReV
Ne
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
S`QQ7X "v /2}MBiBQM Q7 +QM/BiBQM H 2tT2+i iBQM- 7Q` HH A ∈ G
EP [Z | G] dP .
Z dP = A
A
"v /2}MBiBQM Q7 L M/ Q7 +QM/BiBQM H T`Q# #BHBiv ; BM
Z dP = A
EQ [ZL | G] dQ .
ZL dQ = A
A
HbQ
EP [Z | G] dP = A
A
EP [Z | G] L dQ EP [Z | G] EQ [L | G] dQ .
= A
h?2`27Q`2
EQ [ZL | G] dQ = A
EP [Z | G] EQ [L | G] dQ , A
r?B+? Bb 2[mBp H2Mi iQ UjXR8V #2+ mb2 A Bb `#Bi` `v BM GX aBM+2 P Q i?Bb 2[m HBiv HbQ ?QH/b P @ XbX hQ Q#i BM UjXReV- Bi `2K BMb iQ b?Qr i? i P (EQ [L | G] = 0) = 0X AM/22/ P (EQ [L | G] = 0) =
= =
1{EQ [L|G]=0} dP 1{EQ [L|G]=0} LdQ 1{EQ [L|G]=0} EQ [L | G] dQ = 0 .
q2 K v MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K jX8X8X "v G2KK jX8Xe EP g(Λ)L | F N EP g(Λ) | F N = EP [L | F N ] M/ i?2`27Q`2- bBM+2 mM/2` P - N M/ Λ `2 BM/2T2M/2Mi
EP g(Λ) | F
N
=
7`QK r?B+? i?2 `2bmHi 7QHHQrbX
R+
g(λ)λN (E) exp{−(λ − 1)N (E)} F (dλ)
R+
λN (E) exp{−(λ − 1)N (E)} F (dλ)
,
jXeX 1s*h aJSGAL: P6 *Glah1_ SPALh S_P*1aa1a
jXe
Nd
1t +i a KTHBM; Q7 *Hmbi2` SQBMi S`Q+2bb2b
h?2 ;2M2` H b2iiBM; Q7 i?Bb bm#b2+iBQM Bb i? i Q7 +Hmbi2` ` M/QK K2 bm`2 Ub22 am#b2+iBQM RX8V r?Qb2 MQi iBQM Bb /QTi2/X h?2 T`Q#H2K +QMbB/2`2/ Bb i? i Q7 b KTHBM; i?2 /Bbi`B#miBQM Q7 NW - i?2 `2bi`B+iBQM Q7 N iQ +QKT +i bm#b2i W ⊂ E Ui?2 dzrBM/QrǴVX U_2+ HH i? i b KTHBM; i?2 /Bbi`B#miBQM Q7 ` M/QK K2 bm`2 K2 Mb +QMbi`m+iBM; ` M/QK K2 bm`2 rBi? i?2 ;Bp2M /Bbi`B#miBQMXV a KTHBM; N BM i?2 rBM/Qr W `2[mB`2b BM T`BM+BTH2 iQ ;2M2` i2 HH i?2 TQBMib X0,n Q7 i?2 ;2`K TQBMi T`Q+2bb N0 - bBM+2 i?2 bbQ+B i2/ +Hmbi2`b Zn (X0,n , · − X0,n ) + M TQi2MiB HHv +QMi BM TQBMib BM i?2 rBM/Qr W X h?Bb Bb MQi 72 bB#H2 B7 i?2`2 Bb M BM}MBi2 MmK#2` Q7 ;2`KbX PM2 bQHmiBQM Bb iQ TT`QtBK i2 NW #v i FBM; BMiQ ++QmMi QMHv i?2 TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb i? i `2 BM dzbm{+B2MiHvǴ H `;2 rBM/Qr W ⊃ W X "mi i?Bb BMi`Q/m+2b 2/;2 2z2+ib- BM T `iB+mH ` HQbb Q7 K bb BM i?2 rBM/Qr W X h?Bb BHK2Mi }M/b `2K2/v BM i?2 + b2 Q7 i?2 *Qt +Hmbi2` TQBMi T`Q+2bb rBi? i?2 "`BtĜE2M/ HH 2t +i b KTHBM; K2i?Q/Xe h?2 7mM/ K2Mi H B/2 Q7 i?Bb K2i?Q/ + M #2 TTHB2/ iQ KQ`2 ;2M2` H + b2b- b r2 MQr b?Qr- #v K2 Mb Q7 bBKTH2 + H+mH iBQMb QM G TH +2 i` Mb7Q`KbXd q2 +QKTmi2 i?2 G TH +2 i` Mb7Q`K Q7 i?2 +Hmbi2` TQBMi T`Q+2bb M/ i?2 G TH +2 i` Mb7Q`K b BM i?2 +QMbi`m+iBQM T`QTQb2/ BM i?2 "`BtĜE2M/ HH +QMbi`m+iBQM- M/ b?Qr i? i i?2v H2 / iQ i?2 b K2 `2bmHiX RX lbBM; i?2 7 +i i? i {Zn }n∈Z Bb- +QM/BiBQM HHv QM F N0 - M BB/ b2[m2M+2-
LNW (ϕ) := E exp −
n∈N
= E E exp − =E E
ϕ(x) Zn (X0,n , dx − X0,n ) W
n∈N
ϕ(x) Zn (X0,n , dx − X0,n )
|F
N0
W
n∈N
N0 exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F W
N0 . E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F =E
n∈N
W
LQr- rBi? An := {Zn (x, W − X0,n ) > 0}-
e d
("`Bt M/ E2M/ HH- kyyk)X ("`ûK m/- kyRe)X
N3
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F N0 W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) 1An + 1 − 1An | X0,n W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) 1{Zn (W −X0,n )>0} | X0,n W
+ P (Zn (X0,n , W − X0,n ) = 0 | X0,n ) := g1 (X0,n ) + g2 (X0,n ) . h?2`27Q`2
LNW (ϕ) = E
(g1 (X0,n ) + g2 (X0,n ))
n∈N
= E e n∈N log(g1 (X0,n )+g2 (X0,n )) = E e E log(g1 (x)+g2 (x)) N0 (dx) = E e E log(g1 (x)+P (Z1 (x,W −x)>0)) N0 (dx) . r?2`2 Z1 (x, · − x) Bb ivTB+ H +Hmbi2` Zn (·, ·)X LQr g1 (x) + P (Z1 (x, W − x) > 0) g1 (x) = P (Z1 (x, W − x) > 0) + P (Z1 (x, W − x) > 0) . P (Z1 (x, W − x) > 0) P#b2`p2 i? i g1 (x) = E e− W ϕ(x) Nx (dx) 1{Z1 (x,W −x)>0} /P (Z1 (x, W − x) > 0) P (Z1 (x, W − x) > 0) Bb i?2 G TH +2 i` Mb7Q`K Q7 i?2 ` M/QK K2 bm`2 Z1 (x, · − x) +QM/BiBQM2/ iQ ? p2 i H2 bi QM2 TQBMi BM W X W kX h?2 "`BtĜE2M/ HH 2t +i b KTHBM; H;Q`Bi?K +QMbi`m+ib TQBMi T`Q+2bb N QM W b 7QHHQrbX 6B`bi- i?2 TQBMi T`Q+2bb N0 Bb i?BMM2/- TQBMi X0,n #2BM; `2i BM2/ rBi? T`Q# #BHBiv p(X0,n ) r?2`2 p(x) := P (Z1 (x, W − x) > 0)X JQ`2 T`2+Bb2Hv- i?2 0 Bb /2}M2/ #v i?BMM2/ TQBMi T`Q+2bb N 0 (C) := N 1C (X0,n )Yn , n∈N
r?2`2 {Yn }n∈Z Bb- +QM/BiBQM HHv QM FN0 - M BM/2T2M/2Mi b2[m2M+2 rBi? p Hm2b BM {0, 1}- M/ 7Q` 2 +? n ∈ N- P (Yn = 1 | F N0 ) = P (Yn = 1 | X0,n ) = p(X0,n )X h?2M 7Q` 2 +? n ∈ N- H2i Zn #2 TQBMi T`Q+2bb i? i ? b i?2 b K2 /Bbi`B#miBQM b Zn +QM/BiBQM2/ #v Zn (X0,n , W − X0,n ) > 0X Uh?2 TQBMi T`Q+2bb Zn Bb Q#@ i BM2/ #v b KTHBM; BM/2T2M/2Mi TQBMi T`Q+2bb2b Q7 i?2 ivT2 Zn mMiBH i?2 +QM/BiBQM
jXeX 1s*h aJSGAL: P6 *Glah1_ SPALh S_P*1aa1a
NN
Bb i?2M +QMbi`m+i2/ Zn (X0,n , W − X0,n ) > 0 Bb b iBb}2/XV h?2 + M/B/ i2 b KTH2 N b W (C) = Yn Zn (X0,n , C − X0,n ) . N n∈N
Uh?2`27Q`2- QMHv i?2 Zn +Q``2bTQM/BM; iQ TQBMi X0,n i? i ? b #22M `2i BM2/ rBHH W Bb i?2 /2bB`2/ 2t +i b KTH2- Bi M22/ iQ #2 b KTH2/XV AM Q`/2` iQ +?2+F i? i N Kmbi #2 T`Qp2/ i? i Bi ? b i?2 b K2 /Bbi`B#miBQM b NW X h?Bb Bb /QM2 #2HQr #v b?QrBM; i? i i?2v ? p2 i?2 b K2 G TH +2 7mM+iBQM HX q`Bi2 L (ϕ) := E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) NW
W
n∈N
exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) =E
W
n∈Z
=E E =E
exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F N0
W
n∈Z
N0 E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F .
W
n∈N
LQr r`Bi2
N0 E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) Yn | X0,n + E [1 − Yn | X0,n ] W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n E [Yn | X0,n ] + E [1 − Yn | X0,n ] W
= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n p(X0,n )) + (1 − p(X0,n ))) W
= g(X0,n )p(X0,n ) + 1 − p(X0,n ) , r?2`2 g(x) Bb i?2 G TH +2 7mM+iBQM H Q7 Z1 (x, · − x) +QM/BiBQM2/ iQ ? p2 i H2 bi QM2 TQBMi BM W X h?2 `2bi Q7 i?2 p2`B}+ iBQM Bb +QKTH2i2/ #v LNW (ϕ) = E (g(X0,n )p(X0,n ) + 1 − p(X0,n )
n∈N
log(g(x)p(x) + 1 − p(x)) N0 (dx)
= E exp
,
E
M/ i?2 Q#b2`p iBQM g(x)p(x) = g1 (x)X 1t KTH2 jXeXR, 1t +i a KTHBM; Q7 *Qt *Hmbi2` SQBMi S`Q+2bbX AM i?2 + b2 r?2`2 N0 Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 μ - i?2 2t +i b KTHBM; 0 Q7 N0 - BM i?Bb + b2 SQBbbQM T`Q+2/m`2 +QMbBbib Q7 +QMbi`m+iBM; i?BMM2/ p2`bBQM N T`Q+2bb rBi? BMi2MbBiv K2 bm`2 P (Z1 (x, W − x) > 0) μ(dx)- M/ 7`QK 2 +? TQBMi
Ryy
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
0,n Q7 N 0 `2 HBx2 p2`bBQM Q7 Zn X h?2`2 `2 irQ +QM/BiBQMb 7Q` i?Bb iQ T`Q/m+2 M X 2t +i b KTHBM; Q7 NW BM }MBi2 MmK#2` Q7 QT2` iBQMbX h?2 MmK#2` Q7 TQBMib Q7 0 Kmbi #2 }MBi2- bm{+B2Mi +QM/BiBQM 7Q` i?Bb #2BM; i? i N P (Z1 (x, W − x) > 0) μ(dx) < ∞ . () E
6Q` *Qt +Hmbi2` TQBMi T`Q+2bb P (Z1 (x, W − x) > 0) = 1 − e−K(x,W −x) M/ i?2`27Q`2 Rd P (Z1 (x, W − x) > 0) μ(dx) = Rd (1 − e−K(x,W −x) ) μ (dx) < ∞X h?2 H bi BM2[m HBiv 7QHHQrb 7`QK +QM/BiBQM URXR8V M/ i?2 BM2[m HBiv 1−e−x ≤ x Ux ∈ RVX
1ti2MbBQM iQ LQM@SQBbbQMB M :2`K S`Q+2bb2b amTTQb2 ;` Mi2/ i?2 TQbbB#BHBiv Q7 b KTHBM; +Hmbi2` 2 bBHvX h?2M- b i?2 #Qp2 + H+mH iBQMb +QM}`K- i?2`2 `2 irQ BM;`2/B2Mib i? i K F2 i?BM;b rQ`FX 6B`bi Q7 HHi?2 i?BMMBM; T`Q# #BHBiv 7mM+iBQM Kmbi #2 p BH #H2 BM +HQb2/ 7Q`KX h?Bb Bb MQi i?2 + b2 7Q` > rF2b TQBMi T`Q+2bbX AM Q`/2` iQ #vT bb i?Bb /B{+mHiv- M / Ti iBQM Q7 i?2 i?BMMBM; QT2` iBQM rBHH #2 ;Bp2M BM a2+iBQM RkX9X b2+QM/ TQbbB#H2 /B{+mHiv r?2M ii2KTiBM; iQ 2ti2M/ i?2 "`BtĜE2M/ HH K2i?Q/ iQ MQM@SQBbbQMB M ;2`K TQBMi T`Q+2bb2b +QM+2`Mb i?2 Q#i2MiBQM Q7 b KTH2 Q7 i?2 i?BMM2/ ;2`K TQBMi T`Q+2bbX UAM i?2 + b2 Q7 SQBbbQM ;2`K T`Q+2bb- i?2 i?BMM2/ T`Q+2bb Bb HbQ SQBbbQM T`Q+2bb M/ i?2 /B{+mHiv /Q2b MQi 2tBbiXV 1t KTH2 jXeXk, h?BMMBM; i?2 :`B/X >2`2 Bb i?2 # bB+ B/2 X *QMbB/2` TQBMi T`Q+2bb QM N `2T`2b2Mi2/ #v b2[m2M+2 {Xn }n≥0 Q7 BB/ {0, 1}@p Hm2/ ` M/QK p `B@ #H2b- rBi? i?2 +QKKQM /Bbi`B#miBQM ;Bp2M #v P (Xn = 1) = pn Un ≥ 0VX q2 `2 i?2`27Q`2 dzi?BMMBM; i?2 ;`B/Ǵ N U /2i2`KBMBbiB+ TQBMi T`Q+2bbV rBi? i?2 i?BMMBM;
T`Q# #BHBiv 7mM+iBQM pn X amTTQb2 i? i n≥0 pn < ∞- r?B+? ;m ` Mi22b i? i i?2 i?BMM2/ ;`B/ ? b HKQbi bm`2Hv }MBi2 MmK#2` Q7 TQBMib M/ H2i T #2 Bib H bi TQBMiX LQi2 i? i P (T = n) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = pn (1 − pk ) () k≥n+1
M/ i? i- 7Q` 0 ≤ k ≤ n − 1P (Xk = 1, T = n) P (T = n) P (Xk = 1, Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) P (Xk = 1)P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xk = 1) .
P (Xk = 1 | T = n) =
h?2`27Q`2- BM Q`/2` iQ bBKmH i2 i?2 i?BMM2/ ;`B/- QM2 K v bi `i #v b KTHBM; p `B #H2 T rBi? i?2 /Bbi`B#miBQM UV- M/ B7 T = n- b2i Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .- M/ 7Q` 0 ≤ k ≤ n − 1- b KTH2 Xk rBi? i?2 /Bbi`B#miBQM P (Xk = 1) = pk X
jXdX h>1 "PPG1L JP.1G
RyR
h?BMMBM; i?2 irQ@/BK2MbBQM H ;`B/ Z2 Bb +QM+2Tim HHv i?2 b K2X >2`2 i?2 T`Q# #BHBiv Q7 F22TBM; i?2 TQBMi (i, j) ∈ Z2 Bb pi,j r?2`2 Bi Bb bbmK2/ i? i
(i,j)∈Z2 pi,j < ∞ r?2`2#v ;m ` Mi22BM; i? i i?2 MmK#2` Q7 TQBMib Q7 i?2 i?BMM2/ ;`B/ Bb }MBi2X Ai bm{+2b iQ TTHv #BD2+iBp2Hv Z2 QM N #v 2MmK2` iBM; i?2 TQBMib Q7 Z2 b {(in , jn )}n≥0 - /2}MBM; i?Bb #BD2+iBQM #v (in , jn ) → nX h?2 `2bi Bb i?2M Q#pBQmbX h?Bb K2i?Q/ K v #2 mb27mH r?2M b KTH2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb ;Bp2M U2tT2`BK2Mi HHvVX h?2 #Qp2 i?BMMBM; T`Q+2/m`2 Bb 2 bBHv / Ti #H2 iQ i?Bb + b2X h?2`2 Bb biBHH M Bbbm2 H27i bB/2 BM i?2 T`2b2Mi iBQM Q7 i?2 i?BMMBM; T`Q+2/m`2 Q7 i?2 ;`B/ NX * M r2 `2 HHv b KTH2 T \ AM 7 +i QM2 M22/b iQ ? p2 i /BbTQbBiBQM +HQb2/ 2tT`2bbBQM Q7 i?2 /Bbi`B#miBQM Q7 i?Bb p `B #H2- BM T `iB+mH ` Q7 i?2 BM}MBi2 ! T`Q/m+i k≥n+1 (1 − pk )X A7 i?Bb Bb MQi TQbbB#H2- r2 K v
#2 Hm+Fv 2MQm;? iQ }M/ /QKBM iBM; /Bbi`B#miBQM 7mM+iBQM q ≥ p bm+? i? i n n n qn < ∞ M/ bm+? i? i i?2 ! BM}MBi2 T`Q/m+i k≥n+1 (1−qk ) Bb +QKTmi #H2X PM2 rQmH/ i?2M b KTH2 i?2 i?BMM2/ ;`B/ rBi? i?BMMBM; T`Q# #BHBiv 7mM+iBQM qn X TQBMi Q7 i?Bb /QKBM iBM; ;`B/ HQ+ i2/ i k rBHH i?2M #2 F2Ti rBi? T`Q# #BHBiv pk /qk b TQBMi Q7 i?2 /2bB`2/ b KTH2X _2K `F jXeXj MQi?2` 2t KTH2 r?2`2 i?BMMBM; Bb TQbbB#H2 +QM+2`Mb `2M2r H T`Q@ +2bb2bX Ai rBHH #2 i`2 i2/ H i2`- 7i2` i?2 bim/v Q7 `2M2r H T`Q+2bb2b M/ r?2M i?2 iQQHb Q7 biQ+? biB+ BMi2MbBiv i?2Q`v rBHH #2 p BH #H2 U1t KTH2 8XdX8VX a22 HbQ 1t@ KTH2 3X9XReX
jXd
h?2 "QQH2 M JQ/2H
aiQ+? biB+ ;2QK2i`v +QM+2`Mb i?2 bim/v Q7 ` M/QK b? T2bX h?2 2t KTH2b +QMbB/2`2/ #2HQr T2`i BM iQ T `iB+mH ` bQ`i Q7 biQ+? biB+ ;2QK2i`v- r?2`2 i?2 ` M/QKM2bb Q7 i?2 b? T2b Bb /2T2M/2Mi QM i?2 TQbBiBQMb Q7 i?2 TQBMib Q7 M mM/2`HvBM; TQBMi T`Q+2bbX Hi?Qm;? i?2`2 2tBbib bQmM/ K i?2K iB+ H i?2Q`v Q7 ` M/QK b2ib-3 i?Bb i?2Q`v rBHH MQi #2 M2+2bb `v b HQM; b i?2 ` M/QK b2ib +QMbB/2`2/ BM i?2 TTHB+ iBQMb `2 dzMB+2Ǵ b2ib 7mHHv /2b+`B#2/ #v ` M/QK p2+iQ` Q7 }MBi2 /BK2MbBQM U+B`+H2- /BbFTQHv;QM- HBM2- b2;K2Mi- 2i+XVX q2 i?2M `2bQ`i iQ r? i + M #2 + HH2/ i?2 dzTQQ` K MǶb ` M/QK b2i i?2Q`vǴ- BM r?B+? ` M/QK b2i Bb b2i Q7 i?2 7Q`K S(Z) ⊆ Rm r?2`2 Z ∈ Rd Bb ` M/QK p2+iQ` M/ 7Q` 2 +? z- S(z) Bb K2 bm` #H2 b2i M/ S(Z) Bb HbQ K2 bm` #H2 b2iX h?2b2 `2 KBMBK H `2[mB`2K2Mib mbm HHv b iBb}2/ BM TTHB+ iBQMbX q2 b? HH +QMbB/2` `2 H@p Hm2/ 7mM+iBQMb Q7 S- 7Q` BMbi M+2 g(S) = d (S) , g(S) = 1a∈S , 7Q` r?B+? i?2 2tT2+i iBQM Bb r2HH /2}M2/- b g(S(z))P (Z ∈ dz) . E [g(S)] := E [g(S(Z))] = Rd
()
UjXRdV
q2 QMHv M22/ iQ 2Mbm`2 i? i i?2 7mM+iBQM z ∈ Rd → g(S(z)) ∈ R Bb K2 bm` #H2 M/ Bib BMi2;` H rBi? `2bT2+i iQ i?2 /Bbi`B#miBQM Q7 Z r2HH /2}M2/ UBM i?2 2t KTH2b3
(JQH+? MQp- kyy8)X
Ryk
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
i?Bb rBHH ;2M2` HHv i?2 + b2- #2+ mb2 g rBHH #2 MQM@M2; iBp2 7mM+iBQM- b BM UVVX q2 b? HH mb2 7Q` UjXRdV i?2 ##`2pB i2/ MQi iBQM g(s) Q(ds) , S
i?2`2#v T`2i2M/BM; i? i i?2`2 2tBbib b2i S Q7 b? T2b rBi? bmBi #H2 σ@}2H/ G QM Bi- M/ M /2[m i2 T`Q# #BHBiv /Bbi`B#miBQM Q QM (S, G)X 1t KTH2 jXdXR, _ M/QK .BbFX AM i?Bb 2t KTH2- S Bb i?2 +HQb2/ /BbF BM R2 +2Mi2`2/ QM i?2 Q`B;BM M/ rBi? ` /Bmb Z- MQM@M2; iBp2 ` M/QK p `B #H2X q2 ? p2- rBi? g(S) = 2 (S) E [g(S)] = E 2 (S(Z)) = E πZ 2 , M/ rBi? g(S) = 1{a∈S} -
E [g(S)] = E 1{a∈S(Z)} = P (a ∈ S(Z)) = P (Z ≥ a) .
.2}MBiBQM jXdXk h?2 + T +Biv 7mM+iBQM H Q7 i?2 ` M/QK b2i S Bb i?2 7mM+iBQM K → TS (K) UK +QKT +iV /2}M2/ #v TS (K) := P (S ∩ K = ∅) . 1t KTH2 jXdXj, * T +Biv 6mM+iBQM H Q7 SQBMi S`Q+2bbX bBKTH2 TQBMi T`Q+2bb N + M HbQ #2 pB2r2/ b ` M/QK b2i S ≡ N X AM i?Bb + b2 i?2 + T +Biv 7mM+iBQM H Bb i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM, TN (K) := P (N ∩ K = ∅) = P (N (K) = 0) = vN (K) .
1t KTH2 jXdX9, * T +Biv 7mM+iBQM H M/ pQB/ M+2 S`Q# #BHBiv 6mM+@ iBQMX ` M/QK b2i S ⊂ Rm i? i Bb HKQbi bm`2Hv i?2 +HQbm`2 Q7 Bib BMi2`BQ`N K v #2 /2b+`B#2/ b i?2 bmTTQ`i Q7 ` M/QK K2 bm`2- M K2Hv i?2 K2 bm`2 μS rBi? /2MbBiv 1S (x) rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 m μS (C) := m (C ∩ S) , C ∈ B(Rm ). aBM+2 S ∩ K = ∅ Bb 2[mBp H2Mi iQ K 1S (x) dx = 0- i? i Bb μS (K) = 0P (S ∩ K = ∅) = 1 − P (μS (K) = 0) . h?2`27Q`2 1 − TS (K) Bb i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM H Q7 i?2 ` M/QK K2 @ bm`2 μS 2p Hm i2/ i K- i? i Bb- mbBM; i?2 T`2pBQmb MQi iBQM 7Q` i?2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQM HTS (K) = vμS (K) .
N
h?Bb Bb MQi i?2 + b2 B7 N Bb TQBMi T`Q+2bbX
jXdX h>1 "PPG1L JP.1G
Ryj
_2K `F jXdX8 h?2 JƺM+?Ĝ_ûMvB i?2Q`2K ? b ;2M2` HBx iBQM +QM+2`MBM; +HQb2/ ` M/QK b2ib i? i b vb `Qm;?Hv i? i i?2 + T +Biv 7mM+iBQM H Q7 +HQb2/ ` M/QK b2i mMB[m2Hv /2i2`KBM2b i?2 /Bbi`B#miBQM Q7 i?Bb ` M/QK b2iXRy Uh?2 [m HB}+ iBQM dz`Qm;?HvǴ `272`b iQ i?2 7 +i i? i QM2 ? b iQ /2}M2 i?2 /Bbi`B#miBQM Q7 ` M/QK +HQb2/ b2i- M/ BM T `iB+mH ` /2}M2 K2 bm` #H2 bT +2 (F, A) Q7 +HQb2/ b2ibXV q2 MQr BMi`Q/m+2 i?2 "QQH2 M KQ/2HXRR b r2 b? HH b22- i?Bb Q#D2+i Bb +HQb2Hv `2H i2/ iQ +Hmbi2` ` M/QK K2 bm`2bX G2i N #2 SQBbbQM T`Q+2bb QM Rm rBi? MQM@ iQKB+ σ@}MBi2 BMi2MbBiv K2 bm`2 νX .2MQi2 #v {Xn }n∈N Bib b2[m2M+2 Q7 TQBMibX G2i MQr {Sn }n∈N #2 b2[m2M+2 Q7 ` M/QK K `Fb- BB/ M/ BM/2T2M/2Mi Q7 N X 1 +? Sn Bb +QKT +i ` M/QK b2iX h?2 Xn Ƕb `2 + HH2/ i?2 ;2`Kb r?2`2 b i?2 Sn Ƕb `2 + HH2/ i?2 ;` BMbX A7 A M/ B `2 bm#b2ib Q7 Rm M/ x ∈ Rm - x + A := {x + y ; y ∈ A} M/ A ⊕ B := {x + y ; x ∈ A, x ∈ B}X
6B;m`2 jXdXR PM2 Q7 i?2 [m MiBiB2b Q7 BMi2`2bi BM TTHB+ iBQMb Bb i?2 T`Q# #BHBiv Q7 BMi2`b2+iBQM Q7 i?2 ` M/QK b2i Σ = ∪n∈N (Xn + Sn ) rBi? ;Bp2M +QKT +i b2i K ⊂ Rm - i? i Bb TΣ (K) := P (Σ ∩ K = ∅) . AM Q`/2` iQ +QKTmi2 i?Bb [m MiBiv Ui?2 + T +Biv 7mM+iBQM H Q7 ΣV- r2 }`bi Q#b2`p2 i? i TΣ (K) = P (N (K) = 0) , r?2`2
N (K) =
1{(Xn +Sn )∩K=∅} .
n∈N
q2 b?Qr i? i N (K) Bb SQBbbQM p `B #H2 rBi? K2 M θ(K) := P ((x + S1 ) ∩ K = ∅) ν(dx)
()
Rm
M/ i?2`27Q`2
TΣ (K) = 1 − exp −
Rm
Ry RR
P ((x + S1 ) ∩ K = ∅) ν(dx)
a22 7Q` BMbi M+2 (JQH+? MQp- kyy8)- h?K RXjXkyX (J i?2`QM- RNed- RNd8)- (a2`` - RN3k)X
.
UjXR3V
Ry9
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
S`QQ7X h?2 G TH +2 i` Mb7Q`K Q7 i?2 /Bbi`B#miBQM Q7 N (K) Bb ;Bp2M #v E e−tN (K) = E e−t n∈N 1(Xn +Sn )∩K=∅ = E e−t n∈N f (Xn ,Sn ) , r?2`2 t ≥ 0 M/ f (x, s) := 1(x+s)∩K=∅ X hQ b22 i?Bb- mb2 i?2 7Q`KmH BM *Q`QHH `v jXjX9- r?B+? ;Bp2b −tf (x,S1 ) E e−t n∈N f (Xn ,Sn ) = exp E e − 1 ν(dx) m R −t1 (x+S1 )∩K=∅ = exp − 1 ν(dx) E e m R = exp (e−t − 1) P ((x + S1 ) ∩ K = ∅) ν(dx) . Rm
A7 i?2 ;2`K T`Q+2bb Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ- i?2 K2 M p Hm2 θ(K) Q7 N (K) i F2b i?2 7Q`K θ(K) = λE [ m ((−S1 ) ⊕ K)] . AM/22/- 7`QK UV θ(K) = λ Rm
= λE
P ((x + S1 ) ∩ K = ∅) dx
1(x+S1 )∩K=∅ dx = λE
Rm
Rm
1(−S1 )⊕K (x) dx .
h?2`27Q`2- BM i?2 ?QKQ;2M2Qmb + b2TΣ (K) = 1 − exp {λE [ m ((−S1 ) ⊕ K)]} .
UjXRNV
MQi?2` [m MiBiv Q7 BMi2`2bi r?2M i?2 ;2`K TQBMi T`Q+2bb Bb M ?TT rBi? BMi2MbBiv λ Bb i?2 pQHmK2 7` +iBQM p Q7 i?2 ` M/QK b2i Σ- /2}M2/ #v p :=
E [ m (Σ ∩ B)] .
m (B)
Ai Bb BM/2T2M/2Mi Q7 B- M/ #v i` MbH iBQM BMp `B M+2 Q7 i?2 KQ/2H- Bi Bb 2[m H iQ E B 1x∈Σ dx p= = P (0 ∈ Σ) = P (Σ ∩ {0} = ∅) .
m (B) h?2`27Q`2p = p({0}) = 1 − e− = 1 − e−
Rm
Rm
P ((x+S1 )∩{0}=∅) λ dx
P ((x+S1 )∩{0}=∅) λ dx
= 1 − e−
Rm
P (−x∈S1 ) λ dx
= 1 − e−λE[
h?2 +Qp `B M+2 7mM+iBQM C : Rm → R Q7 i?2 ` M/QK b2i Σ Bb /2}M2/ #v
m (S
1 )]
.
jXdX h>1 "PPG1L JP.1G
Ry8
C(x) := P (0 ∈ Σ , x ∈ Σ) . Uh?Bb Bb i?2 +Qp `B M+2 7mM+iBQM- BM i?2 mbm H b2Mb2- Q7 i?2 rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb {1Σ (t)}t∈Rm XV AM i?2 ?QKQ;2M2Qmb + b2C(x) = 2p − 1 − (1 − p)2 exp {λE [ m (S1 ∪ (S1 − x))]} . S`QQ7X C(x) = P (0 ∈ Σ ∩ (Σ − x)) = P (0 ∈ Σ) + P (x ∈ Σ) − P (0 ∈ Σ ∪ (Σ − x)) = 2p − 1 + P (0 ∈ / Σ ∪ (Σ − x)) = 2p − 1 + P (Σ ∩ {o, x} = ∅) = 2p − 1 + TΣ ({0, x}) . 6`QK UjXRNVTΣ ({0, x}) = 1 − exp {λE [ m ((−S1 ) ⊕ {0, x})]} . LQr E [ m ((−S1 ) ⊕ {0, x})] = E [ m ((−S1 ) ∪ (−S1 + x))] = E [ m (−S1 )] + E [ m (−S1 + x)] − E [ m ((−S1 ) ∩ (−S1 + x))] = 2E [ m (−S1 )] − E [ m ((−S1 ) ∩ (−S1 + x))] . *QK#BMBM; i?2 #Qp2 2[m HBiB2b rBi? i?2 Q#b2`p iBQM i? i 1 − p = exp {−λ m (S1 )} ;Bp2b i?2 MMQmM+2/ `2bmHiX 1t KTH2 jXdXe, "QmM/ `v Q7 SQBbbQM *Hmbi2` Q7 .BbFbX G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R2 - Q7 BMi2MbBiv λX G2i {Xn }n∈N #2 Bib b2[m2M+2 Q7 TQBMibX .` r `QmM/ 2 +? TQBMi Xn +HQb2/ /BbF Q7 ` /Bmb aX h?2 `2 BMbB/2 i?2 b[m `2 [0, T ] × [0, T ] i? i Bb MQi +Qp2`2/ #v /BbF Bb /2HBKBi2/ #v +m`p2X q2 b22F iQ +QKTmi2 Bib p2` ;2 H2M;i?- 2t+Hm/BM; i?2 T `ib QM i?2 #QmM/ `B2b Q7 [0, T ] × [0, T ]X
6B;m`2 jXdXkX "QQH2 M b2i M/ Bib #QmM/ `v h?2 MmK#2` Q7 /BbFb +Qp2`BM; ;Bp2M TQBMi y ∈ R2 Bb
Rye
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a Z(y) =
1{||Xn −y|| 0X 6Q` HH t ≥ 0- H2i 2ε2
Rj
+ S0,t := {(s, z) : 0 < s < t, t − s < z} , − S0,t := {(s, z) : −∞ < s < 0, −s < z < t − s}
M/ H2i
+ − Xε (t) := ε N ε (S0,t ) − N ε (S0,t ) .
URV a?Qr i? i Xε (t) Bb r2HH /2}M2/ 7Q` HH t ≥ 0X UkV *QKTmi2 i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 (Xε (t1 ), . . . , Xε (tn )) 7Q` HH 0 ≤ t1 ≤ t 2 . . . ≤ tn X Rj
(*BQ+x2F@:2Q`;2b M/ J M/2H#`Qi- RNN8)X
jX3X 1s1_*Aa1a
RRR
UjV a?Qr i? i 7Q` HH 0 ≤ t1 ≤ t2 . . . ≤ tn - (Xε (t1 ), . . . , Xε (tn )) +QMp2`;2b b ε ↓ 0 BM /Bbi`B#miBQM iQ (BH (t1 ), . . . , BH (tn ))- r?2`2 {BH (t)}t≥0 Bb 7` +i H "`QrMB M KQiBQM rBi? >m`bi T ` K2i2` H = 1−θ M/ p `B M+2 E [BH (1)2 ] = θ−1 (1 − θ)−1 2 i? i Bb- {BH (t)}t≥0 Bb +2Mi2`2/ : mbbB M T`Q+2bb bm+? i? i BH (0) = 0 M/ rBi? +Qp `B M+2 7mM+iBQM 1 2H E [BH (t)BH (s)] = |s| + |t|2H − |s − t|2H E BH (1)2 . 2
+ S0,t
− S0,t
t
0 6B;m`2 jX3XR
1t2`+Bb2 jX3X3X .Bbi`B#miBQM Q7 i?2 J tBKmK LmBb M+2 G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm Q7 TQbBiBp2 BMi2MbBiv λ M/ rBi? TQBMi b2[m2M+2 {Xn }n≥1 X G2i {Zn }n≥1 #2 M BB/ b2[m2M+2 Q7 `2 H MQM@M2; iBp2 ` M/QK p `B #H2b rBi? +QKKQM /Bbi`B#miBQM Q- M/ BM/2T2M/2Mi Q7 N X *QKTmi2 i?2 /Bbi`B#miBQM Q7 i?2 ` M/QK p `B #H2 max Zn e−β||Xn || n≥1
(β > 0) .
Uh?2 iBiH2 Q7 i?2 2t2`+Bb2 `272`b iQ +QKKmMB+ iBQMb +QMi2ti r?2`2 Zn Bb i?2 MQBb2 BMi2MbBiv ;2M2` i2/ i TQBMi Xn - M/ e−β||Xn || Bb M ii2Mm iBQM 7 +iQ` 7Q` `2+2Bp2` HQ+ i2/ i 0XV 1t2`+Bb2 jX3XNX h?2 Jf:Af∞ JQ/2H- h F2 k AM 1t KTH2 jXjXkUBV T`Qp2 i? i i?2 /2T `im`2 T`Q+2bb Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2M@ bBiv λUBBV +QKTmi2 cov(X(t), X(t + τ )) 7Q` HH t, τ ∈ R- τ ≥ 0- M/ UBBBV BMi2`T`2i i?2 T`Q+2bb {X(t)}t∈R b b?Qi MQBb2 BM Q`/2` iQ Q#i BM i?2 `2bmHib Q7 1t KTH2 jXjXk- M/ Q7 UBV M/ UBBV- 7`QK i?2 ;2M2` H `2bmHib Q7 h?2Q`2K jX9XeX 1t2`+Bb2 jX3XRyX GB7iBM; G2i N #2 SQBbbQM T`Q+2bb QM R rBi? i?2 UHQ+ HHv BMi2;` #H2V BMi2MbBiv 7mM+iBQM λ : R → RX G2i {Tn }n∈Z #2 Bib b2[m2M+2 Q7 TQBMib M/ H2i {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N #2 M ?TT QM R × R+ - rBi? BMi2MbBiv 1 M/ BM/2T2M/2Mi Q7 N M/ Q7 {Un }n∈Z X .2}M2 TQBMi QM R × R+ #v T`Q+2bb N
RRk
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a (C) := N
1C ((Tn , Un λ(Tn ))) + N (C ∩ H)
n∈Z
r?2`2 H := {(t, z) ∈ R × R+ ; 0 ≤ z ≤ λ(t)} . Bb M ?TT QM R × R+ rBi? BMi2MbBiv 1X a?Qr i? i N 1t2`+Bb2 jX3XRRX *QKTQmM/ SQBbbQM S`Q+2bb G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ M/ TQBMi b2[m2M+2 {Tn }n≥1 X G2i {Zn }n≥1 #2 M BB/ `2 H@p Hm2/ b2[m2M+2 BM/2T2M/2Mi Q7 N - rBi? +QKKQM +mKmH iBp2 /Bbi`B@ #miBQM 7mM+iBQM F X .2}M2 7Q` HH t ≥ 0 Y (t) = Zn 1(0,t] (Tn ) . n≥1
Uh?2 T`Q+2bb {Y (t)}t≥0 Bb + HH2/ +QKTQmM/ SQBbbQM T`Q+2bbXV a?Qr i? i E e−rY (t) = eλt(1−h(r)) ∞ r?2`2 h(r) := E e−rZ1 = 0 e−rx dF (x)X 1t2`+Bb2 jX3XRkX q i2` "QK#b uQm `2 BMBiB HHv HQ+ i2/ i i?2 Q`B;BM (0, 0) Q7 i?2 TH M2 i r?B+? Bb +2Mi2`2/ /BbF . Q7 ` /Bmb RX uQm `mM BM bi` B;?i HBM2 7`QK i?2 Q`B;BM iQ i?2 dzb?2Hi2` TQBMiǴ (0, R) i +QMbi Mi bT22/ vX h?2 `2 bQM r?v vQm `2 `mMMBM; Bb i? i r i2` #QK#b `2 /`QTT2/ QM i?2 /BbF .X h?2 iBK2b Q7 BKT +i 7Q`K M ?TT Q7 BMi2MbBiv λ- M/ 2 +? BKT +i Bb HQ+ i2/ BM/2T2M/2MiHv Q7 HH i?2 `2bi- mMB7Q`KHv QM i?2 /BbFX uQm rBHH ;2i r2i B7 i?2 BKT +i Q7 i?2 #QK# Bb rBi?BM /Bbi M+2 a Q7 vQm` TQbBiBQM i i?2 iBK2 Q7 BKT +iX PM+2 vQm ``Bp2 i i?2 b?2Hi2` TQBMi (0, R)- i?2 #QK#BM; biQTbX q? i `2 vQm` +? M+2b Q7 MQi ;2iiBM; r2i\ q? i Bb- ;Bp2M i? i vQm /B/ ;2i r2ii?2 2tT2+i2/ iBK2 i? i vQm `2K BM2/ /`v\ 1t2`+Bb2 jX3XRjX aKQFBM; SQi i a BMi J `v@C M2ǰb *QHH2;2 aKQFBM; TQi `2+2MiHv ;Qi # MM2/ QM i?2 a BMi J `v@C M2Ƕb +QHH2;2 + KTmbX h?2 mi?Q`BiB2b MQiB+2/ i? i i?2 pBQH iQ`b Q7 i?2 # M K F2 mb2 Q7 `2bi`QQK BM b2+Hm/2/ rBM; Q7 i?2 + KTmbX h?2v +QMb2[m2MiHv /2pBb2/ bi` i2;v iQ b2M/ dz+QTbǴ iQ + Tim`2 i?2 +mHT`BibX bbmK2 i? i i?2 b+?QQH#QvbǶ ``Bp H iBK2b BM i?2 `2bi`QQK T`2KBb2b 7Q`K SQBbbQM T`Q+2bb rBi? BM/2T2M/2Mi BB/ K `FbX G2i τn /2MQi2 i?2 n@i? ``Bp H iBK2 Q7 b+?QQH#Qv BM i?2 TQi b M+im `v Ui?2 `2bi`QQKbV M/ H2i σn #2 i?2 iBK2 ?2 bT2M/b BM bKQFBM;X h?2 +QTbǶ ``Bp H iBK2b HbQ 7Q`K SQBbbQM T`Q+2bb rBi? BM/2T2M/2Mi BB/ K `FbX .2MQi2 i?2 k@i? ``Bp H iBK2 Q7 +QT i i?2 TQi2MiB H +`BK2 b+2M2 #v Tk M/ H2i Sk `2T`2b2Mi i?2 HBM;2`BM; iBK2 i?2`2 Q7 i?2 +Q``2bTQM/BM; `2T`2b2Mi iBp2 Q7 i?2 +QHH2;2 mi?Q`BivX h?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 σ Bb Qs M/ i? i Q7 S Bb Qc X bbmKBM; i?2 TQBMi T`Q+2bb2b Q7 bim/2Mib M/ Q7 i?2 +QTb iQ #2 ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λs > 0 M/ λc > 0- +QKTmi2 i?2 p2` ;2 MmK#2` Q7 bim/2Mib + m;?i T2` mMBi Q7 iBK2X
jX3X 1s1_*Aa1a
RRj
1t2`+Bb2 jX3XR9X h?B`/ JQK2Mib Q7 a?Qi LQBb2 G2i N #2 M ?TT QM Rd rBi? BMi2MbBiv λ > 0X G2i {X(t)}t∈R #2 i?2 b?Qi MQBb2 +QMbi`m+i2/ QM Bi rBi? M BKTmHb2 7mM+iBQM h : Rd → R i? i Bb #QmM/2/ M/ rBi? +QKT +i bmTTQ`i UMmHH QmibB/2 #QmM/2/ bm#b2i Q7 Rd V, X(t) := h(t − s) N (ds) . R
G2i t1 - t2 - t3 ∈ R X d
U V *QKTmi2 i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 p2+iQ` (X(t1 ), X(t2 ), X(t3 ))X U#V *QKTmi2 E [X(t1 )X(t2 )X(t3 )]X 1t2`+Bb2 jX3XR8X SQBMib BM _ M/QK AMi2`p H G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X G2i Z1 M/ Z2 #2 irQ MQM@M2; iBp2 `2 H ` M/QK p `B #H2b bm+? i? i Z1 ≤ Z2 X :Bp2 i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 ` M/QK p `B #H2 X = N ((Z1 , Z2 ])- bbmKBM; i? i Z1 M/ Z2 `2 BM/2T2M/2Mi Q7 NX 1t2`+Bb2 jX3XReX G TH +2 h` Mb7Q`K Q7 ?QKQ;2M2Qmb *Qt S`Q+2bb G2i N #2 *Qt T`Q+2bb QM Rm rBi? +QMbi Mi BMi2MbBiv T`Q+2bb- i? i Bb ν(dx) := Λ m (dx)- r?2`2 m Bb i?2 G2#2b;m2 K2 bm`2 QM R m M/ Λ Bb MQM@M2; iBp2 ` M/QK p `B #H2 rBi? G TH +2 i` Mb7Q`K LΛ (t) := E e−tΛ X *QKTmi2 Bib G TH +2 i` Mb7Q`KX 1t2`+Bb2 jX3XRdX pQB/ M+2 S`Q# #BHBiv Q7 *Qt S`Q+2bb G2i N #2 *Qt T`Q+2bb QM Rm rBi? ` M/QK BMi2MbBiv K2 bm`2 νX G2i N #2 SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 ν 2[m H iQ i?2 K2 M Q7 ν Ui? i Bb- 7Q` HH C ∈ B(Rm )- ν (C) = E [ν(C)]VX 6Bt Mv bm#b2i C ∈ B(Rm )X a?Qr i? i i?2 T`Q# #BHBiv i? i i?2`2 Bb MQ TQBMi Q7 N BM C Bb ;`2 i2` i? M Q` 2[m H iQ i?2 +Q``2bTQM/BM; T`Q# #BHBiv 7Q` N X 1t2`+Bb2 jX3XR3X GBM2 Q7 aB;?i- h F2 R *QMbB/2` SQBbbQM N QM R2 rBi? MQM@ iQKB+ M/ HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 νX h?2`2 Bb dz` M/QK b? T2Ǵ +2Mi2`2/ `QmM/ 2 +? Q7 Bib TQBMibX G2i i?2 ;2M2`B+ b? T2 S #2 dz/Bbi`B#mi2/ ++Q`/BM; iQ bQK2 T`Q# #BHBiv /Bbi`B#miBQM QS ǴX LQr +QMbB/2` irQ `#Bi` `v TQBMib A- BX q2 b v i? i A M/ B + M +QKKmMB+ i2 B7 i?2 HBM2 +QMM2+iBM; A M/ B /Q2b MQi BMi2`b2+i Mv Q7 i?2 2tBbiBM; b? T2b `QmM/ i?2 TQBMib Q7 i?2 TQBMi T`Q+2bb U7Q` HH n ≥ 1- i?2 dz2tBbiBM; b? T2 `QmM/Ǵ Xn ∈ N Bb Xn + Sn - i? i Bb Sn i` MbH i2/ #v Xn - r?2`2 Sn Bb /Bbi`B#mi2/ ++Q`/BM; iQ QS VX q2 bbmK2 i? i {Sn }n∈N Bb M BB/ b2[m2M+2 BM/2T2M/2Mi Q7 N X *QKTmi2 i?2 T`Q# #BHBiv i? i A M/ B + M bm++2bb7mHHv +QKKmMB+ i2X Smb? i?2 +QKTmi iBQMb r?2M N Bb M ?TT Q7 BMi2MbBiv λ- i?2 ` M/QK b? T2 Bb +B`+H2 Q7 ` /Bmb a- M/ i?2 /Bbi M+2 7`QK A iQ B Bb 2[m H iQ lX 1t2`+Bb2 jX3XRNX GBM2 Q7 aB;?i- h F2 k h?2 bBim iBQM Bb i? i Q7 1t2`+Bb2 jX3XR3X h?2`2 Bb i` p2HH2` ;QBM; 7`QK B iQ i?B`/ TQBMi CX q? i Bb i?2 T`Q# #BHBiv i? i ?2 rBHH ? p2 A BM bB;?i HH i?2 iBK2 /m`BM; ?Bb i` p2H\
RR9
*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a
1t2`+Bb2 jX3XkyX *2HH S?QM2b M/ " b2 ai iBQMb G2i N1 M/ N2 #2 irQ BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM Rm rBi? `2bT2+iBp2 }MBi2 BMi2MbBiv K2 bm`2b ν1 M/ ν2 X *QKTmi2 i?2 p2` ;2 MmK#2` Q7 TQBMib Q7 N1 i? i b22 MQ TQBMi Q7 N2 rBi?BM /Bbi M+2 aX Uh?2 iBiH2 Q7 i?Bb 2t KTH2 `272`b iQ i?2 bBim iBQM r?2`2 i?2 TQBMib Q7 N2 `2 i?2 HQ+ iBQMb Q7 # b2 bi iBQMb BM rB`2H2bb +QKKmMB+ iBQMb M2irQ`F M/ i?Qb2 Q7 N1 `2 i?2 HQ+ iBQMb Q7 +2HH T?QM2bX h?2`27Q`2 i?2 [m MiBiv UV Bb i?2 p2` ;2 MmK#2` Q7 +2HH T?QM2b i? i + MMQi #2 +QMM2+i2/ iQ # b2 bi iBQMXV 1t2`+Bb2 jX3XkRX _ M/QK GBM2b G2i L #2 i?2 SQBbbQM ` M/QK HBM2 KQ/2H Q7 a2+iBQM jXdX G2i D0 := D(r0 , θ0 ) #2 0 i?2 TQBMi T`Q+2bb QM D0 M `#Bi` `v }t2/ ` M/QK HBM2 Q7 R2 M/ /2MQi2 #v N 7Q`K2/ #v i?2 BMi2`b2+iBQMb Q7 i?Bb bi` B;?i HBM2 rBi? i?2 HBM2 T`Q+2bbX a?Qr i? i N0 Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv | sin(θ0 − θ)| G(dθ) . λ0 = [0,π)
*? Ti2` 9 _2M2r H M/ _2;2M2` iBp2 S`Q+2bb2b 6`QK i?2 K i?2K iB+ H TQBMi Q7 pB2r- `2M2r H i?2Q`v Bb +QM+2`M2/ rBi? i?2 `2M2r H 2[m iBQM f (t − s) dF (s) ,
f (t) = g(t) + [0,t]
r?2`2 F Bb i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 }MBi2 K2 bm`2 QM i?2 TQbB@ iBp2 `2 H HBM2X Aib K BM +QM+2`M Bb i?2 bvKTiQiB+ #2? pBQ` Q7 i?2 bQHmiBQM f Ui?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 r?B+? Bb MQi `2 H Bbbm2 mM/2` KBH/ +QM/BiBQMb- b r2 b? HH b22VX PM+2 2K#2//2/ BM i?2 7` K2rQ`F Q7 TQBMi T`Q+2bb2b- `2M2r H i?2Q`v M/ BM T `iB+mH ` "H +Fr2HHǶb i?2Q`2K-R #2+QK2b 7mM/ K2Mi H iQQH Q7 T`Q# #BH@ Biv i?2Q`v- T `iB+mH `Hv mb27mH BM i?2 bim/v Q7 `2;2M2` iBp2 T`Q+2bb2b-k H `;2 M/ BKTQ`i Mi +H bb Q7 biQ+? biB+ T`Q+2bb2b r?B+? BM+Hm/2b 7Q` BMbi M+2 i?2 `2+m``2Mi +QMiBMmQmb@iBK2 ?K+b M/ i?2 b2KB@J `FQp T`Q+2bb2bX
9XR
_2M2r H SQBMi T`Q+2bb2b
*QMbB/2` M BB/ b2[m2M+2 {Sn }n≥1 Q7 MQM@M2; iBp2 ` M/QK p `B #H2b rBi? +QKKQM +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F (x) := P (Sn ≤ x) . h?Bb +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Bb + HH2/ /272+iBp2 r?2M F (∞) := P (S1 < ∞) < 1- M/ T`QT2` r?2M F (∞) = 1X h?2 mMBMi2`2biBM; + b2 r?2`2 P (S1 = 0) = 1 Bb ?2M+27Q`i? 2HBKBM i2/X h?2 #Qp2 b2[m2M+2 Bb + HH2/ i?2 BMi2`@`2M2r H b2[m2M+2X h?2 bbQ+B i2/ `2M2r H b2[m2M+2 {Tn }n≥0 Bb /2}M2/ #v Tn := Tn−1 + Sn
(n ≥ 1) ,
r?2`2 i?2 BMBiB H /2H v T0 Bb 7BMBi2 MQM@M2; iBp2 ` M/QK p `B #H2 BM/2T2M/2Mi Q7 i?2 BMi2`@`2M2r H b2[m2M+2X h?2 iBK2 Tn Bb + HH2/ `2M2r H iBK2- Q` M 2p2MiX q?2M T0 = 0- i?2 `2M2r H b2[m2M+2 Bb + HH2/ mM/2H v2/X h?2 biQ+? biB+ T`Q+2bb R k
("H +Fr2HH- RN93)X (aKBi?- RN88)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_4
RR8
RRe
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a N ([0, t]) :=
(t ≥ 0)
1{Tn ≤t}
n≥0
Bb i?2 +QmMiBM; T`Q+2bb Q7 i?2 `2M2r H b2[m2M+2c N ([0, t]) +QmMib i?2 MmK#2` Q7 2p2Mib BM i?2 +HQb2/ BMi2`p H [0, t]X LQi2 i? i T0 ≥ 0 Ui?Bb +QMp2MiBQM /Bz2`b 7`QK i?2 mbm H QM2- QMHv 7Q` i?Bb +? Ti2`V M/ i? i i?2 TQBMi i 0 Bb +QmMi2/ r?2M i?2`2 Bb QM2X *H2 `Hv- i?2 ` M/QK 7mM+iBQM t → N ([0, t]) Bb HKQbi bm`2Hv `B;?i@+QMiBMmQmb M/ ? b HBKBi QM i?2 H27i 7Q` 2 +? t > 0- M K2Hv N [0, t)X
0
T1
T0
T2
T3
t
h?2Q`2K 9XRXR 6Q` HH t ≥ 0- E[N ([0, t])] < ∞X AM T `iB+mH `- HKQbi bm`2HvN ([0, t]) < ∞ 7Q` HH t ≥ 0X S`QQ7X Ai bm{+2b iQ +QMbB/2` i?2 mM/2H v2/ + b2 U1t2`+Bb2 9XdXRVX "v J `FQpǶb BM2[m HBiv P (Tn ≤ t) = P (e−Tn ≥ e−t ) ≤ et E e−Tn . "mi bBM+2 i?2 `2M2r H b2[m2M+2 Bb BB/n n E e−Sk = αn , E e−Tn = E e− k=1 Sk = k=1
rBi? α = E e−S1 < 1 bBM+2 P (S1 = 0) < 1X h?2`27Q`2 E [N ([0, t])] − 1 = E =
1{Tn ≤t} =
n≥1
n≥1
P (Tn ≤ t) ≤ e
n≥1
E 1{Tn ≤t}
t
αn < ∞ .
n≥1
h?2 7Q`r `/ `2+m``2M+2 {A(t)}t≥0 M/ i?2 # +Fr `/ `2+m``2M+2 {B(t)}t≥0 `2 /2}M2/ b 7QHHQrbX "Qi? T`Q+2bb2b `2 `B;?i@+QMiBMmQmb rBi? H27i@? M/ HBKBibX 6Q` n ≥ 0- i?2v ? p2 HBM2 ` i` D2+iQ`B2b BM (Tn , Tn+1 ) rBi? `2bT2+iBp2 bHQT2b −1 M/ +1- M/ i `2M2r H TQBMi Tn A(Tn ) = Tn+1 − Tn , A(Tn+1 −) = 0 , B(Tn ) = 0 , B(Tn+1 −) = Tn+1 − Tn . 6Q` 0 ≤ t < T0 - A(t) = T0 − t M/ B(t) = tX
9XRX _1L1qG SPALh S_P*1aa1a
0
T1
T0
RRd
T2
t
T3
h?2 7Q`r `/ `2+m``2M+2 iBK2 T`Q+2bb
0
T1
T0
T2
T3
t
h?2 # +Fr `/ `2+m``2M+2 iBK2 T`Q+2bb
.2}MBiBQM 9XRXk h?2 7mM+iBQM R : R+ → R+ /2}M2/ #v R(t) := E[N ([0, t])], r?2`2 N Bb i?2 +QmMiBM; T`Q+2bb Q7 i?2 mM/2H v2/ `2M2r H b2[m2M+2- Bb + HH2/ i?2 `2M2r H 7mM+iBQMX h?2 `2M2r H 7mM+iBQM Bb `B;?i@+QMiBMmQmb U1t2`+Bb2 9XdXjV M/ MQM@/2+`2 bBM;X h?2`27Q`2- QM2 + M bbQ+B i2 rBi? Bi mMB[m2 K2 bm`2 μR QM R+ bm+? i? i μR ([0, a]) = R(a)X h?Bb K2 bm`2- + HH2/ i?2 `2M2r H K2 bm`2- rBHH bQK2iBK2b #2 /2MQi2/ #v R- i?2 +QMi2ti pQB/BM; +QM7mbBQM #2ir22M i?2 K2 bm`2 M/ Bib +mKm@ H iBp2 /Bbi`B#miBQM 7mM+iBQMX LQi2 i? i μR ({0}) = R(0) = 1X 1t KTH2 9XRXj, h?2 SQBbbQM S`Q+2bbX *QMbB/2` i?2 + b2 Q7 2tTQM2MiB H BMi2`@ 2p2Mi iBK2b, F (t) = 1−e−λt Ut ≥ 0VX h?2 mM/2H v2/ `2M2r H T`Q+2bb Bb i?2M M ?TT rBi? BMi2MbBiv λ iQ r?B+? TQBMi i iBK2 0 Bb //2/X h?2`27Q`2- R(t) = 1 + λtX Ai rBHH #2 +QMp2MB2Mi iQ 2tT`2bb i?2 `2M2r H 7mM+iBQM BM i2`Kb Q7 i?2 +QKKQM +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 ` M/QK p `B #H2b Sn X 6Q` i?Bb- Q#b2`p2 i? i BM i?2 mM/2H v2/ + b2 Tn := S1 + · · · + Sn Bb i?2 bmK Q7 n BM/2T2M/2Mi ` M/QK p `B #H2b rBi? +QKKQM +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F M/ i?2`27Q`2 P (Tn ≤ t) = F ∗n (t) , r?2`2 F ∗n Bb i?2 n@7QH/ +QMpQHmiBQM Q7 F - /2}M2/ `2+m`bBp2Hv #v ∗0 ∗n F (t) = 1[0,∞) (t), F (t) = F ∗(n−1) (t − s) dF (s) (n ≥ 1) .
U9XRV
U9XkV
[0,t]
Uh?2 `QH2 Q7 0 BM i?2 BMi2;` iBQM Qp2` [0, t] Bb K /2 T`2+Bb2 #v i?2 7QHHQrBM; 2[m HBiv, ϕ(s) dF (s) = ϕ(0)F (0) + ϕ(s) dF (s).) [0,t]
q`BiBM; i?2 `2M2r H 7mM+iBQM b
(0,t]
RR3
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a E[N ([0, t])] = E[1 +
1{Tn ≤t} ] = 1 +
n≥1
r2 Q#i BM i?2 2tT`2bbBQM, R(t) =
P (Tn ≤ t) ,
n≥1
∞
F ∗n (t) .
U9XjV
n=0
h?2Q`2K 9XRX9 P (S1 < ∞) < 1
⇐⇒
P (N ([0, ∞)) < ∞) = 1
⇐⇒
E[N ([0, ∞))] < ∞ .
S`QQ7X Ai bm{+2b iQ T`Qp2 i?2 i?2Q`2K BM i?2 mM/2H v2/ + b2X 6Q` HH k ≥ 1P (N ([0, ∞)) = k) = P (S1 < ∞, . . . , Sk−1 < ∞, Sk = ∞) = P (S1 < ∞) . . . P (Sk−1 < ∞)P (Sk = ∞) = F (∞)k−1 (1 − F (∞)), M/ P (N ([0, ∞)) < ∞) =
∞
F (∞)k−1 (1 − F (∞)) .
k=1
AM T `iB+mH `- P (N ([0, ∞)) < ∞) = 1 B7 F (∞) < 1 M/ P (N ([0, ∞) < ∞) = 0 B7 F (∞) = 1X HbQ- B7 F (∞) < 1E[N ([0, ∞))] =
∞
kF (∞)k−1 (1 − F (∞)) =
k=1
1 < ∞, 1 − F (∞)
r?2`2 b B7 F (∞) = 1- E[N ([0, ∞))] = ∞X
`2M2r H T`Q+2bb U/2H v2/ Q` MQiV Bb + HH2/ `2+m``2Mi r?2M P (S1 < ∞) = 1 UF Bb T`QT2`V- M/ i` MbB2Mi r?2M P (S1 < ∞) < 1 UF Bb /272+iBp2VX h?2 7QHHQrBM; `2bmHi Bb + HH2/ i?2 2H2K2Mi `v `2M2r H i?2Q`2KX h?2Q`2K 9XRX8 q2 ? p2 lim
t→∞
N ([0, t]) 1 = t E[S1 ]
M/ lim
t→∞
S@ XbX,
E[N ([0, t])] 1 = . t E[S1 ]
U9X9V
U9X8V
S`QQ7X 6Q` i?2 T`QQ7 Q7 U9X9V b22 1t2`+Bb2 9XdXkX S`QQ7 Q7 U9X8V, h?2 i` MbB2Mi + b2 ≤ E[Nt(∞)] - bBM+2 BM i?Bb + b2 E[S1 ] = ∞ 7QHHQrb 7`QK i?2 Q#pBQmb #QmM/ E[N ([0,t])] t M/ E [N (∞)] < ∞X 6Q` i?2 `2+m``2Mi + b2- T`QQ7 Bb `2[mB`2/ Ui?2 +QM/BiBQMb Q7 i?2 /QKBM i2/ +QMp2`;2M+2 i?2Q`2K i? i rQmH/ ;m ` Mi22 i? i U9X9V BKTHB2b U9X8V `2 MQi b iBb}2/VX >Qr2p2`- #v 6 iQmǶb H2KK
9XRX _1L1qG SPALh S_P*1aa1a lim inf E t→∞
RRN
N ([0, t]) N ([0, t]) 1 ≥ E lim inf = t→∞ t t E[S1 ]
M/ i?2`27Q`2 Bi bm{+2b iQ b?Qr i? i lim supt→∞ E[ N ([0,t]) ] ≤ E[S1 1 ] X .2}M2 7Q` }MBi2 t c>0 T0 := T0 , T1 := T0 + S1 ∧ c, T2 := T1 + S2 ∧ c, . . .
r?2`2 Sn := Sn ∧ c Un ≥ 1V- M/ H2i N ([0, t]) := n≥0 1Tn ≤t X aBM+2 N ([0, t]) ≥ N ([0, t]) 7Q` HH t ≥ 0lim sup t→∞
P#b2`p2 i? i 9XdXdV-
S1
E[N ([0, t])] E[N ([0, t])] ≤ lim sup . t t t→∞
+ · · · + SN ([0,t]) ≤ t + c M/ i?2`27Q`2- #v q H/Ƕb H2KK U1t2`+Bb2
E[S1 ]E[N ([0, t])] = E[S1 + · · · + SN ([0,t]) ] ≤ t + c ,
bQ i? i lim sup t→∞
E[N ([0, t])] ≤ lim sup t t→∞
1 c 1 1 + = . E[S1 ] t E[S1 ]
h?2`27Q`2- 7Q` HH c > 0lim sup t→∞
E[N ([0, t])] 1 ≤ . t E[S1 ∧ c]
aBM+2 limc↑∞ E[S1 ∧ c] = E[S1 ]- r2 }M HHv Q#i BM i?2 /2bB`2/ BM2[m HBiv lim sup t→∞
E[N ([0, t])] 1 ≤ . t E[S1 ]
G2i F : R+ → R+ #2 ;2M2` HBx2/ +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM QM R+ i? i Bb- F (x) = c G(x) r?2`2 c > 0 M/ G Bb i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 MQM@M2; iBp2 `2 H ` M/QK p `B #H2 i? i Bb T`QT2` UG(∞) = 1VX h?2 _2M2r H 1[m iBQM h?2 # bB+ Q#D2+i Q7 `2M2r H i?2Q`v Bb i?2 `2M2r H 2[m iBQM f = g + f ∗ F, i? i Bb- #v /2}MBiBQM Q7 i?2 +QMpQHmiBQM bvK#QH ∗ f (t) = g(t) + f (t − s) dF (s)
(t ≥ 0) ,
U9XeV
[0,t]
r?2`2 g : R+ → R Bb K2 bm` #H2 7mM+iBQM + HH2/ i?2 / i X A7 F (∞) = 1 QM2 `272`b iQ i?2 `2M2r H 2[m iBQM b T`QT2` `2M2r H 2[m iBQMQ` Dmbi `2M2r H 2[m iBQMX h?2 `2M2r H 2[m iBQM Bb + HH2/ /272+iBp2 B7 F (∞) < 1 M/ 2t+2bbBp2 B7 F (∞) > 1X
Rky
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
h?2Q`2K 9XRXe h?2 `2M2r H 7mM+iBQM R b iBb}2b i?2 bQ@+ HH2/ 7mM/ K2Mi H `2@ M2r H 2[m iBQM R = 1 + R ∗ F. U9XdV S`QQ7X "v U9XjV ∗n R∗F = ∗F = F (F ∗n ∗ F ) = F ∗n = R − F ∗0 = R − 1 . n≥0
n≥0
n≥1
h?2 7QHHQrBM; bBKTH2 i2+?MB+ H `2bmHi rBHH #2 M22/2/ H i2` QMX G2KK 9XRXd 6Q` HH t ≥ b- R ([t − b, t]) ≤ (1 − F (b))−1 X S`QQ7X "v U9XdV M/ 7Q` t ≥ b F (t − s) dR(s) = (1 − F (t − s)) dR(s) 1 = R(t) − [0,t] [0,t] (1 − F (t − s)) dR(s) ≥ (1 − F (b))R ([t − b, t]) . ≥ [t−b,t]
h?2 }`bi 2t KTH2 72 im`2b i?2 # bB+ K2i?Q/ 7Q` Q#i BMBM; `2M2r H 2[m iBQMbX 1t KTH2 9XRX3, GB72iBK2 Q7 h` MbB2Mi _2M2r H S`Q+2bb- h F2 RX h?2 HB72iBK2 Q7 `2M2r H b2[m2M+2 Bb i?2 ` M/QK p `B #H2 L := sup{Tn ; Tn < ∞}. *H2 `Hv B7 i?2 `2M2r H T`Q+2bb Bb `2+m``2Mi- L Bb HKQbi bm`2Hv BM}MBi2X q2 i?2`27Q`2 +QMbB/2` i?2 i` MbB2Mi + b2- 7Q` r?B+? L Bb HKQbi bm`2Hv }MBi2X AM i?2 mM/2H v2/ + b2- i?2 7mM+iBQM f (t) = P (L > t) b iBb}2b i?2 `2M2r H 2[m iBQM f (t − s) dF (s). f (t) = F (∞) − F (t) + [0,t]
S`QQ7X .2}M2 Sn = Sn+1 (n ≥ 1) M/ H2i {Tn }n≥0 #2 i?2 bbQ+B i2/ mM/2H v2/ `2M2r H T`Q+2bb r?Qb2 HB72iBK2 Bb /2MQi2/ #v LX no more event S1 T0 = 0
S2 = Sˆ1 T1
S3 = Sˆ2
T2
T3 = L ˆ L
t
9XRX _1L1qG SPALh S_P*1aa1a
RkR
*H2 `Hv L M/ L ? p2 i?2 b K2 /Bbi`B#miBQMX HbQ 1{L>t} = 1{tt} + 1{t≥T1 } 1{L>t} . LQr- QM {t ≥ T1 }, {L > t} ≡ {L > t − T1 } M/ i?2`27Q`2 . 1{L>t} = 1{tt} + 1{t≥T1 } 1{L>t−T 1} h FBM; 2tT2+i iBQMbP (L > t) = P (L > t, t < T1 ) + P (L > t − T1 , T1 ≤ t). aBM+2 L M/ T1 `2 BM/2T2M/2Mi UL /2T2M/b QMHv QM S2 , S3 , . . .V P (L > t − s) dF (s) = P (L > t − s) dF (s) , P (L > t − T1 , T1 ≤ t) = [0,t]
[0,t]
r?2`2 r2 ? p2 mb2/ i?2 7 +i i? i L M/ L ? p2 i?2 b K2 /Bbi`B#miBQMX HbQP (L > t, t < T1 ) = P (t < T1 , T1 < ∞) = P (t < T1 < ∞) = F (∞) − F (t) .
1t KTH2 9XRXN, h?2 _BbF JQ/2H- h F2 kX h?Bb 2t KTH2 Bb +QMiBMm iBQM Q7 1t KTH2 kXRX9X q2 }M/ `2M2r H 2[m iBQM 7Q` i?2 T`Q# #BHBiv Q7 `mBM +Q``2bTQM/BM; iQ M BMBiB H + TBi H u, Ψ(u) := P (u + X(t) < 0 7Q` bQK2 t > 0) .
U9X3V
h?Bb 7mM+iBQM Bb MQM@BM+`2 bBM;X Ai Bb +QMp2MB2Mi iQ rQ`F rBi? i?2 MQM@`mBM T`Q#@ #BHBiv Φ(u) := 1 − Ψ(u)X P7 +Qm`b2- Φ(u) = 0 B7 u ≤ 0X A7 i?2 TQBMi T`Q+2bb N Bb bi iBQM `v rBi? p2` ;2 ` i2 λ- i?2 p2` ;2 T`Q}i Q7 i?2 BMbm` M+2 +QKT Mv i iBK2 t Bb E[X(t)] = (c − λμ)t . b 2tT2+i2/- BMbm` M+2 +QKT MB2b T`272` i? i c − λμ > 0 Q`- 2[mBp H2MiHv- i? i ρ :=
c − λμ c = − 1 > 0, λμ λμ
U9XNV
r?2`2 ρ Bb + HH2/ i?2 b 72iv HQ /BM;X AM 7 +i- #v i?2 bi`QM; H r Q7 H `;2 MmK#2`bB7 i?2 b 72iv HQ /BM; Bb M2; iBp2- i?2 T`Q# #BHBiv Q7 `mBM Bb 1 BM/2T2M/2MiHv Q7 i?2 BMBiB H + TBi HX h?2Q`2K 9XRXRy A7 N Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb λ u Φ(u) = Φ(0) + Φ(u − z)(1 − G(z))dz . c 0
U9XRyV
Rkk
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
S`QQ7Xj amTTQb2 u ≥ 0X aBM+2 `mBM + MMQi Q++m` i iBK2 < S1 Φ(u + cs − z)λe−λs ds dG(z) Φ(u) = Φ(u + cT1 − Z1 ) = (0,∞) (0,∞) ∞ Φ(u + cs − z) dG(z) λe−λs ds = 0 (0,u+cs] λ λu/c u Φ(x − z) dG(z) e−λx/c dx . = e c 0 (0,x] h?2 `B;?i@? M/ bB/2 Bb /Bz2`2MiB #H2- M/ /Bz2`2MiB iBQM H2 /b iQ i?2 BMi2;`Q@ /Bz2`2MiB H 2[m iBQM λ λ Φ(u − z) dG(z). U9XRRV Φ (u) = Φ(u) − c c (0,u] h?2`27Q`2Φ(t) − Φ(0) = =
λ c
λ c
t
Φ(u) du + 0
λ c
t Φ(u − z) d(1 − G(z)) du 0
(0,u]
t
Φ(u) du
t u λ + Φ (u − z)(1 − G(z)) dz du Φ(0)(1 − G(u)) − Φ(u) + c 0 0 t t t λ λ Φ (u − z) du (1 − G(z)) dz = Φ(0) (1 − G(u)) du + c c 0 0 z t λ t λ (1 − G(z)(Φ(t − z) − Φ(0)) dz , = Φ(0) (1 − G(u)) du + c c 0 0 0
i? i Bb- U9XRyVX
aBKTH2 `;mK2Mib U1t2`+Bb2 9XdXNV b?Qr i? i BM i?2 + b2 Q7 TQbBiBp2 b 72iv HQ /BM;- Φ(∞) = 1X G2iiBM; u ↑ ∞ BM U9XRyV- r2 ? p2 #v KQMQiQM2 +QMp2`;2M+2 i? i Φ(∞) = Φ(0) + λc Φ(∞) M/ i?2`27Q`2 i?2 T`Q# #BHBiv Q7 `mBM rBi? x2`Q BMBiB H + TBi H Bb λμ . Ψ(0) = c h?2`27Q`2- B7 i?2 b 72iv HQ /BM; Bb TQbBiBp2- r2 Q#i BM U9XRyVX λμ λ u + (1 − Ψ(u − z))(1 − G(z)) dz c c 0 ∞ u λ =1− μ− (1 − G(z))dz + Ψ(u − z)(1 − G(z)) dz , c u 0
1 − Ψ(u) = 1 −
j
(62HH2`- RNdR)- TX R3jX
9XRX _1L1qG SPALh S_P*1aa1a
Rkj
i? i BbΨ(u) =
λ c
∞
λ c
(1 − G(z))dz +
u
u
Ψ(u − z)(1 − G(z)) dz .
U9XRkV
0
1t KTH2 9XRXRR, h?2 GQiF ėoQHi2`` SQTmH iBQM JQ/2HX h?Bb KQ/2H 72 im`2b TQTmH iBQM Q7 rQK2MX rQK M Q7 ;2 a ;Bp2b #B`i? iQ ;B`Hb i i?2 ` i2 λ(a) Ui? i Bb- rQK M Q7 ;2 a rBHH ? p2 QM p2` ;2 λ(a) da / m;?i2`b BM M BM}MBi2bBK H iBK2 BMi2`p H (a, a + da) BM ?2` HB72iBK2VX rQK M Q7 ;2 a Bb HBp2 i iBK2 a + t rBi? T`Q# #BHBiv p(a, t)X i i?2 Q`B;BM Q7 iBK2 i?2`2 `2 f0 (a) da rQK2M Q7 ;2 #2ir22M a M/ a + daX h?2 #B`i? ` i2 f (t) i iBK2 t ≥ 0 Bb i?2 bmK Q7 i?2 #B`i? ` i2 r(t) i iBK2 t /m2 iQ rQK2M #Q`M 7i2` iBK2 0 M/ Q7 i?2 #B`i? ` i2 g(t) /m2 iQ rQK2M #Q`M #27Q`2 iBK2 0X aBM+2 rQK M Q7 ;2 a i iBK2 0 Bb a + t v2 `b QH/ i iBK2 t ∞ g(t) = f0 (a)p(a, t)λ(a + t) da . 0
qQK2M #Q`M i iBK2 t − s ≥ 0 +QMi`B#mi2 #v f (t − s)p(0, s)λ(s) iQ i?2 #B`i? ` i2 i iBK2 t M/ i?2`27Q`2 t r(t) = f (t − s)p(0, s)λ(s) ds . 0
h?2`27Q`2
t
f (t − s)p(0, s)λ(s) ds .
f (t) = g(t) + 0
h?Bb Bb `2M2r H 2[m iBQM U+ HH2/ GQiF Ƕb 2[m iBQMV rBi? / i g M/ +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM t F (t) := p(0, s)λ(s) ds. 0
LQi2 i? i
∞
p(0, s)λ(s) ds
F (∞) = 0
Bb i?2 p2` ;2 MmK#2` Q7 / m;?i2`b 7`QK ;Bp2M KQi?2`- i? i Bb- i?2 `2T`Q/m+iBQM ` i2X _2K `F 9XRXRk h?2 #Qp2 2t KTH2 +QMM2+ib `2M2r H i?2Q`v iQ i?2 i?2Q`v Q7 oQHi2`` BMi2;` H 2[m iBQMb Q7 i?2 b2+QM/ ivT2- i? i Bb- Q7 i?2 7Q`K x ϕ(x) = f (x) + λ K(x, y)ϕ(y) dy , 0
r?2`2 i?2 F2`M2H K b iBb}2b i?2 +QM/BiBQM K(x, y) ≡ 0 B7 y > x .
Rk9
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 bm+? M 2[m iBQM + M #2 T`Qp2/ mM/2` bm{+B2MiHv ;2M2` H bbmKTiBQMbX PM2 Q7 i?2K Bb i? i #Qi? K M/ f #2 HQ+ HHv L2 QM R+ × R+ M/ R+ `2bT2+iBp2HvX h?2 bQHmiBQM + M #2 2tTHB+BiHv ;Bp2M BM i?Bb + b2 b x H(x, y, λ)f (y) dy , ϕ(x) = f (x) − λ 0
r?2`2 H Bb i?2 `2bQHp2Mi F2`M2H H(x, y, λ) :=
∞
λn Kn+1 (x, y)
n=0
M/ r?2`2 i?2 Bi2` i2/ F2`M2Hb {Kn }n≥1 `2 /2}M2/ #v i?2 `2+m``2M+2 7Q`KmH x Kn+1 (x, y) = K(x, z)Kn (z, y) dz (n ≥ 1) 0
bi `iBM; rBi? K1 = KX AM i?2 + b2 Q7 BMi2`2bi BM i?Bb b2+iBQM- K Bb Q7 i?2 dz6 HimM;Ǵ ivT2- i? i Bb K(x, y) = k(x − y)- 7Q` r?B+? i?2 2H2K2Mi `v i?2Q`v Q7 BMi2;` H 2[m iBQMb ? b KQ`2 iQ b vX9 AM bQK2 + b2b- 2tTHB+Bi bQHmiBQMb + M #2 Q#i BM2/X _2M2r H i?2Q`v /2 Hb rBi? KQ`2 ;2M2` H ivT2 Q7 BMi2;` H 2[m iBQMb- M/ Bib ;Q Hb `2 /Bz2`2Mi- +2Mi2`2/ KQbiHv QM i?2 bvKTiQiB+ #2? pBQm` Q7 i?2B` bQHmiBQMbX 1t KTH2 9XRXRj, h?2 1H2K2Mi `v _2M2r H h?2Q`2K, *Q``2+iBQM h2`K- h F2 RX "v i?2 2H2K2Mi `v `2M2r H i?2Q`2K- limt↑∞ R(t) = E[S1 1 ] X AM pB2r t Q7 Q#i BMBM; }M2` bvKTiQiB+ `2bmHib Ub22 1t KTH2 9XkXR9 #2HQrV- r2 b? HH bim/v i?2 7mM+iBQM t f (t) := R(t) − · E[S1 ] AM i?2 + b2 r?2`2 E[S1 ] < ∞- f b iBb}2b i?2 `2M2r H 2[m iBQM rBi? / i ∞ 1 (1 − F (x))dx , g(t) := E[S1 ] t b r2 MQr T`Q+22/ iQ b?QrX G2i E[S1 ] := mX "v U9XdV 1 (f ∗ F )(t) = (R ∗ F )(t) − (t − s) dF (s) m [0,t] 1 (t − s) dF (s) = R(t) − 1 − m [0,t] t 1 t − 1− + (t − s) dF (s) = R(t) − m m m [0,t] ∞ 1 (1 − F (x))dx , = f (t) − m t r?2`2 i?2 H bi 2[m HBiv Bb Q#i BM2/ #v i?2 7QHHQrBM; +QKTmi iBQMbX AMi2;` iBQM #v T `ib Uh?2Q`2K RXeXjV ;Bp2b 9
a22 7Q` BMbi M+2 (h`B+QKB- RN8d- RN38)- RXNX
9XRX _1L1qG SPALh S_P*1aa1a
Rk8
t(1 − F (t)) =
(1 − F (s)) ds − [0,t]
s dF (s) , [0,t]
∞ M/ i?2`27Q`2- bBM+2 m = 0 (1 − F (s)) ds 1 ∞ 1 ∞ 1 t (1 − F (s)) ds = (1 − F (s)) ds − (1 − F (s)) ds m t m 0 m 0 1 t 1 =1− t(1 − F (t)) + (1 − F (s)) ds = 1 − s dF (s) m 0 m [0,t] 1 t 1 t+ + (s − t) dF (s) = 1 − (t − s) dF (s) . =1− m m m [0,t] [0,t]
aQHmiBQM Q7 i?2 _2M2r H 1[m iBQM M 2tT`2bbBQM Q7 i?2 bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM BM i2`Kb Q7 i?2 `2M2r H 7mM+iBQM Bb 2 bv iQ Q#i BMX _2+ HH i?2 7QHHQrBM; /2}MBiBQM, 7mM+iBQM g : R+ → R Bb + HH2/ HQ+ HHv #QmM/2/ B7 7Q` HH a ≥ 0- supt∈[0,a] |g(t)| < ∞X h?2Q`2K 9XRXR9 A7 F (∞) ≤ 1 M/ B7 i?2 / i 7mM+iBQM g : R+ → R Bb HQ+ HHv #QmM/2/- i?2 `2M2r H 2[m iBQM U9XeV /KBib mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM f : R+ → R ;Bp2M #v f = g ∗ R- i? i Bb f (t) = g(t − s)dR(s) . U9XRjV [0,t]
S`QQ7X h?2 7mM+iBQM f = g ∗ R Bb BM/22/ HQ+ HHv #QmM/2/ bBM+2 g Bb HQ+ HHv #QmM/2/ M/ R(t) Bb }MBi2 7Q` HH tX HbQ f ∗ F = (g ∗ R) ∗ F = g ∗ (R ∗ F ) = g ∗ (R − 1) = g ∗ R − g = f − g . h?2`27Q`2 f Bb BM/22/ bQHmiBQM Q7 i?2 `2M2r H 2[m iBQMX G2i f1 #2 MQi?2` HQ+ HHv #QmM/2/ bQHmiBQM M/ H2i h := f − f1 X h?Bb Bb HQ+ HHv #QmM/2/ bQHmiBQM r?B+? b iBb}2b h = h ∗ F X "v Bi2` iBQMh = h ∗ F ∗n . h?2`27Q`2- 7Q` HH t ≥ 0-
|h(t)| ≤
sup |h(s)| F ∗n (t). s∈[0,t]
aBM+2 R(t) = |h(t)| ≡ 0X
n≥0
F ∗n (t) < ∞- r2 ? p2 limn→∞ F ∗n (t) = 0- r?B+? BKTHB2b i? i
h?2 }`bi bvKTiQiB+ `2bmHi QM i?2 bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM +QM+2`Mb i?2 /272+iBp2 + b2,
Rke
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
h?2Q`2K 9XRXR8 A7 F Bb /272+iBp2 M/ B7 i?2 / i 7mM+iBQM g : R+ → R Bb #QmM/2/ M/ ? b HBKBi g(∞) := limt→∞ g(t)- i?2 mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM b iBb}2b lim f (t) =
t→∞
g(∞) . 1 − F (∞)
S`QQ7X 6`QK T`2pBQmb +QKTmi iBQMb- r2 ? p2 E[N ([0, ∞))] = R(∞) = M/ i?2`27Q`2
g(∞) = 1 − F (∞)
1 , 1 − F (∞)
g(∞) dR(s). [0,∞)
HbQ
g(t − s) dR(s) ,
f (t) = [0,t]
M/ i?2`27Q`2 g(∞) = f (t) − 1 − F (∞)
(g(t − s)1{s≤t} − g(∞)) dR(s). [0,∞)
h?2 H ii2` BMi2;` M/ Bb #QmM/2/ BM #bQHmi2 p Hm2 #v 2 × sup |g(t)|- }MBi2 +QM@ bi MiX +QMbi Mi 7mM+iBQM Bb BMi2;` #H2 rBi? `2bT2+i iQ i?2 `2M2r H K2 bm`2 #2@ + mb2- BM i?2 /272+iBp2 + b2- i?2 iQi H K bb Q7 i?2 `2M2r H K2 bm`2 Bb R(∞) = E [N ([0, ∞))] < ∞X LQr 7Q` }t2/ s ≥ 0- limt→∞ (g(t − s)1{s≤t} − g(∞)) = 0X h?2`27Q`2- #v /QKBM i2/ +QMp2`;2M+2- i?2 BMi2;` H +QMp2`;2b iQ 0 b t → ∞X ai iBQM `v _2M2r H S`Q+2bb2b "v T`QT2` +?QB+2 Q7 i?2 BMBiB H /2H v- `2M2r H T`Q+2bb + M #2 K /2 bi iBQM `vBM b2Mb2 iQ #2 K /2 T`2+Bb2X *QMbB/2` `2M2r H T`Q+2bb T0 = S0 , T1 = S0 + S1 , . . . , Tn = S0 + · · · + Sn r?2`2 0 ≤ S0 < ∞X G2i G #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 BMBiB H /2H v S0 := T0 M/ bmTTQb2 i? i P (S1 < ∞) = 1 Ui?2 `2M2r H T`Q+2bb Bb T`QT2`VX b mbm H- 2t+Hm/2 i`BpB HBiB2b #v BKTQbBM; i?2 +QM/BiBQM P (S1 = 0) < 1X 6Q` t ≥ 0/2}M2 S0 (t) := TN ([0,t]) − t M/ Sn (t) := TN ([0,t])+n − TN ([0,t])+n−1 (n ≥ 1) .
U9XR9V
AM T `iB+mH `- Sn (0) = Sn 7Q` HH n ≥ 0X HbQ Q#b2`p2 i? i S0 (t) = A(t)- i?2 7Q`r `/ `2+m``2M+2 iBK2 i tX t S0
S1
S2
S3 S0 (t)
S4
S5
S1 (t)
S2 (t)
9XRX _1L1qG SPALh S_P*1aa1a
Rkd
.2}MBiBQM 9XRXRe h?2 /2H v2/ `2M2r H T`Q+2bb Bb + HH2/ bi iBQM `v B7 i?2 /Bbi`B@ #miBQM Q7 i?2 b2[m2M+2 S0 (t) , S1 (t) , S2 (t) , . . . Bb BM/2T2M/2Mi Q7 iBK2 t ≥ 0X Ai im`Mb Qmi i? i S0 (t) Bb BM/2T2M/2Mi Q7 {Sn (t)}n≥1 M/ i? i i?2 H ii2` b2[m2M+2 ? b i?2 b K2 /Bbi`B#miBQM b {Sn }n≥1 U1t2`+Bb2 9XdXRRVX h?2`27Q`2, G2KK 9XRXRd 6Q` /2H v2/ `2M2r H T`Q+2bb iQ #2 bi iBQM `v Bi Bb M2+2bb `v M/ bm{+B2Mi i? i 7Q` HH t ≥ 0- i?2 /Bbi`B#miBQM Q7 S0 (t) = A(t) #2 i?2 b K2 b i? i Q7 S0 X G2KK 9XRXR3 A7 i?2 /2H v2/ `2M2r H T`Q+2bb Bb bi iBQM `v- i?2M M2+2bb `BHv E[S1 ] < ∞ M/ E[N ([0, t])] = E[St 1 ] X S`QQ7X h?2 K2 bm`2 M QM R+ /2}M2/ #v M (C) := E[N (C)] Bb i` MbH iBQM@ BMp `B Mi M/ i?2`27Q`2 KmHiBTH2 Q7 i?2 G2#2b;m2 K2 bm`2 Uh?2Q`2K XRX3V- i? i Bb- M (C) = K (C) 7Q` bQK2 +QMbi Mi K- r?B+? Bb }MBi2 UM Bb HQ+ HHv }MBi2V M/ TQbBiBp2 Ui?2 `2M2r H T`Q+2bb Bb MQi 2KTivVX "v i?2 2H2K2Mi `v `2M2r H i?2Q`2KK = lim t↑∞
M/ i?2`27Q`2
1 E[S1 ]
E[N ((0, t])] 1 = , t E[S1 ]
> 0X
G2KK 9XRXRN A7 i?2 /2H v2/ `2M2r H T`Q+2bb Bb bi iBQM `v- i?2M M2+2bb `BHv E[S1 ] < ∞ M/ i?2 /Bbi`B#miBQM Q7 i?2 BMBiB H /2H v T0 Bb x 1 F0 (x) := (1 − F (y)) dy , U9XR8V E[S1 ] 0 + HH2/ i?2 bi iBQM `v 7Q`r `/ `2+m``2M+2 iBK2 /Bbi`B#miBQMX S`QQ7X h?2 }MBi2M2bb Q7 E[S1 ] r b T`Qp2/ BM i?2 T`2pBQmb H2KK X 6Q` HH u ∈ R HH t ≥ 0- i?2 7QHHQrBM; 2[m iBQM t d iuA(s) iuA(t) iuA(0) iuA(Tn ) iuA(Tn −) e e 1{Tn ≤t} + =e + −e ds e ds 0 n≥0 Bb Q#i BM2/ #v HQQFBM; i r? i ? TT2Mb i i?2 2p2Mi iBK2b M/ #2ir22M i?2 2p2Mi d iuA(s) iBK2b Ub22 1[MX RXk9VX P#b2`pBM; i? i ds e = −iueiuA(s) - A(0) = S0 - A(Tn −) = 0- A(Tn ) = Sn+1 - r2 i?2`27Q`2 ? p2 t eiuA(t) = eiuS0 + eiuSn+1 − 1 1{Tn ≤t} − iu eiuA(s) ds . n≥0
0
h?2`27Q`2- i FBM; BMiQ ++QmMi i?2 BM/2T2M/2M+2 Q7 Sn+1 M/ Tn -
Rk3
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a E eiuA(t) = E eiuS0 + E eiuS1 − 1 E[N ((0, t])] − iu
t
E eiuA(s) ds .
0
"v i?2 bbmK2/ bi iBQM `Biv- E[N ((0, t])] = E[St 1 ] M/ E eiuA(t) = E eiuS0 - M/ i?2`27Q`2 E eiuS1 − 1 − iuE eiuS0 = 0 , E[S1 ] i? i Bb-
E eiuS1 − 1 . E eiuS0 = iu E[S1 ]
"mi i?2 `B;?i@? M/ bB/2 Bb i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 F0 - b i?2 7QHHQrBM; +QK@ Tmi iBQM b?Qrb, ∞ ∞ 1 1 eiux (1 − F (x)) dx = eiux P (S1 > x) dx E[S1 ] 0 E[S1 ] 0 ∞ 1 eiux E 1{S1 >x} dx = E[S1 ] 0 ∞
1 iux = E e 1{S1 >x} dx E[S1 ] 0 S1 iuS1
e −1 1 1 = E E . eiux dx = E[S1 ] E[S1 ] iu 0 h?2Q`2K 9XRXky 6Q` /2H v2/ `2M2r H T`Q+2bb iQ #2 bi iBQM `v- Bi Bb M2+2bb `v M/ bm{+B2Mi i? i E[S1 ] < ∞ M/ i? i P (T0 ≤ x) = F0 (x) , r?2`2 F0 Bb i?2 bi iBQM `v 7Q`r `/ `2+m``2M+2 iBK2 /Bbi`B#miBQM U9XR8VX S`QQ7X h?2 T`QQ7 Q7 M2+2bbBiv Bb +QMi BM2/ BM G2KK b 9XRXR3 M/ 9XRXRNX 6Q` bm7@ }+B2M+v- r2 }`bi b?Qr i? i RF0 (t) := E[N ([0, t]))] =
t E[S1 ]
Ui?2 MQi iBQM 2KT? bBx2b i?2 `QH2 Q7 i?2 BMBiB H /2H v rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F0 VX q2 ? p2 RF0 (t) = E 1{Tn ≤t} = P (Tn ≤ t) = F0 (t)+(F0 ∗F )(t)+(F0 ∗F ∗2 )(t)+· · · , n≥0
n≥0
i? i Bb- RF0 = F0 + RF0 ∗ F X h?2`27Q`2- #v h?2Q`2K 9XRXR9- RF0 Bb i?2 mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM f = F0 + f ∗ F X Ai i?2M bm{+2b iQ b?Qr i? i f (t) = E[St 1 ] Bb BM/22/ bQHmiBQMX hQ p2`B7v i?Bb- Q#b2`p2 i? i 7Q` bm+? f-
9XkX h>1 _1L1qG h>1P_1J 1 (f ∗ F )(t) = E[S1 ] M/ i?2`27Q`2
RkN (t − s) dF (s) [0,t]
t 1 1 − t dF (s) + s dF (s) m E[S1 ] [0,t] E[S1 ] [0,t] 1 t (1 − F (t)) + s dF (s) . = E[S1 ] E[S1 ] [0,t]
f (t) − (f ∗ F )(t) =
Ai `2K BMb iQ b?Qr i? i i?2 `B;?i@? M/ bB/2 Q7 i?2 #Qp2 2[m HBiv Bb F0 (t)X AMi2;` @ iBQM #v T `ib /Q2b Bi, t 1 1 t(1 − F (t)) + (1 − F (s)) ds = s dF (s) . F0 (t) = E[S1 ] 0 E[S1 ] [0,t] > pBM; T`Qp2/ i? i E[N ([0, t]))] = E[St 1 ] - r2 `2 HKQbi /QM2X 6`QK +QKTmi @ iBQMb BM i?2 T`QQ7 Q7 G2KK 9XRXRN- r2 2ti` +i i?2 B/2MiBiv t t E eiuA(s) ds . − iu E eiuA(t) = E eiuS0 + E eiuS1 − 1 E[S1 ] 0 iuA(t) h?2 7mM+iBQM z(t) := E e Bb i?2`27Q`2 bQHmiBQM Q7 i?2 Q`/BM `v /Bz2`2MiB H 2[m iBQM dz 1 = −iuz + E eiuS1 − 1 dt E[S1 ] rBi? BMBiB H +QM/BiBQM z(0) = E eiuS0 = E[S1 1 ] E eiuS1 − 1 - r?Qb2 mMB[m2 bQHmiBQM Bb E[S1 ] iuS1 E eiuA(t) = E eiuS0 = E e −1 . m h?2`27Q`2- 7Q` HH t ≥ 0- S0 (t) U= A(t)V ? b i?2 b K2 /Bbi`B#miBQM b S0 X h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK G2KK 9XRXRdX
9Xk
h?2 _2M2r H h?2Q`2K
_2M2r H i?2Q`v /2 Hb K BMHv rBi? i?2 HBKBiBM; #2? pBQm` Q7 i?2 bQHmiBQM Q7 i?2 `2M2r H 2[m iBQMX h?2 i?2Q`v Bb ` i?2` bBKTH2 r?2M i?2 `2M2r H T`Q+2bb Bb i` MbB2Mi Uh?2Q`2K 9XRXR8VX Ai #2+QK2b KQ`2 BMpQHp2/ BM i?2 `2+m``2Mi + b2X p2`v bBKTH2 2t KTH2 rBHH ;Bp2 i?2 ~ pQm` Q7 bm+? `2bmHiX 1t KTH2 9XkXR, h?2 _2M2r H h?2Q`2K 7Q` SQBbbQM S`Q+2bbX A7 {Tn }n≥1 Bb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ > 0- r2 FMQr Uh?2Q`2K 9XRXR8V i? i R(t) = 1 + λtX h?2`27Q`2- i?2 bQHmiBQM Q7 i?2 +Q``2bTQM/BM; `2M2r H 2[m iBQM Bbr?2M i?2 / i g Bb MQM@M2; iBp2 t t t f (t) = g(t − s) R(ds) = λ g(t − s)ds = λ g(s)ds. 0
0
0
Rjy
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
>2`2 λ = 1/E[S1 ]X A7 r2 bmTTQb2 i? i g Bb BMi2;` #H2- i?2M ∞ lim f (t) =
t→∞
0
g(s) ds . E[S1 ]
()
.2}MBiBQM 9XkXk G2i F #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 MQM@M2; iBp2 `2 H ` M/QK p `B #H2 S1 X "Qi? F M/ S1 `2 + HH2/ MQM@H iiB+2 B7 i?2`2 Bb MQ bi`B+iHv TQbBiBp2 `2 H a bm+? i? i k≥0 P (S1 = ka) = 1X h?2 E2v _2M2r H h?2Q`2K Ai im`Mb Qmi i? i 7Q` MQM@H iiB+2 /Bbi`B#miBQMb UV Bb [mBi2 ;2M2` H- KQ/mHQ KBH/ i2+?MB+ H bbmKTiBQM QM i?2 / i g, i?Bb 7mM+iBQM ? b iQ #2 /B`2+iHv _B2K MM BMi2;` #H2X h?Bb Bb i?2 bBim iBQM +QMbB/2`2/ BM i?2 F2v `2M2r H i?2Q`2K Uh?2Q`2K 9XkXRj #2HQrVX q2 T`Q+22/ iQ ;Bp2 i?2 # bB+ 7 +ib +QM+2`MBM; /B`2+i _B2K MM BMi2;` #BHBivX G2i g : R+ → R #2 MQMM2; iBp2 HQ+ HHv #QmM/2/ 7mM+iBQMX .2}M2 7Q` 2 +? b > 0 M/ 2 +? t ≥ 0
g b (t) = sup{g(s); nb ≤ s < (n + 1)b} QM [nb, (n + 1)b) g b (t) = inf{g(s); nb ≤ s < (n + 1)b} QM [nb, (n + 1)b) .
h?2 7mM+iBQMb g b M/ g b `2 }MBi2 +QMbi Mib QM i?2 BMi2`p Hb [nb, (n + 1)b)- 7Q` HH n ∈ N- M/ i?mb G2#2b;m2 BMi2;` #H28 QM #QmM/2/ BMi2`p HbX .2}MBiBQM 9XkXj h?2 7mM+iBQM g ≥ 0 Bb b B/ iQ #2 _B2K MM BMi2;` #H2 U_BV a QM i?2 #QmM/2/ BMi2`p H [0, a] B7 7Q` bQK2 U M/ i?2M 7Q` HHV b > 0- i?2 BMi2;` H 0 g b (t)dt Bb }MBi2- M/ a a lim g b (t)dt − g b (t)dt = 0. U9XReV b↓0
0
0
Uh?2 7 +i i? i- r?2M g b Bb BMi2;` #H2 7Q` bQK2 b > 0- i?2M g b Bb HbQ BMi2;` #H2 7Q` HH b > 0- 7QHHQrb 7`QK i?2 BM2[m HBiv g b (t) ≤
+n
g b (t + kb),
k=−n
r?2`2 n = b /b XV 8 h?2 i?2Q`v Q7 i?2 _B2K MM BMi2;` H T`2/ i2b i? i Q7 i?2 G2#2b;m2 BMi2;` HX q2 /Q MQi 7QHHQr i?2 ?BbiQ`B+ H /2p2HQTK2Mi BM i?2 T`2b2Mi i`2 iK2Mi Q7 _B2K MM BMi2;` Hb- M/ mb2 G2#2b;m2 BMi2;` iBQM i?2Q`v Ĝ BM T `iB+mH ` i?2 TQr2`7mH G2#2b;m2Ƕb /QKBM i2/ +QMp2`;2M+2 i?2Q`2KĜ r?B+? HHQrb +QMbB/2` #Hv bBKTH2` `;mK2MibX
9XkX h>1 _1L1qG h>1P_1J
RjR
h?2Q`2K 9XkX9 G2i g #2 _B2K MM BMi2;` #H2 7mM+iBQM QM i?2 #QmM/2/ BMi2`p H [0, a]X h?2M, UBV g Bb #QmM/2/ M/ HKQbi 2p2`vr?2`2 +QMiBMmQmb QM [0, a] M/ UBBV i?2 HBKBi
a
g b (t)dt
lim b↓0
0
2tBbib M/ Bb }MBi2X h?Bb HBKBi Bb #v /2}MBiBQM i?2 _@BMi2;` H U_B2K MM BMi2@ ;` HV Q7 g QM [0, a]X Ai Bb /2MQi2/ #v a a R@ g(t) dt = lim g b (t)dt. b↓0
0
0
Ai +QBM+B/2b rBi? i?2 G2#2b;m2 BMi2;` H Q7 g QM [0, a]X S`QQ7X UBV "QmM/2/M2bb Q7 g Bb +H2 ` bBM+2 supx∈[0,a] g(x) ≤ b−1 }MBi2 #v bbmKTiBQMX
a 0
g b (t)dt- r?B+? Bb
q2 MQr b?Qr i? i i?2 b2i Q7 /Bb+QMiBMmBiv TQBMib Q7 g ? b MmHH G2#2b;m2 K2 bm`2X G2i g(x) = lim supy→x g(y)- M/ g(x) = lim inf y→x g(y)X "Qi? 7mM+iBQMb `2 K2 bm` #H2X AM 7 +i- KQ`2 Bb i`m2, g Bb mTT2` b2KB@+QMiBMmQmb- i? i Bb- 7Q` HH A ∈ R+ - i?2 b2i {x : g(x) ≥ A} Bb +HQb2/- r?BH2 g Bb HQr2` b2KB@+QMiBMmQmb- i? i Bb7Q` HH A ∈ R+ - i?2 b2i {x : g(x) ≤ A} Bb +HQb2/X q2 QKBi i?2 U2 bvV T`QQ7 Q7 i?2b2 7 +ibX h?2 b2i Q7 /Bb+QMiBMmBiv TQBMib Q7 g QM [0, a]- i? i Bb- {x : g(x) > g(x)}- Bb i?2`27Q`2 K2 bm` #H2X amTTQb2 Bi Bb Q7 TQbBiBp2 G2#2b;m2 K2 bm`2X h?2M- i?2`2 2tBbib M > 0 bm+? i? i i?2 b2i {x : g(x) − g(x) > } Bb HbQ Q7 TQbBiBp2 G2#2b;m2 K2 bm`2- b v δX aBM+2 7Q` HH b > 0- M/ bBM+2 7Q` HKQbi 2p2`v t ∈ [0, a] g b (t) ≥ g(t) ≥ g(t) ≥ g b (t), Bi 7QHHQrb i? i 7Q` HH b > 0 a
g b (t)dt −
0
0
a
g b (t)dt ≥ δ > 0,
r?B+? +QMi` /B+ib i?2 bbmKTiBQM Q7 _B2K MM BMi2;` #BHBiv U9XReVX UBBV i +QMiBMmBiv TQBMi x Q7 g- Bi ?QH/b i? i lim g b (x) = g(x) = g(x). b↓0
"v UBV- i?Bb +QMp2`;2M+2 ?QH/b HKQbi 2p2`vr?2`2X "v /QKBM i2/ +QMp2`;2M+2 UrBi? /QKBM iBM; 7mM+iBQM g 1 (x) + g 1 (x − 1) + g 1 (x + 1)V a a lim g b (t)dt = g(t)dt . b↓0
0
0
Rjk
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
.2}MBiBQM 9XkX8 h?2 7mM+iBQM g ≥ 0 Bb b B/ iQ #2 _B2K MM BMi2;` #H2 QM [0, ∞) B7 a lim R@ g(t) dt a↑∞
0
2tBbib M/ i?2 HBKBi Bb i?2M- #v /2}MBiBQM- Bib _B2K MM BMi2;` H QM [0, ∞)X _2K `F 9XkXe K DQ` `2bmHi Q7 _B2K MMǶb BMi2;` iBQM i?2Q`v Bb i? i i?2 _B2@ K MM BMi2;` H QM i?2 }MBi2 BMi2`p H [0, a] Q7 7mM+iBQM 2tBbib B7 M/ QMHv B7 i?Bb 7mM+iBQM Bb HKQbi 2p2`vr?2`2 +QMiBMmQmb M/ #QmM/2/ QM i?Bb BMi2`p H- i? i BbS`QT2`iv UBV BM i?2 #Qp2 T`QTQbBiBQM Bb MQi QMHv M2+2bb `v- #mi HbQ bm{+B2Mi 7Q` _B2K MM@BMi2;` #BHBivX h?Bb `2bmHi Bb MQi M22/2/ BM i?Bb #QQFX q2 MQr im`M iQ i?2 /2}MBiBQM Q7 i?2 /B`2+i _B2K MM BMi2;` HX dz.B`2+iǴ K2 Mb i? i i?Bb BMi2;` H QM [0, ∞) Bb MQi /2}M2/ b HBKBi Q7 BMi2;` Hb Qp2` }MBi2 BMi2`p Hb#mi dz/B`2+iHvǴ QM [0, ∞)X .2}MBiBQM ∞ 9XkXd h?2 7mM+iBQM g ≥ 0 Bb b B/ iQ #2 /B`2+iHv _B2K MM BMi2;` #H2 U/_BV B7 0 g b (t)dt < ∞ 7Q` bQK2 U M/ i?2M 7Q` HHV b > 0- M/ B7
∞
lim b↓0
0
∞
g b (t)dt − 0
g b (t)dt
= 0.
6`QK i?2 /2}MBiBQMb- Bi Bb +H2 ` i? i 7Q` 7mM+iBQMb p MBb?BM; QmibB/2 #QmM/2/ BMi2`p H- i?2 MQiBQMb Q7 _B2K MM BMi2;` #BHBiv M/ Q7 /B`2+i _B2K MM BMi2;` #BH@ Biv `2 i?2 b K2X HbQ- i?2 7QHHQrBM; M HQ;m2 Q7 h?2Q`2K 9XkX9 ?QH/b 7Q` /B`2+i _B2K MM BMi2;` #BHBiv, h?2Q`2K 9XkX3 G2i g ≥ 0 #2 /_BX h?2M, UBV g Bb #QmM/2/- M/ HKQbi 2p2`vr?2`2 +QMiBMmQmb QM R+ X UBBV h?2 HBKBi
∞
g b (t)dt
lim b↓0
0
2tBbib M/ Bb }MBi2X h?Bb HBKBi Bb- #v /2}MBiBQM- i?2 dR@BMi2;` H U/B`2+i _B2@ K MM BMi2;` HV Q7 g QM R+ - M/ Bb /2MQi2/ #v ∞ ∞ dR@ g(t) dt = lim g b (t)dt. 0
b↓0
0
Ai +QBM+B/2b rBi? i?2 G2#2b;m2 BMi2;` H Q7 g QM R+ X h?2 T`QQ7 Bb B/2MiB+ H iQ i? i Q7 h?2Q`2K 9XkX9X h?2 7QHHQrBM; 2t KTH2 72 im`2b 7mM+iBQM i? i Bb _B2K MM BMi2;` #H2- #mi MQi /B`2+iHv _B2K MM BMi2;` #H2X
9XkX h>1 _1L1qG h>1P_1J
Rjj
1t KTH2 9XkXN, *QmMi2`2t KTH2X G2i {an }n≥1 M/ {b n }n≥1 #2 b2[m2M+2b Q7 TQbBiBp2 `2 H MmK#2`b bm+? i? i 1/2 > a > a > · · · 1 2 n≥1 bn = ∞ M/
a b < ∞X G2i g #2 MmHH QmibB/2 i?2 mMBQM Q7 i?2 BMi2`p Hb [n − an , n + an ]n n n≥1 n ≥ 1- M/ bm+? i? i 7Q` HH n ≥ 1- g(n − an ) = g(n + an ) = 0 M/ g(n) = bn - M/ g Bb HBM2 ` BM i?2 BMi2`p Hb [n − an , n] M/ [n, n + an ]X h?2M- g Bb _B2K MM BMi2;` #H2, ∞ g(t) dt = an b n < ∞ . R@ 0
n≥1
Ai Bb ?Qr2p2` MQi /B`2+iHv _B2K MM BMi2;` #H2 bBM+2
∞ 0
g¯b (t)dt = ∞ 7Q` HH b > 0X
h?2`2 2tBbi ?Qr2p2` 72r `2 bbm`BM; `2bmHib, h?2Q`2K 9XkXRy U V A7 g Bb /B`2+iHv _B2K MM BMi2;` #H2- Bi Bb _B2K MM BMi2;` #H2 QM [0, ∞) M/ ∞ ∞ R@ g(t)dt = dR@ g(t)dt . 0
0
U#V LQM@M2; iBp2 MQM@BM+`2 bBM; 7mM+iBQMb `2 /B`2+iHv _B2K MM BMi2;` #H2 B7 M/ QMHv B7 i?2v `2 _B2K MM BMi2;` #H2 QM [0, ∞)X U+V MQM@M2; iBp2 ∞ 7mM+iBQM i? i Bb _B2K MM BMi2;` #H2 QM HH }MBi2 BMi2`p Hb- M/ bm+? i? i 0 g¯1 (t)dt < ∞- Bb /B`2+iHv _B2K MM BMi2;` #H2X AM T `iB+mH `- ∞ MQM@M2; iBp2 HKQbi 2p2`vr?2`2 +QMiBMmQmb 7mM+iBQM bm+? i? i 0 g¯1 (t)dt < ∞ Bb /B`2+iHv _B2K MM BMi2;` #H2X U/V MQM@M2; iBp2 7mM+iBQM i? i Bb _B2K MM BMi2;` #H2 M/ #QmM/2/ #Qp2 #v /B`2+iHv _B2K MM@BMi2;` #H2 7mM+iBQM Bb /B`2+iHv _B2K MM BMi2;` #H2X S`QQ7X U V aBM+2 g Bb /B`2+iHv _B2K MM BMi2;` #H2 ∞ a 0 = lim (¯ gb (t) − g b (t))dt ≥ lim (¯ gb (t) − g b (t))dt, b↓0
b↓0
0
0
BKTHvBM; _B2K MM@BMi2;` #BHBiv QM [0, a]X 6Q` HH a > 0- `2+ HHBM; i? i g Bb UG2#2b;m2V BMi2;` #H2 QM [0, +∞)& & ∞ & ∞ & a a ∞ & & & & &=& &= &dR− g(t)dt − _@ g(t)dt g(t)dt − g(t)dt g(t)dt. & & & & 0
0
0
0
a
h?2 `B;?i@? M/ bB/2 i2M/b iQ x2`Q b a → ∞ #v /QKBM i2/ +QMp2`;2M+2X U#V h?2 M2+2bbBiv 7QHHQrb 7`QK U VX AM pB2r Q7 T`QpBM; bm{+B2M+v- bmTTQb2 i? i i?2 MQM@M2; iBp2 MQM@BM+`2 bBM; 7mM+iBQM g Bb _B2K MM BMi2;` #H2 QM [0, +∞)X Ai Bb BM T `iB+mH ` UG2#2b;m2V BMi2;` #H2 QM [0, a] 7Q` HH }MBi2 a > 0- M/ i?2 BMi2;` H
Rj9
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
a
g(t)dt /KBib }MBi2 HBKBi b a → ∞X h?2`27Q`2- g Bb UG2#2b;m2V BMi2;` #H2 QM 0 R+ - #v KQMQiQM2 +QMp2`;2M+2X aBM+2 Bi Bb MQM@BM+`2 bBM;- 7Q` HH b > 0 ∞ ∞ g¯b (t)dt = bg(nb) ≤ bg(0) + g(t)dt < ∞ . 0
6m`i?2`KQ`2 ∞
0
n≥0
g¯b (t)dt −
0
0
∞
g b (t)dt =
bg(nb) −
bg(nb) = bg(0).
n>0
n≥0
h?2 H ii2` i2`K p MBb?2b b b → 0- 2bi #HBb?BM; i? i g Bb /B`2+iHv _B2K MM BMi2@ ;` #H2X ∞ U+V 6Bt ε > 0 M/ b2H2+i a > 0 bm+? i? i a g¯1 (t)dt ≤ εX 6Q` HH b ∈ (0, 1]g b (t) ≤ g¯b (t) ≤ g¯1 (t − 1) + g¯1 (t) + g¯1 (t + 1). Ai 7QHHQrb i? i ∞ g¯b (t)dt − 0
∞ 0
g b (t)dt ≤
a+1
0
≤ 0
a+1
g¯b (t)dt − g b (t) dt + 3
∞
g¯1 (t)dt a
g¯b (t)dt − g b (t) dt + 3ε .
b g Bb bbmK2/ _B2K MM BMi2;` #H2 QM [0, a + 1]- i?2 `B;?iKQbi BMi2;` H i2M/b iQ x2`Q b b → 0X aBM+2 ε > 0 Bb `#Bi` `v- r2 +QM+Hm/2 i? i i?2 H27i@? M/ bB/2 ;Q2b iQ x2`Q b b → 0X >2M+2- g Bb /B`2+iHv _B2K MM BMi2;` #H2X U/V 6QHHQrb 7`QK U+V bBM+2 g Bb _B2K MM BMi2;` #H2 QM }MBi2 BMi2`p Hb- M/ + HHBM; z i?2 #QmM/BM; 7mM+iBQM- r2 ? p2- bBM+2 g¯1 ≤ z¯1 ∞ ∞ g¯1 (t) dt ≤ z¯1 (t) dt, 0
0
}MBi2 [m MiBiv #2+ mb2 z Bb /B`2+iHv _B2K MM BMi2;` #H2 #v bbmKTiBQMX
h?2 7QHHQrBM; Bb 7`2[m2MiHv 2M+QmMi2`2/ 2t KTH2 Q7 /B`2+iHv _B2K MM@ BMi2;` #H2 7mM+iBQM, 1t KTH2 9XkXRR, h BH /Bbi`B#miBQM Q7 M BMi2;` #H2 p `B #H2X G2i F #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 M BMi2;` #H2 MQM@M2; iBp2 ` M/QK p `B #H2 S1 X h?2M 1 − F Bb /B`2+iHv _B2K MM BMi2;` #H2X h?Bb 7QHHQrb 7`QK U#V Q7 h?2Q`2K ∞ 9XkXRy M/ 0 (1 − F (t)) dt = E[S1 ] < ∞X _2K `F 9XkXRk 1t KTH2 9X9X8 #2HQr 72 im`2b bT2+i +mH ` 2t KTH2 b?QrBM; i? i i?2 /B`2+i BMi2;` #BHBiv +QM/BiBQM + MMQi #2 /BbT2Mb2/ rBi? BM ;2M2` HX h?2Q@ `2K 9XkXRy #Qp2 M/ h?2Q`2K 9X9Xe ;Bp2 T` +iB+ H r vb iQ T`Qp2 /B`2+i _B2K MM BMi2;` #BHBivX
9XkX h>1 _1L1qG h>1P_1J
Rj8
h?2 E2v _2M2r H h?2Q`2K h?2 `2M2r H T`Q+2bb2b +QMbB/2`2/ 7`QK MQr QM MQr `2 i?Qb2 rBi? `2M2r H /Bb@ i`B#miBQM i? i Bb MQM@H iiB+2 U.2}MBiBQM 9XkXkVX h?2Q`2K 9XkXRj G2i F #2 MQM@H iiB+2 /Bbi`B#miBQM 7mM+iBQM bm+? i? i F (∞) = 1 UrBi? TQbbB#Hv BM}MBi2 K2 MV M/ H2i R #2 i?2 bbQ+B i2/ `2M2r H 7mM+iBQMX h?2M, UαV "H +Fr2HHǶb i?2Q`2K,e 7Q` HH τ ≥ 0lim{R(t + τ ) − R(t)} = t↑∞
τ . E[S1 ]
U9XRdV
UβV E2v `2M2r H i?2Q`2K, B7 g : R+ → R Bb MQM@M2; iBp2 /B`2+iHv _B2K MM@ BMi2;` #H2 7mM+iBQM ∞ 1 lim(R ∗ g)(t) = g(y) dy . U9XR3V t↑∞ E[S1 ] 0 AM 7 +i- UαV M/ UβV `2 2[mBp H2MiX S`QQ7X UαV q2 b? HH /KBi Bi 7Q` i?2 iBK2 #2BM;X HH 2tBbiBM; T`QQ7b `2 bQK2r? i i2+?@ MB+ HX T`QQ7- # b2/ QM i?2 bQ@+ HH2/ dz+QmTHBM; K2i?Q/Ǵ- Bb ;Bp2M H i2` Ubi `iBM; rBi? h?2Q`2K 9XjXRV r?2M E[S1 ] < ∞X UβV _2+ HH i? i r?2M g Bb HQ+ HHv #QmM/2/- f = R ∗ g Bb i?2 mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM f (t) = g(t) + f (t − s) dF (s). [0,t]
ah1S RX * b2 g(t) = 1[(n−1)b,nb) (t)X h?2M f (t) = R(t − (n − 1)b) − R(t − nb) M/ i?2 `2bmHi Bb Dmbi "H +Fr2HHǶb i?2Q`2KX
ah1S kX * b2 g(t) = n≥1 cn 1[(n−1)b,nb) (t)- r?2`2 cn ≥ 0- n≥1 cn < ∞- M/ b Bb bm+? i? i F (b) < 1X h?2M f (t) = cn (R(t − (n − 1)b) − R(t − nb)). n≥1
"v G2KK 9XRXdsup(R(t − (n − 1)b) − R(t − nb)) ≤ (1 − F (b))−1 < ∞. t≥0
AM T `iB+mH `- #v /QKBM i2/ +QMp2`;2M+2e
("H +Fr2HH- RN93)X
Rje
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
b 1 = cn (R(t − (n − 1)b) − R(t − nb)) = cn lim t↑∞ E[S ] E[S 1 1] n≥1 n≥1
∞
g(y) dy . 0
ah1S jX A7 g Bb /B`2+iHv _B2K MM BMi2;` #H2- i?2 7mM+iBQMb g¯b M/ g b T`2pBQmbHv /2}M2/ `2 Q7 i?2 ivT2 +QMbB/2`2/ BM ai2T k bBM+2 g b ≤ g¯b < ∞X "mi g b ≤ g ≤ g¯b M/ i?2`27Q`2 1 g b (s) ds = lim g b (t − s) R(ds) t↑∞ E[S1 ] ≤ lim inf g(t − s)R(ds) t↑∞ ≤ lim sup g(t − s) R(ds) t↑∞ 1 ≤ lim g¯b (t − s) R(ds) = g¯b (s) ds. t↑∞ E[S1 ] h?2 `2bmHi 7QHHQrb #v H2iiBM; b i2M/ iQ 0X q2 b?Qr2/ i? i UαV BKTHB2b UβVX h?2 +QMp2`b2 BKTHB+ iBQM 7QHHQrb #v +?QQbBM; g(t) := 1[0,τ ] (t)X 1t KTH2 9XkXR9, 1H2K2Mi `v _2M2r H h?2Q`2K, *Q``2+iBQM h2`Kh F2 kX AM i?2 `2+m``2Mi + b2- i?2 2H2K2Mi `v `2M2r H i?2Q`2K i2HHb mb i? i lim
t→∞
R(t) 1 = . t E[S1 ]
AM Q`/2` iQ Q#i BM KQ`2 BM7Q`K iBQM QM i?2 bvKTiQiB+ #2? pBQm` Q7 i?2 `2M2r H 7mM+iBQM- r2 b? HH bim/v i?2 #2? pBQm` Q7 f (t) := R(t) −
t E[S1 ]
b t ;Q2b iQ ∞ BM i?2 MQM@H iiB+2 + b2 r?2M S1 ? b }MBi2 }`bi M/ b2+QM/ KQK2MibX * HHBM; σ 2 i?2 +QKKQM p `B M+2 Q7 i?2 BMi2`@`2M2r H iBK2b M/ H2iiBM; m := E[S1 ]r2 ? p2 t 1 E[S1 ]2 + σ 2 lim R(t) − = . U9XRNV t↑∞ E[S1 ] 2 E[S1 ]2 S`QQ7X _2+ HH 7`QK 1t KTH2 9XRXRj i? i f b iBb}2b i?2 `2M2r H 2[m iBQM f = g + F ∗ f rBi? / i ∞ 1 g(t) = (1 − F (x)) dx . E[S1 ] t h?2 7mM+iBQM g Bb Q7 i?2 7Q`K 1 − F0 r?2`2 F0 Bb i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+@ iBQM Q7 MQM@M2; iBp2 p `B #H2 i? i Bb BMi2;` #H2X Ai Bb i?2`27Q`2 /B`2+iHv _B2K MM BMi2;` #H2 U1t KTH2 9XkXRRVX h?2 F2v `2M2r H i?2Q`2K i?2M ;Bp2b 1 ∞ t lim R(t) − = g(s) ds. t→∞ m m 0
9XkX h>1 _1L1qG h>1P_1J
Rjd
"mi 1 m
∞ ∞ 1 (1 − F (x)) dx ds m2 0 ∞ s x 1 = 2 (1 − F (x))ds dx m 0 0 ∞ 1 1 ∞ 2 1 x(1 − F (x)) dx = 2 x dF (x), = 2 m 0 m 2 0
∞
g(s)ds = 0
?2M+2 i?2 `2bmHiX US`QQ7 Q7 i?2 H bi 2[m HBiv, 1 2 E X =E 2
X
x dx = E 0
∞
∞
1{x t) = P (eαS1 > eαt ) ≤ e−αt E[eαS1 ]. _2K `F 9XkXRd *H2 `Hv- 7`QK U9XkjV- BM i?2 MQM@H iiB+2 /272+iBp2 + b2 M/ bbmK@ BM; i?2 2tBbi2M+2 Q7 bm+? α- i?2 bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM /2+ vb 2tTQM2M@ iB HHv 7 bi b t → ∞- r?2`2 b BM i?2 MQM@H iiB+2 2t+2bbBp2 + b2 U7Q` r?B+? α Hr vb 2tBbibV i?2 bQHmiBQM Q7 i?2 `2M2r H 2[m iBQM 2tTHQ/2b 2tTQM2MiB HHv 7 bi b t → ∞X 1t KTH2 9XkXR3, bvKTiQiB+b BM i?2 h` MbB2Mi * b2X amTTQb2 i? i F Bb /272+iBp2 UF (∞) < 1V M/ i? i i?2`2 2tBbib M α UM2+2bb `BHv > 0V b iBb7vBM; U9XkkVX "v h?2Q`2K 9XRXR8- r?2M i?2 / i g Bb #QmM/2/ M/ bm+? i? i i?2`2 2tBbib g(∞) = limt→∞ g(t)- i?2 mMB[m2 bQHmiBQM f Q7 i?2 `2M2r H 2[m iBQM f = g + f ∗ F b iBb}2b g(∞) lim f (t) = . U9Xk9V t↑∞ 1 − F (∞) qBi? i?2 ?2HT Q7 i?2 /272+iBp2 `2M2r H i?2Q`2K //BiBQM H BM7Q`K iBQM +QM+2`MBM; i?2 bvKTiQiB+ #2? pBQ` Q7 f + M #2 Q#i BM2/X AM 7 +i- B7 i?2 7mM+iBQM g1 /2}M2/ #v F (t) − F (∞) g1 (t) = g(t) − g(∞) + g(∞) 1 − F (∞)
R9y
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
Bb bm+? i? i i?2 7mM+iBQM t → g1 (t) = eαt g1 (t) Bb /B`2+iHv _B2K MM BMi2;` #H2- i?2M g(∞) lim eαt f (t) − =C, U9Xk8V t→∞ 1 − F (∞) r?2`2
∞ C=
0
eαt [g(t) − g(∞)]dt − ∞ teαt dF (t) 0
S`QQ7X .2}M2 f1 (t) := f (t) −
g(∞) α
.
g(∞) . 1 − F (∞)
ai` B;?i7Q`r `/ +QKTmi iBQMb mbBM; i?2 B/2MiBiv R ∗ F = R − 1 b?Qr i? i f1 = R∗g1 X h?2`27Q`2 f1 Bb bQHmiBQM Q7 i?2 U/272+iBp2V `2M2r H 2[m iBQM f1 = g1 +f1 ∗F X aBM+2 g1 (t) = eαt g1 (t) Bb bbmK2/ iQ #2 /B`2+iHv _B2K MM BMi2;` #H2 ∞ αt e g1 (t)dt αt . lim e f1 (t) = 0∞ αt t→∞ te dF (t) 0 "mi
0
∞
dF (s) dt 0 (t,∞) ∞ t 1 αs e ds dF (t) = (1 − F (∞)), = α 0 0
eαt (F (∞) − F (t))dt =
∞
eαt
7`QK r?B+? i?2 #Qp2 2tT`2bbBQM 7Q` C 7QHHQrbX
1t KTH2 9XkXRN, h?2 _BbF JQ/2H- h F2 jX h?Bb 2t KTH2 Bb +QMiBMm iBQM Q7 1t KTH2 9XRXNX _2+ HH 1[MX U9XRkV BM i?2 + b2 r?2`2 i?2 b 72iv HQ /BM; Bb TQbBiBp2 M/ r`Bi2 i?Bb 2[m iBQM BM i?2 7Q`K u λ ∞ Ψ(u) = (1 − G(z)) dz + Ψ(u − z) dF (z) , c u 0 r?2`2 F (x) :=
λ c
x
(1 − G(z)) dz . 0
∞ < 1 mM/2` i?2 h?Bb `2M2r H 2[m iBQM Bb /272+iBp2 bBM+2 λc 0 (1 − G(z)) dz = λμ c TQbBiBp2 b 72iv HQ /BM; +QM/BiBQMX bbmK2 i?2 2tBbi2M+2 Q7 α > 0 bm+? i? i λ x αz e (1 − G(z)) dz = 1. c 0 "v i?2 /272+iBp2 `2M2r H i?2Q`2K- B7 λ ∞ αu ∞ e (1 − G(z)) dz du < ∞ , c 0 u
9XkX h>1 _1L1qG h>1P_1J i?2M
R9R
∞ αu ∞ e (1 − G(z)) dz du lim e Ψ(u) = C := 0 ∞ uαz u↑∞ ze (1 − G(z)) dz 0 αu
Q`- 2[mBp H2MiHv-
Ψ(u) = Ce−αu + o(u) .
* b2 Q7 .272+iBp2 am#2tTQM2MiB H .Bbi`B#miBQM 6Q` t ≥ 0- γ ∈ (0, ∞)- H2i ∞
Rγ (t) :=
γ n F ∗n (t) ,
n=0
r?2`2 F ∗n Bb i?2 n@i? +QMpQHmiBQM Q7 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 MQM@M2; iBp2 ` M/QK p `B #H2 UrBi? i?2 mbm H +QMp2MiBQM F ∗0 (t) ≡ 1VX A7 γF (0) < 1- i?2M i?2 7mM+iBQM Rγ - + HH2/ i?2 r2B;?i2/ `2M2r H 7mM+iBQM- i F2b }MBi2 p Hm2b QM R+ M/ Bb HQ+ HHv #QmM/2/X S`QQ7X G2i {Sn }n≥1 #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b rBi? +QKKQM +mKm@ H iBp2 /Bbi`B#miBQM 7mM+iBQM F X h?2M- 7Q` θ ≥ 0 γ n F ∗n (t) = γ n P (S1 + · · · Sn ≤ t) = γ n P e−θ(S1 +···Sn ) ≥ e−θt n ≤ γ n eθt E e−θ(S1 +···Sn ) = eθt γe−θS1 . aBM+2 γF (0) < 1- i?2`2 2tBbib θ > 0 bm+? i? i γE e−θS1 < 1 M/ i?2`27Q`2 −1 Rγ (t) ≤ eθt 1 − E e−θS1 . "v bi M/ `/ `;mK2Mib- 7Q` Mv γ ∈ R+ - Rγ b iBb}2b i?2 `2M2r H 2[m iBQM Rγ (t) = 1 + γ Rγ (t − s) F (ds) , [0,t]
M/ i?2 /272+iBp2 `2M2r H 2[m iBQM
f (t − s) F (ds)
f (t) = g(t) + γ [0,t]
/KBib 7Q` mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM g(t − s) dRγ (s) . f (t) = g ∗ Rγ := [0,t]
A7 γ = 1- i?2 +H bbB+ H `2M2r H i?2Q`2K TTHB2bX 6Q` i?2 + b2 γ = 1- r2 ? p2 /272+iBp2 Uγ < 1V Q` 2t+2bbBp2 Uγ > 1V `2M2r H 2[m iBQMX q2 ? p2 b22M ?Qr iQ
R9k
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
`2/m+2 i?2 bvKTiQiB+ bim/v Q7 bm+? mMT`QT2` `2M2r H 2[m iBQMb BM i?2 + b2 QM2 + M }M/ α b iBb7vBM; U9XkkVX q2 ? p2 HbQ b22M i? i r?2`2 b bm+? α Hr vb 2tBbib BM i?2 2t+2bbBp2 + b2- Bib 2tBbi2M+2 Bb MQi Hr vb ;` Mi2/ BM i?2 /272+iBp2 + b2X h?2 bvKTiQiB+ bim/v Bb ?Qr2p2` TQbbB#H2 BM i H2 bi QM2 BKTQ`i Mi + b2- KQ`2 T`2+Bb2Hv r?2M F Bb bm#2tTQM2MiB H +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQMX .2}MBiBQM 9XkXky Ud V /Bbi`B#miBQM 7mM+iBQM F Bb b B/ iQ #2 bm#2tTQM2MiB H Ui?Bb Bb /2MQi2/ #v F ∈ SV B7 1 − F ∗2 (t) lim = 2. U9XkeV t↑∞ 1 − F (t) h? i Bb- B7 S1 M/ S2 `2 BM/2T2M/2Mi ` M/QK p `B #H2b rBi? i?2 +QKKQM /Bbi`B@ #miBQM F P (S1 + S2 > t) lim = 2. t↑∞ P (S1 > t) 1t KTH2 9XkXkR, 1t KTH2bX h?2 ` M/QK p `B #H2b r?Qb2 /Bbi`B#miBQMb ? p2 i BHb rBi? i?2 7QHHQrBM; bvKTiQiB+b `2 bm#2tTQM2MiB H, 1 − F (t) ∼ t−k
(k > 0)
−tβ
∼e
(0 < β < 1) .
h?2 K BM T`QT2`iB2b Q7 bm#2tTQM2MiB H /Bbi`B#miBQMb Q7 /B`2+i BMi2`2bi BM i?Bb #QQF rBHH #2 ;Bp2M rBi?Qmi T`QQ7X3 6B`bi Q7 HH- bm#2tTQM2MiB H /Bbi`B#miBQM F Bb ?2 pv@ i BH2/- Q` KQ`2 T`2+Bb2Hv- Bib i BH Bb dzbm#2tTQM2MiB HǴ BM i?2 b2Mb2 i? i 7Q` HH α > 0lim eαt (1 − F (t)) = ∞ . t↑∞
AM T `iB+mH `- i?2`2 Bb MQ α b iBb7vBM; U9XkkVX G2KK 9XkXkk A7 F ∈ S- 7Q` HH A ∈ (0, ∞)lim t↑∞
1 − F (t − A) = 1. 1 − F (t)
G2KK 9XkXkj A7 F ∈ S- 7Q` HH n ≥ 11 − F ∗n (t) = n. t↑∞ 1 − F (t)
lim
h?2 M2ti `2bmHi Bb FMQrM b E2bi2MǶb #QmM/, d 3
(*?Bbiv FQp- RNe9)X a22 7Q` BMbi M+2 (6Qbb- EQ`b?mMQp M/ w +? `v- kyRR)X
9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha
R9j
G2KK 9XkXk9 A7 F ∈ S- ;Bp2M Mv ε > 0- i?2`2 2tBbib D = D(ε) < ∞ bm+? i? i 7Q` HH n ≥ M/ t ≥ 01 − F ∗n (t) ≤ D(1 + ε)n . 1 − F (t) > pBM; +QHH2+i2/ i?2b2 T`2HBKBM `v H2KK b- i?2 K BM `2bmHi Q7 i?Bb bm#b2+iBQM + M #2 bi i2/ M/ T`Qp2/, h?2Q`2K 9XkXk8 UN V A7 F ∈ S M/ 0 < γ < 1(1 − γ)−1 − Rγ (t) γ = . t↑∞ 1 − F (t) (1 − γ)2
lim
S`QQ7X
(1 − γ)−1 − Rγ (t) n 1 − F ∗n (t) = . γ 1 − F (t) 1 − F (t) n=0 ∞
∗n
(t) ≤ D(γ(1 + ε))n - M/ qBi? ε bm+? i? i γ(1 + ε) < 1- #v E2bi2MǶb #QmM/- γ n 1−F 1−F (t) i?2`27Q`2- QM2 K v i F2 i?2 HBKBi b t ↑ ∞ i2`K #v i2`K BM i?2 `B;?i@? M/ bB/2 iQ Q#i BM- mbBM; G2KK 9XkXkj-
lim t↑∞
∞ n=0
1 − F ∗n (t) n 1 − F ∗n (t) n γ = = γ lim nγ = . t↑∞ 1 − F (t) 1 − F (t) (1 − γ)2 n=0 n=0 ∞
γn
∞
9Xj
"H +Fr2HHǶb h?2Q`2K M/ Bib _2}M2K2Mib
q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 "H +Fr2HHǶb i?2Q`2K BM i?2 MQM@H iiB+2 + b2 M/ mM/2` i?2 //BiBQM H +QM/BiBQM E[S1 ] < ∞X h?2 *QmTHBM; S`QQ7 Q7 "H +Fr2HHǶb h?2Q`2K q2 }`bi ;Bp2 i?`22 2[mBp H2Mi 7Q`KmH iBQMb Q7 "H +Fr2HHǶb i?2Q`2KX G2KK 9XjXR *QMbB/2` `2M2r H T`Q+2bb rBi? BMBiB H /2H v T0 U HKQbi bm`2Hv }MBi2V /Bbi`B#mi2/ ++Q`/BM; iQ GX G2i x F0 (x) = μ−1 (1 − F (y)) dy U9XkdV 0
M/ RG (t) = E
1{Tn ≤t} .
n≥0 N
(i?`2v M/ L2v- RNdk)- h?2Q`2K AoX9Xj- TX R8yX
R99
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
UBV 6Q` HH a > 0- limt↑∞ (RG (t + a) − RG (t)) = μ−1 a U"H +Fr2HHǶb i?2Q`2KVX UBBV 6Q` HH x ≥ 0- limt↑∞ P (B(t) ≤ x) = F0 (x)X UBBBV 6Q` HH x ≥ 0- limt↑∞ P (A(t) ≤ x) = F0 (x)X S`QQ7X UBBV ⇔ UBBBVX Cmbi Q#b2`p2 i? i 7Q` t ≥ 0- x ≥ 0P (A(t) ≤ x) = P (N [0, t + x] − N ([0, t]) ≥ 1) = P (B(t + x) ≤ x). UBV ⇒ UBBVX q?2M T0 ≡ 0- i?2 7mM+iBQM t → P (B(t) ≤ x) b iBb}2b `2M2r H 2[m iBQM rBi? / i g(t) = (1 − F (t))1[0,x] (t)X h?Bb 7mM+iBQM Bb /B`2+iHv _B2K MM BMi2;` #H2- M/ i?2`27Q`2 #v i?2 F2v `2M2r H i?2Q`2K U +QMb2[m2M+2 Q7 "H +Fr2HHǶb i?2Q`2KV ∞
lim P (B(t) ≤ x) = μ−1 t↑∞
g(t)dt = F0 (x). 0
h?2 + b2 Q7 MQM@MmHH M/ T`QT2` BMBiB H /2H v 7QHHQrb #v i?2 mbm H `;mK2Mi Ub22 i?2 T`QQ7 Q7 h?2Q`2K 9X9X9VX UBBBV ⇒ UBVX .2}M2 Gt (x) = P (A(t) ≤ x) M/ Q#b2`p2 i? i RG (t + a) − RG (t) = (Gt ∗ R)(a) a = R(a − s)Gt (ds) 0 a = Gt (a − s)R(ds) 0
M/ i? i bBM+2 #v ?vTQi?2bBb limt↑∞ Gt (x) = F0 (x)- r2 ? p2 #v /QKBM i2/ +QMp2`@ ;2M+2 UR ;Bp2b }MBi2 K bb iQ #QmM/2/ BMi2`p HbV a a lim Gt (a − s)R(ds) = F0 (a − s)R(ds) = μ−1 a. t↑∞
0
0
h?2`27Q`2- 7Q` i?2 T`QQ7 Q7 "H +Fr2HHǶb i?2Q`2K- Bi Bb 2MQm;? iQ T`Qp2 UBBBV Q7 G2KK 9XjXRX q2 /Q i?Bb BM i?2 + b2 r?2`2 μ < ∞XRy >2`2 Bb i?2 +QmTHBM; `;mK2MiXRR *QMbB/2` irQ BM/2T2M/2Mi `2M2r H b2[m2M+2b rBi? i?2 b K2 BMi2` ``Bp H /Bb@ i`B#miBQM F X h?2 }`bi QM2 Bb mM/2H v2/, S0 = 0- S1 - S2 - Ę M/ i?2 b2+QM/ QM2 Bb bi iBQM `v, S0 - S1 - S2 X AM T `iB+mH `- i?2 /Bbi`B#miBQM Q7 S0 Bb F0 ;Bp2M #v U9XkdVX *QMbi`m+i `2M2r H b2[m2M+2 {Sn∗ }n≥1 b 7QHHQrbX h F2 Sn∗ = Sn mMiBH i?2 }`bi iBK2 r?2`2 irQ TQBMib Q7 i?2 iBH/2/ M/ mMiBH/2/ T`Q+2bb2b `2 ε@+HQb2- r?2`2 ε Ry RR
6Q` i?2 2ti2MbBQM iQ μ = ∞- b22 7Q` BMbi M+2 (GBM/p HH- RNNk)- TTX deĜddX (GBM/p HH- RNdd)X
9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha
R98
Bb }t2/X UAM i?Bb + b2 r2 b v i? i ε@+QmTHBM; r b bm++2bb7mH- r?B+? Bb MQi ;` Mi2/ BM ;2M2` HX h?2 i2+?MB+ H T `i Q7 i?2 T`QQ7 Q7 "H +Fr2HHǶb i?2Q`2K Bb iQ b?Qr i? i ε@+QmTHBM; Bb +im HHv `2 HBx #H2 rBi? T`Q# #BHBiv R r?2M i?2 BMi2`p H /Bbi`B#miBQM Bb MQM@H iiB+2XV h?2M 7QHHQr i?2 iBH/2/ T`Q+2bbX 6Q` BMbi M+2- bmTTQb2 i? i T5 M/ ∗ T3 `2 i /Bbi M+2 H2bb i? M εX h?2M Sn∗ = Sn 7Q` n = 1, 2, 3, 4, 5- M/ S5+k = S3+k 7Q` k ≥ 1X .2MQi2 #v T = T ε i?2 }`bi TQBMi Q7 i?2 rB/2iBH/2/ T`Q+2bb r?B+? Bb ε@+HQb2 iQ TQBMi Q7 i?2 mMrB/2iBH/2/ T`Q+2bb UBM i?2 2t KTH2 T = T3 VX T5 S1 = S1∗
S2 = S2∗
S3 = S3∗ S4 = S4∗
S5 = S5∗ ≤
S˜0
S˜1
S˜2
S˜3 S˜6∗
S˜7∗
T˜3
G2KK 9XjXk qBi? i?2 bbmKTiBQMb Q7 h?2Q`2K 9XkXRj- B7 ε@+QmTHBM; ? TT2Mb HKQbi bm`2Hv- i? i Bb- B7 P (T < ∞) = 1- i?2M "H +Fr2HHǶb i?2Q`2K Bb T`Qp2/X S`QQ7X 6Q` bBKTH2` MQi iBQM- H2i T := Tε X G2i {A(t)}t≥0 M/ {A(t)} t≥0 #2 i?2 `2+m``2M+2 iBK2b +Q``2bTQM/BM; iQ i?2 mM/2H v2/ bi ``2/ `2M2r H T`Q+2bb M/ i?2 + D) r?2`2 Ubi iBQM `vV iBH/2/ `2M2r H T`Q+2bbX 6Q` HH t ≥ T - r2 ? p2 A(t) = A(t |D| ≤ εX G2i f #2 +QMiBMmQmb 7mM+iBQM #QmM/2/ #v 1- M/ /2}M2 & & & && . Mε (t) = sup &f (A(t + s)) − f (A(t)) |s|≤ε
LQi2 i? i limε↓0 E [Mε (0)] = 0X LQr& & & & & & & && &E f (A(t)) − f (A(t)) & ≤ E &f (A(t)) − f (A(t)) & & & && 1{t < T } = E &(f (A(t)) − f (A(t)) & & & && 1{t ≥ T } + E &(f (A(t)) − f (A(t)) ≤ 2P (T > t) + E [Mε (t)] = 2P (T > t) + E [Mε (0)] , .2/m+2 7`QK i?Bb i? i- bBM+2 r?2`2 7QHHQrb 7`QK bi iBQM `Biv Q7 AX i?2 H bi 2[m HBiv E f (A(t)) = E f (S0 ) lim E [f (A(t))] = E f (S0 ) . t↑∞
AM Qi?2` rQ`/b- bBM+2 f Bb M `#Bi` `v +QMiBMmQmb 7mM+iBQM #QmM/2/ #v 1- A(t) i2M/b BM /Bbi`B#miBQM iQ S0 b t ↑ ∞X AM T `iB+mH `- bBM+2 i?2 /Bbi`B#miBQM F0 Q7 S0 Bb +QMiBMmQmb- 7Q` HH x ∈ R+ -
R9e
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a lim P (A(t) ≤ x) = F0 (x). t↑∞
h?2 +QM+HmbBQM 7QHHQrb 7`QK UBBBV Q7 G2KK 9XjXRX
AM Q`/2` iQ T`Qp2 @+QmTHBM;- r2 }`bi 2t KBM2 i?2 `QH2 Q7 i?2 MQM@H iiB+2 b@ bmKTiBQMX _2+ HH i? i TQBMi x Bb b B/ iQ #2 BM i?2 bmTTQ`i Q7 i?2 /Bbi`B#miBQM 7mM+iBQM F B7 F (x + ) − F (x − ) > 0 7Q` HH > 0X h?2 b2i Q7 HH bm+? TQBMib Bb + HH2/ i?2 bmTTQ`i Q7 F M/ Bb /2MQi2/ #v bmTT(F )X h?2 F2v BKTHB+ iBQM Q7 i?2 MQM@H iiB+2 bbmKTiBQM Bb i?2 7QHHQrBM;, G2KK 9XjXj G2i F #2 MQM@H iiB+2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQMX G2i G /2@ MQi2 i?2 b2i Q7 }MBi2 HBM2 ` +QK#BM iBQMb Q7 2H2K2Mib Q7 supp(F ) rBi? +Q2{+B2Mib BM N- i? i Bb n G= gi ; g1 , . . . , gn ∈ supp(F ) . U9Xk3V i=1
n∈N
h?2M G Bb bvKTiQiB+ HHv /2Mb2 BM R+ - i? i Bb lim d(x, G) = 0,
x→∞
r?2`2 d(x, G) = inf g∈G |x − g|X P#b2`p2 i? i i?2 b2i G b /2}M2/ #v U9Xk3V Bb i?2 mMBQM Q7 i?2 bmTTQ`ib Q7 i?2 +mKmH iBp2 /Bbi`B#miBQM F ∗n Un ∈ NV Q`- 2[mBp H2MiHv- i?2 bmTTQ`i Q7 i?2
7mM+iBQMb ∗n `2M2r H 7mM+iBQM R = n∈N F bbQ+B i2/ iQ F X S`QQ7X G2iiBM; μ :=
inf
g,h∈G,g>h
{g − h} ,
U9XkNV
r2 }`bi T`Qp2 i? i μ = 0X amTTQb2 BM pB2r Q7 +QMi` /B+iBQM i? i μ > 0X h?2 BM}KmK BM U9XkNV Bb i?2M M2+2bb `BHv ii BM2/- 7Q` Qi?2`rBb2 i?2`2 rQmH/ 2tBbi b2[m2M+2b gn hn BM G bm+? i? i gn − hn > gn+1 − hn+1 - M/ gn − hn → μ b n → ∞X h?2M7Q` n H `;2 2MQm;?- gn − hn < μ + μ/2X *QMb2[m2MiHv- H2iiBM; g := gn + hn+1 M/ h = hn + gn+1 - Bi ?QH/b i? i g − h = (gn − hn ) − (gn+1 − hn+1 ) ∈ (0, μ/2). h?Bb Bb +QMi` /B+iBQM- BM pB2r Q7 i?2 /2}MBiBQM Q7 μ M/ i?2 7 +i i? i g, h ∈ GX h?2`2 Kmbi i?2`27Q`2 2tBbi g, h ∈ G bm+? i? i g − h = μX aBM+2 F Bb MQM@H iiB+2i?2`2 2tBbib z ∈ bmTT(F ) bm+? i? i- 7Q` bQK2 k ∈ Nkμ < z < (k + 1)μ. .2}M2 i?2M g := z + kh M/ h := kgX "Qi? g M/ h #2HQM; iQ GX 6m`i?2`KQ`2g − h = z − kμ ∈ (0, m)- ; BM +QMi` /B+iBQMX L2+2bb `BHv i?2M- μ = 0X h?2`27Q`2- 7Q` Mv > 0- i?2`2 2tBbi g, h ∈ G bm+? i? i g − h ∈ (0, )X *QMbB/2` i?2 bm#b2i G Q7 G +QMbBbiBM; Q7 i?2 2H2K2Mib kg + h Uk, ∈ NVX q2 `;m2 i? i
9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha
R9d
limx→∞ d(x, G ) ≤ X AM/22/- H2i m = h/ X G2i x > mhX q`Bi2 x = nh + r- rBi? n ∈ N- n ≥ m- M/ r ∈ [0, h)X G2i k ∈ N #2 bm+? i? i (n − k)h + kg ≤ x < (n − k)h + kg + (g − h). L2+2bb `BHv k ≤ m bBM+2 r < hX h?2 i2`K (n − k)h + kg i?mb #2HQM;b iQ G - b n − k ≥ 0X 6m`i?2`KQ`2- Bi Bb i KQbi T `i 7`QK x- #v i?2 T B` Q7 BM2[m HBiB2b /BbTH v2/ #Qp2X Ai 7QHHQrb i? i lim sup d(x, G) ≤ lim sup d(x, G ) ≤ . x→∞
x→∞
b Bb `#Bi` `v- i?Bb +QM+Hm/2b i?2 T`QQ7 Q7 i?2 i?2Q`2KX
PM2 b vb i? i @+QmTHBM; ?QH/b 7Q` `2M2r H T`Q+2bb2b rBi? BMi2`@`2M2r H +mKm@ H iBp2 /Bbi`B#miBQM 7mM+iBQM F B7 7Q` > 0 M/ }t2/ BMBiB H /2H vb t1 , t1 - QM2 + M +QMbi`m+i DQBMiHv irQ `2M2r H T`Q+2bb2b rBi? i?2 +Q``2bTQM/BM; /2H vb bm+? i? irBi? T`Q# #BHBiv R- i?2`2 `2 BM/B+2b m, n bm+? i? i i?2 +Q``2bTQM/BM; `2M2r H iBK2b Tm - Tn `2 H2bb i? M T `iX G2KK 9XjX9 qBi? i?2 bbmKTiBQMb Q7 h?2Q`2K 9XkXRj- ε@+QmTHBM; ? TT2Mb H@ KQbi bm`2HvX S`QQ7X G2i
Zi := min{Tj − Ti ; Tj − Ti ≥ 0}
6Q` }t2/ ε > 0- H2i h?2M
(i ≥ 0) .
Ai := {Zj < ε 7Q` bQK2 j ≥ i} .
A0 ⊇ · · · ⊇ A ⊇ · · · ⊇ ∩∞ i=0 Ai = A∞ := {Zi < ε i.o.} .
aBM+2 i?2 b2[m2M+2 {Ti+n − Ti }n≥1 ? b /Bbi`B#miBQM BM/2T2M/2Mi Q7 i M/ Bb BM@ Bb bi iBQM `v- i?2 b2[m2M+2 {Zi }i≥0 Bb ≡ {Tn }n≥0 - M/ bBM+2 N /2T2M/2Mi Q7 N HbQ bi iBQM `vX h?2`27Q`2 i?2 2p2Mib Ai Ui ≥ 0V ? p2 i?2 b K2 T`Q# #BHBiv- M/ BM T `iB+mH ` P (A0 ) = P (A∞ ) . *QM/BiBQM HHv QM S0 - i?2 2p2Mi A∞ Bb M 2t+? M;2 #H2 2p2Mi Q7 i?2 bvKK2i`B+ b2@ [m2M+2 {(Sn , Sn )}n≥1 X h?2`27Q`2- #v i?2 >2rBiiĜa p ;2 y@R H r Uh?2Q`2K XRXkdV7Q` HH t > 0 P (A∞ | S0 = t) = 0 Q` 1 . () G2KK 9XjXj ;m ` Mi22b i? i 7Q` bm{+B2MiHv H `;2 u M/ }t2/ εP (u − t < Tj − S0 < u − t + ε 7Q` bQK2 j) > 0 . h?2`27Q`2 P (A0 | S0 = t) > 0 7Q` HH t ≥ 0 M/ BM T `iB+mH ` ∞ P (A0 ) = λ P (A0 | S0 = t)(1 − F (t)) dt > 0 , 0
R93
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
r?B+? BKTHB2b bBM+2 P (A0 ) = P (A∞ )
∞
P (A∞ ) = λ
P (A∞ | S0 = t)(1 − F (t)) dt > 0 .
0
AM pB2r Q7 UV- i?Bb BKTHB2b i? i P (A∞ | S0 = t) = 1 7Q` HH t bm+? i? i F (t) < 1 . h?2`27Q`2 P (A∞ ) = 1 = P (A0 )- bQ i? i P (Zi < ε 7Q` bQK2 i) = 1X
h?2 aT`2 /@Qmi * b2 h?2 /B`2+i _B2K MM BMi2;` #BHBiv +QM/BiBQM 7Q` i?2 / i 7mM+iBQM + M #2 `2TH +2/ #v r2 F2` `2[mB`2K2Mi B7 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 BMi2`@`2M2r H iBK2b Bb bT`2 /@Qmi Ui?2 T`2+Bb2 bi i2K2Mi Bb ;Bp2M BM h?2Q`2K 9XjXd #2HQrVX .2}MBiBQM 9XjX8 h?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F Bb b B/ iQ #2 bT`2 /@Qmi B7 i?2`2 2tBbib M BMi2;2` n0 bm+? i? i i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F ∗n0 ? b M #bQHmi2Hv +QMiBMmQmb +QKTQM2Mi- i? i BbF ∗n0 (t) = Σ(t) + A(t) r?2`2 Σ Bb i?2 /Bbi`B#miBQM 7mM+iBQM Q7 /272+iBp2 ` M/QK p `B #H2 Uσ := Σ(∞) < 1V M/ t
A(t) =
a(s)ds 0
7Q` bQK2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM a : R+ → RX h?2`2 `2 Q7 +Qm`b2 BM}MBi2Hv K Mv i`BTH2b n0 , Σ, a rBi? i?2 #Qp2 T`QT2`iB2b b bQQM b i?2`2 Bb QM2X q2 b? HH K F2 mb2 Q7 i?Bb iQ T`Qp2 i? i- rBi?Qmi HQbb Q7 ;2M2` HBiv- a + M #2 i F2M #QmM/2/- +QMiBMmQmb- rBi? +QKT +i bmTTQ`i M/ bm+? i? i a(0) = 0X S`QQ7X 6B`bi r2 b?Qr i? i a + M #2 bbmK2/ #QmM/2/ BMi2;` #H2 M/ +QMiBMmQmbX A7 MQi- `2TH +2 n0 #v 2n0 - M/ mb2 i?2 7 +i i? i F ∗2n0 ? b M #bQHmi2Hv +QMiBMmQmb +QKTQM2Mi rBi? /2MbBiv a ∗ a- M/ i? i B7 a Bb #QmM/2/ M/ BMi2;` #H2- i?2M a ∗ a Bb #QmM/2/ BMi2;` #H2 M/ +QMiBMmQmbX hQ b?Qr i?Bb H bi bb2`iBQM Q#b2`p2 i? i Mv #QmM/2/ M/ BMi2;` #H2 7mM+iBQM g + M #2 TT`QtBK i2/ #v b2[m2M+2 Q7 #QmM/2/ M/ BMi2;` #H2 +QMiBMmQmb 7mM+iBQMb gn bm+? i? i ||g − gn ||L1 → 0X "v /QKBM i2/ +QMp2`;2M+2- g ∗ gn Bb +QMiBMmQmbX aBM+2 ||g ∗ gn − g ∗ g||L∞ ≤ ||g||L∞ × ||g − gn ||L1 i?2M g ∗ g Bb +QMiBMmQmb b mMB7Q`K HBKBi Q7 +QMiBMmQmb 7mM+iBQMbX q2 ? p2 i?2 7QHHQrBM; /2+QKTQbBiBQM,Rk Rk
(aiQM2- RNej)X
9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha
R9N
h?2Q`2K 9XjXe amTTQb2 i? i i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 `2@ M2r H T`Q+2bb Bb bT`2 /@Qmi rBi? K2 M mX G2i R #2 i?2 `2M2r H 7mM+iBQM Q7 i?2 mM/2H v2/ `2M2r H T`Q+2bb- M/ H2i RG = G ∗ R #2 i?2 `2M2r H 7mM+iBQM Q7 i?2 /2H v2/ `2M2r H T`Q+2bb rBi? }MBi2 /2H v Q7 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM GX h?2M RG = R1G + R2G U9XjyV r?2`2 i?2 7mM+iBQMb BM i?2 `B;?i@? M/ bB/2 `2 /Bbi`B#miBQM 7mM+iBQMb- R2G (∞) < ∞ M/ R1G Bb #bQHmi2Hv +QMiBMmQmb- rBi? /2MbBiv u1G b iBb7vBM; lim u1G (t) = t↑∞
1 . E[S1 ]
U9XjRV
S`QQ7X AM i?2 +QKTmi iBQMb #2HQr- i?2 bvK#QH 7Q` /Bbi`B#miBQM 7mM+iBQM Bb HbQ mb2/ iQ `2T`2b2Mi i?2 bbQ+B i2/ K2 bm`2XRj AM T `iB+mH `- i?2 .B` + K2 bm`2 i 0 #2BM; /2MQi2/ #v ε0 - r2 ? p2- rBi? i?Bb +QMp2MiBQM- F ∗0 = ε0 - M/ i?2 7mM/ K2Mi H `2M2r H 2[m iBQM R = 1 + R ∗ F `2 /b R = ε0 + R ∗ F X .2}M2 i?2 K2 bm`2b F ∗nn0 M/ ρ := F ∗n . R3 := 0≤n≤n0 −1
n≥0
P#b2`p2 i? i i?2 iQi H K bb Q7 ρ Bb }MBi2- 2[m H iQ n0 - M/ i? i R = ρ ∗ R3 .
U9XjkV
HbQ- R3 Bb i?2 `2M2r H K2 bm`2 Q7 F ∗n0 M/ i?2`27Q`2 b iBb}2b i?2 `2M2r H 2[m iBQM R3 = ε0 + F ∗n0 ∗ R3 = ε0 + (Σ + A) ∗ R3 M/ i?2`27Q`2
R3 ∗ (ε0 − Σ) = ε0 + A ∗ R3 .
h?2 K2 bm`2 Rσ :=
U9XjjV
Σ∗n
n≥0
#2BM; i?2 `2M2r H K2 bm`2 Q7 /272+iBp2 `2M2r H T`Q+2bb ? b }MBi2 iQi H K bb 2[m H iQ (1 − σ)−1 - M/ (ε0 − Σ) ∗ Rσ = ε0 . h?2 UmMB[m2V bQHmiBQM Q7 U9XjjV BM R3 Bb R3 = R σ + A ∗ R σ ∗ R 3 . AMb2`iBM; i?Bb 2tT`2bbBQM BMiQ U9XjkV ;Bp2b R = ρ ∗ Rσ + A ∗ ρ ∗ R σ ∗ R3 , M/ }M HHv RG = R1G + R2G , Rj
h?2 +QMi2ti rBHH /BbbBT i2 i?2 +QM7mbBQMX
R8y
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
r?2`2
R1G := A ∗ G ∗ ρ ∗ Rσ ∗ R3 M/ R2G := G ∗ ρ ∗ Rσ .
aBM+2 ρ- Rσ M/ G ? p2 }MBi2 iQi H K bb2b- R2G HbQ ? b }MBi2 iQi H K bbX b +QMpQHmiBQM Q7 M #bQHmi2Hv +QMiBMmQmb K2 bm`2 A rBi? HQ+ HHv }MBi2 K2 bm`2 G ∗ ρ ∗ Rσ ∗ R3 - R1G ? b HQ+ HHv BMi2;` #H2 /2MbBiv u1G := H ∗ R3 ∗ a ,
U9Xj9V
r?2`2 H := G ∗ ρ ∗ Rσ ? b i?2 U}MBi2V iQi H K bb H(∞) = 1 × n0 × (1 − σ)−1 . Ai `2K BMb iQ T`Qp2 U9XjRVX 6Q` i?Bb r2 TTHv i?2 F2v `2M2r H i?2Q`2K iQ i?2 /B`2+iHv _B2K MM BMi2;` #H2 7mM+iBQM a iQ Q#i BM ∞ 1 1 lim(R3 ∗ a)(t) = (1 − σ), a(t) dt = t↑∞ n0 E[S1 ] 0 n0 E[S1 ] r?2`2 r2 ? p2 Q#b2`p2/ i? i R3 Bb i?2 `2M2r H 7mM+iBQM bbQ+B i2/ iQ i?2 T`QT2` M/ MQM@H iiB+2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F ∗n0 rBi? K2 M n0 E[S1 ]X h?2 K2 bm`2 H BM U9Xj9V ? pBM; }MBi2 iQi H K bb 2[m H iQ 1 × n0 × (1 − σ)−1 - Bi 7QHHQrb i? i- #v /QKBM i2/ +QMp2`;2M+2lim u1G (t) = lim(H ∗ R3 ∗ a)(t) = H(∞) lim(R3 ∗ a)(t) t↑∞
t↑∞
t↑∞
1 1 (1 − σ) = . = H(∞) n0 E[S1 ] E[S1 ] *Q`QHH `v 9XjXd UR9 V amTTQb2 i? i i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 `2M2r H T`Q+2bb Bb bT`2 /@Qmi rBi? }MBi2 K2 M E[S1 ]X G2i R #2 i?2 `2M2r H 7mM+iBQM Q7 i?2 mM/2H v2/ `2M2r H T`Q+2bb- M/ H2i RG = G ∗ R #2 i?2 `2M2r H 7mM+iBQM Q7 i?2 /2H v2/ `2M2r H T`Q+2bb rBi? }MBi2 /2H v rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM GX G2i h : R+ → R #2 MQM@M2; iBp2 #QmM/2/ BMi2;` #H2 7mM+iBQM bm+? i? i limt↑∞ h(t) = 0X h?2M & & ∞ & & 1 & g(x) dx&& = 0 . U9Xj8V lim sup &RG ∗ g(t) − t↑∞ 0≤g≤h E[S1 ] 0 S`QQ7X aBM+2 R1G ? b }MBi2 K bb-
lim sup |(R1G ∗ g)(t)| ≤ lim R1G ∗ h(t) = lim t↑∞ 0≤g≤h
t↑∞
t↑∞
R+
h(t − s)1{s≤t} R1G (ds) = 0 ,
#v /QKBM i2/ +QMp2`;2M+2X Ai i?2`27Q`2 `2K BMb iQ b?Qr i? i & t & ∞ & & 1 lim sup && u1G (t − x)g(x) dx − g(x) dx&& = 0 . t↑∞ 0≤g≤h E[S1 ] 0 0 R9
(`D b- LmKK2HBM M/ hr22/B2- RNd3)X
9X9X _1:1L1_hAo1 S_P*1aa1a "mi
R8R
& t & ∞ & & 1 sup && u1G (t − x)g(x) dx − g(x) dx&& E[S1 ] 0 0≤g≤h 0 & ∞& & 1 && & ≤ &u1G (t − x) − E[S1 ] & h(x) dx . 0
h?2 7mM+iBQM |u1G (t) − E[S1 1 ] | i2M/b iQ 0 b t ↑ ∞- M/ i?2`27Q`2 i?2`2 2tBbib C < ∞ bm+? i? i 7Q` bm{+B2MiHv H `;2 t- b v t ≥ T - Bi Bb #QmM/2/ #v CX q`Bi2 7Q` t ≥ T - rBi? u1G (x) = 0 B7 x < 0& ∞& & & &u1G (t − x) − 1 & h(x) dx & & E[S ] 1 0 & & t−T & ∞& & & & & &u1G (t − x) − 1 & h(x) dx + &u1G (t − x) − 1 & h(x) dx = & & & E[S1 ] E[S1 ] & 0 t−T & & t−T & T& & & & & &u1G (t − x) − 1 & h(x) dx + &u1G (x) − 1 & h(t − x) dx . = & & & E[S1 ] E[S1 ] & 0 0 "v /QKBM i2/ +QMp2`;2M+2- #Qi? BMi2;` Hb BM i?2 & `B;?i@? M/ bB/2 &i2M/ iQ 0 b & & t ↑ ∞X h?2 }`bi QM2 #2+ mb2 i?2 7mM+iBQM x → &u1G (t − x) − E[S1 1 ] & 1{x≤t−T } h(x) Bb #QmM/2/ #v C iBK2b i?2 BMi2;` #H2 7mM+iBQM h M/ i2M/b iQ 0 b t → ∞X h?2 & & & & b2+QM/ #2+ mb2 QM [0, T ] i?2 7mM+iBQM x → &u1G (x) − E[S1 1 ] & Bb BMi2;` #H2 U`2+ HH i? i R1G Bb `2M2r H 7mM+iBQM M/ i?2`27Q`2 HQ+ HHv }MBi2V M/ limt↑∞ h(t − x) = 0X h?2 bBim iBQM r?2`2 2t +i +QmTHBM; Q++m`b Bb MQi 2t+2TiBQM HX AM 7 +i,R8 h?2Q`2K 9XjX3 MQM@H iiB+2 `2M2r H T`Q+2bb rBi? E[S1 ] < ∞ /KBib 2t +i +Qm@ THBM; B7 M/ QMHv B7 F Bb bT`2 /@QmiX
9X9
_2;2M2` iBp2 S`Q+2bb2b
G2i (E, E) #2 K2 bm` #H2 bT +2X .2}MBiBQM 9X9XR G2i {X(t)}t≥0 #2 K2 bm` #H2 E@p Hm2/ biQ+? biB+ T`Q+2bb M/ H2i {Tn }n≥0 #2 T`QT2` `2+m``2Mi `2M2r H T`Q+2bb- TQbbB#Hv /2H v2/ U`2+ HH- ?Qr2p2`i? i i?2 BMBiB H /2H v T0 Bb Hr vb bbmK2/ }MBi2VX h?2 T`Q+2bb {X(t)}t≥0 Bb b B/ iQ #2 `2;2M2` iBp2 rBi? `2bT2+i iQ {Tn }n≥0 B7 7Q` HH n ≥ 0U V i?2 /Bbi`B#miBQM Q7 i?2 TQbi@Tn T`Q+2bb Sn+1 , Sn+2 , . . . , {X(t + Tn )}t≥0 Bb BM@ /2T2M/2Mi Q7 n ≥ 0- M/ U#V i?2 TQbi@Tn T`Q+2bb Bb BM/2T2M/2Mi Q7 T0 , . . . , Tn X h?2 iBK2b Tn `2 + HH2/ `2;2M2` iBQM iBK2b Q7 i?2 `2;2M2` iBp2 T`Q+2bbX R8
a22 (bKmbb2M- RN3d)- *? Ti2` oA- h?K kXjX
R8k
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
1t KTH2 9X9Xk, *QMiBMmQmb@iBK2 J `FQp *? BMbX G2i {X(t)}t≥0 #2 `2@ +m``2Mi +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J `FQp +? BM i FBM; Bib p Hm2b BM i?2 bi i2 bT +2 E = NX amTTQb2 i? i Bi bi `ib 7`QK bi i2 0 i iBK2 t = 0X "v i?2 bi`QM; J `FQp T`QT2`iv- {X(t)}t≥0 Bb `2;2M2` iBp2 rBi? `2bT2+i iQ i?2 b2[m2M+2 {Tn }n≥0 r?2`2 Tn Bb i?2 n@i? iBK2 Q7 pBbBi iQ bi i2 0 Q7 i?2 +? BMX _2;2M2` iBp2 T`Q+2bb2b `2 i?2 K BM bQm`+2b Q7 `2M2r H 2[m iBQMbX h?2Q`2K 9X9Xj G2i {X(t)}t≥0 M/ {Tn }n≥0 #2 b BM .2}MBiBQM 9X9XR 2t+2Ti 7Q` i?2 //BiBQM H bbmKTiBQM T0 ≡ 0 UmM/2H v2/ `2M2r H T`Q+2bbV M/ H2i h : E → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX h?2 7mM+iBQM f : R+ → R /2}M2/ #v f (t) := E [h(X(t))] b iBb}2b i?2 `2M2r H 2[m iBQM rBi? / i g(t) = E h(X(t))1{t t)X "v h?2Q`2K 9XkXRy- T `i U#V- Bi Bb i?2`27Q`2 2MQm;? iQ T`Qp2 i? i Bi Bb _B2K MM BMi2;` #H2 QM }MBi2 BMi2`p HbX AM 7 +i- b r2 MQr b?Qr- Bi Bb HKQbi
R89
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
2p2`vr?2`2 +QMiBMmQmb Ur?B+? BKTHB2b i? i Bi Bb _B2K MM BMi2;` #H2 QM }MBi2 BM@ i2`p Hb bBM+2 Bi Bb #QmM/2/VX aBM+2 |K(t, A) − K(s, A)| ≤ P ({X(t) ∈ A, t < S1 } # {X(s) ∈ A, s < S1 }), Bi bm{+2b iQ b?Qr i? i i?2 H ii2` i2M/b iQ 0 b s → t ∈ / D- r?2`2 D Bb i?2 mMBQM Q7 i?2 b2i Q7 }t2/ /Bb+QMiBMmBiB2b Q7 i?2 T`Q+2bb U bbmK2/ +QmMi #H2V M/ Q7 i?2 U+QmMi #H2V b2i Q7 /Bb+QMiBMmBiB2b Q7 F X q2 i?2`27Q`2 ? p2 iQ T`Qp2 i? i P ({X(t) ∈ A, t < S1 } ∩ {X(s) ∈ A, s < S1 }) → 0 M/ P ({X(s) ∈ A, s < S1 } ∩ {X(t) ∈ A, t < S1 }) → 0X q2 i F2 + `2 Q7 i?2 }`bi QM2- i?2 Qi?2` QM2 #2BM; bBKBH `X q2 ? p2 P ({X(t) ∈ A, t < S1 } ∩ {X(s) ∈ A, s < S1 }) = P ({X(t) ∈ A, t < S1 , S1 ≤ s} ∪ {X(t) ∈ A, t < S1 , X(s) ∈ / A}) ≤ P ({X(t) ∈ A, t < S1 , S1 ≤ s}) + P ({X(t) ∈ A, t < S1 , X(s) ∈ / A}). h?2 }`bi i2`K QM i?2 `B;?i@? M/ bB/2 Bb #QmM/2/ #Qp2 #v F (s∨t)−F (s∧t)- r?B+? i2M/b iQ 0 bBM+2 t ∈ / D Bb TQBMi Q7 +QMiBMmBiv Q7 F X h?2 b2+QM/ i2`K Bb #QmM/2/ #Qp2 #v P ({X(t) ∈ A, X(s) ∈ / A})X aBM+2 t Bb MQi }t2/ /Bb+QMiBMmBiv TQBMi Q7 i?2 T`Q+2bb- X(s) → X(t)X h?2`27Q`2- bBM+2 A Bb QT2M- lims→t 1A (X(t)1Ac (X(s)) = 0r?B+? ;Bp2b lims→t P ({X(t) ∈ A, X(s) ∈ / A}) = 0X _2K `F 9X9Xd h?2 ?vTQi?2bBb i? i {X(t)}t≥0 ? b }t2/ /Bb+QMiBMmBiB2b 7Q`KBM; i KQbi +QmMi #H2 b2i Bb Hr vb b iBb}2/ B7 i?2 i` D2+iQ`B2b Q7 i?Bb T`Q+2bb `2 HKQbi bm`2Hv BM D([0, ∞))- i?2 b2i Q7 7mM+iBQMb i? i `2 `B;?i@+QMiBMmQmb QM [0, ∞) M/ ? p2 H27i@? M/ HBKBib QM (0, ∞)X AM i?2 7QHHQrBM; 2t KTH2b- /B`2+i _B2K MM BMi2;` #BHBiv Q7 i?2 / i g Bb +QM@ b2[m2M+2 Q7 h?2Q`2K 9X9XeX 1t KTH2 9X9X3, h?2 6mM/ K2Mi H _2HB #BHBiv h?2Q`2KX *QMbB/2` i?2 bBim iBQM Q7 `2M2r H T`Q+2bb 7Q` r?B+? i?2 BMi2``2M2r H b2[m2M+2 {Sn }n≥1 Bb Q7 i?2 7Q`K Sn = Un +Vn - r?2`2 {Un }n≥1 M/ {Vn }n≥1 `2 BM/2T2M/2Mi BB/ b2[m2M+2b M/ /2}M2 1 B7 t ∈ (Tn , Tn + Un ], X(t) = 0 B7 t ∈ (Tn + Un , Tn+1 + Un + Vn ]. h?2 BMi2`T`2i iBQM Q7 i?2 T`Q+2bb {X(t)}t≥0 BM i2`Kb Q7 `2HB #BHBiv Bb i? i X(t) = 1 r?2M i iBK2 t ;Bp2M K +?BM2 Bb +m``2MiHv BM rQ`FBM; +QM/BiBQM- r?2`2 b B7 X(t) = 0 Bi Bb BM `2T B`X G2i a M/ b #2 i?2 `2bT2+iBp2 K2 Mb Q7 U1 M/ V1 X AM i?2 `2+m``2Mi MQM@H iiB+2 + b2- r2 ? p2 i? i lim P (X(t) = 1) =
t→∞
a . a+b
9X9X _1:1L1_hAo1 S_P*1aa1a
R88
Ai bm{+2b iQ TTHv aKBi?Ƕb `2;2M2` iBp2 7Q`KmH U9X9yV rBi? A = {1}- M/ iQ ∞ Q#b2`p2 i? i E[S1 ] = E[U1 ] + E[V1 ] = a + b M/ 0 P (X(s) = 1, s < S1 ) ds = ∞ a P (U1 > s) ds = E[U1 ] = aX AM `2HB #BHBiv +QMi2ti- a+b `2T`2b2Mib i?2 p BH@ 0 #BHBiv Q7 ;Bp2M K +?BM2 rBi? K2 M HB72iBK2 a M/ K2 M `2T B` iBK2 bX 1t KTH2 9X9XN, 6Q`r `/ M/ " +Fr `/ _2+m``2M+2 hBK2bX G2i {Tn }n≥0 #2 M mM/2H v2/ UT0 = 0V `2M2r H T`Q+2bbX *H2 `Hv i?2 7Q`r `/ M/ # +Fr `/ `2+m``2M+2 iBK2b `2 `2;2M2` iBp2 rBi? `2bT2+i iQ i?2 `2M2r H T`Q+2bb {Tn }n≥0 X 6`QK aKBi?Ƕb `2;2M2` iBp2 7Q`KmH U9X9yV- BM i?2 MQM@H iiB+2 + b2 1 ∞ lim P (A(t) > x) = P (A(s) > x, s < S1 ) ds . t→∞ m 0 aBM+2 P (A(s) > x, s < S1 ) = P (S1 > s + x) = 1 − F (s + x)- r2 ? p2 ∞ ∞ P (A(s) > x, s < S1 ) ds = (1 − F (s + x)) ds 0 0 ∞ = (1 − F (s)) ds , x
M/ i?2`27Q`2 lim P (A(t) > x) =
t→∞
1 m
∞
(1 − F (s)) ds .
U9X9kV
x
aBKBH ` `;mK2Mib vB2H/ 7Q` i?2 # +Fr `/ `2+m``2M+2 iBK2 1 ∞ (1 − F (s))ds. lim P (B(t) > y) = t→∞ m y
U9X9jV
Uh?Bb iBK2- i?2 / i 7mM+iBQM Bb (1 − F (t))1{t>y} XV PM2 + M T`Qp2 /B`2+iHv i?2 /B`2+i _B2K MM BMi2;` #BHBiv Q7 i?2 / i 7mM+iBQMb Q7 i?Bb 2t KTH2- Q` mb2 h?2Q`2K 9X9XeX 1t KTH2 9X9XRy, h?2 "mb S ` /QtX h?2 bmK A(t) + B(t) Bb i?2 BMi2`@2p2Mi BMi2`p H `QmM/ iBK2 tX AMi2`T`2iBM; t b i?2 iBK2 i r?B+? vQm ``Bp2 i #mb biQT- M/ i?2 b2[m2M+2 {Tn }n≥1 b i?2 b2[m2M+2 Q7 iBK2b i r?B+? #mb2b ``Bp2 i U M/ BKK2/B i2Hv /2T `i 7`QKV i?2 #mb biQT- A(t) Bb vQm` r BiBM; iBK2X A7 t Bb H `;2 2MQm;?- QM2 + M- BM pB2r Q7 U9X9kV- bbmK2 i? i A(t) Bb /Bbi`B#mi2/ b ` M/QK p `B #H2 A rBi? i?2 /Bbi`B#miBQM 1 ∞ P (A > x) = (1 − F (s)) ds . U9X99V m x aBKBH `Hv- i?2 iBK2 B(t) #v r?B+? vQm KBbb2/ i?2 T`2pBQmb #mb Bb TT`QtBK i2Hvr?2M t Bb H `;2- /Bbi`B#mi2/ b ` M/QK p `B #H2 B rBi? i?2 b K2 /Bbi`B#miBQM b AX h?2 #mb T ` /Qt + M #2 bi i2/ BM b2p2` H r vbX PM2 Q7 i?2K Bb, i?2 K2 M iBK2 BMi2`p H #2ir22M i?2 #mb vQm KBbb2/ M/ i?2 #mb vQm rBHH + i+? Bb bvKTiQiB+ HHv b t → ∞ 2[m H iQ E[A + B] = 2E[A]- M/ Bb BM ;2M2` H /Bz2`2Mi 7`QK i?2 K2 M Q7 i?2 ivTB+ H BMi2`p H E[S1 ] #2ir22M irQ bm++2bbBp2 #mb2bX
R8e
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
aKBi?Ƕb `2;2M2` iBp2 i?2Q`2K rBHH MQr #2 `2pBbBi2/ BM i?2 bT`2 /@Qmi + b2X *QM@ bB/2` K2 bm` #H2 biQ+? biB+ T`Q+2bb {X(t)}t∈R+ rBi? p Hm2b BM bQK2 K2 bm` #H2 bT +2 (E, E) M/ `2;2M2` iBp2 rBi? `2bT2+i iQ `2M2r H T`Q+2bb rBi? bT`2 /@Qmi /Bbi`B#miBQM F Q7 }MBi2 K2 MX h?2 7mM+iBQM h(t) = P (S1 > t) ? b i?2 T`QT2`iB2b `2@ [mB`2/ BM *Q`QHH `v 9XjXdX 6Q` Mv A ∈ E- i?2 7mM+iBQM gA : t → P (X(t) ∈ A, S1 > t) Bb #QmM/2/ #v h- M/ i?2`27Q`2 & & ∞ & & 1 & gA (x) dx&& = 0 , lim sup &RG ∗ gA (t) − t↑∞ A∈E E[S1 ] 0 i? i Bb iQ b v lim sup |P (X(t) ∈ A) − Q∗ (A)| = 0 , t↑∞ A∈E
r?2`2 Q∗ (A) :=
1 E0 E[S1 ]
S1
1A (X(s)) ds ,
0
r?2`2 E0 /2MQi2b 2tT2+i iBQM rBi? `2bT2+i iQ i?2 mM/2H v2/ `2;2M2` iBp2 T`Q+2bbX h?2`27Q`2- b t → ∞- i?2 /Bbi`B#miBQM Q7 X(t) +QMp2`;2b BM p `B iBQM iQ Q∗ X G2i MQr {X(t)}t∈R #2 biQ+? biB+ T`Q+2bb i FBM; Bib p Hm2b BM K2i`B+ bT +2 E- ? pBM; `B;?i@+QMiBMmQmb T i?b M/ #2BM; `2;2M2` iBp2 `2H iBp2 iQ i?2 UTQbbB#Hv /2H v2/V `2M2r H b2[m2M+2 {Tn }n≥0 rBi? MQM@H iiB+2 M/ }MBi2 K2 M BMi2`@2p2Mi /Bbi`B#miBQM μX G2i P0 M/ E0 bvK#QHBx2 `2bT2+iBp2Hv i?2 T`Q# #BHBiv M/ i?2 2t@ T2+i iBQM +Q``2bTQM/BM; iQ i?2 mM/2H v2/ p2`bBQM Q7 i?2 `2M2r H b2[m2M+2X h?2M S1
1 1A (X(s)) ds P ∗ (A) := E0 μ 0 /2}M2b T`Q# #BHBiv K2 bm`2 QM (E, B(E))X h?2Q`2K 9X9XRR lM/2` i?2 #Qp2 +QM/BiBQMb- X(t) +QMp2`;2b BM /Bbi`B#miBQM iQ P ∗ b t ↑ ∞X S`QQ7X q2 Kmbi b?Qr i? i 7Q` HH #QmM/2/ Ub v- #v 1V +QMiBMmQmb 7mM+iBQMb h : E → R
S1 1 h(X(s)) ds . U9X98V lim E [h(X(t))] = E0 t↑∞ μ 0 "v i?2 mbm H `2M2r H `;mK2Mi U+QM/BiBQMBM; QM T0 - r?Qb2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Bb /2MQi2/ #v FT0 V f (t − s) dFT0 (s) , () E [h(X(t))] = E h(X(t))1{tSh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
S`QQ7X h?2 7 +i i? i R∗g Bb r2HH /2}M2/- HQ+ HHv #QmM/2/- M/ b iBb}2b i?2 `2M2r H 2[m iBQM Bb T`Qp2/ BM i?2 b K2 r v b BM i?2 mMBp `B i2 + b2X G2i MQr f M/ f #2 irQ p2+iQ`b Q7 HQ+ HHv #QmM/2/ 7mM+iBQMb b iBb7vBM; i?2 `2M2r H 2[m iBQM- M/ H2i h := f − fX h?2M h = R ∗ h- M/ Bi2` iBp2Hv- h = F ∗n ∗ h 7Q` HH n ≥ 1- bQ i? i |h| ≤ F ∗n ∗ |h|X G2i supi |hi (t)| ≤ M (a) < ∞ QM [0, a]- b v M (a) = 1- rBi?Qmi HQbb Q7 ;2M2` HBiv- bQ i? i |h| ≤ F ∗n ∗ 1 QM [0, a] M/ i?2`27Q`2- QM [0, a] Fij∗n (t) = Pi (S1 + · · · + Sn ≤ t) , |hi (t)| ≤ j∈E
[m MiBiv i? i i2M/b iQ 0 b n ↑ ∞ bBM+2 T∞ = ∞ mM/2` i?2 T`2p BHBM; +QM/BiBQMb Ui?2 2K#2//2/ +? BM Bb B``2/m+B#H2 `2+m``2MiVX h?2`27Q`2 h ≡ 0 QM HH [0, a] Ua ∈ R+ V M/ i?2`27Q`2 QM R+ X Ai 7QHHQrb 7`QK U9X9dV i? i B7 gi Bb /B`2+iHv _B2K MM BMi2;` #H2 νj ∞ gj (s) ds , Rij ∗ gj (t) → μ 0 M/ 7`QK i?Bb- BM i?2 + b2 r?2`2 E Bb }MBi2fi (t) =
Rij ∗ gj (t) →
j∈E
1 νj μ j∈E
∞
gj (s) ds . 0
h?2 + b2 r?2M E Bb BM}MBi2 `2[mB`2b 7m`i?2` +QM/BiBQMb M/ rBHH MQi #2 i`2 i2/ ?2`2XR3 AKT`QT2` JmHiBp `B i2 _2M2r H 1[m iBQMb h?2 #Qp2 `2bmHib +QM+2`M i?2 + b2 r?2`2 Q := {||Fij ||}i,j∈E Bb biQ+? biB+ K i`Bti? i Bb- i?2 i` MbBiBQM K i`Bt Q7 ?QKQ;2M2Qmb J `FQp +? BM UM K2Hv- i?2 i` M@ bBiBQM K i`Bt P Q7 i?2 2K#2//2/ J `FQp +? BMVX >Qr2p2` i?2 `2M2r H 2[m iBQMb U9X9eV K F2 b2Mb2 2p2M B7 i?Bb Bb MQi i?2 + b2X q2 MQr ;Bp2 `2bmHibRN Q7 i?2 b K2 FBM/ b i?2 QM2b BM i?2 /272+iBp2 Q` 2t+2bbBp2 mMBp `B i2 `2M2r H 7mM+iBQMb r?2M i?2 bi i2 bT +2 Bb }MBi2X h?2 K i`Bt Q Bb MQ HQM;2` biQ+? biB+ K i`Bt- #mi biBHH bbmK2/ B``2/m+B#H2X .2}M2 7Q` bQK2 `2 H β i?2 K i`Bt A := {aij }i,j∈E ∞ aij := eβt dFij (t) . 0
bbmK2 i? i β + M #2 +?Qb2M bm+? i? i A ? b bT2+i` H ` /Bmb 1X AM T `iB+mH `i?2`2 2tBbib irQ TQbBiBp2 p2+iQ`b ν M/ h bm+? i? i ν T A = ν M/ Ah = h . h?2 2tBbi2M+2 Q7 ν M/ h Bb 2Mbm`2/ #v i?2 S2``QMĜ6`ƺ#2MBmb i?2Q`2KX h?2 7QHHQr@ BM; 7 +ib `2 2 bvX 6B`bi i?2 K i`Bt R3 RN
a22- 7Q` BMbi M+2- (ÎBMH `- RNd8)X (bKmbb2M M/ >2`BM;- RNdd)X
9XdX 1s1_*Aa1a
ReR := Q
hj aij hi
i,j∈E
Bb M UB``2/m+B#H2V biQ+? biB+ K i`Bt /KBiiBM; i?2 BMp `B Mi K2 bm`2 ν ;Bp2M #v νi = νi hi X G2i hj Fij (t) := hi
t
eβs dFij (s) . 0
= {||Fij ||}i,j∈E Bb B``2/m+B#H2 M/ h?Bb /2}M2b b2KB@J `FQp F2`M2H 7Q` r?B+? Q `2+m``2MiX .2}MBM; fi (t) := eβt fi (t)/hi M/ gi := eβt gi (t)/hi , r2 b22- M HQ;QmbHv iQ i?2 mMBp `B i2 + b2- i? i f = g + F ∗ f. h?2`27Q`2- B7 F Bb MQM@H iiB+2 M/ i?2 gi Ƕb `2 HQ+ HHv #QmM/2/ M/ BMi2;` #H2 ∞ 1 νj gj (s) ds , lim fi (t) = t↑∞ μ j∈E 0 i? i Bb hi lim e fi (t) = βt
t↑∞
j∈E
k,j∈E
νj
ν k hj
∞ 0∞ 0
eβs gj (s) ds seβs dFkj (s)
.
h?2 2tBbi2M+2 Q7 β Bb ;m ` Mi22/- 7Q` BMbi M+2-ky r?2M i?2 bT2+i` H ` /Bmb Q7 {||Fij ||}i,j∈E Bb bi`B+iHv H2bb i? i 1X
9Xd
1t2`+Bb2b
1t2`+Bb2 9XdXRX h?2Q`2K 9XRXR i`m2 BM i?2 mM/2H v2/ + b2 S`Qp2 i? i b h?2Q`2K 9XRXR Bb i`m2 BM i?2 mM/2H v2/ + b2- Bi Bb i?2M i`m2 BM i?2 /2H v2/ + b2 U`2+ HH, rBi? }MBi2 /2H vVX 1t2`+Bb2 9XdXkX h?2 bvKTiQiB+ +QmMiBM; ` i2 S`Qp2 U9X9V- i? i BbN ([0, t]) 1 = - S@ XbX lim t→∞ t E[S1 ]
1t2`+Bb2 9XdXjX _B;?i@+QMiBMmBiv Q7 i?2 `2M2r H 7mM+iBQM a?Qr i? i i?2 `2M2r H 7mM+iBQM R Bb `B;?i@+QMiBMmQmbX ky
a22 (bKmbb2M- RN3d)- S`Q#H2K kXj- +? TX sX
Rek
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1t2`+Bb2 9XdX9X #Qmi i?2 /Bbi`B#miBQM Q7 N ([0, t]) AM i?2 mM/2H v2/ + b2- +QKTmi2 P (N ([0, t]) = n) 7Q` n ≥ 1 BM i2`Kb i?2 βNQ7 +QM@ ([0,t]) pQHmiBQM Bi2` i2b Q7 i?2 BMi2`@`2M2r H /Bbi`B#miBQM M/ b?Qr i? i E e X} . :Bp2 M 2tT`2bbBQM Q7 i?2 +mKmH iBp2 /Bbi`B#miBQM Q7 T BM i2`Kb Q7 G- i?2 `2M2r H 7mM+iBQM R M/ i?2 pQB/ M+2 7mM+iBQM v U/2}M2/ BM a2+iBQM RXjVX 1t2`+Bb2 9XdXeX 6Q`r `/ `2+m``2M+2 iBK2 q? i Bb i?2 HBKBi /Bbi`B#miBQM Q7 i?2 7Q`r `/ `2+m``2M+2 iBK2 Q7 `2M2r H T`Q+2bb UTQbbB#Hv /2H v2/V r?2M S1 Bb /2i2`KBMBbiB+- 2[m H iQ a\ 1t2`+Bb2 9XdXdX q H/ǰb H2KK 7Q` `2M2r H T`Q+2bb2b G2i {Sn }n≥1 #2 `2M2r H b2[m2M+2 M/ H2i {N ([0, t])}t≥0 #2 i?2 +QmMiBM; T`Q+2bb Q7 i?2 +Q``2bTQM/BM; UTQbbB#Hv /2H v2/V `2M2r H T`Q+2bbX bbmK2 i? i E [S1 ] < ∞X a?Qr i? i 7Q` HH t ≥ 0E[S1 + · · · + SN ([0,t]) ] = E[S1 ]E[N ([0, t])].
1t2`+Bb2 9XdX3X 1tT2+i2/ HB72iBK2 AM 1t KTH2 9XRX3- +QKTmi2 i?2 2tT2+i iBQM Q7 i?2 HB72iBK2 L BM i?2 i` MbB2Mi + b2X 1t2`+Bb2 9XdXNX a 72iv HQ / h?Bb 2t2`+Bb2 `272`b iQ 1t KTH2 kXRX9X S`Qp2 i? i BM i?2 + b2 Q7 TQbBiBp2 b 72iv HQ /BM;- Φ(∞) = 1X 1t2`+Bb2 9XdXRyX h?2 #mb T ` /Qt a22 1t KTH2 9X9XRy 7Q` i?2 +QMi2tiX *QMbB/2` M mM/2H v2/ `2M2r H T`Q+2bb rBi? }MBi2 K2 M BMi2`@`2M2r H iBK2 E[S1 ]X 6Q` t ≥ 0- +QMbB/2` i?2 BMi2`p H #2ir22M i?2 H bi `2M2r H iBK2 #27Q`2 t M/ i?2 }`bi `2M2r H iBK2 7i2` tX a?Qr i? i BM i?2 SQBbbQM + b2 Ui?2 BMi2` ``Bp H /Bbi`B#miBQM Bb 2tTQM2MiB HV i?2 K2 M H2M;i? Q7 i?Bb BMi2`p H Bb bvKTiQiB+ HHv 2E[S1 ]X a?Qr i? i Bi Bb 2[m H iQ E[S1 ] B7 M/ QMHv S1 Bb +QMbi MiX 1t2`+Bb2 9XdXRRX 6B`bi 2p2Mi 7i2` 7Bt2/ iBK2 _272` iQ .2}MBiBQM 9XRXRe 7Q` i?2 MQi iBQMX S`Qp2 i? i S0 (t) Bb BM/2T2M/2Mi Q7 {Sn (t)}n≥1 M/ i? i i?2 H ii2` b2[m2M+2 ? b i?2 b K2 /Bbi`B#miBQM b {Sn }n≥1 X
9XdX 1s1_*Aa1a
Rej
1t2`+Bb2 9XdXRkX HBKBi i?2Q`2K 7Q` +QMiBMmQmb@iBK2 ?K+b G2i {X(t)}t≥0 - #2 TQbBiBp2 `2+m``2Mi +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J `FQp +? BM i FBM; Bib p Hm2b BM i?2 bi i2 bT +2 E = NX G2i P0 M/ E0 /2MQi2 `2bT2+iBp2Hv T`Q# #BHBiv M/ 2tT2+i iBQM ;Bp2M X(0) = 0X G2i T0 #2 i?2 `2im`M iBK2 iQ 0 UT0 := inf{t > 0 ; X(t) = 0, X(t−) = 0}VX _2+ HH i? i BM i?2 TQbBiBp2 `2+m``2Mi + b2E0 [T0 ] < ∞X a?Qr i? i T E0 0 0 1{X(s)=i} ds lim P (X(t) = i) = . t↑∞ E0 [T0 ]
1t2`+Bb2 9XdXRjX GQiF ėoQHi2`` bvKTiQiB+b AM i?2 GQiF ĜoQHi2`` KQ/2H- ;Bp2 i?2 /2i BHb +QM+2`MBM; i?2 bvKTiQiB+b Q7 i?2 #B`i? ` i2 f BM i?2 + b2b F (∞) = 1 M/ F (∞) > 1X q? i + M vQm b v #Qmi i?2 /272+iBp2 + b2 F (∞) < 1\ 1t2`+Bb2 9XdXR9X " +Fr `/ M/ 7Q`r `/ `2+m``2M+2 T`Q+2bb2b _272` iQ .2}MBiBQM 9X9XNX *QKTmi2 limt→∞ P (A(t) > x, B(t) > y) 7Q` x, y ≥ 0X 1t2`+Bb2 9XdXR8X bvKTiQiB+ p `B M+2 Q7 N ((0, t]) 6Q` T`QT2` `2M2r H T`Q+2bb rBi? M BMi2` ``Bp H /Bbi`B#miBQM Q7 }MBi2 p `B M+2b?Qr i? i o `N ((0, t]) o `S1 lim = . t↑∞ t E[S1 ] 1t2`+Bb2 9XdXReX h?2 ;2 `2TH +2K2Mi TQHB+v q2 BMi2`T`2i i?2 ` M/QK p `B #H2b S1 , S2 , . . . b i?2 HB72iBK2b Q7 K +?BM2b bm++2b@ bBp2Hv Tmi BMiQ b2`pB+2- M2r K +?BM2 BKK2/B i2Hv `2TH +BM; 7 BH2/ QM2X Ai rBHH #2 bbmK2/ i? i E[S1 ] < ∞- M/ i?2`27Q`2- #v U9X9V- E[S1 1 ] Bb i?2 bvKTiQiB+ 7 BH@ m`2 ` i2 T2` mMBi iBK2X AM bQK2 bBim iBQMb- i?2 BM+QMp2MB2M+2 + mb2/ #v 7 BHm`2 Bb iQQ BKTQ`i Mi- M/ i?2 7 BHm`2 ` i2 Kmbi #2 +QMi`QHH2/X h?2 ;2 `2TH +2K2Mi TQHB+v bm;;2bib i? i M 2M;BM2 b?QmH/ #2 `2TH +2/ i 7 BHm`2 iBK2 Q` i }t2/ iBK2 T > 0r?B+?2p2` Q++m`b }`biX q? i Bb i?2 bvKTiQiB+ 7 BHm`2 ` i2\ U `2TH +2K2Mi Bb MQi +QMbB/2`2/ b 7 BHm`2XV 1t2`+Bb2 9XdXRdX MQi?2` K BMi2M M+2 TQHB+v ;Bp2M K +?BM2 + M #2 BM 2Bi?2` QM2 Q7 i?`22 bi i2b, : U;QQ/V- J UBM K BMi2@ M M+2V- Q` _ UBM `2T B`VX Aib bm++2bbBp2 T2`BQ/b r?2`2 Bi Bb BM bi i2 : U`2bTX- J- _V 7Q`K M BM/2T2M/2Mi M/ B/2MiB+ HHv /Bbi`B#mi2/ b2[m2M+2 {Sn }n≥0 U`2bTX- {Un }n≥0 {Vn }n≥0 V rBi? }MBi2 K2 MX HH i?2b2 b2[m2M+2b `2 bbmK2/ Kmim HHv BM/2T2M/2MiX h?2 K BMi2M M+2 TQHB+v mb2b MmK#2` T > 0X A7 i?2 K +?BM2 ? b ;2 T M/ ? b MQi 7 BH2/- Bi ;Q2b iQ bi i2 JX A7 Bi 7 BHb #27Q`2 Bi ? b `2 +?2/ ;2 T - Bi 2Mi2`b bi i2 _X 6`QK bi i2b J M/ _- i?2 M2ti bi i2 Bb :X 6BM/ i?2 bi2 /v bi i2 T`Q# #BHBiv i? i i?2 K +?BM2 Bb QT2` iBQM HX ULQi2 i? i dz;QQ/Ǵ /Q2b MQi K2 M dzQT2` iBQM HǴX h?2 K +?BM2 + M #2 dz;QQ/Ǵ #mi- /m2 iQ i?2 QT2` iBQMb TQHB+v- BM K BMi2M M+2- M/
Re9
*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a
i?2`27Q`2 MQi QT2` iBQM HX >Qr2p2`- 7i2` T2`BQ/ Q7 K BMi2M M+2 Q` Q7 `2T B`- r2 +QMbB/2` i? i i?2 K +?BM2 bi `ib M2r- M/ 2Mi2`b : T2`BQ/XV 1t2`+Bb2 9XdXR3X irQ bi i2 b2KB@J `FQp T`Q+2bb G2i α 1−α P= , 1−β β r?2`2 α, β ∈ (0, 1)- M/ H2i G1 M/ G2 #2 irQ T`QT2` +mKmH iBp2 /Bbi`B#miBQM 7mM+@ iBQMbX G2i {X(t)}t≥0 #2 i?2 biQ+? biB+ T`Q+2bb 2pQHpBM; b 7QHHQrbX q?2M BM bi i2 i Ui = 1, 2V Bi bi vb i?2`2 7Q` ` M/QK iBK2 rBi? /Bbi`B#miBQM Gi Ui = 1, 2V 7i2` r?B+? Bi KQp2b iQ bi i2 j UTQbbB#Hv i?2 b K2 bi i2V rBi? i?2 T`Q# #BHBiv pij Ui?2 Ui, jV@2Mi`v Q7 PVX h?2 bm++2bbBp2 bQDQm`M iBK2b UBM 2Bi?2` bi i2V `2 BM/2T2M/2Mi ;Bp2M i?2 FMQrH2/;2 Q7 i?2 bi i2 i?2 T`Q+2bb Bb BM Ui?2 `2 /2` rBHH +H `B7v i?Bb BK@ T`2+Bb2 b2Mi2M+2VX q? i Bb i?2 bvKTiQiB+ /Bbi`B#miBQM Q7 i?2 T`Q+2bb- i? i Bb- r? i Bb limt↑∞ P (X(t) = 1)\
*? Ti2` 8 SQBMi S`Q+2bb2b rBi? aiQ+? biB+ AMi2MbBiv G2i {Ft }t∈R #2 bQK2 ?BbiQ`v Q7 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb N QM R- i? i Bb- MQM@/2+`2 bBM; 7 KBHv Q7 σ@}2H/b bm+? i? i 7Q` HH t ∈ R M/ HH a ≤ b ≤ tN ((a, b]) Bb Ft @K2 bm` #H2X A7 Bi ?QH/b i? i 7Q` HH t ∈ Rlim h↓0
1 E[N (t, t + h] | Ft ] = λ(t) , h
P @ XbX,
()
7Q` bQK2 MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 Ft @ / Ti2/ biQ+? biB+ T`Q+2bb {λ(t)}t∈R i?2 H ii2` Bb + HH2/ biQ+? biB+ Ft @BMi2MbBiv Q7 N X h?Bb /2}MBiBQM BMpQHpBM; HBK@ BiBM; /2`Bp iBp2@ivT2 T`Q+2/m`2 Bb /p Mi ;2QmbHv `2TH +2/ #v ;HQ# H /2}MBiBQM BM i2`Kb Q7 K `iBM; H2b i? i MQi QMHv HHQrb QM2 iQ +H `B7v 72r K i?2K iB+ H Bbbm2b #mi- KQ`2 BKTQ`i MiHv- Bb i i?2 Q`B;BM Q7 M2r T ` /B;K R BM i?2 bim/v Q7 biQ+? biB+ bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b M/ `2p2 Hb T`Q/m+iBp2 M HQ;v rBi? i?2 i?2Q`v Q7 biQ+? biB+ bvbi2Kb /`Bp2M #v r?Bi2 MQBb2k i? i rBHH #2 2tTHQBi2/ BM *? Ti2`b Ry M/ RkX
8XR
h?2 aKQQi?BM; 6Q`KmH b
6Q` SQBbbQM T`Q+2bb N QM i?2 `2 H HBM2 rBi? i?2 HQ+ HHv BMi2;` #H2 BMi2MbBiv 7mM+iBQM λ(t)- Bi ?QH/b i? i 7Q` HH BMi2`p Hb [c, d] ⊂ R E N (c, d] | FcN =
d
λ(s) ds c
Q`- 2[mBp H2MiHv bBM+2 i?2 `B;?i@? M/ bB/2 Bb /2i2`KBMBbiB+ [m MiBiv
E N (c, d] |
FcN
d
λ(s) ds |
=E
FcN
.
c
h?Bb KQiBp i2b i?2 7QHHQrBM; /2}MBiBQM Q7 biQ+? biB+ BMi2MbBivX R k
("`ûK m/- RNdk)X h?Bb rBHH #2 #`B2~v 2tTH BM2/ BM a2+iBQM 8X3X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_5
Re8
Ree *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu .2}MBiBQM 8XRXR G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R- H2i {Ft }t∈R #2 ?BbiQ`v Q7 N M/ H2i {λ(t)}t∈R #2 MQM@M2; iBp2 XbX HQ+ HHv BMi2;` #H2 `2 H@ p Hm2/ Ft @T`Q;`2bbBp2Hv K2 bm` #H2 biQ+? biB+ T`Q+2bbX A7 7Q` HH a ∈ R- HH BMi2`p Hb (c, d] ⊂ (a, ∞) M/ HH BMi2;2`b n ≥ 1 (a) d∧Tn
E[N (c ∧ Tn(a) , d ∧ Tn(a) ]|Fc ] = E
(a)
λ(s)ds | Fc ,
U8XRV
c∧Tn (a)
(a)
b2[m2M+2 Q7 Ft @biQTTBM; iBK2b bm+? i? i Tn > r?2`2 {Tn }n≥1 Bb MQM@/2+`2 bBM; (a) (a) a- limn↑∞ Tn = ∞ M/ E N (a, Tn ] < ∞- N Bb i?2M b B/ iQ /KBi i?2 biQ+? biB+ (P, Ft )@BMi2MbBiv {λ(t)}t∈R X h?2 /2}MBiBQM Q7 biQ+? biB+ BMi2MbBiv QT2Mb +QMM2+iBQM rBi? i?2 i?2Q`v Q7 K `iBM; H2bX AM 7 +i- .2}MBiBQM 8XRXR Q7 biQ+? biB+ BMi2MbBiv BKTHB2b i? i t M (t) := N (0, t] − λ(s) ds 0
Bb HQ+ H (P, Ft )@K `iBM; H2 U1t2`+Bb2 8XNXRV- + HH2/ i?2 7mM/ K2Mi H UHQ+ HV Ft @ K `iBM; H2 Q7 i?2 TQBMi T`Q+2bb N X q?2M i?2 +?QB+2 Q7 T`Q# #BHBiv P Bb +H2 ` 7`QK i?2 +QMi2ti- QM2 b vb dzi?2 Ft @BMi2MbBivǴ BMbi2 / Q7 dzi?2 (P, Ft V@BMi2MbBivǴX _2K `F 8XRXk h?2 `2[mB`2K2Mi Q7 Ft @T`Q;`2bbBp2M2bb Q7 i?2 biQ+? biB+ BMi2MbBiv t ;m ` Mi22b i? i i?2 BMi2;` i2/ BMi2MbBiv T`Q+2bb { 0 λ(s) ds}t∈R Bb K2 bm` #H2 M/ Ft @ / Ti2/ U1t2`+Bb2 8XNX9VX _2K `F 8XRXj q?2M +QMbB/2`BM; TQBMi T`Q+2bb2b QM i?2 TQbBiBp2 ? H7@HBM2- i?2 BMi2`p2MiBQM Q7 i?2 aǶb Bb bmT2`~mQmb- M/ Bi bm{+2b iQ `2[mB`2 i? i U8XRV ?QH/b 7Q` a = 0X >Qr2p2`- i?2 bHB;?iHv KQ`2 +QKTHB+ i2/ /2}MBiBQM ;Bp2M #Qp2 Bb M22/2/ BM Q`/2` iQ ? M/H2 TQBMi T`Q+2bb2b QM i?2 r?QH2 `2 H HBM2- 2bT2+B HHv bi iBQM `v TQBMi T`Q+2bb2bX _2K `F 8XRX9 h?2 `2 bQM r?v `2[mB`2K2Mi U8XRV + MMQi #2 `2TH +2/ #v i?2 bBK@ TH2` QM2
d
E [N (c, d] | Fc ] = E
λ(s) ds | Fc ,
(†)
c (a)
MQi BMpQHpBM; i?2 biQTTBM; iBK2b Tn - Bb i? i Bi K v Q++m` i? i #Qi? bB/2b BM U†V `2 BM}MBi2- BM r?B+? + b2 i?2 BM7Q`K iBQM +QMi BM2/ BM U†V Bb MBHX h?Bb ? TT2Mb 7Q` BMbi M+2 r?2M N Bb ?QKQ;2M2Qmb *Qt T`Q+2bb r?Qb2 ` M/QK BMi2MbBiv Λ ? b M BM}MBi2 2tT2+i iBQM Ub22 1t KTH2 8XRX8 #2HQrVX 1t KTH2 8XRX8, SQBbbQM M/ *Qt S`Q+2bb2bX G2i N #2 *Qt T`Q+2bb QM ν R+ rBi? +QM/BiBQM H BMi2MbBiv K2 bm`2 ν rBi? `2bT2+i iQ G ⊇ F Ub22 .2}MBiBQM RXRXRyV M/ bmTTQb2 i? i ν(C) := C λ(s) ds UC ∈ B(R+ )V r?2`2 {λ(t)}t≥0 Bb
8XRX h>1 aJPPh>AL: 6P_JlGa
Red
HQ+ HHv BMi2;` #H2 MQM@M2; iBp2 T`Q+2bbX h?2M N /KBib i?Bb T`Q+2bb b Ft @BMi2MbBivr?2`2 Ft := FtN ∨ G Ut ≥ 0V U1t2`+Bb2 8XNXjVX h?2Q`2K 8XRXRR #2HQr ;Bp2b 7Q`KmH i? i + M #2 +QMbB/2`2/ #Qi? `2}M2@ K2Mi Q7 * KT#2HHǶb 7Q`KmH M/ M 2ti2MbBQM Q7 i?2 bKQQi?BM; 7Q`KmH 7Q` ?TTb Uh?2Q`2K kXkXRVX h?2 MQiBQM Q7 T`2/B+i #H2 T`Q+2bb rBHH #2 M22/2/X .2}MBiBQM 8XRXe G2i T = R Q` R+X G2i {Ft }t∈T #2 ?BbiQ`vX h?2 T`2/B+i #H2 σ@}2H/ P (F· ) QM T × Ω Bb i?2 σ@}2H/ ;2M2` i2/ #v i?2 +QHH2+iBQM Q7 b2ib (a, b] × A
([a, b] ⊂ T, A ∈ Fa ) ,
U8XkV
iQ r?B+? QM2 Kmbi //- BM i?2 + b2 r?2`2 T = R+ - i?2 b2ib {0} × A (A ∈ F0 )X biQ+? biB+ T`Q+2bb {X(t)}t∈T i FBM; Bib p Hm2b BM K2 bm` #H2 bT +2 (E, E) Bb + HH2/ M Ft @T`2/B+i #H2 T`Q+2bb B7 i?2 K TTBM; (t, ω) → X(t, ω) Bb P(F· )@K2 bm` #H2X 6Q` b?Q`i- QM2 i?2M b vb, {X(t)}t∈T Bb BM P (F· )X _2K `F 8XRXd M Ft @T`2/B+i #H2 T`Q+2bb Bb Ft @T`Q;`2bbBp2 U1t2`+Bb2 8XNXeVX .2}MBiBQM 8XRX3 G2i T = R Q` R+ M/ H2i {Ft }t∈T #2 ?BbiQ`vX G2i (K, K) #2 bQK2 `#Bi` `v K2 bm` #H2 bT +2X G2i H : (T×Ω×K, P (F· )⊗K) → (R, B(R))X PM2 i?2M b vb i? i {H(t, z)}t∈T,z∈K Bb M Ft @T`2/B+i #H2 biQ+? biB+ T`Q+2bb BM/2t2/ #v KX 6Q` b?Q`i QM2 r`Bi2b, H ∈ P (F· ) ⊗ KX
1t KTH2 8XRXN, G27i@+QMiBMmBiv M/ S`2/B+i #BHBivX +QKTH2t@p Hm2/ biQ+? biB+ T`Q+2bb {X(t)}t∈R / Ti2/ iQ {Ft }t∈R M/ rBi? H27i@+QMiBMmQmb i` D2+iQ@ `B2b Bb Ft @T`2/B+i #H2X AM 7 +i- #v H27i@+QMiBMmBiv- X(t, ω) = limn↑∞ Xn (t, ω)- r?2`2 Xn (t, ω) :=
n +n2
X(k2−n , ω)1(k2−n ,(k+1)2−n ] (t) ,
k=−n2n
M/ bBM+2 X(k2−n ) Bb Fk2−n @K2 bm` #H2- (t, ω) → Xn (t, ω) Bb P(F· )@K2 bm` #H2X
1t KTH2 8XRXRy, MQi?2` hvTB+ H Ft @T`2/B+i #H2 S`Q+2bbX G2i S M/ τ #2 irQ Ft @biQTTBM; iBK2b bm+? i? i S ≤ τ - M/ H2i ϕ : R+ ×R → R #2 K2 bm` #H2 7mM+iBQMX h?2M X(t, ω) := ϕ(S(ω), t)1{S(ω)Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu h?2Q`2K 8XRXRR G2i N #2 bBKTH2 TQBMi T`Q+2bb N QM R M/ H2i {Ft }t∈R #2 ?BbiQ`v Q7 N X G2i {λ(t)}t∈R #2 MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 Ft @T`Q;`2bbBp2Hv K2 bm` #H2 biQ+? biB+ T`Q+2bbX UBV A7 7Q` HH MQM@M2; iBp2 H27i@+QMiBMmQmb Ft @ / Ti2/ T`Q+2bb2b {H(t)}t∈R
E H(t) N (dt) = E H(t)λ(t) dt , U8XjV R
R
i?2M {λ(t)}t∈R Bb (P, Ft )@BMi2MbBiv Q7 N X UBBV A7 {λ(t)}t∈R Bb (P, Ft )@BMi2MbBiv Q7 N - U8XjV ?QH/b i`m2 7Q` HH MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb2b {H(t)}t∈R X S`QQ7X UBV G2i a ∈ R M/ H2i Tn(a)
:= inf t ≥ a ;
t
λ(s)ds ≥ n
.
a (a)
(a)
aBM+2 {λ(t)}t∈R Bb HQ+ HHv BMi2;` #H2- limn↑∞ Tn = +∞X HbQ- 7Q` 2 +? n ≥ 1- Tn Bb M Ft @biQTTBM; iBK2X G2i 7Q` [c, d] ⊆ [a, +∞) M/ C ∈ Fc H(t, ω) := 1C (ω)1(c,d] (t)1(a,Tn(a) (ω)] (t) .
h?Bb /2}M2b H27i@+QMiBMmQmb Ft @ / Ti2/ biQ+? biB+ T`Q+2bbX AMb2`iBM; i?Bb T`Q+2bb BMiQ U8XjV ;Bp2b E[1C N (c ∧
Tn(a) , d
∧
Tn(a) ]]
(a)
d∧Tn
= E 1C
(a)
λ(s)ds .
c∧Tn
aBM+2 C Bb `#Bi` `v BM Fc - i?Bb Bb 2[mBp H2Mi iQ U8XRVX (a)
UBBV G2i a ∈ R M/ H2i Tn #2 b BM .2}MBiBQM 8XRXRX .2}M2 7Q` 2 +? n ≥ 1 i?2 TQBMi T`Q+2bb Nn #v Nn (C) := N (C ∩ (a, Tn(a) ]) M/ H2i H #2 i?2 +QHH2+iBQM Q7 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb2b {H(t)}t∈R bm+? i? i (a)
Tn
H(t)Nn (dt) = E
E [a,+∞)
H(t)λ(t)dt .
U8X9V
a
H Bb d@bvbi2K Q7 7mM+iBQMb U.2}MBiBQM XRX8V +QMi BMBM; i?2 +QMbi Mi 1 b r2HH b i?2 7mM+iBQMb Q7 i?2 7Q`K 1D UD ∈ SV r?2`2 S := {C ×(c, d]; a < c < d < +∞, C ∈ Fc }X JQ`2Qp2`- S Bb π@bvbi2K ;2M2` iBM; P (F· )X h?2`27Q`2- #v .vMFBMǶb i?2Q`2K Uh?2Q`2K XRX9V- H +QMi BMb HH i?2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb2bX G2iiBM; n ↑ +∞ BM U8X9V M/ i?2M a ↓ −∞ ;Bp2b i?2 `2bmHi #v KQMQiQM2 +QMp2`;2M+2X
8XRX h>1 aJPPh>AL: 6P_JlGa
ReN
_2K `F 8XRXRk 6Q`KmH U8XjV Bb + HH2/ i?2 bKQQi?BM; 7Q`KmH X Ai b vb i? i i?2 dzMQM@bKQQi?Ǵ K2 bm`2 P (dω) N (ω, dt) M/ i?2 dzbKQQi?Ǵ K2 bm`2 P (dω)λ(t, ω) dt ;`22 QM P(F· )X 1t KTH2 8XRXRj, h` MbBiBQM hBK2b Q7 *QMiBMmQmb@iBK2 ?K+- h F2 RX _272` iQ i?2 +QMbi`m+iBQM BM a2+iBQM kX8 Q7 +QMiBMmQmb@iBK2 ?K+ {X(t)}t≥0 rBi? +QmMi #H2 bi i2 bT +2 E M/ r?Qb2 BM}MBi2bBK H ;2M2` iQ` A = {qij }i,j∈E Bb bi #H2 M/ +QMb2`p iBp2- M/ bmTTQb2 i? i i?Bb ?K+ Bb `2;mH ` UMQ ++mKmH iBQM TQBMi Q7 i?2 i` MbBiBQM iBK2bVX h?Bb +QMbi`m+iBQM Bb # b2/ QM 7 KBHv Nij Ui = jV Q7 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiB2b qij Ui = jV ij ;Bp2M #v M/ i?2 i` MbBiBQM iBK2b 7`QK bi i2 i iQ bi i2 j 7Q`K TQBMi T`Q+2bb N ij (t) = Zi (s−) Nij (ds) , N (0,t]
r?2`2 Zi (t) := 1{X(t)=i} X "v h?2Q`2K kXkXR- 7Q` HH MQM@M2; iBp2 H27i@+QMiBMmQmb FtX @ / Ti2/ T`Q+2bb2b {H(t)}t∈R
ij (dt) = E E H(t)N H(t)Zi (t−) Nij (dt) R R
=E H(t)Zi (t−) qij dt = E H(t)Zi (t) qij dt . R
R
ij (t) = Zi (t) qij X ij /KBib i?2 F X @BMi2MbBiv λ h?2`27Q`2- #v UBV Q7 h?2Q`2K 8XRXRR- N t
1tiBM+iBQM M/ S2`bBbi2M+2 1t KTH2 8XRXR9, aiQ+? biB+ 6 BHm`2 _ i2X 6Q` TQBMi T`Q+2bb N QM R M/ t Mv t ∈ R- H2i N−∞ ∪ ∅∞ t /2MQi2 i?2 TQBMi T`Q+2bb +QBM+B/BM; rBi? N QM (−∞, t] M/ rBi?Qmi TQBMib QM (t, +∞)X G2i N #2 bBKTH2 TQBMi T`Q+2bb QM R rBi? i?2 FtN @BMi2MbBiv {λ(t)}t∈R Q7 i?2 7Q`K t λ(t, ω) = v(t, (N−∞ ∪ ∅∞ t )(ω)) ,
r?2`2 v : R×M (R) → R+ Bb K2 bm` #H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(R)⊗M(R) M/ B(R+ )X h?2M- 7Q` HH [a, b] ⊂ R b a ∞ P (N ((a, b]) = 0 | Fa ) = exp − v(s, N−∞ ∪ ∅a ) ds , a
M/ BM T `iB+mH `- H2iiBM; b ↑ ∞-
P (N ((a, ∞)) = 0|Fa ) = exp −
∞ a
a v(s, N−∞ ∪ ∅∞ a ) ds
S`QQ7X G2i Z(t) := 1{N ((a,t])=0} 7Q` t ≥ aX h?2M U1t KTH2 RXeXRyV-
.
U8X8V
Rdy *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu Z(t) = 1 −
Z(s−)N (ds) . (a,t]
6Q` HH A ∈ FaN - #v h?2Q`2K 8XRXRR UbBM+2 (t, ω) → 1A (ω)Z(t−, ω) Bb P (F·N )@ K2 bm` #H2V
E[1A Z(t)] = P (A) − E 1A Z(s−)λ(s) ds (a,t]
1A Z(s)λ(s) ds = P (A) − E (a,t] t
s ∞ = P (A) − E 1A Z(s)v(s, N−∞ ∪ ∅s ) ds
a t a 1A Z(s)v(s, N−∞ ∪ ∅∞ ) ds = P (A) − E a
a t a E[Z(s) | Fa ]v(s, N−∞ ∪ ∅∞ ) ds . = P (A) − E 1A a a
h?2`27Q`2
t
E[Z(t) | Fa ] = 1 − a
M/ }M HHv
a E[Z(s) | Fa ]v(s, N−∞ ∪ ∅∞ a ) ds
t a v(s, N−∞ ∪ ∅∞ ) ds . E[Z(t) | Fa ] = exp − a a
h?2 b K2 T`QQ7- i?Bb iBK2 rBi? Z(t) := 1{N ((Tn ,Tn +t])=0} Ut > 0V- Bb 2 bBHv 2ti2M/2/ U1t2`+Bb2 8XNX3V iQ T`Qp2 i? i QM {Tn < ∞} Tn +h Tn ∞ P (N ((Tn , Tn + h]) = 0 | FTn ) = exp − v(s, N−∞ ∪ ∅Tn ) ds , Tn
i? i Bb P (Tn+1 − Tn >
h | FTNn )
= exp −
Tn +h Tn
Tn v(s, N−∞
∪
∅∞ Tn ) ds
.
h?Bb ;Bp2b i?2 BMi2`T`2i iBQM Q7 biQ+? biB+ BMi2MbBiv b biQ+? biB+ 7 BHm`2 ` i2X
1t KTH2 8XRXR8, h BH .Bbi`B#miBQM Q7 i?2 hBK2 iQ 1tiBM+iBQMX h?Bb 2t KTH2 +QMiBMm2b 1t KTH2 8XRXR9X Aib Tm`TQb2 Bb iQ ;Bp2 2biBK i2b Q7 i?2 i BH Q7 i?2 /Bbi`B#miBQM Q7 iBK2 T iQ 2tiBM+iBQM Q7 i?2 TQBMi T`Q+2bb i?2`2Q7X q2 ? p2 7Q` HH a ≥ a P (T ≥ a) = P (N ((a, ∞) > 0) = 1 − P (N ((a, ∞) = 0) , M/- #v U8X8V-
8XRX h>1 aJPPh>AL: 6P_JlGa
RdR
t a v(s, N−∞ ∪ ∅∞ ) ds . P (N ((a, ∞) = 0) = E exp − a a
h?2`27Q`2- bBM+2
1 x − x2 ≤ 1 − e−x ≤ x (x ≥ 0) 2
r2 Q#i BM i?2 #QmM/b ∞
a E v(s, N−∞ ∪ ∅∞ ) ds ≥ P (T ≥ a) a a
≥E
∞
a
a v(s, N−∞
∪
∅∞ a ) ds
∞
−E a
2 a v(s, N−∞
∪
∅∞ a ) ds
. U8XeV
h?2Q`2K XjXN ? b b +Q`QHH `v i?2 7QHHQrBM; p i ` Q7 i?2 "Q`2HĜ* Mi2HHB i?2Q`2KX h?2Q`2K 8XRXRe G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ft @BMi2MbBiv {λ(t)}t≥0 X h?2M ∞ λ(s) ds < ∞ P @ XbX N (∞) < ∞ ⇐⇒ 0
S`QQ7X TTHv h?2Q`2K XjXN U1t2`+Bb2 8XNXRjVX AM i?2 b2[m2H- QMHv T `iB H `2bmHi rBHH #2 M22/2/, ∞ P (N (0, ∞) = ∞) = 1 ⇐⇒ P λ(s) ds = ∞ = 1 . 0
6Q` b2H7@+QMi BM2/M2bb- Bi rBHH #2 T`Qp2/ rBi?Qmi `272`2M+2 iQ h?2Q`2K XjXNX ∞ X amTTQb2 i? i N (0, ∞) = ∞ XbX M/ i? i BiBb MQi i`m2 i? i 0 λ(s) ds = ∞ ∞ XbX h?2M- i?2`2 2tBbib c < ∞ bm+? i? i P 0 λ(s) ds ≤ c > 0X "v i?2 bKQQi?BM; 7Q`KmH ∞
∞
E 1{0s λ(u) du≤c} N (ds) = E 1{0s λ(u) du≤c} λ(s) ds . 0
0
h?2 `B;?i@? M/ bB/2 Bb #QmM/2/ #Qp2 #v c- r?2`2 b i?2 H27i@? M/ bB/2 Bb #QmM/2/ #2HQr #v E 1{0∞ λ(u) du≤c} N (0, ∞) = ∞ , ∞ +QMi` /B+iBQMX h?2`27Q`2 0 λ(s) ds = ∞ XbX ∞ "X amTTQb2 i? i 0 λ(s) ds = ∞ XbX M/ i? i Bi Bb MQi i`m2 i? i N (0, ∞) = ∞ XbX h?2M i?2`2 2tBbib c < ∞ bm+? i? i P (N (0, ∞) ≤ c) > 0X "v i?2 bKQQi?BM; 7Q`KmH ∞
∞
E 1{N ((0,s))≤c} N (ds) = E 1{N ((0,s))≤c} λ(s) ds 0
0
Rdk *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu Ur i+? i?2 T `2Mi?2b2b 7Q` H27i@+QMiBMmBivVX h?2 H27i@? M/ bB/2 Bb #QmM/2/ #v c + 1r?2`2 b i?2 `B;?i@? M/ bB/2 Bb #QmM/2/ #2HQr #v
∞ E 1{N (0,∞)≤c} λ(u) du = ∞ , 0
+QMi` /B+iBQMX h?2`27Q`2 N (0, ∞) = ∞ XbX
aiQ+? biB+ AMi2MbBiv E2`M2Hb h?2 MQiBQM Q7 biQ+? biB+ BMi2MbBiv rBHH MQr #2 2ti2M/2/ iQ TQBMi T`Q+2bb2b QM R×K r?2`2 (K, K) Bb bQK2 K2 bm` #H2 bT +2X h?2 K BM `QH2 Bb ii`B#mi2/ iQ i?2 }`bi UdziBK2ǴV +QQ`/BM i2 tBb RX G2i (Ω, F, P ) #2 T`Q# #BHBiv bT +2- M/ H2i (K, K) #2 K2 bm` #H2 bT +2X G2i λ : R × Ω × K → R+ #2 K TTBM; bm+? i? i • 7Q` HH ω ∈ Ω- HH t ∈ R- C → λ(ω, t, C) /2}M2b σ@}MBi2 K2 bm`2 QM (K, K)• 7Q` HH L ∈ K- i?2 K TTBM; (t, ω) → λ(t, ω, L) Bb K2 bm` #H2X h?2M λ(·, ·) Bb + HH2/ biQ+? biB+ F2`M2H 7`QK (R × Ω, B(R) ⊗ F) iQ (K, K)X A7 KQ`2Qp2`- 7Q` bQK2 ?BbiQ`v {Ft }t∈R M/ 7Q` HH L ∈ K- i?2 biQ+? biB+ T`Q+2bb {λ(t, L)}t∈R Bb Ft @ / Ti2/ U`2bTX Ft @T`Q;`2bbBp2- Ft @T`2/B+i #H2V- i?2 biQ+? biB+ F2`M2H λ Bb + HH2/ Ft @ / Ti2/ U`2bTX Ft @T`Q;`2bbBp2- Ft @T`2/B+i #H2VX A7 i?2`2 2tBbib b2[m2M+2 {Lk }k≥1 Q7 b2ib Q7 K BM+`2 bBM; iQ K M/ bm+? i? i 7Q` HH #QmM/2/ b BMi2`p Hb [a, b]- HH k ≥ 1- a λ(s, Lk )ds < ∞ XbX- i?2 F2`M2H λ Bb + HH2/ HQ+ HHv BMi2;` #H2X G2i MQr N #2 TQBMi T`Q+2bb QM R × KX .2}M2 7Q` HH t ∈ R FtN := σ{N (A) ; A ∈ B(R) ⊗ K, A ⊆ (−∞, t] × K} .
U8XdV
h?2 7 KBHv {FtN }t∈R Bb i?2 BMi2`M H ?BbiQ`v Q7 N M/ Mv ?BbiQ`v {Ft }t∈R bm+? i? i FtN ⊆ Ft 7Q` HH t ∈ R Bb + HH2/ ?BbiQ`v Q7 N X q2 b? HH HbQ mb2 i?2 MQi iBQM N L (C) := N (C × L)X .2}MBiBQM 8XRXRd TQBMi T`Q+2bb N QM R × K Bb + HH2/ bBKTH2 M/ HQ+ HHv }MBi2 B7 i?2`2 2tBbib b2[m2M+2 {Lk }k≥1 BM K BM+`2 bBM; iQ K M/ bm+? i? i 7Q` HH k ≥ 1N Lk Bb bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM RX .2}MBiBQM 8XRXR3 G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R × K M/ H2i {Ft }t∈R #2 ?BbiQ`v Q7 N X G2i λ(·, ·) #2 M Ft @ / Ti2/ HQ+ HHv BMi2;` #H2 biQ+? biB+ F2`M2H 7`QK (R × Ω, B ⊗ F ) iQ (K, K)X h?2 TQBMi T`Q+2bb N Bb b B/ iQ /KBi i?2 biQ+? biB+ (P, Ft )@BMi2MbBiv F2`M2H λ(·, ·) B7 7Q` HH [a, b] ⊂ R- HH L ∈ K M/ HH k ≥ 1- i?2 TQBMi T`Q+2bb N L∩Lk /KBib i?2 biQ+? biB+ (P, Ft V@BMi2MbBiv {λ(t, L∩Lk )}t∈R X q2 b? HH Q7i2M r`Bi2 dzi?2 Ft @BMi2MbBiv F2`M2H λ(t, dz)Ǵ BMbi2 / Q7 dzi?2 Ft @ BMi2MbBiv F2`M2H λ(·, ·)ǴX
8XRX h>1 aJPPh>AL: 6P_JlGa
Rdj
_2K `F 8XRXRN LQi2 i? i i?2 b2[m2M+2b {Lk }k≥1 Q7 i?2 /2}MBiBQM Q7 HQ+ HHv BMi2;` #H2 F2`M2H M/ Q7 .2}MBiBQMb 8XRXRd M/ 8XRXR3 + M Hr vb #2 i F2M B/2MiB+ H rBi?Qmi HQbb Q7 ;2M2` HBivX h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb M 2 bv / Ti iBQM Q7 i? i Q7 h?2Q`2K 8XRXRRX h?2Q`2K 8XRXky G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R×KX Ai /KBib i?2 biQ+? biB+ Ft @BMi2MbBiv F2`M2H λ(·, ·) B7 M/ QMHv B7
H(t, z)N (dt × dz) = E H(t, z)λ(t, dz)dt U8X3V E R×K
R×K
7Q` HH MQM@M2; iBp2 P(F· ) ⊗ K@K2 bm` #H2 7mM+iBQMb H : R × Ω × K → RX 6Q` i?2 dzB7Ǵ T `i r2 M22/ iQ b iBb7v U8X3V QMHv 7Q` HH MQM@M2; iBp2 B(R)⊗F ⊗K@K2 bm` #H2 7mM+iBQMb i? i `2 / Ti2/ iQ {Ft }t∈R M/ H27i@+QMiBMmQmb BM i?2 t@p `B #H2X h?2 7Q`KmH U8X3V Bb + HH2/ i?2 bKQQi?BM; 7Q`KmH 7Q` iBK2@bT +2 TQBMi T`Q+2bb2bX M BKK2/B i2 +Q`QHH `v Q7 h?2Q`2K 8XRXky +QM+2`Mb +QKTH2t@p Hm2/ BMi2;` M/b, *Q`QHH `v 8XRXkR G2i N #2 b BM h?2Q`2K 8XRXky- M/ H2i H ∈ P(F· ) ⊗ K #2 +QKTH2t@p Hm2/ 7mM+iBQM bm+? i? i i H2 bi QM2 Q7 i?2 7QHHQrBM; bi i2K2Mib Bb i`m2,
|H(t, z)|N (dt × dz) < ∞ , E R×K
|H(t, z)|λ(t, dz)dt < ∞ .
E R
K
h?2M i?2 Qi?2` bi i2K2Mi Bb HbQ i`m2- i?2 BMi2;` Hb R K H(t, z)N (dt × dz) M/ H(t, z)λ(t, dz)dt `2 r2HH /2}M2/ M/ }MBi2- M/ 2[m HBiv U8X3V ?QH/bX R K h?2 * b2 Q7 J `F2/ SQBMi S`Q+2bb2b aQ 7 `- i?2 T`QD2+iBQM QM R Q7 N M22/ MQi #2 HQ+ HHv }MBi2 TQBMi T`Q+2bb- i? i BbN (C) := N (C × K) K v #2 BM}MBi2 7Q` bQK2 #QmM/2/ "Q`2H b2ib C U7Q` BMbi M+2 B7 N Bb M ?TT Q7 BMi2MbBiv 1VX h?Bb Bb r?v QM2 M22/b i?2 Lk ǶbX h?2 + b2 Q7 K `F2/ TQBMi T`Q+2bb Bb /Bz2`2Mi 7`QK i?Bb TQBMi Q7 pB2r- #mi Bi Bb M2p2`i?2H2bb T `iB+mH ` + b2 iQ r?B+? i?2 ;2M2` H `2bmHib TTHvX _2+ HH i?2 7QHHQrBM; /2}MBiBQM, .2}MBiBQM 8XRXkk G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R- + HH2/ i?2 # b2- rBi? iBK2 b2[m2M+2 {Tn }n∈Z X G2i Z := {Zn }n∈Z #2 ` M/QK b2[m2M+2 rBi? p Hm2b BM bQK2 K2 bm` #H2 bT +2 (K, K)X q2 i?2M b v i? i (N, Z) Bb bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb rBi? K `Fb BM K- M/ /2}M2 i?2 bbQ+B i2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × K #v 1A ((Tn , Zn )) (A ∈ B ⊗ K) . NZ (A) := n∈Z
Rd9 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu U_2K2K#2` i?2 +QMp2MiBQM ++Q`/BM; iQ r?B+? i?2 bmK 2ti2M/b QMHv iQ i?Qb2 BM/B+2b n 7Q` r?B+? |Tn | < ∞XV _2TH +BM; N #v NZ BM U8XdV- r2 Q#i BM i?2 /2}MBiBQM Q7 FtNZ - r?B+? r2 b? HH HbQ r`Bi2 b FtN,Z X .2}MBiBQM 8XRXkj G2i (N, Z) #2 bBKTH2 M/ HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb rBi? K `Fb BM K M/ rBi? bbQ+B i2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × KX G2i {Ft }t∈R #2 ?BbiQ`v Q7 NZ X M Ft @ / Ti2/ biQ+? biB+ F2`M2H 7`QK (R × Ω, B(R) ⊗ F) iQ (K, K) Bb + HH2/ (P, Ft )@BMi2MbBiv F2`M2H Q7 (N, Z) B7 Bi Bb (P, Ft )@BMi2MbBiv F2`M2H Q7 NZ X LQi2 i? i- BM i?Bb + b2- i?2`2 Bb MQ M22/ 7Q` i?2 b2[m2M+2 {Lk }k≥1 BM .2}MBiBQM 8XRXR3 bBM+2 i?2 TQBMi T`Q+2bb NZL QM R /2}M2/ #v NZL (C) := NZ (C × L) =
1C (Tn )1L (Zn )
(C ∈ B, L ∈ K)
n∈Z
Bb bBKTH2 M/ HQ+ HHv }MBi2X G2i (N, Z) #2 bBKTH2 M/ HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb rBi? K `Fb BM K- rBi? bbQ+B i2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × K- M/ bmTTQb2 i? i Bi /KBib i?2 Ft @biQ+? biB+ F2`M2H λ(t, dz)X AM T `iB+mH `- i?2 biQ+? biB+ T`Q+2bb λ(t) := λ(t, K) Bb i?2 Ft @BMi2MbBiv Q7 N X q2 MQr /2}M2 Φ(t, C) := rBi? i?2 +QMp2MiBQM
0 0
λ(t, C) , λ(t, K)
= 0- bQ i? i λ(t, dz) = λ(t)Φ(t, dz).
h?2 K `F2/ TQBMi T`Q+2bb (N, Z) Bb i?2M b B/ iQ ? p2 i?2 HQ+ H Ft @+? ` +i2`BbiB+b (λ(t), Φ(t, dz))X A7 Ft = FtN,Z 7Q` HH t ≥ 0- i?2 BMi2`T`2i iBQM Q7 Φ Bb i? i QM {Tn < ∞}Φ(Tn , L) = P (Zn ∈ L | FTn −) Uh?2Q`2K 8XkX9 #2HQrV r?2`2 r2 `2+ HH i? i FTn − := σ(Tk , Zk ; k ≤ n − 1) ∨ σ(Tn )X 1t KTH2 8XRXk9, SQBbbQM rBi? AM/2T2M/2Mi BB/ J `FbX A7 i?2 # b2 TQBMi T`Q+2bb N Bb SQBbbQM T`Q+2bb rBi? U/2i2`KBMBbiB+V BMi2MbBiv λ(t)- M/ B7 i?2 b2@ [m2M+2 Q7 K `Fb Bb BB/ M/ BM/2T2M/2Mi Q7 N - (N, Z) /KBib i?2 FtN,Z @BMi2MbBiv F2`@ M2H λ(t)Q(dz) Q`- 2[mBp H2MiHv- i?2 HQ+ H FtN,Z @+? ` +i2`BbiB+b (λ(t), Q(dz))- r?2`2 Q Bb i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 K `Fb U1t2`+Bb2 8XNXdVX
8XRX h>1 aJPPh>AL: 6P_JlGa
Rd8
1t KTH2 8XRXk8, JmHiBp `B i2 SQBMi S`Q+2bb2b- h F2 RX h?2 /2}MBiBQMb M/ `2bmHib #Qp2 rBHH MQr #2 bT2+B HBx2/ iQ KmHiBp `B i2 TQBMi T`Q+2bb2b #v H2iiBM; K = E- /2MmK2` #H2 b2iX 6Q` 2 +? i ∈ E- /2}M2 TQBMi T`Q+2bb Ni QM R #v Ni (C) := N (C × {i}) 7Q` HH C ∈ B(R)- M/ H2i λi (t) := λ(t, {i})X h?Bb /2}M2b KmHiBp `B i2 TQBMi T`Q+2bb {Ni }i∈E - 2 +? Ni ? pBM; i?2 Ft @BMi2MbBiv {λi (t)}t∈R r?2`2 {Ft }t∈R Bb +QKKQM ?BbiQ`v Q7 i?2 Ni ǶbX
1t KTH2 8XRXke, MQi?2` HQQF i i?2 GQiF ėoQHi2`` TQTmH iBQM KQ/2HX M p i ` Q7 i?2 GQiF ĜoQHi2`` KQ/2H U1t KTH2 9XRXRRV Bb b 7QHHQrbX _2+ HH i? i i?Bb KQ/2H +QM+2`Mb TQTmH iBQM Q7 rQK2M 7i2` ;Bp2M iBK2 Ub v- iBK2 0VX rQK M #Q`M i iBK2 a > 0 ;Bp2b #B`i? /m`BM; ?2` HB72iBK2 iQ ;B`Hb ++Q`/BM; iQ SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM μ(t − a)X h?2 ` M/QK HB72iBK2b Q7 rQK2M `2 BM/2T2M/2Mi M/ ? p2 +QKKQM /Bbi`B#miBQM QX h?2 72KBMBM2 T`Q;2Mv Q7 HH rQK2M #Q`M #27Q`2 i?2 Q`B;BM Q7 iBK2 7Q`Kb QM i?2 TQbBiBp2 ? H7@HBM2 SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ0 (t)X G2iiBM; σn #2 i?2 HB72iBK2 Q7 i?2 rQK M #Q`M i iBK2 Tn > 0 Un ≥ 1V- i?2 TQBMi T`Q+2bb Q7 #B`i?b QM R+ ? b 7Q` BMi2MbBiv μ(t − Tn )1{σn >t−Tn } . λ(t) = λ0 (t) + n≥1
1t KTH2 8XRXkd, ai`2bb _2H2 b2 JQ/2HX h?2 Q`B;BM H TT2HH iBQM Q7 i?Bb KQ/2H Bb dzb2H7@+Q``2+iBM; TQBMi T`Q+2bbǴXj Aib biQ+? biB+ Ft @BMi2MbBiv Bb N (0,t)
λ(t) := eX(0)+ct−
i=1
Zi
,
r?2`2 X(0) Bb ;Bp2M ` M/QK p `B #H2- c > 0- M/ {Zn }n≥1 Bb M BB/ b2[m2M+2 Q7 MQM@M2; iBp2 `2 H ` M/QK p `B #H2bX >2`2 (N,Z)
Ft := σ(X(0)) ∨ Ft
,
r?2`2 (N, Z) Bb i?2 K `F2/ TQBMi T`Q+2bb rBi? # b2 N M/ K `F b2[m2M+2 {Zn }n≥1 X h?Bb KQ/2H M/ Bib `2}M2K2Mib `2 mb2/ iQ /2b+`B#2 b2BbKB+ +iBpBivX 1t KTH2 8XRXk3, h` MbBiBQM hBK2b Q7 *QMiBMmQmb@iBK2 ?K+- h F2 , Z) kX h?2 ?K+ Q7 1t KTH2 8XRXRj + M #2 pB2r2/ b K `F2/ TQBMi T`Q+2bb (N r?2`2 N +QmMib HH i?2 i` MbBiBQMb M/ Zn := X(Tn )X G2i Nij #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 i` MbBiBQMb 7`QK bi i2 i iQ jX _2+ HH i?2 MQi iBQM Zi (t) = 1{X(t)=i} X /KBib i?2 FtN @BMi2MbBiv h?2 # b2 TQBMi T`Q+2bb N ij (t) = = qij Zi (t) = qi Zi (t) = qX(t) . λ λ(t) i,j∈E ; i=j
i∈E j , j=i
i∈E
j (Ab? K M/ q2bi+Qii- RNdN)X T`QQ7 Q7 2tBbi2M+2 Q7 bi iBQM `v p2`bBQM Bb ;Bp2M BM (G bikyyy)X
Rde *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu i (t) #2 i?2 F X @BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb N i +QmMiBM; i?2 i` MbBiBQMb G2i λ t
#`BM;BM; i?2 ?K+ BMiQ bi i2 iX b Ni = k∈E ; k=i Nki
i (t) = λ
Zk (t) qki = qX(t),i .
k∈E ; k=i
h?2 7QHHQrBM; +Q`QHH `v Dmbi `2T?` b2b h?2Q`2K 8XRXkyX h?2Q`2K 8XRXkN G2i (N, Z) #2 bBKTH2 K `F2/ TQBMi T`Q+2bb QM R rBi? K `Fb BM K M/ /KBiiBM; i?2 Ft @biQ+? biB+ F2`M2H λ(t, dz)X h?2M
H(Tn , Zn ) = E H(t, z)λ(t, dz) E R
n∈Z
K
7Q` HH MQM@M2; iBp2 P(F· ) ⊗ K@K2 bm` #H2 7mM+iBQMb H : R × Ω × K → RX
1t KTH2 8XRXjy, J2 M "mbv S2`BQ/ BM i?2 M/GI/1/∞ Zm2m2X G2i i?2 b2[m2M+2 {(Tn , σn )}n∈N #2 M M/GI BMTmi ~QrX h?Bb K2 Mb i? i {Tn }n∈N\{0} Bb i?2 2p2Mi@iBK2b b2[m2M+2 Q7 M ?TT A rBi? BMi2MbBiv λ- T0 := 0- M/ {σn }n∈N Bb M BB/ b2[m2M+2 Q7 MQM@M2; iBp2 ` M/QK p `B #H2b rBi? +QKKQM +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM G- BM/2T2M/2Mi Q7 AX AM [m2m2BM; i?2Q`v- A +QmMib i?2 ``Bp H iBK2b BM dzbvbi2KǴ U+ HH2/ i?2 [m2m2V M/ σn Bb i?2 b2`pB+2 `2[mB`2/ #v i?2 +mbiQK2` ``BpBM; i iBK2 Tn X G2i W (t) #2 i?2 rQ`FHQ / i iBK2 t- i? i Bb- i?2 iQi H KQmMi Q7 b2`pB+2 `2K BMBM; iQ #2 /QM2 i iBK2 t- bbmKBM; i? i b2`pB+2 Bb T`QpB/2/ i mMBi ` i2 b HQM; i? i i?2`2 Bb bQK2 rQ`F iQ #2 /QM2X JQ`2 2tTHB+BiHv- 7Q` t ≥ 0 t W (t) = σ0 + σn 1(0,t] (Tn ) − 1{W (s)>0} ds . 0
n≥1
G2i R1 #2 i?2 }`bi bi`B+iHv TQbBiBp2 iBK2 i r?B+? i?2 bvbi2K Bb 2KTiv U∞ B7 i?2 bvbi2K M2p2` 2KTiB2bVX W (t)
σ1
σ2
σ4 σ3
σ0 0
t T1
T2
T3
T4
R1
*H2 `Hv R1 Bb M FtN @biQTTBM; iBK2- r?2`2 N Bb i?2 TQBMi T`Q+2bb QM R+ × R+ rBi? TQBMi b2[m2M+2 {(Tn , σn )}n∈N X 6Q` HH M > 0 ∞ R1 ∧ M ≤ σ0 + σk 1(0,R1 ∧M ] (Tk ) = σ0 + σ1(0,R1 ∧M ] (t)N (dt × dσ) . U8XNV k≥1
0
E
8XRX h>1 aJPPh>AL: 6P_JlGa h?2 K TTBM;
Rdd
(t, ω, σ) → H(t, ω, σ) := σ1(0,R1 (ω)∧M ] (t),
Bb BM P(F·N ) ⊗ B(R+ ) UMQiBM; i? i Bi Bb i?2 FtN @BMi2MbBiv F2`M2H Q7 N Bb λG(dz)-
H27i@+QMiBMmQmb BM i?2 t@ `;mK2MiVX aBM+2 r2 Q#i BM 7`QK U8XNV M/ h?2Q`2K 8XRXkN ∞
σ1(0,R1 ∧M ] (t)λG(dσ) dt E[R1 ∧ M ] ≤ E[σ0 ] + E R+
0
= E[σ0 ] + λE[σ0 ]E[R1 ∧ M ].
U8XRyV
AM T `iB+mH `- E[R1 ∧ M ](1 − λE[σ0 ]) ≤ E[σ0 ]- M/ i?2`27Q`2- B7 λE[σ0 ] < 1E[R1 ∧ M ] ≤
E[σ0 ] 1 − λE[σ0 ]
E[σ0 ] < ∞X _2T`Q/m+BM; i?2 + H+mH iBQM 7Q` HH M > 0- M/ i?2`27Q`2 E [R1 ] ≤ 1−λE[σ 0] rBi? R1 `2TH +BM; R1 ∧ M - r2 ? p2- bBM+2 R1 Bb HKQbi bm`2Hv }MBi2- i?2 2[m HBiv
E[R1 ] =
E[σ0 ] . 1 − λE[σ0 ]
AM bmKK `v, λE[σ0 ] < 1 Bb bm{+B2Mi +QM/BiBQM 7Q` E[R1 ] iQ #2 }MBi2 BM M/GI/1/∞ [m2m2X Ai Bb HbQ M2+2bb `v +QM/BiBQM B7 E[σ0 ] > 0- #2+ mb2 r?2M R1 Bb }MBi2 R1 = σ0 + σk 1(0,R1 ] (Tk ) k≥1
M/ i?2`27Q`2 E[R1 ] = E[σ0 ]+λE[σ0 ]E[R1 ]- i? i Bb E[R1 ](1−λE[σ0 ]) = E[σ0 ]- r?B+? BKTHB2b i? i 1 − λE[σ0 ] > 0X G2i (N, Z) #2 bBKTH2 HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K) M/ H2i {Ft }t≥0 #2 ?BbiQ`v Q7 (N, Z)X G2i (N, Z) /KBi i?2 Ft @BMi2MbBiv F2`M2H λ(t, dz)X AM i?2 b2[m2H- i?2 7QHHQrBM; MQi iBQM Bb mb2/ H(s, z) MZ (ds × dz) (0,t]×K H(s, z) NZ (ds × dz) − H(s, z) λ(s, dz) ds , := (0,t]×K
(0,t]×K
T`QpB/2/ i?2 `B;?i@? M/ bB/2 Bb r2HH /2}M2/X h?2`27Q`2- 7Q`K HHvMZ (ds × dz) := NZ (ds × dz) − λ(s, dz) ds . aiQ+? biB+ AMi2;` Hb M/ J `iBM; H2b h?2Q`2K 8XRXjR G2i {H(t, z)}t≥0 #2 `2 H@p Hm2/ Ft @T`2/B+i #H2 biQ+? biB+ T`Q@ +2bb BM/2t2/ #v K bm+? i? i 7Q` HH t ≥ 0
E |H(s, z)|λ(s, dz) ds < ∞ . U8XRRV (0,t]×K
Rd3 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu h?2M i?2 biQ+? biB+ T`Q+2bb H(s, z) MZ (ds × dz)
M (t) :=
U8XRkV
(0,t]×K
Bb r2HH@/2}M2/ +2Mi2`2/ Ft @K `iBM; H2X |H(s, z)| NZ (ds × dz) < ∞ 7Q` HH t ≥ 0 Uh?2Q`2K 8XRXkNVX h?2`27Q`2- 7Q` HH t ≥ 0- (0,t]×K |H(s, z)|λ(s, dz) ds < ∞ M/ (0,t]×K |H(s, z)| NZ (ds × dz) < ∞- S@ XbX AM T `iB+mH `- {M (t)}t≥0 Bb S@ XbX r2HH@/2}M2/ M/ }MBi2 biQ+? biB+ T`Q+2bbX 6Q` HH a, b ∈ R+ U0 ≤ a ≤ bV M/ 7Q` HH A ∈ Fa
E [1A (M (b) − M (a)] = E H (t, z) MZ (ds × dz) , S`QQ7X *QM/BiBQM U8XRRV Bb 2[mBp H2Mi iQ E
(0,t]×K
R+ ×K
r?2`2 H (t, z) := H(t, z)1A 1(a,b] (t) /2}M2b M Ft @T`2/B+i #H2 `2 H@p Hm2/ biQ+? biB+ T`Q+2bb BM/2t2/ #v KX "v h?2Q`2K 8XRXkN
E H (t, z) NZ (ds × dz) = E H (t, z) λ(s, dz) ds R+ ×K
R+ ×K
M/ i?2`27Q`2- 7Q` HH A ∈ Fa - E [1A (M (b) − M (a)] = 0X
*Q`QHH `v 8XRXjk _2TH +BM; bbmKTiBQM U8XRRV Q7 h?2Q`2K 8XRXjR #v i?2 +QM/B@ iBQM i? i P @ HKQbi bm`2Hv |H(s, z)|λ(s, dz) ds < ∞ 7Q` HH t ≥ 0 , U8XRjV (0,t]×K
i?2 biQ+? biB+ T`Q+2bb {M (t)}t≥0 /2}M2/ #v U8XRkV Bb i?2M HQ+ H Ft @K `iBM; H2X S`QQ7X G2i 7Q` 2 +? BMi2;2` n ≥ 1 i?2 Ft @biQTTBM; iBK2b Sn := inf{t > 0 ; |H(s, z)|λ(s, dz) ds ≥ n} , (0,t]×K
rBi? i?2 mbm H +QMp2MiBQM inf ∅ = +∞X h?2M limn↑∞ Sn = ∞- M/ KQ`2Qp2` H(t, z)1{t≤Sn } b iBb}2b +QM/BiBQM U8XRRVX h?2`27Q`2 #v h?2Q`2K 8XRXjR{M (t ∧ Sn )}t≥0 Bb r2HH@/2}M2/ Ft @K `iBM; H2X h?2Q`2K 8XRXjj G2i H #2 M Ft @T`2/B+i #H2 `2 H@p Hm2/ biQ+? biB+ T`Q+2bb BM/2t2/ #v Kbm+? i? i 7Q` HH t ≥ 0- S@ XbX
E |H(s, z)|2 λ(s, dz) ds < ∞ . U8XR9V (0,t]×K
8XRX h>1 aJPPh>AL: 6P_JlGa
RdN
h?2M i?2 biQ+? biB+ T`Q+2bb H(s, z) MZ (ds × dz)
M (t) := (0,t]×K
Bb r2HH /2}M2/ M/ b[m `2@BMi2;` #H2 Ft @K `iBM; H2X JQ`2Qp2`
|H(s, z)|2 λ(s, dz) ds . E M (t)2 = E
U8XR8V
(0,t]×K
S`QQ7X G2i Tn #2 i?2 n@i? 2p2Mi iBK2 Q7 i?2 # b2 TQBMi T`Q+2bbX h?2 T`QQ7 i? i M (t) Bb r2HH /2}M2/ 7QHHQrb 7`QK h?2Q`2K 8XRXjRX AM 7 +i- Q#b2`pBM; i? i E (0,Tn ] λ(s) ds = E [N (0, Tn ]] = n E
|H(s, z)|λ(s, dz) ds ≤ E (1 + |H(s, z)|2 )λ(s, dz) ds (0,t∧Tn ]×K (0,t∧Tn ]×K
|H(s, z)|2 λ(s, dz) ds < ∞ = E [N (0, Tn ]] + E (0,t∧Tn ]×K
2 |H(s, z)| λ(s, dz) ds < ∞ . =n+E (0,t∧Tn ]×K
h?2`27Q`2 {M (t∧Tn )}t≥0 Bb r2HH /2}M2/- M/ bQ Bb {M (t)}t≥0 bBM+2 limn↑∞ Tn = ∞X q2 MQr im`M iQ i?2 T`QQ7 Q7 U8XR8VX "v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb M (t)2 = M (t−) dM (t) + H(s, z)2 NZ (ds × dz) . (0,t]
aBM+2
(0,t]×K
M (s−)H(s, z) MZ (ds × dz)
M (t−) dM (t) =
m(t) := (0,t]
(0,t]×K
Bb HQ+ H Ft @K `iBM; H2 rBi? `2bT2+i iQ i?2 HQ+ HBxBM; biQTTBM; iBK2b Vn := inf t ≥ 0 ; |M (t−)| + |H(s, z)|λ(s, dz) ds ≥ n ∧ Tn , (0,t]×K
r2 ? p2 i? i
H(s, z)2 NZ (ds × dz) (0,t∧Vn ]×K
2 H(s, z) λ(s, dz) ds , =E
E M (t ∧ Vn )2 = E
(0,t∧Vn ]×K
7`QK r?B+? U8XR8V 7QHHQrb- B7 r2 + M b?Qr i? i limn↑∞ E [M (t ∧ Vn )2 ] = E [M (t)2 ]X h?Bb rBHH #2 i?2 + b2 #2+ mb2 U b r2 b? HH bQQM T`Qp2V B7 M (t ∧ Vn ) +QMp2`;2b
R3y *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu BM L2C (P ) iQ bQK2 HBKBiX h?Bb HBKBi Bb M2+2bb `BHv M (t)- i?2 HKQbi bm`2 HBKBi Q7 M (t ∧ Vn )X h?2 L2C (P )@+QMp2`;2M+2 Q7 M (t ∧ Vn ) iQ #2 T`Qp2/ 7QHHQrb 7`QK i?2 * m+?v +`Bi2`BQM bBM+2- #v +QKTmi iBQM bBKBH ` iQ i?2 QM2 #Qp2- rBi? m ≥ n E (M (t ∧ Vm ) − M (t ∧ Vn ))2
H(s, z)2 λ(s, dz) ds = E [m(t ∧ Vm ) − m(t ∧ Vn )] + E (t∧Vn ,t∧Vm ]×K
H(s, z)2 λ(s, dz) ds , =E (t∧Vn ,t∧Vm ]×K
[m MiBiv i? i p MBb?2b b m, n ↑ ∞X
*Q`QHH `v 8XRXj9 A7 i?2 bbmKTiBQM U8XR9V Q7 h?2Q`2K 8XRXjj Bb `2TH +2/ #v |H(s, z)|2 λ(s, dz) ds < ∞ , P @ XbX (t ≥ 0) , U8XReV (0,t]×K
i?2 biQ+? biB+ T`Q+2bb H(s, z) MZ (ds × dz)
M (t) := (0,t]×K
Bb b[m `2@BMi2;` #H2 HQ+ H Ft @K `iBM; H2X S`QQ7X h?Bb 7QHHQrb 7`QK h?2Q`2K 8XRXjj BM i?2 b K2 K MM2` b *Q`QHH `v 8XRXjk Bb /2`Bp2/ 7`QK h?2Q`2K 8XRXjRX .2T2M/2M+2 mTQM i?2 >BbiQ`v AM i?2 /2}MBiBQM Q7 i?2 Ft @biQ+? biB+ BMi2MbBiv {λ(t)}t∈R Q7 i?2 bBKTH2 TQBMi T`Q+2bb N - i?2 KBMBK H `2[mB`2K2Mi QM i?2 ?BbiQ`v {Ft }t∈R Bb Ft ⊇ Ftλ ∨ FtN
U8XRdV
#2+ mb2 i?2 BMi2MbBiv T`Q+2bb M/ i?2 TQBMi T`Q+2bb Bib2H7 `2 Ft @ / Ti2/X q? i ? TT2Mb B7 r2 KQ/B7v i?2 ?BbiQ`v\ q2 }`bi Mbr2` i?2 7QHHQrBM; [m2biBQM, hQ r? i 2ti2Mi + M r2 +? M;2 i?2 ?BbiQ`v M/ `2i BM i?2 b K2 biQ+? biB+ BMi2MbBiv\ h?2Q`2K 8XRXj8 G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2M, UBV A7 {Gt }t∈R Bb ?BbiQ`v bm+? i? i G∞ Bb BM/2T2M/2Mi Q7 F∞ - {λ(t)}t∈R Bb HbQ M Ft ∨ Gt @BMi2MbBiv Q7 N X UBBV A7 {Ft }t∈R Bb ?BbiQ`v Q7 N bm+? i? i Ftλ ∨ FtN ⊆ Ft ⊆ Ft - {λ(t)}t∈R Bb HbQ M Ft @BMi2MbBiv Q7 N X
8XRX h>1 aJPPh>AL: 6P_JlGa
R3R
S`QQ7X 1t2`+Bb2 8XNX8X
q2 MQr +QMbB/2` bBim iBQMb r?2`2 i?2 biQ+? biB+ BMi2MbBiv Bb KQ/B}2/ #v +? M;2 Q7 ?BbiQ`vX G2i N ? p2 i?2 Ft @BMi2MbBiv {λ(t)}t∈R - r?2`2 i?2 ?BbiQ`v {Ft }t∈R b iBb}2b i?2 KBMBK H `2[mB`2K2Mi U8XRdVX amTTQb2 i? i i?2 M2r ?BbiQ`v {Ft }t∈R Bb bK HH2`- i? i BbFt ⊇ Ft ⊇ FtN . U8XR3V Ai Bb TQbbB#H2 i? i {λ(t)}t∈R Bb MQi Ft @ / Ti2/ M/ i?2`27Q`2 + MMQi #2 i?2 Ft @ BMi2MbBivX L2p2`i?2H2bb- i?2`2 biBHH 2tBbib M Ft @BMi2MbBiv- M/ i?Bb BMi2MbBiv Bb Q#@ i BM2/ #v dzT`QD2+iBQMǴ Q7 i?2 BMBiB H biQ+? biB+ BMi2MbBiv QM i?2 bK HH2` ?BbiQ`v- BM b2Mb2 iQ #2 /2b+`B#2/ MQrX *QMbB/2` i?2 ?BbiQ`B2b {Ft }t∈R M/ {Ft }t∈R b iBb7vBM; +QM/BiBQM U8XR3V 7Q` HH t ∈ RX G2i {Y (t)}t∈R #2 MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 K2 bm` #H2 T`Q+2bbX G2i i?2 σ@}MBi2 K2 bm`2b μ1 M/ μ2 QM (R × Ω, P(F· )) #2 /2}M2/ `2bT2+iBp2Hv #v,
μ1 (H) := E H(t)Y (t)dt M/ μ2 (H) := E H(t)dt R
R
7Q` HH MQM@M2; iBp2 H : R × Ω i? i `2 P(F)@K2 bm` #H2X LQi2 i? i μ2 Bb i?2 T`Q/m+i K2 bm`2 P × QM (Ω × R, P(F· ))X *H2 `Hv μ1 μ2 - M/ i?2`27Q`2 i?2`2 1 2tBbib _ /QMĜLBFQ/ɷK /2`Bp iBp2 Y (t, ω) = dμ (t, ω) i? i Bb P(F· )@K2 bm` #H2 dμ2 U M/ i?2`27Q`2 /2}M2b Ft ĜT`2/B+i #H2 T`Q+2bb {Y (t)}t∈R V bm+? i? i μ1 (H) = μ2 (Y H) = E
H(t)Y (t) dt . R
JQ`2Qp2`- i?Bb _ /QMĜLBFQ/ɷK /2`Bp iBp2 Bb μ2 @mMB[m2- i? i Bb iQ b v- B7 i?2`2 2tBbib MQi?2` bm+? _ /QMĜLBFQ/ɷK /2`Bp iBp2- b v Y - i?2M Y (t, ω) = Y (t, ω) ,
P (dω) × dt
X2X
U8XRNV
.2}MBiBQM 8XRXje h?2 #Qp2 biQ+? biB+ T`Q+2bb {Y (t)}t∈R Bb + HH2/ i?2 T`2/B+i #H2 T`QD2+iBQM Q7 {Y (t)}t∈R QM {Ft }t∈R - Q` i?2 Ft @T`2/B+i #H2 T`QD2+iBQM9 Q7 {Y (t)}t∈R X h?2Q`2K 8XRXjd G2i i?2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb N QM R ? p2 i?2 biQ+? biB+ Ft @BMi2MbBiv {λ(t)}t∈R X G2i {Ft }t∈R #2 MQi?2` ?BbiQ`v- b iBb7vBM; +QM/B@ t @BMi2MbBiv {λ(t)} iBQM U8XR3VX h?2M N /KBib i?2 biQ+? biB+ F t∈R - i?2 Ft @T`2/B+i #H2 T`QD2+iBQM Q7 {λ(t)}t∈R X S`QQ7X G2i {H(t)}t∈R #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bbX Ai Bb 7Q`iBQ`B M Ft @T`2/B+i #H2 T`Q+2bb- M/ i?2`27Q`2 9
h?Bb i2`KBMQHQ;v Bb MQi i?2 bi M/ `/ QM2 BM i?2 ;2M2` H i?2Q`v Q7 biQ+? biB+ T`Q+2bb2bX
R3k *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
dt . H(t) N (dt) = E H(t) λ(t) dt = E H(t) λ(t)
E R
R
R
6Q` ;Bp2M ?BbiQ`v {Ft }t∈R M/ ;Bp2M T`Q# #BHBiv P - i?2`2 2tBbi K Mv p2`bBQMb Q7 i?2 (P, Ft )@BMi2MbBiv (λ(t))t∈R X am+? M BMi2MbBiv + M Hr vb #22M +?Qb2M iQ #2 Ft @T`2/B+i #H2- bBM+2 B7 Bi Bb MQi- QM2 + M `2TH +2 Bi #v Bib Ft @T`2/B+i #H2 T`QD2+iBQMr?B+? 2tBbib M/ Bb 2bb2MiB HHv mMB[m2 BM b2Mb2 iQ #2 /2b+`B#2/ #2HQrX h?2Q`2K 8XRXj3 V h?2 Ft @T`2/B+i #H2 p2`bBQM Bb P (dω) dt@mMB[m2 M/ i?2`27Q`2 P (dω) N (ω, dt) mMB[m2X AM T `iB+mH `- B7 {λ (t)}t∈R Bb MQi?2` Ft @T`2/B+i #H2 p2`@ bBQM- i?2M- P @ HKQbi bm`2Hv- λ (Tn ) = λ(Tn ) 7Q` HH n ∈ ZX U#V A7 {λ(t)}t∈R Bb T`2/B+i #H2 p2`bBQM Q7 i?2 Ft @BMi2MbBiv Q7 i?2 HQ+ HHv }MBi2 bBKTH2 TQBMi T`Q+2bb N - i?2M λ(t, ω) > 0
λ(t, ω)P (dω)dt M/ N (dt, ω)P (dω)@ X2X
U8XkyV
AM T `iB+mH `- 7Q` HH nλ(Tn ) > 0
P @ XbX
U8XkRV
S`QQ7X U V 7QHHQrb 7`QK i?2 mMB[m2M2bb T`QT2`iv U8XRNVX 6Q` U#V- MQi2 i? i H(t) = 1{λ(t)=0} Bb M Ft @T`2/B+i #H2 T`Q+2bbX AMb2`iBM; i?Bb BM i?2 bKQQi?BM; 7Q`KmH - r2 Q#i BM
E 1{λ(Tn )=0} = E 1{λ(t)=0} λ(t) dt = 0, n∈Z
R
r?B+? BKTHB2b U8XkyVX
AM T `iB+mH `- B7 i?2 HQ+ HHv }MBi2 bBKTH2 K `F2/ TQBMi T`Q+2bb (N, Z) ? b i?2 Ft @T`2/B+i #H2 biQ+? biB+ BMi2MbBiv F2`M2H λ(t)Φ(t, dz)- i?2M- 7Q` HH L ∈ K- QM {Tn < ∞}λ(Tn )Φ(Tn , L) > 0 . h?2 #Qp2 `2bmHib 2ti2M/ bi` B;?i7Q`r `/Hv iQ K `F2/ TQBMi T`Q+2bb2b M/ i?2B` biQ+? biB+ BMi2MbBiv F2`M2Hb b 7QHHQrbX G2i i?2 bBKTH2 HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb (N, Z) ? p2 i?2 Ft @BMi2MbBiv F2`M2H λ(t)Φ(t, dz) r?2`2 Φ(t, dz) = λ(t)μ(t, z)Q(t, dz)
U8XkkV
7Q` bQK2 FtN,Z @T`2/B+i #H2 F2`M2H Q(t, dz)X G2i {Ft }t∈R #2 ?BbiQ`v bm+? i? iFt ⊇ Ft ⊇ FtN,Z .
U8XkjV
Ai K v ? TT2M i? i i?2 biQ+? biB+ F2`M2H λ(t)Φ(t, dz) Bb MQi Ft @ / Ti2/ M/ i?2`2@ 7Q`2 + MMQi #2 i?2 Ft @BMi2MbBiv F2`M2HX L2p2`i?2H2bb- i?2`2 biBHH 2tBbib biQ+? biB+
8XRX h>1 aJPPh>AL: 6P_JlGa
R3j
Ft @BMi2MbBiv F2`M2H- M/ Bi Bb Q#i BM2/ #v dzT`QD2+iBQMǴ Q7 i?2 BMBiB H biQ+? biB+ BM@ i2MbBiv F2`M2H QM i?2 bK HH2` ?BbiQ`v- BM b2Mb2 iQ #2 /2b+`B#2/ MQrX _2+ HH i?2 i2`KBMQHQ;v, B7 i?2 K TTBM; Y : (t, ω, z) → Y (t, ω, z) ∈ R Bb B(R+ )⊗ F ⊗ K- QM2 b vb i? i Y (t, z) Bb K2 bm` #H2 T`Q+2bb BM/2t2/ #v KX Ai Bb b B/ iQ #2 Ft @ / Ti2/ U`2bTX Ft @T`2/B+i #H2V B7 KQ`2Qp2` 7Q` HH z ∈ K- i?2 biQ+? biB+ T`Q+2bb {Y (t, z)}t∈R Bb Ft @ / Ti2/ U`2bTX Ft @T`2/B+i #H2VX *QMbB/2` i?2 ?BbiQ`B2b {Ft }t∈R M/ {Ft }t∈R b iBb7vBM; +QM/BiBQM U8XkjV 7Q` HH t ∈ RX G2i {Y (t, z)}t∈R #2 MQM@M2; iBp2 K2 bm` #H2 T`Q+2bb BM/2t2/ #v KX G2i i?2 sigma@}MBi2 K2 bm`2b μ1 M/ μ2 QM (R × Ω × K, P(F· ) ⊗ K) #2 /2}M2/ `2bT2+iBp2Hv #v,
μ1 (H) := E
H(t, z)Y (t, z) Q(t, dz) dt R
K
M/
μ2 (H) := E
H(t, z) Q(t, dz) dt R
K
⊗ K@K2 bm` #H2X LQi2 7Q` HH MQM@M2; iBp2 K TTBM;b H : R × Ω × K i? i `2 P(F) i? i μ2 Bb i?2 T`Q/m+i K2 bm`2 P (dω) × Q(t, ω, dz) dt QM (Ω × R × K, P(F· ) ⊗ K)X *H2 `Hv μ1 μ2 - M/ i?2`27Q`2 i?2`2 2tBbib _ /QMĜLBFQ/ɷK /2`Bp iBp2 1 (t, ω, z) i? i Bb P(F· ) ⊗ K@K2 bm` #H2 M/ i?2`27Q`2 /2}M2b M Y (t, ω, z) = dμ dμ2 Ft @T`2/B+i #H2 T`Q+2bb BM/2t2/ #v K- Y (t, z)- bm+? i? i
μ1 (H) = μ2 (Y H) = E H(t, z)Y (t, z) Q(t, dz) dt . R
K
JQ`2Qp2`- i?Bb _ /QMĜLBFQ/ɷK /2`Bp iBp2 Bb μ2 @mMB[m2- i? i Bb iQ b v- B7 i?2`2 2tBbib MQi?2` bm+? _ /QMĜLBFQ/ɷK /2`Bp iBp2- b v Y - i?2M Y (t, ω, z) = Y (t, ω, z) ,
P (dω)Q(t, ω, dz)dt
X2X
U8Xk9V
.2}MBiBQM 8XRXjN h?2 #Qp2 biQ+? biB+ T`Q+2bb Y (t, z) BM/2t2/ #v K Bb + HH2/ i?2 T`2/B+i #H2 T`QD2+iBQM Q7 Y (t, z) QM {Ft }t∈R - Q` i?2 Ft @T`2/B+i #H2 T`QD2+iBQM Q7 Y (t, z)X h?2Q`2K 8XRX9y G2i i?2 bBKTH2 HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb (N, Z) QM R ? p2 i?2 biQ+? biB+ Ft @BMi2MbBiv F2`M2H U8XkkV 7Q` bQK2 FtN,Z @T`2/B+i #H2 F2`M2H Q(t, dz)X G2i {Ft }t∈R #2 MQi?2` ?BbiQ`v- b iBb7vBM; +QM/BiBQM U8XkjVX h?2M (N, Z) t @BMi2MbBiv F2`M2H λ(t) ? b i?2 biQ+? biB+ F h(t, z)Q(t, dz) r?2`2 {λ(t)} t∈R Bb i?2 Ft @ T`2/B+i #H2 T`QD2+iBQM Q7 {λ(t)}t∈R M/ h(t, z) Bb i?2 Ft @T`2/B+i #H2 T`QD2+iBQM Q7 h(t, z)X S`QQ7X G2i H(t, z) #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 BM/2t2/ biQ+? biB+ T`Q+2bbX Ai Bb 7Q`iBQ`B M Ft @T`2/B+i #H2 BM/2t2/ biQ+? biB+ T`Q+2bb- M/ i?2`27Q`2
E H(t, z)) N (dt × dz) = E H(t, z)) λ(t)h(t, z) Q(t, dz) dt R K R K
=E H(t, z)) v(t, z) Q(t, dz) dt , R
K
R39 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu r?2`2 v(t, z) Bb i?2 Ft @T`2/B+i #H2 T`QD2+iBQM Q7 λ(t)h(t, z)X G2i MQr v(t, ω, z) h(t, ω, z) := , ω) λ(t, [m MiBiv i? i Bb P (dω) × N (ω, dt × dz)@ M/ P (dω) × λ(t, ω)Q(t, ω, dz)dt@r2HH /2}M2/ BM pB2r Q7 h?2Q`2K 8XRXj3X q2 ? p2 i? i 7Q` HH MQM@M2; iBp2 Ft @T`2/B+i #H2 BM/2t2/ T`Q+2bb2b H
v (t, z) H(t, z) N (dt × dz) = E H(t, z)λ(t) E Q(t, dz) dt λ(t) R K R K
=E H(t, z)λ(t) h(t, z) Q(t, dz) dt . R
K
1[mBp H2MiHv E R
v(t, z) dt H(t, z) Q(t, dz) λ(t) λ(t) K
=E H(t, z) h(t, z) Q(t, dz) λ(t) dt . R
LQr
K
E R
K
H(t, z) h(t, z) Q(t, dz) λ(t) dt
=E H(t, z) h(t, z) Q(t, dz) N (dt) R K
=E H(t, z) h(t, z) Q(t, dz) λ(t) dt , R
K
M/ i?2`27Q`2 v(t, z) H(t, z) E Q(t, dz) λ(t) dt λ(t) R K
dt . =E H(t, z) h(t, z) Q(t, dz) λ(t) R
_2TH +BM; H(t, z) #v
K
H(t,z) λ(t)
E R
v(t, z) H(t, z) Q(t, dz) dt λ(t) K
=E H(t, z) h(t, z) Q(t, dz) dt , R
r?B+? b?Qrb i? i
v (t,z) λ(t)
= h(t, z)X
K
8XkX _1:1L1_hAo1 6P_J P6 h>1 ALh1LaAhu E1_L1G
8Xk
R38
_2;2M2` iBp2 6Q`K Q7 i?2 AMi2MbBiv E2`M2H
1t KTH2 8XRXR9 ? b b?QrM i? i i?2 MQiBQM Q7 biQ+? biB+ BMi2MbBiv Bb ;2M2` HBx iBQM Q7 i? i Q7 ? x `/ ` i2X am+? `2bmHi + M #2 2ti2M/2/ iQ K `F2/ TQBMi T`Q+2bb2b- biBHH BM i?2 bT2+B H + b2 r?2`2 i?2 ?BbiQ`v 7Q` r?B+? i?2 biQ+? biB+ BMi2MbBiv UF2`M2HV Bb /2}M2/ Bb i?2 BMi2`M H ?BbiQ`v dzTHmb T`2?BbiQ`vǴX h?Bb Bb i?2 + b2 7Q` *Qt T`Q+2bb2b M/ b2KB@*Qt T`Q+2bb2b iQ #2 /2}M2/ bQQMX a2H7@2t+BiBM;- *Qt M/ b2KB@*Qt SQBMi S`Q+2bb2b AM i?Bb bm#b2+iBQM r2 //`2bb [m2biBQM Q7 i2`KBMQHQ;v BM i2`Kb Q7 mMK `F2/ TQBMi T`Q+2bb2b- #mi i?Bb /Bb+mbbBQM HbQ +QM+2`Mb Kmi iBb Kmi M/Bb i?2 K `F2/ TQBMi T`Q+2bb2bX G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM i?2 `2 H HBM2 rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X _2K `F 8XkXR A7 i?2 ?BbiQ`v Bb i?2 BMi2`M H ?BbiQ`v Ui? i Bb- Ft = FtN 7Q` HH t ∈ RV- i?2 TQBMi T`Q+2bb Bb Q7i2M + HH2/ b2H7@2t+BiBM; TQBMi T`Q+2bbX h?Bb Bb M mM7Q`imM i2 U Hi?Qm;? ;2M2` HHv ? `KH2bbV ? #BiX AM 7 +i Mv TQBMi T`Q+2bb rBi? M Ft @BMi2MbBiv {λ(t)}t∈R i? i Bb MQi FtN @ / Ti2/ HbQ /KBib M BMi2`M H /2b+`BTiBQM BM i2`Kb Q7 Bib FtN @BMi2MbBiv b h?2Q`2K 8XRX9y b?Qrb r?2M bT2+B HBxBM; Bi iQ i?2 + b2 r?2`2 Ft = FtN Ut ∈ RVX AM bmKK `v- TQBMi T`Q+2bb rBi? biQ+? biB+ BMi2MbBiv Bb MQi T2` b2 b2H7@2t+BiBM;- #mi Bib /2b+`BTiBQM K v Q` K v MQi #2 dzb2H7@ 2t+BiBM;Ǵ- /2T2M/BM; QM r?2i?2` i?2 biQ+? biB+ BMi2MbBiv Bb Q` Bb MQi rBi? `2bT2+i iQ Bib BMi2`M H ?BbiQ`vX amTTQb2 MQr i? i Ft = G0 ∨ FtN Ut ∈ RV r?2`2 G0 Bb bQK2 σ@}2H/ bm+? i? i 7Q` HH t ∈ R- λ(t) Bb G0 @K2 bm` #H2X h?2 TQBMi T`Q+2bb Bb i?2M + HH2/ *Qt@T`Q+2bbX q2 ? p2 H`2 /v 2M+QmMi2`2/ bm+? TQBMi T`Q+2bb2b- r?B+? `2 T `iB+mH ` + b2b Q7 i?2 b2KB@*Qt TQBMi T`Q+2bb2b r?B+? r2 MQr /2}M2X h?2 i?B`/ bBim iBQM Q++m``BM; Bb r?2M Ft = G0 ∨ FtN Ut ∈ RV M/ λ(t) Bb G0 ∨ FtN @K2 bm` #H2- #mi MQi Hr vb G0 @K2 bm` #H2X am+? TQBMi T`Q+2bb2b rBHH #2 + HH2/ b2KB@*Qt TQBMi T`Q+2bb2bX am+? TQBMi T`Q+2bb2b K v #2 +QMbi`m+i2/ BM /Qm#Hv biQ+? biB+ K MM2`- b 7QHHQrbX G2i (Ω1 , G1 , P1 ) #2 bQK2 T`Q# #BHBiv bT +2 M/ /2MQi2 #v (Ω2 , G2 ) i?2 + MQMB+ H K2 bm` #H2 bT +2 (Mp (R+ ), Mp (R+ )) Q7 TQBMi T`Q+2bb2b QM R+ - r?Qb2 + MQMB+ H +QQ`/BM i2 TQBMi T`Q+2bb Bb ω2 := N2 (ω2 )X G2i v : (t, ω1 , μ) → v(t, ω1 , μ) #2 MQM@M2; iBp2 7mM+iBQM 7`QK R+ × Ω1 × Mp (R+ ) iQ R+ i? i Bb K2 bm` #H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(R+ ) × G1 × Mp (R+ ) M/ B(R+ )X G2i P2 (ω1 , ·) #2 7Q` 2 +? ω1 ∈ Ω1 T`Q# #BHBiv K2 bm`2 QM (Ω2 , G2 ) bm+? i? i 7Q` HH A2 ∈ G2 - i?2 K TTBM; ω1 → P2 (ω1 , A2 ) Bb F1 @K2 bm` #H2X amTTQb2 i? i 7Q` 2 +? ω1 ∈ Ω1 - N2 /KBib i?2 FtN2 @BMi2MbBiv {v(t, ω1 , N2 (ω2 )|[0,t) )}t∈R+ X .2}M2 MQr (Ω, F, P ) #v Ω = Ω1 × Ω2 F = G1 ⊗ G2 M/ P (A1 × A2 ) = P2 (ω1 , A2 ) P1 (dω1 ) (A1 ∈ G1 , A2 ∈ G2 ) . A1
HbQ H2i N (ω) := N2 (ω2 )- Ft := G1 ⊗FtN2 M/ G0 := G1 ⊗(∅, Ω2 )- v(t, ω, N (ω)|[0,t) := v(t, ω1 , N2 (ω2 ) |[0,t) )X PM2 p2`B}2b i? i N Bb TQBMi T`Q+2bb QM (Ω, F) rBi? i?2
R3e *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu (P, Ft )@BMi2MbBiv {v(t, N |[0,t) )}t∈R+ M/ i? i Bi Bb b2KB@*Qt TQBMi T`Q+2bb BM i?2 b2Mb2 ;Bp2M #Qp2X h?2 ai`m+im`2 Q7 SQBMi S`Q+2bb >BbiQ`B2b 72r `2bmHib QM i?2 bi`m+im`2 Q7 TQBMi T`Q+2bb ?BbiQ`B2b rBHH #2 M22/2/X h?2b2 `2 BMimBiBp2 `2bmHib r?Qb2 i2+?MB+ H T`QQ7b ? p2 #22M `2H2; i2/ iQ a2+iBQM X9 Q7 i?2 TT2M/BtX _2+ HH i?2 ;2M2` H b2iiBM;X G2i {Tn }n≥1 #2 bBKTH2 TQBMi T`Q+2bb QM (0, +∞]i? i Bb- MQM@/2+`2 bBM; b2[m2M+2 Q7 TQbBiBp2 ` M/QK p `B #H2b TQbbB#Hv i FBM; i?2 p Hm2 +∞- M/ bi`B+iHv BM+`2 bBM; QM R+ UTn < ∞ ⇒ Tn < Tn+1 VX AM T `iB+mH `Bi K v #2 }MBi2 TQBMi T`Q+2bb UB7 Tn = ∞ 7Q` bQK2 n ∈ NV M/ Bi M22/ MQi #2 HQ+ HHv }MBi2 UT∞ := limn↑∞ K v #2 }MBi2VX a2i T0 ≡ 0X G2i {Zn }n≥1 #2 b2[m2M+2 Q7 ` M/QK p `B #H2b i FBM; i?2B` p Hm2b BM i?2 K2 bm` #H2 bT +2 (K, K)X h?2 b2[m2M+2 {(Tn , Zn )}n≥1 Bb + HH2/ K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM KX 6Q` 2 +? L ∈ K- /2}M2 i?2 UbBKTH2V TQBMi T`Q+2bb N L QM R+ #v N L (C) := 1C (Tn ) 1L (Zn ) (C ∈ B(R+ )) . n≥1
LQi2 i? i N ({0}) = 0X .2}M2 i?2 UbBKTH2V TQBMi T`Q+2bb NZ QM R+ × K #v L
NZ (C × L) := N L (C) .2}M2 i?2 BMi2`M H ?BbiQ`v FtN,Z
{FtN,Z }t≥0
(C ∈ B(R+ ), L ∈ K) .
Q7 NZ #v
:= σ (NZ (C × L) ; C ∈ (0, t], L ∈ K) ,
M/ i?2 ?BbiQ`v {Ft }t≥0 #v Ft = σ(Z0 ) ∨ FtN,Z ,
U8Xk8V
r?2`2 Z0 Bb ` M/QK 2H2K2Mi i FBM; p Hm2b BM M `#Bi` `v K2 bm` #H2 bT +2 (L0 , K0 ) U7Q` BMbi M+2 bT +2 Q7 7mM+iBQMbVX Ai `2T`2b2Mib dzT`2?BbiQ`vǴ Q7 i?2 K `F2/ TQBMi T`Q+2bb- BM i?2 b2Mb2 i? i F0 = σ(Z0 ) +QMi BMb i?2 BM7Q`K iBQM H`2 /v ; i?2`2/ i iBK2 0 i? i K v BM~m2M+2 Bib 7mim`2 #2? pBQm`X *QM/BiBQM H AMi2`p H .Bbi`B#miBQMb M/ aiQ+? biB+ AMi2MbBiv E2`M2Hb amTTQb2 i? i 7Q` HH n ≥ 0- HH L ∈ K M/ HH C ∈ B(R+ ) P (Sn+1 ∈ C , Zn+1 ∈ L | FTn ) (ω) = g (n+1) (ω, x, L) dx := G(n+1) (ω, C, L) , C
r?2`2 7Q` HH L ∈ K- i?2 K TTBM; (ω, x) → g (n+1) (ω, x, L) Bb FTn ⊗ B(R+ )@ K2 bm` #H2- M/ r?2`2 7Q` HH (ω, x)- i?2 K TTBM; L → g (n+1) (ω, x, L) Bb σ@}MBi2 K2 bm`2 QM (K, K)X AM T `iB+mH ` P (Sn+1 ∈ C | FTn ) (ω) = g (n+1) (ω, x) dx := G(n+1) (ω, C) , C
r?2`2 g
(n+1)
(ω, x) := g
(n+1)
(ω, x, K)) M/ G(n+1) (ω, C) := G(n+1) (ω, C, K)X
8XkX _1:1L1_hAo1 6P_J P6 h>1 ALh1LaAhu E1_L1G
R3d
h?2Q`2K 8XkXk 6Q` L ∈ K M/ t ≥ 0- H2i g (n+1) (t − Tn , L) λ(t, L) := 1{Tn ≤t 0- Q7 i?2 7Q`K, λ(n) (t, ω) 1{Tn (ω)1 ALh1LaAhu E1_L1G
R3N
1{N ((Tn ,t])=0} = 1 −
1{N ((Tn ,s))=0} N (ds) , (Tn ,t]
M/ i?2`27Q`2 E 1A 1{N ((Tn ,t])=0} 1{t≥Tn }
1{N ((Tn ,s))=0} N (ds) 1{t≥Tn } . = E 1A 1 − (Tn ,t]
h?2 T`Q+2bb H(t) := 1A 1{N ((Tn ,s))=0} 1{s>Tn }
(t ≥ 0)
Bb Ft @T`2/B+i #H2 U#2BM; Ft @ / Ti2/ M/ H27i@+QMiBMmQmbV- M/ i?2`27Q`2- #v i?2 bKQQi?BM; 7Q`KmH
1{N ((Tn ,s))=0} N (ds) 1{t≥Tn } E 1A 1 − (Tn ,t]
t = E 1A 1 − 1{N ((Tn ,s])=0} λ(s) ds 1{t≥Tn } T
nt (n) 1{N ((Tn ,s])=0} λ (s) ds 1{t≥Tn } . = E 1A 1 − Tn
h?2 H ii2` [m MiBiv 2[m Hb- 7i2` +QM/BiBQMBM; rBi? `2bT2+i iQ FTn
t (n+1) (n) E 1A 1 − G ([s − Tn , ∞]) λ (s) ds 1{t≥Tn } . Tn
*QK#BMBM; i?2 #Qp2 2[m HBiB2b- r2 Q#i BM E 1A G(n+1) ((t − Tn , ∞]) 1{t≥Tn }
t = E 1A 1 − G(n+1) ([s − Tn , ∞]) λ(n) (s) ds 1{t≥Tn } . Tn
h?Bb T`Qp2b i?2 2tBbi2M+2 Q7 i?2 +QM/BiBQM H T`Q# #BHBiv /2MbBiv g (n+1) M/ UVX G2i NZ /KBi i?2 biQ+? biB+ (P, Ft )@BMi2MbBiv F2`M2H λ(t, dz)X .2MQi2 #v {λ(t)}t≥0 M Ft @T`2/B+i #H2 biQ+? biB+ BMi2MbBiv Q7 i?2 # bB+ TQBMi T`Q+2bb N := N K - M/ 7Q` 2 +? L ∈ K- HH t ≥ 0- /2}M2 Φ(t, L) #v λ(t, L) := λ(t)Φ(t, L) B7 λ(t) > 0- M/ #v Φ(t, L) := 0 B7 λ(t) = 0X aBM+2 λ(Tn ) > 0 P @ XbX QM {Tn < ∞}Φ(Tn , L)1{Tn ahP*>ahA* ALh1LaAhu S`QQ7X "v h?2Q`2K X9XR- Φ(Tn , L)1{Tn BH#2`i bm#bT +2 Q7 L2R (F1N,Z , P ) +QMbBbiBM; Q7 i?2 +2Mi2`2/ ` M/QK p `B #H2b i?2`2Q7X
8XjX J_hAL:G1a a ahP*>ahA* ALh1:_Ga
RNR
h?2Q`2K 8XjXR h?2 K TTBM; ϕ : H → M20 ;Bp2M #v ϕ(C)(t) := C(t, z) MZ (dt × dz) [0,t]×K
/2}M2b M BbQK2i`v #2ir22M H M/ M20 - M/ i?2 K TTBM; ψ : H → L2R (F1N,Z , P ) ;Bp2M #v C(t, z) MZ (dt × dz) ψ(C) = [0,1]×K
/2}M2b M BbQK2i`v #2ir22M H M/ L2R,0 (F1N,Z , P )X Mv +2Mi2`2/ b[m `2@BMi2;` #H2 FtN,Z @K `iBM; H2 {M (t)}t∈[0,1] /KBib i?2 `2T`2@ b2Mi iBQM C(s, z) MZ (ds × dz) , M (t) = [0,t]×K
r?2`2 C ∈ HX q2 }`bi ;Bp2 irQ T`2HBKBM `v `2bmHibX G2i f : [0, 1] × K → R #2 K2 bm` #H2 7mM+iBQM bm+? i? i 2 f (s,z) − 1 λ(t, z) dt Q(dz) < ∞ . e [0,1]×K
Uh?Bb Bb i?2 + b2 B7 f Bb MQM@TQbBiBp2 Q` #QmM/2/VX G2i 7Q` t ∈ [0, 1] f (s,z) Mf (t) := exp e f (s, z) NZ (ds × dz) + − 1 λ(s, z) Q(dz) . [0,t]×K
[0,t]×K
G2KK 8XjXk lM/2` i?2 #Qp2 +QM/BiBQMbU V,
Mf (s−) ef (s,z) − 1 MZ (ds × dz) ,
Mf (t) = 1 +
()
[0,t]×K
U#V, {Mf (t)}t∈[0,1] Bb b[m `2 BMi2;` #H2 K `iBM; H2- M/ 2 U+V, E [Mf (t)2 − 1] = E [0,t]×K Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) (t ∈ [0, 1])X S`QQ7X U V, Ai bm{+2b iQ Q#b2`p2 i? i i M 2p2Mi@iBK2 t Q7 i?2 # b2 TQBMi T`Q+2bb rBi? +Q``2bTQM/BM; K `F z ∈ K Mf (t) − Mf (t−) = Mf (t−) ef (t,z) − 1 , M/ i? i i iBK2 t #2ir22M irQ 2p2Mi iBK2b f (t,z) dMf (t) = Mf (t) e − 1 λ(t, z) Q(dz) . dt K U#V, Ai bm{+2b iQ b?Qr- BM pB2r Q7 h?2Q`2K 8XRXjj- i? i
RNk *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
1
2 Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) ds < ∞ .
E 0
U8XkdV
K
h?Bb Bb i`m2 r?2M Mf (t) Bb `2TH +2/ #v Mf (t ∧ Sn ) r?2`2 Sn := inf{t ≥ 0 ; Mf (t−) ≥ n} . AM T `iB+mH ` E (Mf (t ∧ Sn ) − 1)2 = E
t 0
2 Mf (s)2 1{s≤Sn } ef (s,z) − 1 λ(s, z) Q(dz) ds . K
U8Xk3V LQr-
E (Mf (t ∧ Sn ) − 1)2 = E Mf (t ∧ Sn )2 − 1
M/ KQ`2Qp2`
E Mf (t ∧ Sn )2 ≥ E Mf (t)2 1{t≤Sn } .
h?2`27Q`2 t
2 Mf (s)2 1{s≤Sn } ef (s,z) − 1 λ(s, z) Q(dz) E Mf (t)2 1{t≤Sn } ≤ 1 + E 0 K t f (s,z) 2 2 e E Mf (s) 1{s≤Sn } − 1 λ(s, z) Q(dz) ds . =1+ 0
K
"v :`QMr HHǶb H2KK - 7Q` HH t ≥ 0 t f (s,z) 2 2 E Mf (t) 1{t≤Sn } ≤ exp e − 1 λ(s, z) Q(dz) ds < ∞ . K
0
aBM+2 limn↑∞ Sn = ∞- #v KQMQiQM2 +QMp2`;2M+2 E Mf (t)2 ≤ exp
1
ef (s,z) − 1
0
M/ i?2`27Q`2 E
1 0
2
λ(s, z) Q(dz)
:= C < ∞ ,
K
2 Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) K 2 f (s,z) ≤C − 1 λ(t, z) dt Q(dz) , e [0,1]×K
}MBi2 [m MiBiv #v ?vTQi?2bBbX h?2`27Q`2 U8XkdV Bb T`Qp2/X U+V, ai `i 7`QK U8Xk3V M/ H2i n ↑ ∞ iQ Q#i BM t
f (s,z) 2 2 2 E (Mf (t)) = 1 + E Mf (s) e − 1 λ(s, z) Q(dz) ds . 0
K
h?Bb Bb /Bz2`2MiB H 2[m iBQM BM E [Mf (t)2 ] r?Qb2 bQHmiBQM ;Bp2b U+VX
8XjX J_hAL:G1a a ahP*>ahA* ALh1:_Ga
RNj
G2KK 8XjXj X h?2 +QHH2+iBQM Q7 ` M/QK p `B #H2b H := {eX ; X ∈ U} r?2`2 U Bb i?2 +QHH2+iBQM Q7 ` M/QK p `B #H2b k
aj NZ (Cj × Lj ) (k ∈ N, Cj ∈ B([0, 1]), Lj ∈ K, a1 , . . . , ak ∈ R)
j=1 (N,Z)
Bb iQi H BM i?2 >BH#2`i bT +2 L2R (F1
, P )X
"X h?2 +QHH2+iBQM Q7 ` M/QK p `B #H2b i? i `2 HBM2 ` +QK#BM iBQMb Q7 2H2K2Mib Q7 , , - K0 := Mf (1) := exp [0,1]×K f (s, z) NZ (ds × dz) + [0,t]×K ef (s,z) − 1 λ(s, z) Q(dz) ,
r?2`2 f : [0, 1] × K → R Bb K2 bm` #H2 7mM+iBQM bm+? i? i f (s,z) 2 e − 1 λ(t, z) dt Q(dz) < ∞ , [0,1]×K (N,Z)
Bb /2Mb2 BM i?2 >BH#2`i bT +2 L2R (F1
, P )X
S`QQ7X S `i " Bb M BKK2/B i2 +QMb2[m2M+2 Q7 S `i X 6Q` i?2 T`QQ7 Q7 - Q#b2`p2 i? i H ⊂ L2R (F 1N,Z , P ) M/ i? i 1 ∈ HX q2 ? p2 iQ T`Qp2 i? i B7 Y ∈ L2R (F1N,Z , P ) Bb bm+? i? i E Y eX = 0 7Q` HH X ∈ U - i?2M P (Y = 0) = 1X b 1 ∈ H- E [Y ] = 0X JmHiBTHvBM; Y #v +QMbi Mi B7 M2+2bb `v- r2 K v bmTTQb2 i? i E [|Y |] = 2 M/ BM T `iB+mH `- bBM+2 E[Y ] = 0- E [Y + ] = 1 M/ E [Y − ] = 1X h?2`27Q`2 Q+ := Y + P − X M/ Q− := Y +P X `2 T`Q# #BHBiv − X K2 bm`2bX aBM+2 E Y e X = 0 7Q` HH X ∈ U - r2 ? p2 i? i E Y e = E Y e Q`- 2[mBp H2MiHv- EQ+ e = EQ− eX X h?2`27Q`2 i?2 G TH +2 i` Mb7Q`Kb Q7 i?2 p2+iQ`b Q7 i?2 ivT2 (NZ (C1 × L1 ), . . . , NZ (Ck × Lk )) `2 i?2 b K2 mM/2` Q + M/ Q− X h?Bb BKTHB2b i? i Q+ M/ Q− ;`22 BM T `iB+mH ` QM F1W X h?2`27Q`2 E 1{Y + >Y − } (Y + − Y − ) = 0 M/ E 1{Y + Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu h?2 T`QQ7 Bb QKBii2/X8 h?2 ?vTQi?2bBb Q7 `B;?i@+QMiBMmBiv BM h?2Q`2K 8XjX9 Bb bmT2`~mQmb B7 i?2 7QH@ HQrBM; bQ@+ HH2/ mbm H +QM/BiBQMb `2 b iBb}2/, UBV (Ω, F, P ) Bb +QKTH2i2UBBV F0 +QMi BMb HH i?2 P @M2;HB;B#H2 b2ib- M/ UBBBV {Ft }t≥0 Bb `B;?i@+QMiBMmQmbX h?Bb 7QHHQrb 7`QK i?2 7QHHQrBM; 7mM/ K2Mi H `2bmHi Q7 K `iBM; H2 i?2Q`v, h?2Q`2K 8XjX8 G2i (Ω, F, P ) #2 bQK2 T`Q# #BHBiv bT +2 M/ H2i {Ft }t≥0 #2 bQK2 ?BbiQ`v /2}M2/ QM BiX A7 i?2 mbm H +QM/BiBQMb `2 b iBb}2/- Mv HQ+ H Ft @K `iBM; H2 {M (t)}t≥0 /KBib `B;?i@+QMiBMmQmb KQ/B}+ iBQM rBi? H27i@? M/ HBKBibX AM h?2Q`2K 8XjX9- B7 (Ω, F, P ) Bb +QKTH2i2 M/ B7 F0 +QMi BMb HH i?2 P @M2;HB;B#H2 b2ib- i?2M i?2 mbm H +QM/BiBQMb `2 b iBb}2/ bBM+2 i?2 BMi2`M H ?BbiQ`v {FtN,Z }t≥0 Bb `B;?i@+QMiBMmQmbX 1t KTH2 8XjXe, JmHiBp `B i2 SQBMi S`Q+2bb2b- h F2 kX h?2 #Qp2 `2bmHi + M #2 bT2+B HBx2/ iQ i?2 bBim iBQM /2b+`B#2/ BM 1t KTH2 8XRXk8- iQ r?B+? r2 `272` 7Q` i?2 MQi iBQMbX h?2 ?BbiQ`v i?2`2Q7 Bb MQr bbmK2/ iQ #2 Q7 i?2 7Q`K Ft = G∨FtN X 6Q` 2 +? i ∈ E- H2i {λi (t)}t≥0 #2 i?2 F t @BMi2MbBiv Q7 Ni X h?2M Mv `B;?i@+QMiBMmQmb HQ+ H Ft @K `iBM; H2 {M (t)}t≥0 /KBib i?2 `2T`2b2Mi iBQM Hi (s)Mi (ds) M (t) = M (0) + i∈E
(0,t]
K
r?2`2 7Q` 2 +? i ∈ E- Mi (ds) = Ni (ds) − λi (s)ds M/ Hi ∈ P(F· ) Bb bm+? i? i 7Q` HH t ≥ 0 t
|Hi (s)|λi (s) ds < ∞ . 0
K
8X9 hBK2 a+ HBM; _2+ HH i?2 7QHHQrBM; irQ r2HH@FMQrM `2bmHib +QM+2`MBM; SQBbbQM T`Q+2bb2b, #2 M ?TT QM i?2 TQbBiBp2 ? H7@HBM2 rBi? BMi2MbBiv 1 U dzbi M/ `/Ǵ U_RV G2i N ?TTV- M/ H2i t → λ(t) #2 U/2i2`KBMBbiB+V HQ+ HHv BMi2;` #H2 7mM+iBQMX h?2 TQBMi ((0, t λ(s) ds]) Bb SQBbbQM T`Q+2bb rBi? T`Q+2bb N /2}M2/ #v N ((0, t]) := N 0 BMi2MbBiv λ(t)X U_kV *QMp2`b2Hv- B7 N Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t)- /2}MBM; τ (t) τ (t) /2}M2/ #v N ((0, t]) = N ((0, τ (t)]) Bb #v 0 λ(s) ds = t- i?2 TQBMi T`Q+2bb N bi M/ `/ ?TTX 8 6Q` i?2 T`QQ7 Q7 KQ`2 ;2M2` H 7Q`K Q7 i?2 K `iBM; H2 `2T`2b2Mi iBQM i?2Q`2K- b22- 7Q` mMK `F2/ TQBMi T`Q+2bb2b, (*?Qm M/ J2v2`- RNd9)- M/ 7Q` K `F2/ TQBMi T`Q+2bb2b- (C +Q/RNd8) M/ ("Q2H- o ` Bv M/ qQM;- RNd8)X
8X9X hAJ1 a*GAL:
RNd
h?2b2 `2bmHib ? p2 M im` H 2ti2MbBQMb iQ TQBMi T`Q+2bb2b rBi? biQ+? biB+ BM@ i2MbBivX h?2 2ti2MbBQM Q7 _k Bb Q7 BMi2`2bi BM BM7Q`K iBQM i?2Q`v M/ rBHH #2 ;Bp2M BM a2+iBQM RyXjX h?2 2ti2MbBQM Q7 _R #2HQM;b iQ i?2 7QHFHQ`2 Q7 TQBMi T`Q+2bb i?2Q`v- M/ T `iB+mH `Hv BMi2`2biBM; TTHB+ iBQM `2 r b ;Bp2M BM a2+iBQM kXe UEm`ixǶb h?2Q`2K kXeXkVX q2 MQr ;Bp2 ;2M2` H +QMbi`m+iBQM Q7 TQBMi T`Q+2bb rBi? ;Bp2M biQ+? biB+ BMi2MbBiv 7`QK bi M/ `/ ?TTX JQ`2 T`2+Bb2Hv- H2i (Ω, F, P ) #2 T`Q# #BHBiv bT +2 QM r?B+? HBp2 Q7 BMi2MbBiv 1 M/ U V M ?TT N X U#V bm#@σ@}2H/ G ⊂ F BM/2T2M/2Mi Q7 N G2i #2 ;Bp2M MQM@M2; iBp2 K2 bm` #H2 K TTBM; ϕ : (R+ × Ω × Mp (R), B(R+ ) ⊗ G ⊗ Mp (R)) → (R, B(R)) . .2}M2 `2+m`bBp2Hv i?2 b2[m2M+2 {Tn }n≥1 Q7 TQBMi T`Q+2bb N QM (0, ∞) #v t ϕ(s, ω, ∅) ds = 1} T1 (ω) := inf{t > 0 ; N 0
r?2`2 ∅ /2MQi2b i?2 2KTiv TQBMi T`Q+2bb U M/ rBi? i?2 mbm H +QMp2MiBQM i? i i?2 BM}KmK Q7 M 2KTiv b2i Bb ∞V- M/ 7Q` n ≥ 1 t Tn+1 (ω) := inf{t > Tn (ω) ; N ϕ(s, ω, N(0,Tn ] ) ds = 1} Tn (ω)
r?2`2 7Q` HH c > 0- N(0,c] Bb i?2 TQBMi T`Q+2bb i? i +QBM+B/2b rBi? N QM (0, c] M/ Bb 2KTiv QM (c, ∞)X G2i T∞ := limn↑∞ Tn X h?2 TQBMi T`Q+2bb N Bb /2}M2/ #v i?2 #Qp2 +QMbi`m+iBQM mT iQ T∞ UMQi BM+Hm/2/VX Ai ? b- #v /2}MBiBQM- MQ TQBMib QM [T∞ , ∞)X h?2Q`2K 8X9XR .2}M2 7Q` 2 +? t ∈ (0, T∞ (ω)) λ(t, ω) := ϕ(t, ω, N(0,t] (ω)) M/ 7Q` t ≥ T∞ (ω)- λ(t, ω) := 0X AM T `iB+mH ` t (0, λ(s) ds] . N ((0, t]) = N 0
h?2M {λ(t)}t≥0 Bb i?2 biQ+? biB+ Ft @BMi2MbBiv Q7 N X c N λ(s) ds Bb M F S`QQ7X 6Q` HH c > 0- τ(c) := 0 t ∨ G@biQTTBM; iBK2 bBM+2 7Q` HH c t > 0- i?2 2p2Mi { τ (c) ≤ t} = 0 λ(s) ds ≤ t + M #2 K2 bm` #Hv 2tT`2bb2/ BM (0,t] M/ GX h?2`27Q`2 Ft := F N ∨ G ⊆ F N ∨ G := Ft X 6Q` HH 0 ≤ a ≤ bi2`Kb Q7 N t τ(t) HH A ∈ Fa (( E [1A N ((a, b])] = E 1A N τ (a), τ(b)]) . (†)
RN3 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu HbQ- A ∈ Fa - M/ i?2`27Q`2 i?2 H27i@+QMiBMmQmb biQ+? biB+ T`Q+2bb (t, ω) → 1A (ω)1(τ (ω)(a),τ (ω,b)] (t) Bb Ft @T`2/B+i #H2X h?2`27Q`2 #v i?2 bKQQi?BM; 7Q`KmH - i?2 `B;?i@? M/ bB/2 Q7 U†V Bb 2[m H iQ
b
E 1A 1(τ (a),τ (b)] (t) dt = E [1A ( τ (b) − τ(a))] = E 1A λ(s) ds . R
a
8X8 *QMiBMmQmb *? M;2 Q7 S`Q# #BHBiv q2 MQr +QMbB/2` +? M;2b Q7 BMi2MbBiv 2Mi BH2/ #v M #bQHmi2Hv +QMiBMmQmb +? M;2 Q7 T`Q# #BHBiv K2 bm`2X G2i (N, Z) #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? K `Fb BM K M/ bbQ+B i2/ HB7i2/ T`Q+2bb NZ QM R+ × KX G2i {Ft }t≥0 #2 ?BbiQ`v Q7 NZ M/ bmTTQb2 i? i NZ /KBib i?2 (P, Ft )@HQ+ H +? ` +i2`BbiB+b (λ(t), Φ(t, dz))X G2i {μ(t)}t≥0 #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb M/ H2i {h(t, z)}t≥0,z∈K #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 K@BM/2t2/ biQ+? biB+ T`Q+2bb bm+? i? i 7Q` HH t≥0 t
λ(s)μ(s) ds < ∞
P
U8XjRV
=1
0
M/
P
h(t, z)Φ(t, dz) dz = 1
U8XjkV
= 1.
K
.2}M2 7Q` 2 +? t ≥ 0 ⎛ ⎞ L(t) := L(0) ⎝ μ(Tn )h(Tn , Zn )⎠ × · · · Tn ∈(0,t]
exp −
(μ(s)h(s, z) − 1)λ(s)Φ(s, dz) ds (0,t]
, U8XjjV
K
r?2`2 L(0) Bb MQM@M2; iBp2 F0 @K2 bm` #H2 ` M/QK p `B #H2- E[L(0)] = 1X h?2Q`2K 8X8XR Ue V lM/2` i?2 #Qp2 +QM/BiBQMbURV {L(t)}t≥0 Bb MQM@M2; iBp2 (P, Ft )@HQ+ H K `iBM; H2X A7 KQ`2Qp2` E[L(t) = 1 7Q` HH t ≥ 0- Bi Bb MQM@M2; iBp2 (P, Ft )@K `iBM; H2X e bBKBH ` `2bmHi TT2 `2/ BM i?2 2M;BM22`BM; HBi2` im`2 BM i?2 mMK `F2/ + b2- 7Q` BMbi M+2 BM (_m#BM- RNdk)X ("`ûK m/- RNdk) +QMi BMb i?2 }`bi dz:B`b MQp@ivT2Ǵ `2bmHi Q7 i?Bb FBM/ rBi? T`QQ7 # b2/ QM i?2 K `iBM; H2 + H+mHmbX 6Q` i?2 2ti2MbBQM iQ K `F2/ TQBMi T`Q+2bb- b22 (C +Q/RNd8) M/ ("Q2H- o ` Bv M/ qQM;- RNd8)X
8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu
RNN
UkV A7 E[L(T )] = 1 7Q` bQK2 T > 0- M/ B7 r2 /2}M2 i?2 T`Q# #BHBiv Q #v i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 T`Q+2bb dQ = L(T ) , dP
U8Xj9V
i?2 K `F2/ TQBMi NZ /KBib i?2 (Q, Ft )@HQ+ H +? ` +i2`BbiB+b (μ(t)λ(t), h(t, z)Φ(t, dz)) QM [0, T ]X
S`QQ7X URV "v i?2 2tTQM2MiB H `mH2 Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb Uam#b2+iBQM RXe- 1t@ KTH2 RXeXNV
(μ(s)h(s, z) − 1)L(s−)MZ (ds × dz) ,
L(t) = L(0) + (0,t]
K
r?2`2 MZ (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) dsX G2i 7Q` n ≥ 1 t Sn = inf t ; L(t−) + μ(s)λ(s) ds ≥ n .
U8Xj8V
0
h?2M- #v h?2Q`2K 8XRXkN- {L(t ∧ Sn )}t≥0 Bb (P, Ft ) K `iBM; H2- M/ bBM+2 mM/2` +QM/BiBQMb U8XjRV M/ U8XjkV- P (limn↑∞ Sn = ∞) = 1- {L(t)}t≥0 Bb (P, Ft )@HQ+ H K `iBM; H2X "2BM; MQM@M2; iBp2- Bi Bb HbQ (P, Ft )@bmT2`K `iBM; H2X "mi bmT2`@ K `iBM; H2 rBi? +QMbi Mi K2 M Bb K `iBM; H2 U1t2`+Bb2 8XNXRdVX UkV q2 ? p2 iQ T`Qp2 i? i 7Q` Mv MQM@M2; iBp2 Ft @T`2/B+i #H2 K@BM/2t2/ biQ+? biB+ T`Q+2bb {H(t, z)}t≥0,z∈K M/ HH t ∈ [0, T ]
EQ (0,t]
H(s, z)NZ (ds × dz) K
H(s, z)μ(s)λ(s)h(s, z)Φ(s, dz) ds . = EQ (0,t]
K
h?Bb Bb /QM2 i?`Qm;? i?2 7QHHQrBM; b2[m2M+2 Q7 2[m HBiB2b UrBi? TT`QT`B i2 DmbiB@ }+ iBQMb i i?2 2M/V
kyy *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
EQ (0,t]
H(s, z)NZ (ds × dz) K
H(s, z)NZ (ds × dz) = E L(t) (0,t] K
L(s)H(s, z)NZ (ds × dz) =E (0,t] K
L(s−)H(s, z)μ(s)h(s, z)NZ (ds × dz) =E (0,t] K
L(s−)H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds =E (0,t] K
L(s)H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds =E (0,t] K
H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds = E L(t) (0,t] K
H(s, z)μ(s)λ(s)h(s, z)Φ(s, dz) ds . = EQ (0,t]
K
h?2 }`bi 2[m HBiv 7QHHQrb 7`QK U8Xj9V M/ i?2 7 +i i? i 7Q` MQMM2; iBp2 Ft @ K2 bm` #H2 ` M/QK p `B #H2 V (t)- EQ [V (t)] = EP [L(t)V (t)]X h?2 b2+QM/ 2[m HBiv 7QHHQrb 7`QK h?2Q`2K XjXd- i?2 i?B`/ QM2 7`QK i?2 Q#b2`p iBQM i? i i TQBMi Tn < ∞ Q7 N - L(Tn ) = L(Tn −)μ(Tn )h(Tn , Zn )X h?2 i?B`/ 2[m HBiv mb2b i?2 bKQQi?@ BM; i?2Q`2K- i?2 }7i? Bb #v h?2Q`2K XjXd M/ i?2 H bi QM2 mb2b U8Xj9VX _2K `F 8X8Xk h?2 K BM +QM/BiBQM iQ p2`B7v r?2M mbBM; h?2Q`2K 8X8XR Bb EP [L(T )] = 1X ;2M2` H K2i?Q/ iQ /Q i?Bb +QMbBbib BM }M/BM; bQK2 γ > 1 bm+? i? i 7Q` i?2 b2[m2M+2 Q7 biQTTBM; iBK2b {Sn }n≥1 /2}M2/ #v U8Xj8Vsup EP [L(T ∧ Sn )γ ] < ∞ . n≥1
h?Bb BKTHB2b i? i i?2 b2[m2M+2 {L(T ∧ Sn )}n≥1 Bb mMB7Q`KHv BMi2;` #H2- M/ i?2`2@ 7Q`2
lim E [L(T ∧ Sn )] = E lim L(T ∧ Sn ) . n↑∞
n↑∞
"mi- #v S `i URV Q7 h?2Q`2K 8X8XR- E [L(T ∧ Sn )] = 1 M/ Sn ↑ ∞X h?2`27Q`2 E [L(T )] = 1X
1t KTH2 8X8Xj, GBF2HB?QQ/ _ iBQ 7Q` aBKTH2 SQBMi S`Q+2bb QM 6BMBi2 AMi2`p HX G2i N #2 mM/2` T`Q# #BHBiv P M Ft @SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X G2i {λ(t)}t≥0 #2 MQM@M2; iBp2 #QmM/2/ Ft @T`2/B+i #H2 T`Q+2bbX .2}M2 7Q` HH t≥0 t λ(Tn ) exp (λ(s) − 1) ds , L(t) = L(0) n≥1
0
8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu
kyR
r?2`2 L(0) Bb MQM@M2; iBp2 b[m `2 BMi2;` #H2 ` M/QK p `B #H2 bm+? i? i E [L(0)2 ] < ∞X "v i?2 2tTQM2MiB H 7Q`KmH Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb L(t) = L(0) + L(s−)(λ(s) − 1) (N (ds) − ds) . (0,t]
"v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb L(s−) dL(s) + L(s−)ΔL(s) L(t)2 = L(0)2 + 2
(0,t]
(0,t]
= L(0)2 + 2
s≤t
L(s−)2 (λ(s) − 1) N (ds)
L(s−) dL(s) +
= L(0)2 + 2
(0,t]
L(s−)2 (λ(s) − 1) (N (ds) − ds)
L(s−) dL(s) + (0,t]
(0,t]
U8XjeV
L(s)2 (λ(s) − 1) ds)
+
U8XjdV
(0,t]
UMQiBM; i? i 7Q` G2#2b;m2@ HKQbi HH t- L(t) = L(t−)VX .2}M2 7Q` 2 +? n ≥ 1 t λ(s) ds ≥ n ∧ Tn , Sn := inf t ; L(t−) + 0
M Ft @biQTTBM; iBK2 bm+? i? i limn↑∞ Sn = ∞X AM T `iB+mH ` L(s−) dL(s) = L(s−)2 (λ(s) − 1) (N (ds) − ds) (0,t]
(0,t]
Bb M Ft @HQ+ H K `iBM; H2 UrBi? HQ+ HBxBM; b2[m2M+2 {Sn }n≥1 V Q7 K2 M 0X _2TH +BM; BM U8XjdV t #v t ∧ Tn M/ i FBM; 2tT2+i iBQMb E L(t ∧ Sn )2 = E L(0)2 + L(s ∧ Sn )2 (λ(s) − 1) ds) . (0,t∧Sn ]
AM T `iB+mH `- BM pB2r Q7 i?2 #QmM/2/M2bb bbmKTiBQM QM i?2 λ(t)
2 2 2 L(s ∧ Sn ) (λ(s) + 1) ds E L(t ∧ Sn ) ≤ E L(0) + E (0,t] E L(s ∧ Sn )2 (C2 + 1) ds = C1 + (0,t]
7Q` bQK2 }MBi2 TQbBiBp2 C1 M/ C2 X h?Bb BKTHB2b- #v :`QMr HHǶb H2KK - i? i t E L(t ∧ Sn )2 ≤ C1 exp (C2 + 1) ds . 0
AM T `iB+mH `- 7Q` Mv T < ∞- supn≥1 E [L(T ∧ Sn )2 ] < ∞X
kyk *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
1t KTH2 8X8X9, .2`Bp iBp2 Q7 i?2 1tT2+i iBQM Q7 TQBMi T`Q+2bb 6mM+iBQM HX h?2 b2iiBM; Bb +HQb2 iQ i? i Q7 1t KTH2b jXRX8 M/ jX8Xj- rBi? i?2 b K2 Q#D2+iBp2X h?2 T`Q# #BHBiv Pα Bb Q#i BM2/ 7`QK i?2 T`Q# #BHBiv P1 K FBM; Q7 N SQBbbQM T`Q+2bb QM [0, 1] rBi? T`2/B+i #H2 FtN @BMi2MbBiv {λ(t, α)} U bbmK2/ #QmM/2/ 7Q` bBKTHB+Biv BM i?Bb 2t KTH2V pB i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 1 dPα = exp λ(t, α) N (dt) − (λ(t, α) − 1) dt . dP1 0 (0,1] bbmKBM; / ?Q+ ?vTQi?2b2b Q7 /Bz2`2MiB #BHBiv M/ BMi2;` #BHBiv- 7Q`K H +QKTm@ i iBQMb ;Bp2 7Q` #QmM/2/ 7mM+iBQM H f (N )
dEα [f (N )] ∂ = Eα f (N ) λ(t, α)(N (dt) − dt) . dα (0,1] ∂α
1t KTH2 8X8X8, GBF2HB?QQ/ _ iBQb 7Q` *QMiBMmQmb@iBK2 ?K+bX G2i {X(t)}t≥0 #2- mM/2` P - `2;mH ` +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J `FQp +? BM rBi? bi i2 bT +2 E M/ bi #H2 M/ +QMb2`p iBp2 BM}MBi2bBK H ;2M2` iQ` {qij }i,j∈E X G2i α ij Ui = j ∈ EV #2 MQM@M2; iBp2 MmK#2`b bm+? i? i 7Q` HH i ∈ Eqi := j=i∈E, αij qij < ∞X 6Q` HH t ≥ 0- H2i Ft := FtX M/ H2i L(t) := L(0)
N (t) αij i,j
exp
t
(αij − 1)qij 1{X(s)=i} ds
.
0 i,j;i=j
i,j;i=j
amTTQb2 i? i EP [L(0)] = 1- i? i EP [L(0)2 ] < ∞ M/ i? i j=i αij qij < ∞ Ui ∈ EVX h?2M EP [L(T )] = 1 M/ mM/2` T`Q# #BHBiv Q /2}M2/ #v dQ = L(T )dP i?2 T`Q+2bb {X(t)}t≥0 Bb QM i?2 BMi2`p H (0, T ] `2;mH ` bi #H2 M/ +QMb2`p iBp2 +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J `FQp +? BM rBi? bi i2 bT +2 E M/ BM}MBi2bBK H T ` K2i2`b q˜ij := αij qij Uj = iVX S`QQ7X i /Bb+QMiBMmBiv iBK2 t Q7 i?2 +? BM L(t−)(αij − 1)ΔNi,j (t) , L(t) − L(t−) = i,j ; j=i
r?2`2 b 7Q` t bi`B+iHv #2ir22M irQ DmKTb Q7 i?2 +? BM dL(t) = L(t) (αij − 1)qij 1{X(t)=i} . dt i,j ; j=i h?2`27Q`2
L(s−)
L(t) = L(0) + (0,t]
i=j
(αij − 1)(Nij (ds) − qij 1{X(s)=i} ds) .
8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu
kyj
"v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb L(s−) dL(s) + L(s−)ΔL(s) L(t)2 = L(0)2 + 2
(0,t]
2
= L(0) + 2
s≤t
L(s−) dL(s) (0,t]
L(s−)2
+ (0,t]
+
i=j
(αij − 1)(Nij (ds) − qij 1{X(s)=i} ds)
(αij − 1)qij 1{X(s)=i} ds .
(0,t] i=j
lbBM; i?2 biQTTBM; iBK2b Q7 ivT2 U8Xj8V- r2 ? p2 i? i L(s)2 (αij − 1)qij 1{X(s)=i} ds . L(t)2 = L(0)2 + HQ+ H K `iBM; H2 + (0,t] i=j
h?2M
E L(t ∧ Sn )
2
= E L(0)
2
+E
L(s ∧ Sn ) (αij − 1)qij 1{X(s)=i} ds 2
(0,t∧Sn ] i=j
M/ BM T `iB+mH ` E L(t ∧ Sn )2 ≤ E L(0)2 +
E L(s ∧ Sn )2 |αij − 1|qij ds . (0,t]
i=j
h?2`27Q`2- #v :`QMr HHǶb H2KK
E L(t ∧ Sn )
2
≤ E L(0) exp 2
q˜i +
i
M/ }M HHv
qi t
i
sup E L(T ∧ Sn )2 < ∞ . n≥1
h?2 #Qp2 irQ 2t KTH2b + M #2 ;2M2` HBx2/ b 7QHHQrbX 1t KTH2 8X8Xe, GBF2HB?QQ/ _ iBQb 7Q` J `F2/ SQBMi S`Q+2bb2bX *QM@ bB/2` i?2 ;2M2` H bBim iBQM Q7 h?2Q`2K 8X8XRX G2i (N, Z) #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? K `Fb BM K M/ bbQ+B i2/ HB7i2/ T`Q+2bb NZ QM R+ × KX G2i {Ft }t≥0 #2 ?BbiQ`v Q7 (N, Z) M/ bmTTQb2 i? i (N, Z) /KBib i?2 (P, Ft )@HQ+ H +? ` +i2`BbiB+b (λ(t), Φ(t, dz))X G2i {μ(t)}t≥0 #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb M/ H2i {h(t, z)}t≥0,z∈K #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 K@BM/2t2/ biQ+? biB+ T`Q+2bb bm+? i? i 7Q` HH t ≥ 0
ky9 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
t
λ(s)μ(s) ds < ∞
P
U8Xj3V
=1
0
M/
h(t, z)Φ(t, dz) dz = 1
P
U8XjNV
= 1.
K
6Q` 2 +? t ≥ 0- /2}M2 L(t) b BM U8XjjV- rBi? L(0) MQM@M2; iBp2 F0 @K2 bm` #H2 ` M/QK p `B #H2 bm+? i? i E[L(0)] = 1X q2 bmTTQb2 BM //BiBQM i? i L(0) Bb b[m `2 BMi2;` #H2 M/ i? i (μ(t) + 1)h(t, z)λ(t) ≤ K(t) , T r?2`2 K : R+ → R+ Bb /2i2`KBMBbiB+ 7mM+iBQM bm+? i? i 0 K(s) ds < ∞X h?2M E [L(T )] = 1X h?2 T`QQ7 7QHHQrb i?2 b K2 HBM2b b i?2 T`QQ7 BM 1t KTH2 8X8Xj M/ Bb H27i b M 2t2`+Bb2X
a2T ` iBQM Q7 .2i2+iBQM M/ 6BHi2`BM; h?2 b2iiBM; Bb i? i Q7 i?2 +m``2Mi b2+iBQM- #mi BM //BiBQM- r2 bmTTQb2 i? i mM/2` T`Q# #BHBiv Q- (N, Z) Bb bi M/ `/ SQBbbQM T`Q+2bb rBi? BMi2MbBiv 1 M/ BB/ K `Fb Q7 +QKKQM /Bbi`B#miBQM q- i? i Bb- i?2 +Q``2bTQM/BM; TQBMi T`Q+2bb N Z QM R+ × K Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 × qX h?2 +QM/BiBQMb T`2p BHBM; BM h?2Q`2K 8X8XR `2 `2i BM2/ rBi? i?2 //BiBQM H bbmKTiBQM i? i EQ L(T )2 < ∞ r?B+? ;m ` Mi22b i? i L(T ) Bb +im HHv _ /QMĜLBFQ/ɷK /2`Bp iBp2 U_2K `F 8X8XkVX h?2Q`2K 8X8Xd lM/2` i?2 #Qp2 +QM/BiBQMb- 7Q` HH t ∈ [0, T ]⎞ ⎛ μ(Tn )h(Tn , Zn )⎠ × · · · L(t) := ⎝ Tn ∈(0,t]
exp −
(μ(s)h(s, z) − 1)λ(s)q(dz) ds (0,t]
,
U8X9yV
K
:= EQ L(t) | FtN,Z X Bb p2`bBQM Q7 L(t) S`QQ7X aBM+2 {L(t)}t∈[0,T ] Bb b[m `2@BMi2;` #H2 (Q, Ft )@K `iBM; H2- {L(t)} t∈[0,T ] Bb N,Z b[m `2@BMi2;` #H2 (Q, Ft )@K `iBM; H2X "2BM; b[m `2@BMi2;` #H2 K `iBM; H2- Bi Bb mMB7Q`KHv BMi2;` #H2- r?B+? HHQrb mb iQ mb2 h?2Q`2K XjXd b i?2 M22/ `Bb2bX "v h?2Q`2K 8XjXR QM i?2 bi`m+im`2 Q7 SQBbbQMB M b[m `2@BMi2;` #H2 (Q, FtN,Z )@ K `iBM; H2b EQ L(t) | FtN,Z = E L(T ) | FtN,Z = 1 + K(s, z)M (ds × dz) (0,t]
K
8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu
ky8
Ur?2`2 M (ds × dz) := M (ds × dz) − Q(dz) dsV 7Q` bQK2 FtN,Z @T`2/B+i #H2 BM/2t2/ T`Q+2bb K(t, z) bm+? i? i
EQ |K(s, z)|2 q(dz) ds < ∞ . (0,T ]
K
hQ T`Qp2 i?2 i?2Q`2K- Bi bm{+2b iQ b?Qr i? i 7Q` Mv b[m `2@BMi2;` #H2 FtN,Z @ K `iBM; H2 {m(t)}t∈[0,1] () EQ [L(t)m(t)] = EQ L(t)m(t) . "mi Uh?2Q`2K 8XjXRV
H(s, z)M (ds × dz) ,
m(t) = m(0) + (0,t]
K
7Q` bQK2 FtN,Z @T`2/B+i #H2 BM/2t2/ T`Q+2bb H(t, z) bm+? i? i
EQ |H(s, z)|2 q(dz) ds < ∞ . (0,T ]
K
6`QK UV r2 i?2M ;2i
EQ L(s−)(μ(s)h(s, z) − 1) M (ds × dz) H(s, z)M (ds × dz) (0,t] K (0,t] K
K(s, z) M (ds × dz) H(s, z)M (ds × dz) . EQ (0,t]
K
(0,t]
K
"v BbQK2i`v M/ BM pB2r Q7 h?2Q`2K XjXd
EQ L(s−)(μ(s)h(s, z) − 1) M (ds × dz) H(s, z)M (ds × dz) (0,t] K (0,t] K
L(s−)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ (0,t] K
L(s)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ (0,t] K
(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(t) (0,t] K
(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EP (0,t] K
(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EP (0,t] K
μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(t) (0,t] K
μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(s) (0,t] K
L(s−) μ(s)h(s, z) − 1)H(s, z) ds q(dz) . = EQ (0,t]
K
kye *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu amKK `BxBM;,
EQ (0,t]
L(s−)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) K
K(s, z)H(s, z) ds q(dz) = EQ (0,t]
K
7Q` HH #QmM/2/ FtN,Z @T`2/B+i #H2 BM/2t2/ T`Q+2bb2b H(t, z)X h?2`27Q`2 L(s−)( μ(s)h(s, z) − 1) = K(s, z) M/ =1+ L(t)
K
L(s−)( μ(s)h(s, z) − 1) M (ds × dz) , (0,t]
7`QK r?B+? i?2 `2bmHi 7QHHQrbX
8Xe 1ti2MbBQM Q7 i?2 h?2Q`v Q7 aiQ+? biB+ AMi2MbBiv t h?Bb 2ti2MbBQM +QMbBbib BM `2TH +BM; i?2 BMi2;` i2/ BMi2MbBiv T`Q+2bb { 0 λ(s) ds}t≥0 Ud V #v +Q`HQH MQM@/2+`2 bBM; T`2/B+i #H2 T`Q+2bb {A(t)}t≥0 bm+? i? i {N (0, t] − A(t)}t≥0 Bb HQ+ H K `iBM; H2X h? i bm+? T`Q+2bb /Q2b 2tBbi mM/2` [mBi2 ;2M2` H +QM/BiBQMb Bb /B`2+i +QMb2[m2M+2 Q7 `2bmHi 7`QK i?2 ;2M2` H i?2Q`v Q7 biQ+? biB+ T`Q+2bb2b Uh?2Q`2K 8XeXj #2HQrVX AM Q`/2` iQ bi i2 i?Bb `2bmHi BM 7mHH `B;Q`- /2}MBiBQM Bb BM Q`/2`X .2}MBiBQM 8XeXR ?BbiQ`v {Ft }t≥0 /2}M2/ QM i?2 T`Q# #BHBiv bT +2 (Ω, F, P ) Bb b B/ iQ b iBb7v i?2 mbm H +QM/BiBQMb B7 Bi Bb `B;?i@+QMiBMmQmb M/ F0 +QMi BMb HH i?2 P @M2;HB;B#H2 b2ibX h?2 +QM/BiBQM +QM+2`MBM; i?2 M2;HB;B#H2 b2ib ? b MQ BKT +i BM TTHB+ iBQMb bBM+2 bm+? 2p2Mib `2 MQi Q#b2`p #H2- r?2`2 b i?2 `B;?i@+QMiBMmBiv bbmKTiBQM 7Q` i?2 ?BbiQ`v Bb ;` Mi2/ BM i?2 + b2 Q7 BMi2`2bi BM i?Bb #QQF- b i?2 M2ti 2t KTH2 `2KBM/b mbX 1t KTH2 8XeXk, h?2 >BbiQ`v Q7 J `F2/ SQBMi T`Q+2bbX A7 (N, Z) Bb K `F2/ TQBMi T`Q+2bb M/ F0 Bb M `#Bi` `v σ@}2H/ +QMi BMBM; HH i?2 P @M2;HB;B#H2 (N,Z) b2ib- i?2 ?BbiQ`v {F0 ∨ Ft }t≥0 b iBb}2b i?2 mbm H +QM/BiBQMb Uh?2Q`2K XkXR8VX h?2 M2ti i?2Q`2K Bb QM2 Q7 i?2 7mM/ K2Mi H `2bmHib Q7 i?2 ;2M2` H i?2Q`v Q7 biQ+? biB+ T`Q+2bb2bX d
AM i?Bb b2+iBQM- i?2 iBK2 BM/2t Bb i?2 b2i Q7 MQM@M2; iBp2 MmK#2`bX
8XeX 1sh1LaAPL P6 h>1 h>1P_u P6 ahP*>ahA* ALh1LaAhu
kyd
h?2Q`2K 8XeXj G2i {N (t)}t≥0 #2 `B;?i@+QMiBMmQmb MQM@/2+`2 bBM; T`Q+2bb / Ti2/ iQ i?2 ?BbiQ`v {Ft }t≥0 b iBb7vBM; i?2 mbm H +QM/BiBQMbX h?2M- i?2`2 2tBbib mMB[m2 `B;?i@+QMiBMmQmb MQM@/2+`2 bBM; Ft @T`2/B+i #H2 T`Q+2bb {A(t)}t≥0 bm+? i? i {N (t) − A(t)}t≥0 Bb HQ+ H (P, Ft )@K `iBM; H2- + HH2/ i?2 Ft @+QKT2Mb iQ` Q7 {N (t)}t≥0 X 6`QK MQr QM i?2 dzmbm H +QM/BiBQMbǴ `2 Hr vb bbmK2/X h?2Q`2K 8XeXj TTHB2b BM T `iB+mH ` iQ i?2 +QmMiBM; T`Q+2bb {N (t)}t≥0 Q7 TQBMi T`Q+2bbX q?2M i?Bb +QKT2Mb iQ` ? b i?2 7Q`K A(t) =
t
λ(s) ds , 0
r?2`2 {λ(t)}t≥0 Bb MQM@M2; iBp2 Ft @T`Q;`2bbBp2 biQ+? biB+ T`Q+2bb- r2 `2i`B2p2 i?2 /2}MBiBQM Q7 i?2 Ft @BMi2MbBiv Q7 TQBMi T`Q+2bbX G2i (N, Z) #2 K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K)- M/ H2i NL #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 TQBMib rBi? K `F BM L ∈ KX "v h?2Q`2K 8XeXj- i?2`2 2tBbib 7Q` HH L ∈ K mMB[m2 `B;?i@+QMiBMmQmb MQM@/2+`2 bBM; Ft @T`2/B+i #H2 T`Q+2bb {AL (t)}t≥0 bm+? i? i {NL (t) − AL (t)}t≥0 Bb HQ+ H (P, Ft )@K `iBM; H2X AM 7 +i- i?2 T`Q+2bb2b AL UL ∈ KV + M #2 +?Qb2M3 BM bm+? r v i? i i?2`2 2tBbib ` M/QK K2 bm`2 A QM (R+ × K, B(R+ ) ⊗ K) bm+? i? i 7Q` 2 +? L ∈ K- {NZ ((0, t] × L) − A((0, t] × L)}t≥0 Bb HQ+ H Ft @K `iBM; H2X h?Bb Bb bmKK `Bx2/ BM i?2 7QHHQrBM; 7Q`K, h?2Q`2K 8XeX9 G2i (N, Z) #2 K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K)- M/ H2i NL #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 TQBMib rBi? K `F BM L ∈ KX h?2M i?2`2 2tBbib ` M/QK K2 bm`2 A QM (R+ × K, B(R+ ) ⊗ K) bm+? i? i 7Q` 2 +? L ∈ K- {A((0, t] × L)}t≥0 Bb Ft @T`2/B+i #H2 M/ {NZ ((0, t] × L) − A((0, t] × L)}t≥0 Bb HQ+ H Ft @K `iBM; H2X .2}MBiBQM 8XeX8 h?2 ` M/QK K2 bm`2 A BM i?2 bi i2K2Mi Q7 h?2Q`2K 8XeX9 Bb + HH2/ i?2 BMi2;` i2/ Ft @BMi2MbBiv F2`M2H Q7 i?2 K `F2/ TQBMi T`Q+2bb (N, Z)X h?2 M2ti `2bmHi Bb i?2 M im` H 2ti2MbBQM Q7 h?2Q`2K 8XRXjR M/ Bb T`Qp2/ BM 2t +iHv i?2 b K2 r vX h?2Q`2K 8XeXe G2i (N, Z) #2 K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K) M/ /KBiiBM; i?2 BMi2;` i2/ Ft @BMi2MbBiv F2`M2H A M/ H2i H #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb BM/2t2/ #v KX h?2M
H(s, z) NZ (ds × dz) = E
E R+ 3
(C +Q/- RNd8)X
K
H(s, z) A(ds × dz) .
R+
K
ky3 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu amTTQb2 i? i 7Q` HH n ≥ 0- HH L ∈ K M/ HH C ∈ B(R+ )P (Sn+1 ∈ C , Zn+1 ∈ L | FTn ) (ω) := G(n+1) (ω, C, L) , r?2`2 7Q` HH C ∈ B(R+ ) M/ HH L ∈ K- i?2 K TTBM; ω → G(n+1) (ω, C, L) Bb FTn @K2 bm` #H2- M/ r?2`2 7Q` HH ω ∈ Ω M/ HH L ∈ K- i?2 K TTBM; C → G(n+1) (ω, C, L) Bb bm#@T`Q# #BHBiv K2 bm`2 QM (R+ , B(R+ ))X G2i G(n+1) (ω, C) := G(n+1) (ω, C, K)X h?2Q`2K 8XeXd h?2 biQ+? biB+ Ft @F2`M2H A Q7 i?2 K `F2/ TQBMi T`Q+2bb (N, Z) Bb ;Bp2M #v G(n+1) (dx, L) A((0, t], L) = A((0, Tn ], L) + (n+1) ([x − Tn , +∞]) (0,t−Tn ] G (t ∈ (Tn , Tn+1 ]L ∈ K) . h?2 T`QQ7 7QHHQrb i? i Q7 h?2Q`2K 8XkXj- Kmi iBb Kmi M/BbX q2 b? HH MQi bT2M/ KQ`2 iBK2 QM i?Bb M im` H 2ti2MbBQM Q7 i?2 i?2Q`v Q7 biQ+? b@ iB+ BMi2MbBivXN
8Xd :`B;2HBQMBbǶ _2T`2b2Mi iBQM MQM@?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? U/2i2`KBMBbiB+V BMi2MbBiv 7mM+iBQM λ(t) + M #2 Q#i BM2/ #v T`QD2+iBM; QMiQ i?2 iBK2 tBb i?2 TQBMib Q7 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R2 Q7 BMi2MbBiv 1 r?B+? HB2 #2ir22M i?2 +m`p2 y = λ(t) M/ i?2 iBK2 tBb U1t2`+Bb2 jX3XRyVX AM 7 +i- Mv TQBMi T`Q+2bb rBi? biQ+? biB+ Ft @BMi2MbBiv {λ(t)}t∈R MQi QMHv + M #2 Q#i BM2/ BM i?Bb r v Ui?2 /B`2+i BK#2//BM; i?2Q`2KbV #mi + M Hr vb #22M i?Qm;?i Q7 b ? pBM; #22M Q#i BM2/ BM i?Bb r v Ui?2 BMp2`b2 BK#2//BM; i?2Q`2KbVXRy h?2 2t +i 7Q`KmH iBQM M/ i?2 K i?2K iB+ H /2i BHb rBHH #2 ;Bp2M BM i?2 M2ti b2+iBQMX 6Q` i?Bb i?2 7QHHQrBM; T`2HBKBM `B2b Q7 BMi`BMbB+ BMi2`2bi `2 M22/2/X M 1ti2MbBQM Q7 q i M #2Ƕb h?2Q`2K h?2 Q`B;BM H p2`bBQM Q7 q i M #2Ƕb i?2Q`2K +QM+2`Mb ?QKQ;2M2Qmb SQBbbQM T`Q@ +2bb2b QM i?2 HBM2X Ai rBHH #2 2ti2M/2/- rBi? T`QQ7 bBKBH ` iQ i? i Q7 h?2Q`2K kX9XR- iQ SQBbbQM T`Q+2bb2b QM T`Q/m+i bT +2b Q7 i?2 ivT2 R × KX h?Bb M2r p2`bBQM rBHH TH v +2Mi` H `QH2 BM i?2 T`QQ7 Q7 i?2 BK#2//BM; i?2Q`2Kb BM i?Bb b2+iBQMX G2i N #2 TQBMi T`Q+2bb QM R × KX _2+ HH i?2 MQi iBQM St N Ut ∈ RV 7Q` i?2 TQBMi T`Q+2bb Q#i BM2/ #v b?B7iBM; N #v t iQ i?2 H27i U H;2#` B+ HHvVX G2i St N + /2MQi2 i?2 TQBMi T`Q+2bb Q#i BM2/ #v `2bi`B+iBM; i?2 b?B7i2/ T`Q+2bb St N iQ R+ ×KX GQQb2Hv bT2 FBM;- St N + Bb i?2 7mim`2 Q7 N 7i2` iBK2 t- M/ KQ`2 7Q`K HHvSt N + ([a, b] × L) := N (R+ ∩ [a + t, b + t] × L) N Ry
6Q` +QKTH2i2 ++QmMi- b22 (G bi M/ "` M/i- RNN8)X (:`B;2HBQMBb- RNdR)X
([a, b] ⊂ R, L ∈ K) .
8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL
kyN
PM2 + M bBKBH `Hv /2}M2 St N − - i?2 `2bi`B+iBQM Q7 St N iQ R− × KX h?2 7QHHQrBM; p2`bBQM Q7 q i M #2Ƕb i?2Q`2K +QM+2`Mb BM T `iB+mH ` *Qt T`Q+2bb2b QM R2 X h?2Q`2K 8XdXR G2i (K, K) #2 M `#Bi` `v K2 bm` #H2 bT +2X G2i G #2 bQK2 σ@ }2H/ Q7 FX G2i λ(t, dz) #2 HQ+ HHv BMi2;` #H2 F2`M2H 7`QK R × Ω iQ (K, K) bm+? i? i 7Q` HH t ∈ R M/ HH L ∈ K- λ(t, L) Bb G@K2 bm` #H2X G2i N #2 TQBMi T`Q+2bb QM R × K i? i /KBib i?2 Ft @BMi2MbBiv F2`M2H λ(t, ·)- r?2`2 Ft = Ht ∨ G M/ {Ht }t∈R Bb ?BbiQ`v Q7 N X h?2M N Bb- +QM/BiBQM HHv QM G- SQBbbQM T`Q+2bb rBi? i?2 BMi2MbBiv K2 bm`2 ν(dt × dz) = λ(t, dz) × dtX 6m`i?2`KQ`2- +QM/BiBQM HHv QM G- 7Q` HH t ∈ R- i?2 7mim`2 St N + Q7 N 7i2` iBK2 t Bb BM/2T2M/2Mi Q7 Ht X S`QQ7X Ai bm{+2b iQ b?Qr i? i 7Q` HH a ∈ R- 7Q` Mv }MBi2 7 KBHv Q7 /BbDQBMi K2 bm` #H2 b2ib C1 , . . . , Cm ⊂ (a, +∞) × K- HH t1 , . . . , tm ∈ R+ m m E exp − tj N (Cj ) |Fa = exp ν(Cj )(e−tj − 1) . U8X9RV j=1
j=1
PM2 K v bbmK2 i? i C1 , . . . , Cm ⊂ (a, b] × Lk 7Q` bQK2 Lk b BM .2}MBiBQM 8XRXR3X Pi?2`rBb2- `2TH +2 i?2 Cj Ƕb #v Cj ∩ ((a, b] × Lk ) M/ H2i b M/ k ;Q iQ BM}MBivX .2MQi2 i?2 #Qp2 Lk #v LX 6Q` HH j U1 ≤ j ≤ mV M/ HH t ∈ R- H2i Cj (t) := Cj ∩ {(−∞, t] × K} M/ Cjt := {z ∈ K; (t, z) ∈ Cj }. .2}M2 7Q` t ≥ a
Z(t) := exp −
m
.
U8X9kV
− 1)1Cjs (z) N (ds × dz) .
U8X9jV
tj N (Cj (t))
j=1
AM T `iB+mH `-
Z(b) = exp −
m
tj N (Cj )
.
j=1
HbQ- bBM+2 Z(a) = 1
Z(s−)
Z(t) = 1 + (a,t]
K
m
(e
−tj
j=1
6Q` i?2 T`QQ7 Q7 i?Bb 2[m HBiv- Q#b2`p2 i? i Mv i` D2+iQ`v t → Z(t) Bb TB2+2@ rBb2 +QMbi Mi rBi? /Bb+QMiBMmBiv iBK2b i? i `2 TQBMib Q7 i?2 bBKTH2 TQBMi T`Q+2bb NL (·) := N (· × L)X h?2`27Q`2 Z(t) = Z(a) + (Z(s) − Z(s−)) , s∈(a,t]
kRy *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu r?2`2 Z(s) = Z(s−) QMHv B7 i?2`2 Bb TQBMi (s, z) Q7 N i? i #2HQM;b iQ U i KQbiV QM2 Q7 i?2 Cj ǶbX A7 (s, z) ∈ Cj M/ Bb BM N - Z(s) = Z(s−)e−tj X LQr b vBM; i? i (s, z) ∈ Cj Bb 2[mBp H2Mi iQ b vBM; i? i z ∈ Cjs - M/ i?2`27Q`2- B7 (s, z) Bb TQBMi Q7 N Z(s) − Z(s−) =
m
Z(s−)(e−tj − 1)1Cjs (z),
j=1
r?B+? i?2M ;Bp2b U8X9jV bBM+2 Z(a) = 1X G2i MQr A ∈ Fa X q2 ? p2
1A Z(t) = 1A +
1A Z(s−) (a,t]×K
m
(e
−tj
− 1)1Cjs (z) N (ds × dz) .
j=1
h?2 biQ+? biB+ T`Q+2bb BM/2t2/ #v K H(t, ω, z) := 1A (ω)1(a,t] (t)Z(t− , ω) ×
m
(e−tj − 1)1Cjt (z)
j=1
Bb P(F· ) ⊗ K K2 bm` #H2 M/ Q7 +QMbi Mi bB;M UM2; iBp2VX h?2`27Q`2 m −tj 1A Z(s−) (e − 1)1Cjs (z) N (ds × dz) E[1A Z(t)] = P (A) + E (a,t]
K
t = P (A) + E
1A Z(s)
a
K
(e
−tj
j=1
t
= P (A) + E 1A
Z(s) a
j=1 m
m
K
− 1)1Cjs (z) λ(s, dz) ds
(e−tj − 1)1Cjs (z) λ(s, dz) ds .
j=1
aBM+2 A Bb `#Bi` `v BM Fa m t −tj Z(s) (e − 1)1Cjs (z) λ(s, dz) ds|Fa E[Z(t)|Fa ] = 1 + E a
K
j=1
t = 1+
E[Z(s)|Fa ] a
K
m
(e
−tj
− 1)1Cjs (z) λ(s, dz) ds .
j=1
h?2`27Q`2 E[Z(t)|Fa ] = exp
m
(e
−tj
t − 1)
j=1
G2iiBM; t = b ;Bp2b i?2 MMQmM+2/ `2bmHiX
a
K
1Cjs (z)λ(s, dz)ds
.
8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL
kRR
:`B;2HBQMBbǶ AK#2//BM; h?2Q`2Kb G2i (K, K) #2 bQK2 K2 bm` #H2 bT +2 M/ H2i {Ft }t∈R #2 ?BbiQ`vX bHB;?i 2ti2MbBQM Q7 i?2 /2}MBiBQM Q7 SQBbbQM T`Q+2bb rBHH #2 M22/2/, .2}MBiBQM 8XdXk h?2 TQBMi T`Q+2bb N QM R × K Bb + HH2/ M Ft @SQBbbQM T`Q+2bb B7 i?2 7QHHQrBM; +QM/BiBQMb `2 b iBb}2/, UBV {Ft }t∈R Bb ?BbiQ`v Q7 N c UBBV N Bb SQBbbQM T`Q+2bbc M/ UBBBV 7Q` Mv t ≥ 0- St N + M/ Ft `2 BM/2T2M/2MiX h?2 M2ti `2bmHi Bb i?2 /B`2+i BK#2//BM; i?2Q`2K, h?2Q`2K 8XdXj G2i (K, K) #2 bQK2 K2 bm` #H2 bT +2 M/ H2i Q #2 bQK2 T`Q# @ #BHBiv K2 bm`2 QM BiX G2i N #2 M Ft @SQBbbQM T`Q+2bb QM R × K × R+ rBi? BMi2MbBiv K2 bm`2 dt × Q(dz) × dsX G2i f : Ω × R × K → R #2 MQM@M2; iBp2 7mM+iBQM i? i Bb P(F· ) ⊗ K@K2 bm` #H2 M/ bm+? i? i i?2 F2`M2H λ(t, dz) := f (t, z)Q(dz) Bb HQ+ HHv BMi2;` #H2X h?2 K `F2/ TQBMi T`Q+2bb (N, Z) rBi? K `Fb BM K /2}M2/ #v N (dt × dz) := N ((dt × dz × [0, f (t, z)]) /KBib i?2 Ft @biQ+? biB+ BMi2MbBiv F2`M2H λ(t, dz)X S`QQ7X h?2 T`QQ7 Bb /QM2 7Q` i?2 mMK `F2/ + b2X h?2 T`QQ7 BM i?2 K `F2/ + b2 7QHHQrb 2t +iHv i?2 b K2 HBM2b M/ Bb H27i 7Q` i?2 `2 /2`X G2i N #2 ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb QM R × R+ rBi? p2` ;2 BMi2M@ bBiv RX G2i {λ(t)}t∈R #2 MQM@M2; iBp2 HQ+ HHv BMi2;` #H2 Ft @T`2/B+i #H2 biQ+? biB+ T`Q+2bbX .2}M2 i?2 TQBMi T`Q+2bb N QM R #v i?2 7Q`KmH N (C) = 1(0,λ(t)] (z)N (dt × dz) U8X99V C
R+
7Q` HH C ∈ BX h?2M- N ? b i?2 Ft @BMi2MbBiv {λ(t)}t∈R X RX q2 }`bi b?Qr i? i U8X99V /2}M2b HQ+ HHv }MBi2 TQBMi T`Q+2bb- i? i Bb N ((0, b]) < ∞ XbX 7Q` HH b ∈ RX .2}M2 7Q` HH n ≥ 1 t λ(s)ds ≥ n τn = inf t ≥ 0 ; 0
U4∞ B7 {. . . } = ∅VX "v i?2 HQ+ H BMi2;` #BHBiv bbmKTiBQM- limn↑∞ τn = ∞- XbX HbQ τn Bb Ft @biQTTBM; iBK2X "v i?2 bKQQi?BM; 7Q`KmH Q7 h?2Q`2K 8XRXky-
kRk *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu
E [N ((0, τn ∧ b])] = E
1(0,τn ] (s)1(0,λ(s)] (z) N (ds × dz) (0,b]×R
1(0,τn ] (s)1(0,λ(s)] (z) ds dz =E
(0,b]×R τn ∧b
λ(s) ds < ∞ .
=E 0
h?2`27Q`2- XbX- 7Q` HH n ≥ 1N (0, τn ∧ b] < ∞ . kX h?2 bBKTHB+Biv Bb H27i b M 2t2`+Bb2 7Q` i?2 `2 /2`X jX AM Q`/2` iQ T`Qp2 i? i N ? b i?2 Ft @BMi2MbBiv {λ(t)}t∈R Bi bm{+2b iQ b?Qr i? i 7Q` HH H ∈ P(F· )- H ≥ 0
H(t)N (dt) = E H(t)λ(t)dt . E R
R
"mi i?2 H27i@? M/ bB/2 Q7 i?Bb 2[m HBiv `2 /b
E H(t)1(0,λ(t)] (z)N (dt × dz) . R
R+
aBM+2 N Bb bbmK2/ Ft @SQBbbQM- Bi /KBib i?2 Ft @BMi2MbBiv F2`M2H λ(t, dz) = dzX h?mb- #v h?2Q`2K 8XRXky- i?Bb Bb HbQ 2[m H iQ
E H(t)1(0,λ(t)] (z)dtdz = E H(t)λ(t)dt . R
R+
R
1t KTH2 8XdX9, *QmTHBM; Q7 .2H v2/ M/ lM/2H v2/ _2M2r H S`Q+2bb2bX G2i N #2 `2M2r H T`Q+2bb rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 BMi2`@`2M2r H iBK2b /KBiiBM; T`Q# #BHBiv /2MbBiv f rBi? mM#QmM/2/ bmTTQ`i M/ /2+`2 bBM; 7 BHm`2 ` i2 U/7`V r(t) :=
f (t) . 1 − F (t)
ULQi2 i? i i?2 /7` T`QT2`iv BKTHB2b mM#QmM/2/M2bb Q7 i?2 bmTTQ`i Q7 F c 1t2`+Bb2 8XNXkjXVX h?2 +QMbi`m+iBQMRR mb2b ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Π QM R+ × R+ Q7 BMi2MbBiv 1X 6B`bi 7Q` i?2 mM/2H v2/ + b2 UT0 ≡ 0V i?2 TQBMib T1 - T2 - . . . Q7 i?2 mM/2H v2/ `2M2r H T`Q+2bb `2 Q#i BM2/ `2+m`bBp2Hv b 7QHHQrb, 7Q` n ≥ 1, Tn = inf{t > Tn−1 ; i?2`2 Bb TQBMi Q7 Π Q7 i?2 7Q`K (t, z), z ≤ r(t − Tn−1 )} , r?2`2 T0 = 0X h?2 /2H v2/ TQBMi T`Q+2bb 7i2` Bib }`bi TQBMi T0 - T1 - T2 - . . . Bb +QMbi`m+i2/ BM bBKBH ` r v- mbBM; i?2 b K2 SQBbbQM T`Q+2bb Π, 7Q` n ≥ 1, RR
(GBM/p HH- RNNk)- p kR@kk UTX RNjzVX
8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL
kRj
Tn = inf{t > Tn−1 ; i?2`2 Bb TQBMi Q7 Π Q7 i?2 7Q`K (t, z), z ≤ r(t − Tn−1 )} ,
r?2`2 T0 ≥ 0 Bb i?2 BMBiB H /2H vX G2i V0 #2 i?2 H bi TQBMi Q7 i?2 mM/2H v2/ T`Q+2bb #27Q`2 T0 M/ H2i V1 i?2 }`bi TQBMi Q7 i?2 mM/2H v2/ T`Q+2bb 7i2` V0 U M/ i?2`27Q`2 7i2` T0 VX V1 rBHH HbQ #2 TQBMi Q7 i?2 /2H v2/ TQBMi T`Q+2bb bBM+2 7Q` t ≥ T0 r(t − V0 ) ≤ r(t − T0 ) M/ 7`QK i?2M QM i?2 irQ `2M2r H T`Q+2bb2b +QBM+B/2X h?2 7 +i i? i i?2`2 2tBbib M 2t +i +QmTHBM; Bb FMQrM Ur2 `2 BM i?2 bT`2 /@Qmi + b2V#mi ?2`2 r2 + M BM T`BM+BTH2 +QKTmi2 i?2 /Bbi`B#miBQM Q7 i?2 +QmTHBM; iBK2X qBi?Qmi +QKTmi iBQM- r2 BKK2/B i2Hv `2K `F i? i V1 Bb biQ+? biB+ HHv bK HH2` i? i T0 + X r?2`2 X Bb BM/2T2M/2Mi Q7 T0 M/ Q7 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F X 1t KTH2 8XdX8, h?BMMBM; _2M2r H SQBMi S`Q+2bbX G2i N0 #2 M mM/2@ H v2/ `2M2r H b2[m2M+2 QM (0, +∞), 7Q` n ≥ 1- X0,n = S1 + · · · + Sn r?2`2 i?2 b2[m2M+2 Q7 MQM@M2; iBp2 ` M/QK p `B #H2b {Sn }n≥1 Bb BB/- rBi? +QKKQM /Bbi`B@ #miBQM /KBiiBM; /2MbBiv f M/ +Q``2bTQM/BM; 7 BHm`2 ` i2 r(t) := 1− ft (t) 0 f (s) ds mMB7Q`KHv #QmM/2/ #v M < ∞X h?2 biQ+? biB+ BMi2MbBiv Q7 bm+? TQBMi T`Q+2bb Bb λ(t) = r(t − θt ) r?2`2 θt Bb i?2 TQbBiBQM Q7 i?2 H bi TQBMi Q7 N0 i? i Bb < tX A7 QM2 ? b i /BbTQbBiBQM ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM i?2 bi`BT (0, +∞) × [0, M ] rBi? BMi2MbBiv 1i?2 bi M/ `/ `2+m`bBp2 T`Q+2/m`2 iQ ;2M2` i2 N0 - # b2/ QM i?2 `2T`2b2Mi iBQM UVBb i?2 7QHHQrBM;X :Bp2M Xn - Xn+1 Bb i?2 bK HH2bi t > Xn bm+? i? i N ? b TQBMi #2HQr i?2 +m`p2 y = r(t − Xn )- Ut ≥ Xn VX Pm` ;Q H Bb iQ b KTH2 i?2 #Qp2 `2M2r H T`Q+2bb i?BMM2/ rBi? i?2 i?BMMBM; ∞ T`Q# #BHBiv 7mM+iBQM p(t) bm+? i? i 0 p(t) dt < ∞X 6Q` i?Bb- i?2 i?BMMBM; rBHH #2 /QM2 #27Q`2 i?2 +QMbi`m+iBQM Q7 i?2 # bB+ `2M2r H T`Q+2bbX h?Bb Bb ?QrX 6B`bi +QMbi`m+i }MBi2 U/m2 iQ i?2 BMi2;` #BHBiv +QM/BiBQM QM i?2 i?BMMBM; T`Q# #BHBiv 7mM+iBQMV SQBbbQM TQBMi T`Q+2bb QM i?2 TQbBiBp2 HBM2 Q7 BMi2MbBiv M p(t)- r?Qb2 TQBMib `2 t1 , . . . , tk BM i?Bb Q`/2`X // iQ i?2b2 TQBMib i?Qb2 Q7 SQBbbQM T`Q+2bb Q7 BMi2MbBiv M (1 − p(t)) iQ Q#i BM b2[m2M+2 t1 , t2 , . . . Ui?2`2 Bb M BM}MBiv Q7 i?2K #mi QMHv i?Qb2 #27Q`2 tk rBHH #2 mb2/VX J2`;2 i?Bb b2[m2M+2 rBi? i?2 b2[m2M+2 t1 , . . . , tk X h?Bb K2`;BM; T`Q/m+2b b KTH2 Q7 SQBbbQM T`Q+2bb Q7 BMi2MbBiv M QM i?2 iBK2 tBb, T1 , T2 , . . .X A7 Tn ∈ {t1 , . . . , tk }- b2i Xn = 1- Qi?2`rBb2 b2i Xn = 0X :Bp2M T1 , T2 , . . .- i?2 b2[m2M+2 {Xn }n≥1 Bb BM/2T2M/2Mi M/ i?2 T`Q# #BHBiv i? i Xn = 1 Bb p(Tn )X LQr H2i {Wn }n≥1 #2 M BB/ b2[m2M+2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1)X h?2 b2@ [m2M+2 {Tn , Wn }n≥1 Q7 TQBMib Q7 R2 7Q`K SQBbbQM T`Q+2bb N Q7 BMi2MbBiv 1 QM i?2 bi`BT R+ × (0, M )X *QMbi`m+i i?2 `2M2r H TQBMi T`Q+2bb, N0 (dt) := N (dt × (0, r(t − θt )) , r?2`2 θt Bb i?2 H bi TQBMi Q7 N0 i? i Bb < t Q` 0 B7 N0 ((0, t)) = 0X TQBMi Q7 N0 Bb bQK2 Tn X E22T Bi B7 M/ QMHv B7 i?2 +Q``2bTQM/BM; Xn Bb 1X h?2 bm`pBpBM; TQBMib `2 i?2 TQBMib Q7 `2M2r H T`Q+2bb rBi? 7 BHm`2 ` i2 r(t) i?BMM2/ rBi? i?2 T`Q# #BHBiv 7mM+iBQM p(t)X
kR9 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu h?2 M2ti `2bmHi Bb i?2 BMp2`b2 BK#2//BM; i?2Q`2K, h?2Q`2K 8XdXe G2i (N, Z) #2 HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K) M/ Ft @biQ+? biB+ BMi2MbBiv F2`M2H Q7 i?2 7Q`K λ(t, dz) = f (t, z)Q(dz) , r?2`2 f : Ω × R × K → R Bb MQM@M2; iBp2 7mM+iBQM i? i Bb P(F· ) ⊗ K@K2 bm` #H2 M/ Q Bb T`Q# #BHBiv K2 bm`2 QM (K, K)X h?2M- i?2 T`Q# #BHBiv bT +2 K v #2 2MH `;2/ iQ ++QKKQ/ i2 M Ft @SQBbbQM T`Q+2bb N QM R × K × R+ rBi? BMi2MbBiv K2 bm`2 dt × Q(dz) × ds bm+? i? i N (dt × dz) := N (dt × dz × [0, f (t, z)]) . S`QQ7X h?2 `2bmHi rBHH #2 T`Qp2/ 7Q` i?2 mMK `F2/ + b2- i?2 ;2M2` H + b2 7QHHQrBM; 2t +iHv i?2 b K2 HBM2bX h?2 i?2Q`2K BM i?Bb bBKTHB}2/ 7Q`K Bb b 7QHHQrbX G2i N #2 bBKTH2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @T`2/B+i #H2 BMi2MbBiv {λ(t)}t∈R X h?2M- i?2`2 2tBbib ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R × R+ rBi? i?2 p2` ;2 BMi2MbBiv Rbm+? i? i U8X99V ?QH/bX JQ`2Qp2`- i?Bb T`Q+2bb N Bb M Ft ∨ FtN ĜSQBbbQM T`Q+2bbX b bm+?- 7Q` HH a ∈ R- Sa N + Bb BM/2T2M/2Mi Q7 Fa X G2i {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]- M/ H2i N 1 #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R × R+ - Q7 BMi2MbBiv Rbm+? i? i {Un }n∈Z - N 1 M/ F∞ `2 BM/2T2M/2MiX .2}M2 N #v N (A) = 1(λ(t),∞) (σ)N 1 (dt × dσ) + 1A ((Tn , Un λ(Tn ))) A
n∈Z
7Q` HH A ∈ B ⊗ B(R+ )X A7 H Bb MQM@M2; iBp2 7mM+iBQM 7`QK R × Ω × R+ iQ R H(t, σ)N (dt × dσ) = R
R
R+
R
H(t, σ)1(λ(t),∞) (σ)N 1 (dt × dσ) +
H(Tn , Un λ(Tn )) .
n∈Z
.2MQi2 #v N U i?2 K `F2/ TQBMi T`Q+2bb Q#i BM2/ #v ii +?BM; i?2 K `F Un iQ TQBMi Tn Q7 N X G2i U Gt = Ft ∨ FtN 1 ∨ FtN M/ bmTTQb2 i? i H Bb P(Gt ) ⊗ B(R+ )@K2 bm` #H2 M/ MQM@M2; iBp2X ×
×
× U0 λ(T0 )
U2 λ(T2 )
U−1 λ(T−1 ) U1 λ(T1 ) T−1
T0
0
T1
T2
t
8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL
kR8
aBM+2 N 1 ? b i?2 FtN 1 @BMi2MbBiv F2`M2H λ1 (t, dσ) = dσ- i?2 H ii2` Bb HbQ i?2 Gt Ĝ U N1 M/ F∞ `2 BM/2T2M/2Mi Q7 F∞ VX "v i?2 BMi2MbBiv F2`M2H Q7 N 1 U`2+ HH i? i F∞ bKQQi?BM; i?2Q`2K E R
R+
1(λ(t),∞) (σ)H(t, σ)N 1 (dt × dσ) = E R
R+
1(λ(t),∞) (σ)H(t, σ)dt × dσ
7Q` Mv MQM@M2; iBp2 H ∈ P(F· ) ⊗ B(R+ )X LQr- 7Q` Mv MQM@M2; iBp2 H ∈ P(F· ) ⊗ B(R+ ) E
H(Tn , Un λ(Tn ))
H(t, uλ(t))N U (dt × du)
= E R
n∈Z
[0,1] 1
= E R
H(t, uλ(t)) dt du .
0
UAM 7 +i- N U /KBib i?2 Ft ∨ FtN @BMi2MbBiv F2`M2H λ(t)1[0,1] (u) duX h?Bb Bb HbQ Gt @biQ+? biB+ F2`M2H 7Q` N U bBM+2 N 1 Bb BM/2T2M/2Mi Q7 N U M/ F∞ X h?2 K T (t, ω, u) → H(t, uλ(t)) Bb P(Gt ) ⊗ B(R)@K2 bm` #H2 BM pB2r Q7 i?2 Ft @T`2/B+i #BHBiv Q7 {λ(t)}t∈R - M/ Q7 i?2 K2 bm` #BHBiv bbmKTiBQMb QM HX h?Bb DmbiB}2b i?2 #Qp2 mb2 Q7 i?2 bKQQi?BM; i?2Q`2KXV h?Bb i2`K Bb HbQ 2[m H iQ U
E R
R+
H(t, σ)1(0,λ(t)] (σ)dtdσ ,
#v i?2 +? M;2 Q7 p `B #H2b σ = uλ(t)X h?2`27Q`2
H(t, σ)N (dt × dσ) R R+
=E H(t, σ)1(0,λ(t)] (σ)dtdσ H(t, σ)1(λ(t),∞) (σ)dtdσ + E R R R×R+ +
=E H(t, σ)dtdσ
E
R
R+
7Q` HH MQM@M2; iBp2 H ∈ P(G) ⊗ B+ X h?2`27Q`2- #v h?2Q`2K 8XdXR- N Bb ?QKQ@ ;2M2Qmb Gt @SQBbbQM T`Q+2bb QM (R × R+ , B(R) ⊗ B(R+ )) rBi? BMi2MbBiv RX Ai Bb 7Q`iBQ`B M Ft ∨ FtN @SQBbbQM T`Q+2bbX
o `B Mib Q7 i?2 AK#2//BM; h?2Q`2Kb h?2 7QHHQrBM; `2bmHib `2 Q7 i?2 b K2 FBM/ b i?2 T`2pBQmb QM2b- #mi i?2v bbmK2 #QmM/2/M2bb Q7 i?2 BMi2MbBiv F2`M2HX q2 bi i2 i?2K BM i?2 mMK `F2/ + b2X h?2B` T`QQ7b `2 H27i b M 2t2`+Bb2 U1t2`+Bb2 8XNXkRVX
kRe *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu M/ H2i {Un }n∈Z #2 M BB/ #2 M ?TT QM R rBi? BMi2MbBiv λ h?2Q`2K 8XdXd G2i N U #2 i?2 bbQ+B i2/ HB7i2/ b2[m2M+2 Q7 K `Fb- mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N U - M/ H2i 7Q` TQBMi T`Q+2bb QM R × [0, 1]X G2i {Gt }t∈R #2 ?BbiQ`v BM/2T2M/2Mi Q7 N HH t ∈ R Ft := Gt ∨ FtNU . < ∞- M/ G2i {λ(t)}t∈R #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb #QmM/2/ #v λ /2}M2 TQBMi T`Q+2bb N QM (R, B(R)) #v N (C) = 1C (Tn )10≤U ≤ λ(Tn ) (C ∈ B(R)) . U8X98V n
n∈Z
λ
h?2M N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2Q`2K 8XdX3 G2i N #2 bBKTH2 TQBMi T`Q+2bb QM (R, B(R)) rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R U bbmK2/ Ft @T`2/B+i #H2- rBi?Qmi HQbb Q7 ;2M2` HBivVX amTTQb2 i? i i?2`2 < ∞ bm+? i? i P @ XbX 2tBbib +QMbi Mi λ λ(t, ω) ≤ λ
(t ∈ R).
, U ) QM (R, B(R)) rBi? K `Fb h?2M- i?2`2 2tBbib +QKTQmM/ SQBbbQM T`Q+2bb (N BM ([0, 1]- B([0, 1]))- +? ` +i2`BbiB+b (λ, Q) r?2`2 Q Bb i?2 mMB7Q`K /Bbi`B#miBQM QM [0, 1]- M/ bm+? i? i U8X98V ?QH/bX AK#2//BM; i?2Q`2Kb + M #2 mb2/ BM K Mv r vbX h?2v b2`p2 7Q` BMbi M+2 b # bBb 7Q` i?2 +QMbi`m+iBQM Q7 irQ TQBMi T`Q+2bb2b rBi? biQ+? biB+ BMi2MbBiv QM i?2 b K2 T`Q# #BHBiv bT +2- bQ i? i i?2v + M #2 +QKT `2/ i` D2+iQ`vrBb2X 6Q` BMbi M+2/2}MBM; i?2 BM2[m HBiv N1 ≥ N2 `2H iBp2 iQ TQBMi T`Q+2bb2b QM R iQ bB;MB7v i? i N1 (C) ≥ N2 (C) 7Q` HH C ∈ B(R)- Bi Bb +H2 ` i? i B7 i?2b2 TQBMi T`Q+2bb2b /KBi biQ+? biB+ BMi2MbBiB2b λ1 (t, N1t ) M/ λ2 (t, N2t ) Ur?2`2 N t Bb i?2 `2bi`B+iBQM Q7 N iQ (−∞, t]V rBi? `2bT2+i iQ i?2B` `2bT2+iBp2 BMi2`M H ?BbiQ`B2b- M/ B7 7Q` HH t ∈ RN1t ≥ N2t ⇒ λ1 (t, N1t ) ≥ λ2 (t, N2t ) , i?2M N1 ≥ N2 X :`B;2HBQMBbǶ +QMbi`m+iBQM Bb Q7 T `iB+mH ` BKTQ`i M+2 BM i?2 T`QQ7b Q7 +QmTHBM; Q7 TQBMi T`Q+2bb2b M/ rBHH #2 TTHB2/ BM a2+iBQM RkXe iQ i?2 bim/v Q7 i?2 E2`bi M TQBMi T`Q+2bbX aBKmH iBQM H;Q`Bi?Kb M Q#pBQmb TTHB+ iBQM Q7 T` +iB+ H BKTQ`i M+2 Q7 i?2 /B`2+i BK#2//BM; i?2Q`2Kb Bb iQ i?2 bBKmH iBQM Q7 TQBMi T`Q+2bb2b rBi? biQ+? biB+ BMi2MbBivX q2 bi `i rBi? bBKTH2 2t KTH2 Q7 i?2 K2i?Q/QHQ;vXRk amTTQb2 i? i r2 rBb? iQ bBKmH i2 TQBMi T`Q+2bb QM i?2 TQbBiBp2 ? H7@HBM2 rBi? biQ+? biB+ BMi2MbBiv Q7 i?2 7Q`K λ(t, ω) = v(t, N[0,t) (ω)) , Rk
(G2rBb M/ a?2/H2`- RNde)- (P; i - RN3R)X
8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kRd r?2`2 v : R+ × Mp (R+ ) → R+ Bb K2 bm` #H2 rBi? `2bT2+i iQ B(R+ ) ⊗ Mp (R+ ) M/ B(R+ ) M/ #QmM/2/- b v- #v K < ∞X 6Q` i?Bb r2 + M mb2 h?2Q`2K 8XdX3 QM R+ r?B+? i2HHb mb i? i Bi bm{+2b iQ i?BM ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N rBi? BMi2MbBiv K- 2t KBMBM; Bib TQBMib b2[m2MiB HHv- F22TBM; TQBMi Tn B7 M/ QMHv v(Tn ,N
)
[0,Tn ) B7 Un ≤ r?2`2 {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv K X /Bbi`B#mi2/ QM [0, 1] M/ BM/2T2M/2Mi Q7 N
h?Bb K2i?Q/ Bb 2 bBHv / Ti2/ iQ i?2 + b2 Q7 TQBMi T`Q+2bb2b rBi? M FtN ∨ G0 @ BMi2MbBiv Q7 i?2 7Q`K λ(t, ω) = v(t, X(t−, ω), N[0,t) (ω)) ,
()
r?2`2 UBV {X(t)}t≥0 Bb bQK2 +Q`HQH biQ+? biB+ T`Q+2bb rBi? p Hm2b BM bQK2 K2 bm`@ X #H2 bT +2 (K, K) M/ G0 = F∞ := ∨t≥0 FtX UBBV v : R+ ×Mp (R+ ) → R+ Bb K2 bm` #H2 rBi? `2bT2+i iQ B(R+ )⊗K⊗Mp (R+ ) M/ B(R+ )- M/ Bb #QmM/2/ #v KUAM Qi?2` rQ`/b- i?2 TQBMi T`Q+2bb r2 b22F iQ bBKmH i2 Bb b2KB@*Qt TQBMi T`Q+2bbXV - M/ UBBBV {X(t)}t≥0 Bb BM/2T2M/2Mi Q7 N UBpV r2 bmTTQb2 i? i r2 ? p2 i /BbTQbBiBQM i Mv iBK2 t i?2 p Hm2 X(t)X Bb F2Ti B7 M/ QMHv B7 Un ≤ TQBMi Tn Q7 N
v(Tn ,X(Tn −),N(0,T ) ) n
K
X
q2 + M K F2 MQi?2` bi2T iQr `/b ;2M2` HBx iBQM bbmKBM; M FtN ∨ FtX @ BMi2MbBiv Q7 i?2 7Q`K UVX "mi i?Bb iBK2 r2 Kmbi // i?2 +QM/BiBQM i? i- /2MQiBM; #v Tn i?2 n@i? TQBMi Q7 N - QM2 + M +QMbi`m+i X(t) 7`QK Tn QMr `/ # b2/ QM i?2 FMQrH2/;2 Q7 N[0,t) X PM2 2t KTH2 Bb r?2M i?2 T`Q+2bb {X(t)}t≥0 Bb Q7 i?2 7Q`K t t ϕ(N (t−)) dt + σ(N (t−)) dW (t) , X(t) = X(0) + 0
0
X r?2`2 {W (t)}t≥0 Bb ;Bp2M qB2M2` T`Q+2bb BM/2T2M/2Mi Q7 N Pi?2` 2t KTH2b- +QM+2`MBM; 7Q` BMbi M+2 Kmim HHv 2t+BiBM; TQBMi T`Q+2bb2b- `2 H27i iQ i?2 BK ;BM iBQM Q7 i?2 `2 /2`X h?2 `2H t iBQM Q7 i?2 ?vTQi?2bBb Q7 #QmM/@ 2/M2bb Q7 i?2 biQ+? biB+ BMi2MbBiv Bb MQi T`Q#H2K bBM+2 QM2 + M Hr vb p `v i?2 r?2M M22/2/X BMi2MbBiv Q7 i?2 SQBbbQM T`Q+2bb N
8X3 P`B;BM M/ JQiBp iBQM Q7 i?2 J `iBM; H2 TT`Q +? h?Bb b2+iBQM T`2b2Mib i?2 i?2Q`v Q7 biQ+? biB+ BMi2MbBiv BM #`Q /2` T2`bT2+iBp2 M/ ;Bp2b i?2 KQiBp iBQMb M/ i?2 B/2 b i i?2 Q`B;BM Q7 i?2 K `iBM; H2 TT`Q +? iQ TQBMi T`Q+2bb2bXRj h?2 bivH2 Q7 2tTQbBiBQM Q7 i?Bb b2+iBQM rBHH #2 Tm`TQb2Hv HQQb2 BM Rj
("`ûK m/- RNdk)X
kR3 *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu Q`/2` MQi iQ Q#b+m`2 i?2 bBKTH2 B/2 b rBi? i?2 T2iiv UM2p2`i?2H2bb BM/BbT2Mb #H2V /2i BHb i? i bi`B+iHv K i?2K iB+ H i`2 iK2Mi rQmH/ `2[mB`2X J `iBM; H2 *? ` +i2`Bx iBQMb UGûpv M/ q i M #2V Ai ? b #22M FMQrM 7Q` HQM; iBK2 i? i 7Q` ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N Q7 BMi2MbBiv 1- {N (t) − t}t≥0 Bb K `iBM; H2X AM 7 +i- 7Q` bBKTH2 TQBMi T`Q+2bb N iQ #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1 Bi Bb HbQ bm{+B2Mi i? i {N (t)−t}t≥0 #2 K `iBM; H2 Uq i M #2Ƕb i?2Q`2Kc *Q`QHH `v kX9XkVX h?Bb r b i?2 }`bi bB;MB}+ Mi i?2Q`2iB+ H `2bmHi Q7 K `iBM; H2 i?2Q`v +QM+2`MBM; TQBMi T`Q+2bb2bX h?Bb +? ` +i2`Bx iBQM BM i2`Kb Q7 K `iBM; H2b Bb iQ i?2 SQBbbQM T`Q+2bb r? i GûpvǶb +? ` +i2`Bx iBQM Bb iQ i?2 qB2M2` T`Q+2bbR9 Bb iQ i?2 qB2M2` T`Q+2bbX AM i?2 K `iBM; H2 TT`Q +? iQ TQBMi T`Q+2bb2b- q i M #2Ƕb i?2Q`2K + M #2 Q#i BM2/ BM irQ r vb, UBV, b +Q`QHH `v Q7 i?2 KQ`2 ;2M2` H `2bmHi bi iBM; BM T `iB+mH ` i? i i?2 /Bbi`B#miBQM Q7 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb Bb +? ` +i2`Bx2/ #v Bib FtN @BMi2MbBiv r?2M Bi 2tBbibX UBBV, pB i?2 bKQQi?BM; 7Q`KmH Q7 SQBbbQM + H+mHmb- b r b /QM2 BM i?2 T`QQ7 Q7 h?2Q`2K kX9XRX LQi2 i? i i?Bb i?2Q`2K r b Q`B;BM HHv bi i2/ BM i2`Kb Q7 i?2 BMi2`M H ?BbiQ`v Q7 N - BM +QMi` bi rBi? i?2 p2`bBQM ;Bp2M BM i?Bb #QQFX h?Bb Bb M BKTQ`i Mi Q#b2`@ p iBQM #2+ mb2 r? i +? ` +i2`Bx2b i?2 biQ+? biB+ BMi2MbBiv i?2Q`v Bb T`2+Bb2Hv i?2 7 +i i? i Bi i F2b BMiQ ++QmMi i?2 BM7Q`K iBQM i? i Bb MQi bQH2Hv +QMi BM2/ BM i?2 TQBMi T`Q+2bb Bib2H7- 7mM/ K2Mi H bT2+i BM K Mv TTHB+ iBQMb- 2bT2+B HHv BM i?2 bvbi2Kb M/ +QKKmMB+ iBQMb b+B2M+2b- r?B+? `2 +QM+2`M2/ rBi? i?2 T`Q#H2Kb Q7 2biBK iBQM- /2i2+iBQM M/ +QMi`QHX AM i?Bb `2 Q7 TTHB2/ K i?2K iB+b- i?2`2 Bb H `;2 +Q`Tmb Q7 FMQrH2/;2 `QmM/ i?2b2 i?2K2b- r?B+? K v #2 + HH2/ i?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+? biB+ bvbi2Kb- r?2`2 i?2 # bB+ Q#D2+i Bb i?2 biQ+? biB+ T`Q+2bb+ HH2/ i?2 dzQ#b2`p iBQMǴ
t
X(t) =
ϕ(s) ds + W (t) ,
(†)
0
r?2`2 {W (t)}t≥0 Bb qB2M2` T`Q+2bb UBM T `iB+mH ` T`Q+2bb rBi? Q`i?Q;QM H BM+`2K2MibV- + HH2/ i?2 dzBMi2;` i2/ MQBb2Ǵ- M/ {ϕ(t)}t≥0 + ``B2b i?2 BM7Q`K iBQM Q7 BMi2`2bi Ui?2 dzbB;M HǴVX AM i?2 2M;BM22`BM; HBi2` im`2- UV Bb i?2 BMi2;` i2/ 7Q`K Q7 x(t) = ϕ(t) + b(t) , (††) r?2`2 {b(t)}t≥0 Bb dzr?Bi2 MQBb2Ǵ U7Q`K HHv i?2 /2`Bp iBp2 Q7 i?2 qB2M2` T`Q+2bb#mi b Bb r2HH FMQrM- bm+? /2`Bp iBp2 /Q2b MQi 2tBbi- M/ i?Bb Bb r?v i?2 BMi2;` i2/ 7Q`K Bb mb2/XV R9 ai iBM; i? i +QMiBMmQmb `2 H@p Hm2/ T`Q+2bb {W (t)}t≥0 bm+? i? i {W (t)}t≥0 M/ {W (t)2 − t}t≥0 `2 K `iBM; H2b Bb bi M/ `/ qB2M2` T`Q+2bbX
8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kRN h?2 K `iBM; H2 TT`Q +? iQ TQBMi T`Q+2bb2b r b Q`B;BM HHv KQiBp i2/ #v i?2 +QMbi`m+iBQM Q7 i?2Q`v Q7 bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b M HQ;Qmb iQ i?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+? biB+ bvbi2KbX Ai im`Mb Qmi i? i i?2 ` M;2 Q7 i?2 7Q`K H M HQ;v TT `2Mi BM i?2 +QKT `BbQM Q7 i?2 irQ 7mM/ K2Mi H i?2Q`2Kb Q7 q i M #2 M/ Gûpv BM 7 +i 2ti2M/b 7 ` #2vQM/ i?2b2 irQ `2bmHibX AM i?2 i?2Q`v Q7 TQBMi T`Q+2bb /`Bp2M biQ+? biB+ bvbi2Kb- i?2 # bB+ Q#D2+i Bb TQBMi T`Q+2bb N i? i + M HbQ #2 `2T`2b2Mi2/ BM i?2 7Q`K dzbB;M H THmb MQBb2Ǵ, t N (t) = λ(s) ds + M (t) , () 0
r?2`2- b v- {M (t)}t≥0 Bb M Ft @K `iBM; H2 M/ i?2`27Q`2- BM T `iB+mH `- biQ+? biB+ T`Q+2bb rBi? Q`i?Q;QM H BM+`2K2MibX 6`QK i?2 BM7Q`K iBQM i?2Q`2iB+ TQBMi Q7 pB2r- i?2 KQ/2H U††V T2`i BMb iQ KTHB@ im/2 KQ/mH iBQM i?2Q`v- i?2 mb27mH BM7Q`K iBQM #2BM; +QMi BM2/ BM i?2 KTHBim/2 Q7 i?2 bB;M H {ϕ(t)}t≥0 M/ i?Bb BM7Q`K iBQM Bb iQ #2 2ti` +i2/ 7`QK +Q``mTi2/ p2`bBQM Q7 Bi- i?2 +Q``mTiBQM #2BM; /m2 iQ M //BiBp2 : mbbB M MQBb2X AM i?2 /B7@ 72`2MiB i2/ p2`bBQM Q7 UV- i?2 BM7Q`K iBQM p BH #H2 Bb i?2 bTBF2 i` BM Δ(t) = δ(t − Tn ) . n≥1
h?Bb +Q``2bTQM/b BM +QKKmMB+ iBQMb i?2Q`v iQ 7`2[m2M+v KQ/mH iBQM- r?2`2 i?2 ` i2 Udz7`2[m2M+vǴV Q7 TQBMib Bb T`QTQ`iBQM H iQ i?2 bB;M H KTHBim/2X h?2 /Bz2`2M@ iB i2/ p2`bBQM Q7 i?2 KQ/2H UV Bb ˙ Δ(t) = λ(t) + p(t) ,
()
˙ r?2`2 Δ(t) := 1N ({t}=1 M/ p(t) := 1N ({t}=1 − λ(t)X h?Bb Bb Q7 +Qm`b2 Tm`2Hv 7Q`K H /2b+`BTiBQM- M/ Bi Bb i?2 BMi2;` i2/ p2`bBQM U†V r?B+? Bb i?2 2{+B2Mi QM2- Dmbi b i?2 BMi2;` i2/ p2`bBQM UV Bb i?2 T` +iB+ H QM2 7Q` i?2 bB;M H THmb MQBb2 KQ/2HX h?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+? biB+ bvbi2Kb Bb +QM+2`M2/ rBi? i?2 BMiB@ K i2Hv +QMM2+i2/ T`Q#H2Kb Q7 2biBK iBQM- /2i2+iBQM M/ +QMi`QH- M/ r b /2p2H@ QT2/ BM T `iB+mH ` 7Q` i?2 M22/b Q7 bT +2 2tTHQ` iBQMX 6`QK i?2 K i?2K iB+ H TQBMi Q7 pB2r- Bi `2bib QM i?`22 TBHH `b, ∞ UBV i?2 AiƬ biQ+? biB+ BMi2;` H 0 ϕ(s) dW (s) /2}M2/ 7Q` dzbmBi #H2Ǵ BMi2;` M/bt UBBV i?2 Q#b2`p iBQM i? i i?2 T`Q+2bb { 0 ϕ(s) dW (s)}t≥0 Bb K `iBM; H2- M/ UBBBV AiƬǶb /Bz2`2MiB iBQM `mH2, t 1 t F (W (t)) = F (W (0)) + F (W (s)) ds + F (W (s)) dW (s) , 2 0 0 r?2`2 F Bb dzbmBi #H2Ǵ irB+2 +QMiBMmQmbHv /Bz2`2MiB #H2 7mM+iBQMX
kky *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu aQK2r? i M HQ;QmbHv- i?2 biQ+? biB+ BMi2MbBiv T ` /B;K 7Q` TQBMi T`Q+2bb /`Bp2M bvbi2Kb `2bib QM, ∞ U V i?2 biQ+? biB+ BMi2;` H 0 ϕ(s) N (ds) 7Q` bmBi #H2 BMi2;` M/b∞ U#V i?2 Q#b2`p iBQM i? i { 0 ϕ(s) (N (ds) − λ(s)ds)}t≥0 Bb K `iBM; H2 B7 {ϕ(t)}t≥0 Bb M Ft @T`2/B+i #H2 T`Q+2bb- i?2M- M/ U+V i?2 aiB2HiD2bĜG2#2b;m2 + H+mHmbX q2 b? HH MQr HBbi b2`B2b Q7 2t KTH2b BHHmbi` iBM; i?2 M HQ;v #2ir22M i?2 bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b M/ i?Qb2 /`Bp2M #v qB2M2` T`Q+2bbX 1ti2MbBQMb Q7 i?2 /2}MBiBQMb Q7 #Qi? i?2 qB2M2` T`Q+2bb M/ i?2 SQBbbQM T`Q+2bb rBHH #2 M22/2/X .2}MBiBQM 8X3XR G2i {Ft }t≥0 #2 bQK2 ?BbiQ`v QM i?2 T`Q# #BHBiv bT +2 (Ω, F, Q) M/ H2i {W (t)}t≥0 #2 +QMiBMmQmb Ft @ / Ti2/ biQ+? biB+ T`Q+2bb bm+? i? i 7Q` HH 0 ≤ a ≤ b < ∞- W (b)−W (a) Bb +2Mi2`2/ : mbbB M ` M/QK p `B #H2 rBi? p `B M+2 b − a BM/2T2M/2Mi Q7 Fa X h?2M {W (t)}t≥0 Bb + HH2/ (Q, Ft )@qB2M2` T`Q+2bbX .2}MBiBQM 8X3Xk G2i {Ft }t≥0 #2 bQK2 ?BbiQ`v Q7 bBKTH2 TQBMi T`Q+2bb N QM i?2 T`Q# #BHBiv bT +2 (Ω, F, Q) M/ bmTTQb2 i? i 7Q` HH 0 ≤ a ≤ b < ∞- N (a, b] Bb SQBbbQM p `B #H2 rBi? K2 M b − a BM/2T2M/2Mi Q7 Fa X h?2M N Bb + HH2/ M (Q, Ft )@SQBbbQM T`Q+2bbX :B`b MQp@ivT2 _2bmHib q2 #2;BM rBi? i?2 Q`B;BM H :B`b MQp i?2Q`2KX G2i {Ft }t≥0 M/ {X(t)}t≥0 #2 b BM .2}MBiBQM 8X3XR M/ H2i {ϕ(t)}t≥0 #2 #QmM/2/ U7Q` bBKTHB+BivV Ft @ / Ti2/ biQ+? biB+ T`Q+2bbX h?2 T`Q+2bb t 1 t L(t) := exp ϕ(s) dX(s) − ϕ(s)2 ds 2 0 0 Bb bQHmiBQM Q7 i?2 2[m iBQM
t
L(s)ϕ(s) dX(s)
L(t) = 1 + 0
M/ M Ft @K `iBM; H2X JQ`2Qp2`- 7Q` Mv T > 0- i?2 T`Q# #BHBiv P QM (Ω, F) /2}M2/ #v dP = L(T ) dQ Bb bm+? i? i QM i?2 BMi2`p H [0, T ] t ϕ(s) ds + W (t) X(t) =
(t ∈ [0, T ])
0
r?2`2 {W (t)}t≥0 Bb (P, Ft )@qB2M2` T`Q+2bbX q2 MQr `2+ HH i?2 M HQ;Qmb `2bmHi 7Q` SQBbbQM@/`Bp2M TQBMi T`Q+2bb2bX
8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kkR G2i N #2 bBKTH2 TQBMi T`Q+2bb / Ti2/ iQ bQK2 ?BbiQ`v {Ft }t≥0 M/ H2i Q #2 T`Q# #BHBiv- #Qi? QM i?2 b K2 K2 bm` #H2 bT +2 (Ω, F)X amTTQb2 i? i N Bb (Q, Ft ) SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X G2i {λ(t)}t≥0 #2 TQbBiBp2 M/ #QmM/2/ U7Q` bBKTHB+BivV Ft @ / Ti2/ biQ+? biB+ T`Q+2bbX h?2 T`Q+2bb t t log(λ(s) N (ds) − (λ(s) − 1) ds L(t) := exp 0
Bb bQHmiBQM Q7 i?2 2[m iBQM
0
t
L(s−)(λ(s) − 1) (N (ds) − ds)
L(t) = 1 + 0
M/ M Ft @K `iBM; H2X JQ`2Qp2`- 7Q` Mv T > 0- i?2 T`Q# #BHBiv P QM (Ω, F) /2}M2/ #v dP = L(T ) dQ Bb bm+? i? i QM i?2 BMi2`p H [0, T ]- N /KBib i?2 (P, Ft ) BMi2MbBiv {λ(t)}t∈[0,T ] X AM bvK#QHb t
λ(s) ds + M (t) (t ∈ [0, T ]) ,
N (t) = 0
r?2`2 {M (t)}t∈[0,T ] Bb (P, Ft )@K `iBM; H2X AMMQp iBQMb AM i?2 bB;M H THmb MQBb2 KQ/2H- 7Q` dzbmBi #H2Ǵ p2`bBQM Q7 i?2 2biBK i2/ bB;M H T`Q+2bb ϕ(t) := E ϕ(t) | FtX (t ≥ 0) , i?2 Q#b2`p iBQM /KBib i?2 `2T`2b2Mi iBQM t 2(t) , ϕ(s) ds + W X(t) = 0 2(t)}t≥0 Bb (P, FtW )@qB2M2` T`Q+2bbX h?2 bBKBH ` `2bmHi 7Q` TQBMi T`Q@ r?2`2 {W +2bb2b Bb i? i t 2(t) , N (t) = λ(s) ds + M 0
2(t)}t≥0 Bb M F N @K `iBM; H2- + HH2/ i?2 BMMQp iBQMb T`Q+2bb- #2+ mb2 r?2`2 {M t 2(t) Bb 2[m H iQ dzr? i vQm ;2iǴ UdN (t)V BM i?2 BM}MBi2bBK H BMi2`p H dt KBMmb dM dzr? i vQm 2tT2+i2/Ǵ UE dN (t) | FtN VX *QM+2`MBM; i?2 TQBMi T`Q+2bb N - ` i?2` i`BpB H Q#b2`p iBQM Bb i? i 7`QK i?2 2(t)}t≥0 QM2 + M BKK2/B i2Hv Q#i BM i?2 TQBMi T`Q+2bb Bib2H7X AM MQBb2 T`QD2+iBQM {M bvK#QHb, FtN ≡ FtM . bBKBH ` U Hi?Qm;? i?Bb iBK2 MQi Q#pBQmbV `2bmHi 7Q` i?2 bB;M H THmb r?Bi2 MQBb2 KQ/2H i2HHb mb i? i- mM/2` dzbmBi #H2Ǵ +QM/BiBQMb
FtX ≡ FtW . h?Bb Bb + HH2/ i?2 BMMQp iBQMb 2[mBp H2M+2X
kkk *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu J `iBM; H2b b aiQ+? biB+ AMi2;` Hb G2i {W (t)}t≥0 #2 M FtW @qB2M2` T`Q+2bbX h?2M Mv +2Mi2`2/ b[m `2@BMi2;` #H2 FtW @K `iBM; H2 Bb Q7 i?2 7Q`K t m(t) = H(s) dW (s) , 0
t r?2`2 {H(t)}t≥0 Bb T`2/B+i #H2 T`Q+2bb bm+? i? i 0 H(s)2 ds < ∞ Ut ≥ 0VX aBKBH `Hv- B7 N Bb bBKTH2 TQBMi T`Q+2bb rBi? i?2 FtN @ BMi2MbBiv {λ(t)}t≥0 - i?2M Mv +2Mi2`2/ b[m `2@BMi2;` #H2 FtN @K `iBM; H2 Bb Q7 i?2 7Q`K t 2(s) , m(t) = H(s) dM 0
r?2`2 {H(t)}t≥0 Bb T`2/B+i #H2 T`Q+2bb bm+? i? i
t 0
H(s)2 ds < ∞ Ut ≥ 0VX
JQ`2 2t KTH2b rBHH #2 ;Bp2M b i?2 QTTQ`imMBiv `Bb2b- BM T `iB+mH ` BM *? Ti2` Ry QM i?2 BM7Q`K iBQM@i?2Q`2iB+ H bT2+ib Q7 TQBMi T`Q+2bb2b Ua2+iBQMb RyXR M/ RyXk QM i?2 }Hi2`BM; T`Q#H2K- M/ 1t KTH2 RyXjX8VX
8XN 1t2`+Bb2b 1t2`+Bb2 8XNXRX h?2 6mM/ K2Mi H J `iBM; H2 t a?Qr i? i .2}MBiBQM 8XRXR BKTHB2b i? i 7Q` HH a ∈ R- Ma (t) := N (a, t] − a λ(s) ds Ut ≥ aV Bb M Ft @HQ+ H K `iBM; H2X 1t2`+Bb2 8XNXkX *QMM2+iBM; iQ i?2 AMimBiBp2 .27BMBiBQM G2i i?2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb N QM R ? p2 i?2 Ft @BMi2MbBiv {λ(t)}t∈R - M/ bmTTQb2 i? i t → λ(t, ω) Bb 7Q` HH ω ∈ Ω `B;?i@+QMiBMmQmb 7mM+iBQM M/ i? i (t, ω) → λ(t, ω) Bb mMB7Q`KHv #QmM/2/X a?Qr i? i lim h↓0
1 E[N (t, t + h]|Ft ] = λ(t) , h
P @ XbX
1t2`+Bb2 8XNXjX *Qt S`Q+2bb2b G2i N #2 /Qm#Hv biQ+? biB+ SQBbbQM T`Q+2bb U*Qt T`Q+2bbV rBi? `2bT2+i iQ i?2 σ@}2H/ G- rBi? i?2 HQ+ HHv BMi2;` #H2 biQ+? biB+ BMi2MbBiv {λ(t)}t∈R X h?Bb K2 Mb λ U.2}MBiBQM RXRXRyV i? i G ⊇ F∞ := σ (λ(s), s ∈ R) M/ i? i r?2M2p2` C1 , . . . , CK `2 #QmM/2/ /BbDQBMi K2 bm` #H2 bm#b2ib Q7 R K K iuj E exp i uj N (Cj ) | G = exp (e − 1) λ(s)ds j=1
j=1
Cj
7Q` HH u1 , . . . , uK ∈ RX a?Qr i? i N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R - r?2`2 Ft = G ∨ FtN X
8XNX 1s1_*Aa1a
kkj
1t2`+Bb2 8XNX9X Ft @T`Q;`2bbBp2M2bb G2i {X(t)}t∈R+ #2 HQ+ HHv BMi2;` #H2 Ft @T`Q;`2bbBp2 T`Q+2bbX S`Qp2 i? i i?2 t biQ+? biB+ T`Q+2bb { 0 X(s) ds}t∈R+ Bb K2 bm` #H2 M/ Ft @ / Ti2/X 1t2`+Bb2 8XNX8X Pi?2` >BbiQ`B2b G2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X S`Qp2 i?2 7QHHQrBM;, UBV A7 {Gt }t∈R Bb ?BbiQ`v bm+? i? i G∞ Bb BM/2T2M/2Mi Q7 F∞ - i?2M {λ(t)}t∈R Bb M Ft ∨ Gt @BMi2MbBiv Q7 N X UBBV A7 {Ft }t∈R Bb ?BbiQ`v Q7 N bm+? i? i Ftλ ∨ FtN ⊆ Ft ⊆ Ft - i?2M {λ(t)}t∈R Bb M Ft @BMi2MbBiv Q7 N X 1t2`+Bb2 8XNXeX #Qmi S`2/B+i #BHBiv U V a?Qr i? i /2i2`KBMBbiB+ K2 bm` #H2 T`Q+2bb Bb Ft @T`2/B+i #H2- M/ i? i M Ft @ T`2/B+i #H2 T`Q+2bb Bb Ft @T`Q;`2bbBp2Hv K2 bm` #H2- M/ BM T `iB+mH ` K2 bm` #H2 M/ Ft @ / Ti2/X U#V G2i S M/ τ #2 irQ Ft @biQTTBM; iBK2b bm+? i? i S ≤ τ - M/ H2i ϕ : R+ ×R → R #2 K2 bm` #H2 7mM+iBQMX h?2M X(t, ω) = ϕ(S(ω), t)1{S(ω)Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu 1t2`+Bb2 8XNXRyX JmHiBp `B i2 q i M #2ǰb h?2Q`2K G2i Nj Uj ≥ 1V #2 HQ+ HHv }MBi2 bBKTH2 TQBMi T`Q+2bb2b rBi?Qmi +QKKQM TQBMib M/ H2i Ft #2 Q7 i?2 7Q`K U8X9eV rBi? FtN =
∞ 3
Nj
Ft
j=1
UN := (N1 , N2 , . . .)VX 6Q` HH j ≥ 1- H2i {λj (t)}t≥0 #2 G@K2 bm` #H2 HQ+ HHv BMi2@ ;` #H2 biQ+? biB+ T`Q+2bb M/ bmTTQb2 i? i 7Q` HH (a, b] ⊆ R
b
E[Nj (a, b] | Fa ] =
λj (s) ds . a
a?Qr i? i Nj Bb G@+QM/BiBQM H SQBbbQM T`Q+2bb rBi? G@+QM/BiBQM H bbQ+B i2/ K2 bm`2 C → E[Nj (C)|G] = C λj (t)dt M/ i? i i?2 Nj Ƕb `2 BM/2T2M/2Mi +QM/B@ iBQM HHv rBi? `2bT2+i iQ GX 1t2`+Bb2 8XNXRRX ai`QM; J `FQp S`QT2`iv Q7 ?TTb AM i?2 b2iiBM; Q7 1t2`+Bb2 8XNXRy rBi? TQbbB#Hv K = ∞- H2i T #2 }MBi2 Ft @biQTTBM; iBK2 M/ H2i 7Q` HH t ≥ 0 Ft := FTN+t . .2}M2 i?2 TQBMi T`Q+2bb2b Nj U1 ≤ j ≤ KV #v Nj (a, b] = Nj (a + T, b + T ] .
AM T `iB+mH ` Ft = G ∨ FtN - r?2`2 G = G ∨ FTN X S`Qp2 i? i 7Q` HH j- Nj /KBib i?2 Ft @BMi2MbBiv {λj (t + T )}t≥0 X *QKK2Mi, M BMi2`2biBM; bT2+B H + b2 Bb r?2M 7Q` HH j U1 ≤ j ≤ KV- λj (t) ≡ λj 7Q` HH t- i? i Bb- i?2 Q`B;BM H TQBMi T`Q+2bb2b Nj `2 SQBbbQM T`Q+2bb2b rBi? p2` ;2 BMi2MbBiB2b λj - Kmim HHv BM/2T2M/2Mi M/ BM/2T2M/2Mi Q7 GX AM i?Bb + b2- i?2 /2H v2/ TQBMi T`Q+2bb2b ST Nj U/2}M2/ #v ST Nj (C) = Nj (C + τ )V `2 BM/2T2M/2Mi SQBbbQM Nj T`Q+2bb2b rBi? BMi2MbBiB2b λj - M/ i?2v `2 BM/2T2M/2Mi Q7 G M/ Q7 FTN = ∨K j=1 FT X h?2 H ii2` BM/2T2M/2M+2 T`QT2`iv Bb i?2 bi`QM; J `FQp T`QT2`iv Q7 KmHiBp `B i2 SQBbbQM T`Q+2bb2bX Ai r b T`Qp2/ #v Qi?2` K2 Mb BM h?2Q`2K kXRXdX t 1t2`+Bb2 8XNXRkX M (t)2 − 0 λ(s) ds G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+rBi? i?2 UHQ+ HHv BMi2;` #H2V Ft @ t BMi2MbBiv {λ(t)}t≥0 X G2i 7Q` t ≥ 0- M (t) := N (t) − 0 λ(s) dsX S`Qp2 i? i {M (t)2 − t λ(s) ds}t≥0 Bb M Ft @HQ+ H K `iBM; H2X 0 1t2`+Bb2 8XNXRjX "Q`2Hė* Mi2HHB 7Q` SQBMi S`Q+2bb2b S`Qp2 i? i B7 N Bb bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ft @BMi2MbBiv {λ(t)}t≥0 ∞
N (∞) < ∞ ⇐⇒ 0
λ(s) ds < ∞
P @ XbX
8XNX 1s1_*Aa1a
kk8
1t2`+Bb2 8XNXR9X SQBbbQM rBi? AM/2T2M/2Mi BB/ J `Fb a?Qr i? i B7 N Bb SQBbbQM T`Q+2bb QM R × K rBi? BMi2MbBiv K2 bm`2 ν(dt × dz) = λ(t)dt × ν(dz) , r?2`2 ν Bb σ@}MBi2 K2 bm`2 QM (K, K) M/ t → λ(t) Bb /2i2`KBMBbiB+ MQM@ M2; iBp2 HQ+ HHv BMi2;` #H2 7mM+iBQM- i?2M N Bb HQ+ HHv }MBi2 bBKTH2 TQBMi T`Q+2bb rBi? i?2 biQ+? biB+ FtN @BMi2MbBiv F2`M2H λ(t, dz) = λ(t)ν(dz). 1t2`+Bb2 8XNXR8X BB/ J `Fb AM/2T2M/2Mi Q7 i?2 " bB+ SQBMi S`Q+2bb G2i (N, Z) #2 bBKTH2 M/ HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb rBi? K `Fb BM KrBi? bbQ+B i2/ UHB7i2/V TQBMi T`Q+2bb NZ QM R × KX amTTQb2 i? i N ? b i?2 Ft @BMi2MbBiv {λ(t)}t∈R M/ i? i i?2 K `F b2[m2M+2 {Zn }n∈Z Bb BB/- rBi? +QKKQM /Bbi`B#miBQM P (Z1 ∈ C) = Q(C)- M/ BM/2T2M/2Mi Q7 F∞ X a?Qr i? i NZ ? b i?2 (FtNZ ∨ Ft )@BMi2MbBiv F2`M2H λ(t, dz) = λ(t)Q(dz). 1t2`+Bb2 8XNXReX GBF2HB?QQ/ _ iBQ S`Qp2 i?2 bb2`iBQM Q7 1t KTH2 8X8XeX 1t2`+Bb2 8XNXRdX *QMbi Mi J2 M amT2`K `iBM; H2 S`Qp2 i? i bmT2`K `iBM; H2 rBi? +QMbi Mi K2 M Bb M2+2bb `BHv K `iBM; H2X 1t2`+Bb2 8XNXR3X J `iBM; H2b b aiQ+? biB+ AMi2;` Hb G2i {M (t)}t≥0 #2 `B;?i@+QMiBMmQmb HQ+ H Ft @K `iBM; H2X amTTQb2 BM //BiBQM i? i Bi Bb b[m `2@BMi2;` #H2- i? i Bb, supt≥0 E [|M (t)|2 ] < ∞X S`Qp2 i? i H(s, z)M Z (ds × dz) , M (t) = M (0) + (0,t]
K
r?2`2 M Z (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) ds M/ H ∈ P(F· ) ⊗ K Bb bm+? i? i
t |H(s, z)|2 λ(s) Φ(s, dz) ds < ∞ E 0
K
7Q` HH t ≥ 0X 1t2`+Bb2 8XNXRNX h?2 bBim iBQM Bb i? i Q7 h?2Q`2K XkXR8X a?Qr i? i 7Q` 2 +? n ≥ 0- Tn Bb M Ft @biQTTBM; iBK2 M/ i? i FTn ⊇ σ (T0 , Z0 , T1 , Z1 , . . . , Tn , Zn ) , M/ i? i 7Q` 2 +? t ≥ 0-
kke *>Sh1_ 8X SPALh S_P*1aa1a qAh> ahP*>ahA* ALh1LaAhu Ft = FtY := σ(Y (s) ; s ∈ [0, t]) , r?2`2- 7Q` HH t ≥ 0- HH n ∈ N- Y (t) = Zn QM [Tn , Tn+1 ) M/ Y (t) = Δ 7Q` t ≥ T∞ r?2`2 Δ Bb M `#Bi` `v dz/mKKvǴ 2H2K2MiX 1t2`+Bb2 8XNXkyX 1tTHB+BiHv /2b+`B#2 i?2 `2bmHi Q7 h?2Q`2K X9Xj 7Q` S = T5 M/ 7Q` S = inf{Tn > 0 ; Tn − Tn−1 ≥ 3}X 1t2`+Bb2 8XNXkRX MQi?2` hvT2 Q7 1K#2//BM; S`Qp2 i?2 7QHHQrBM; `2bmHib, M/ H2i #2 M ?TT QM R rBi? BMi2MbBiv λh?2Q`2K R, /B`2+i BK#2//BM;X G2i N U #2 {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 K `Fb- mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N i?2 bbQ+B i2/ HB7i2/ TQBMi T`Q+2bb QM R×[0, 1]X G2i {Gt }t∈R #2 ?BbiQ`v BM/2T2M/2Mi U - M/ /2}M2 7Q` HH t ∈ RQ7 N
Ft = Gt ∨ FtNU . < ∞- M/ G2i {λ(t)}t∈R #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bb #QmM/2/ #v λ /2}M2 TQBMi T`Q+2bb N QM (R, B) #v N (C) = 1C (Tn )10≤U ≤ λ(Tn ) (C ∈ B) . () n∈Z
n
λ
h?2M N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2Q`2K k, BMp2`b2 BK#2//BM;X G2i N #2 bBKTH2 TQBMi T`Q+2bb QM (R, B(R)) rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R U bbmK2/ Ft @T`2/B+i #H2- rBi?Qmi HQbb Q7 ;2M2` HBivVX < ∞ bm+? i? i amTTQb2 i? i i?2`2 2tBbib +QMbi Mi λ , λ(t, ω) ≤ λ
∀t ∈ R ,
P @ XbX
, U ) QM (R, B(R)) rBi? K `Fb h?2M- i?2`2 2tBbib +QKTQmM/ SQBbbQM T`Q+2bb (N BM ([0, 1]- B([0, 1]))- +? ` +i2`BbiB+b (λ, Q) r?2`2 Q Bb i?2 mMB7Q`K /Bbi`B#miBQM QM [0, 1]- M/ bm+? i? i UV ?QH/bX 1t2`+Bb2 8XNXkkX ∞ G2i h : R → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM bm+? i? i 0 th(t) dt < ∞X G2i N #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM R+ rBi? i?2 FtN @BMi2MbBiv λ(t) = (−∞,t) h(t − s) N (ds) M/ p2` ;2 BMi2MbBiv λX S`Qp2 i? i B7 λ < ∞- i?2M P (N (R+ ) = 0) > 0X .2/m+2 7`QK i?Bb i? i i?2 QMHv bBKTH2 bi iBQM `v 2`;Q/B+ TQBMi T`Q+2bb N QM R+ rBi? i?2 FtN @BMi2MbBiv λ(t) = (−∞,t) h(t − s) N (ds) Bb i?2 2KTiv TQBMi T`Q+2bbX 1t2`+Bb2 8XNXkjX /7` AMi2``2M2r H .Bbi`B#miBQM a?Qr i? i i?2 /7` T`QT2`iv Q7 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F BKTHB2b mM@ #QmM/2/M2bb Q7 i?2 bmTTQ`i Q7 F X Ua22 1t KTH2 8XdX9XV
8XNX 1s1_*Aa1a
kkd
1t2`+Bb2 8XNXk9X aiQ+? biB+ F2`M2H Q7 i?2 bi iBQM `v ;`B/ QM R+ *QMbB/2` i?2 7QHHQrBM; TQBMi T`Q+2bb QM R+ , Tn = (n − 1) + U
(n ≥ 1) ,
r?2`2 U Bb mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X *QKTmi2 Bib FtN @+QKT2Mb iQ`X
*? Ti2` e 1tpBbB#H2 AMi2MbBiv Q7 6BMBi2 SQBMi S`Q+2bb2b h?Bb +? Ti2` BMi`Q/m+2b i?2 MQiBQM Q7 2tpBbB#H2 BMi2MbBivXR h?Bb Bb /QM2 BM i?2 7` K2rQ`F Q7 }MBi2 TQBMi T`Q+2bb2b MQi QMHv #2+ mb2 i?2 i?2Q`v Bb 2 bB2` i? M BM i?2 ;2M2` H + b2- #mi HbQ #2+ mb2 i?Bb `2bi`B+iBQM 7mH}HHb i?2 M22/b Q7 K Mv TTHB+ iBQMb- BM 2+QHQ;v M/ 2Hb2r?2`2X Hi?Qm;? i?2 i?2Q`v Q7 2tpBbB#H2 BMi2MbBiv 7Q` bT iB H TQBMi T`Q+2bb2b K v #2 i }`bi bB;?i +QMbB/2`2/ M 2ti2MbBQM Q7 i?2 i?2Q`v Q7 biQ+? biB+ BMi2MbBiv QM i?2 HBM2 U/m2 iQ 72r 7Q`K H M HQ;B2b i? i rBHH #2 2t?B#Bi2/ H i2`V- Bi #2HQM;b BM 7 +i iQ i?2 S HK i?2Q`v BM bT +2 Q7 *? Ti2` 3X >Qr2p2`- KQ`2 2H2K2Mi `v TT`Q +? Bb /QTi2/ BM i?Bb +? Ti2`X
eXR
h?2 C MQbbv .2MbBiv
6BMBi2 TQBMi T`Q+2bb2b rBHH #2 /2b+`B#2/ BM K MM2` bHB;?iHv /Bz2`2Mi 7`QK i?2 QM2 /QTi2/ T`2pBQmbHv 7Q` ;2M2` H TQBMi T`Q+2bb2bX G2i W #2 K2 bm` #H2 bm#b2i Q7 Rm UBM TTHB+ iBQMb- Bi Bb mbm HHv +QKT +i b2iV M/ H2i W #2 i?2 "Q`2H σ@}2H/ QM BiX G2i Mpf (W )- + HH2/ i?2 +QM};m` iBQM bT +2 QM W - #2 i?2 +QHH2+iBQM Q7 }MBi2 b2ib x Q7 TQBMib Q7 W - rBi? TQbbB#H2 `2T2iB@ iBQMbX JQ`2 T`2+Bb2Hv- +QM};m` iBQM x ∈ Mpf (W ) Q7 + `/BM HBiv n Bb M mMQ`/2`2/ b2[m2M+2 Q7 TQBMib Q7 W x = {x1 , x2 , . . . , xn } , r?2`2 KmHiBTH2 TQBMib Ui? i Bb- xi = xj 7Q` i = jV `2 HHQr2/X G2i Mfp (W ) #2 i?2 σ@}2H/ QM Mpf (W ) ;2M2` i2/ #v i?2 K TTBM;b x → |x ∩ C| 7Q` HH K2 bm` #H2 b2ib Ur?2`2 |A| /2MQi2b i?2 + `/BM HBiv Q7 i?2 b2i AV C ∈ WX }MBi2 TQBMi T`Q+2bb X QM W Bb ` M/QK 2H2K2Mi X : (Ω, F) → (Mpf (W ), Mfp (W )) . hQ +QMM2+i rBi? i?2 T`2pBQmb MQi iBQM N 7Q` TQBMi T`Q+2bb- Dmbi H2i 7Q` Mv C ∈ WN (C) := |X ∩ C| . R
(S T M;2HQm- RNdk)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_6
kkN
kjy *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a PM2 r v Q7 /2}MBM; }MBi2 TQBMi T`Q+2bb QM W Bb i?2 7QHHQrBM;, RX a2H2+i i?2 MmK#2` Q7 TQBMib ++Q`/BM; iQ +QmMi /Bbi`B#miBQM {pn }n≥0 QM N, P (N (W ) = n) = pn
(n ∈ N) .
kX :Bp2M i?2 MmK#2` Q7 TQBMib n ≥ 1- TH +2 i?2K QM W #v /` rBM; (X1 , . . . , Xn ) ∈ W n ++Q`/BM; iQ P ((X1 , . . . , Xn ) ∈ C | N (W ) = n) =
jn (x1 , . . . , xn ) λ(dx1 ) · · · λ(dxn ) , C
r?2`2 λ Bb bQK2 }MBi2 K2 bm`2 QM W M/ jn (x1 , . . . , xn ) Bb bvKK2i`B+ 7mM+iBQM+ HH2/ i?2 /2MbBiv Q7 Q`/2` nX UAi ? b iQ #2 bvKK2i`B+ bBM+2 (X1 , . . . , Xn ) Bb M Q`/2`2/ p2+iQ`- r?2`2 b i?2 +QM};m` iBQM {X1 , . . . , Xn } Bb mMQ`/2`2/XV P7 +Qm`b2i?2 /2MbBiB2b Kmbi b iBb7v i?2 B/2MiBiB2b jn (x1 , . . . , xn ) λ(dx1 ) · · · λ(dxn ) = 1
(n ∈ N) .
Wn
h?2 `2bmHiBM; TQBMi T`Q+2bb Bb bBKTH2 B7 λ Bb MQM@ iQKB+X AM i?Bb b2+iBQM- i?Bb rBHH #2 bbmK2/X G2i {πn }n≥0 #2 T`Q# #BHBiv /Bbi`B#miBQM QM N M/ H2i λ #2 MQM@ iQKB+ T`Q# #BHBiv K2 bm`2 QM (W, W)X G2i Q #2 i?2 /Bbi`B#miBQM Q7 }MBi2 TQBMi T`Q+2bb QM W Q7 i?2 7Q`K ∞ π n λn , n=0
r?2`2 λn := λ⊗n Bb i?2 T`Q# #BHBiv K2 bm`2 QM Mpf (W ) +Q``2bTQM/BM; iQ 2t +iHv n TQBMib TH +2/ BM/2T2M/2MiHv ++Q`/BM; iQ i?2 /Bbi`B#miBQM λ- M/ r?2`2 λ0 Bb i?2 T`Q# #BHBiv K2 bm`2 QM Mpf (W ) rBi? HH Bib K bb QM i?2 2KTiv TQBMi K2 bm`2X h?mb- mM/2` Q- ` M/QK +QM};m` iBQM Bb ;2M2` i2/ b 7QHHQrbX qBi? T`Q# #BHBiv π0 Bi Bb i?2 2KTiv +QM};m` iBQM- M/ rBi? T`Q# #BHBiv πn Un ≥ 1V- Bi +QMbBbib Q7 n TQBMib /` rM BM/2T2M/2MiHv QM W ++Q`/BM; iQ i?2 T`Q# #BHBiv /Bbi`B#miBQM λX h?2 2tT2+i iBQM rBi? `2bT2+i iQ Q Bb /2MQi2/ #v EQ X h?2 7mM+iBQM x → jn (x1 , . . . , xn ) := n!pn jn (x1 , . . . , xn ) Bb + HH2/ i?2 C MQbbv k /2MbBiv Q7 Q`/2` nX Aib BMi2`T`2i iBQM Bb i? i jn (x1 , . . . , xn )λ(dx1 ) · · · λ(dxn ) Bb i?2 T`Q# #BHBiv Q7 }M/BM; 2t +iHv n TQBMib- 1 BM dx1 - Ę- 1 BM dxn - r?2`2 i?2 BM}MBi2bBK H b2ib dx1 , . . . , dxn `2 Kmim HHv /BbDQBMiX k
(C MQbbv- RN93)X
eXRX h>1 CLPaau .1LaAhu
kjR
.2}MBiBQM BM h2`Kb Q7 _ /QMĜLBFQ/ɷK .2`Bp iBp2 G2i : Mpf (W ) → [0, ∞] #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM bm+? i? i EQ [ (X)] = 1 .
UeXRV
h?Bb +QM/BiBQM HHQrb mb iQ /2}M2 T`Q# #BHBiv P QM Mpf (W ) #v Bib _ /QMĜ LBFQ/ɷK /2`Bp iBp2 rBi? `2bT2+i iQ Qc dP (x) = (x) . dQ
UeXkV
S`Q# #BHBiv Q Bb + HH2/ i?2 `272`2M+2 K2 bm`2X h?2Q`2K eXRXR h?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 (x) Bb `2H i2/ iQ i?2 +QmMi /Bb@ i`B#miBQM {pn }n∈N M/ i?2 /2MbBiB2b {jn }n≥1 b 7QHHQrb, p0 = π0 0 (∅) M/ 7Q` n ≥ 1
n ({x1 , . . . , xn }) λ(dx1 ) . . . λ(dxn ) pn = πn Wn
Ui?2 MQi iBQM n ({x1 , . . . , xn }) bi M/b 7Q` (x) r?2M x = {x1 , . . . , xn }V M/ jn (x1 , . . . , xn ) =
n ({x1 , . . . , xn }) .
({x , 1 . . . , xn }) λ(dx1 ) . . . λ(dxn ) Wn n
S`QQ7X h?2 + b2 n = 0 Bb +H2 `X 6Q` n ≥ 1pn = P (N (W ) = n) = EQ [1{N (W )=n} (X)] = EQ [1{N (W )=n} n ({X1 , . . . , Xn })] = Q(N (W ) = n) × EQ [ (X) | N (W ) = n] = πn EQ [ n ({X1 , . . . , Xn }) | N (W ) = n] = πn
n ({x1 , . . . , xn }) λ(dx1 ) · · · λ(dxn ) . Wn
G2i MQr ϕ : Mpf (W ) → [0, ∞] #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX 6Q` n ≥ 1E[ϕ(X) | N (W ) = n] =
E[1{N (W )=n} ϕ(X)] P (N (W ) = n)
M/ E[ϕ(X)1{N (W )=n} ] = EQ [1{N (W )=n} ϕ(X) (X)] = Q(N (W ) = n) EQ [ϕ(X) (X) | N (W ) = n] ϕ({x1 , . . . , xn }) n ({x1 , . . . , xn }) λ(dx1 ) · · · λ(dxn ). = πn Wn
h?2`27Q`2
ϕ({x1 , . . . , xn }) jn (x1 , . . . , xn ) λ(dx1 ) . . . λ(dxn ) ,
E[ϕ(X) | N (W ) = n] = Wn
r?2`2 jn Bb b /2}M2/ BM h?2Q`2K eXRXRX
kjk *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a _2K `F eXRXk LQi2 i? i +QM/BiBQM UeXRV `2 /b π0 (∅) +
∞ n=1
i? i Bb-
∞ n=0
n ({x1 , . . . , xn }) λ(dx1 ) . . . λ(dxn ) = 1 ,
πn Wn
pn = 1X
_2K `F eXRXj AM T`BM+BTH2- i?2 KQ/2HH2` ? b BM KBM/ U V, bT2+B}+ +QmMi /Bbi`B@ #miBQM {pn }n≥0 M/- T2`? Tb KQ`2 BKTQ`i MiHv- U#V, i?2 dzb? T2Ǵ Q7 i?2 /2MbBiB2b jn +Q``2bTQM/BM; iQ i?2 K2 bm`2 λX "v i?2 b? T2 Q7 jn - r2 K2 M 7mM+iBQM jn 2[m H iQ +QMbi Mi UTQbbB#Hv /2T2M/BM; QM nV iBK2b jn X h?2 b? T2 Q7 i?2 /2M@ bBiB2b `2T`2b2Mib i?2 ivT2 Q7 BMi2` +iBQM #2ir22M TQBMib /QTi2/ #v i?2 KQ/2HH2`X b i?2 2tT`2bbBQM Q7 i?2 /2MbBiv BM h?2Q`2K eXRXR b?Qrb- i?2 b? T2 Q7 jn Bb i?2 b K2 b i? i Q7 n X h?2`27Q`2- KmHiBTHvBM; n - 7Q` HH n- #v TQbBiBp2 +QMbi Mi Kn /Q2b MQi +? M;2 i?2 /2MbBiB2b jn X >Qr2p2`- Bi +? M;2b i?2 +QmMi /Bbi`B#miBQM 7`QK pn iQ pn = Kn pn X h?Bb 7`22/QK + M MQr #2 mb2/ BM i?2 2tT`2bbBQM Q7 pn BM h?2Q`2K eXRXR iQ Q#i BM i?2 /2bB`2/ +QmMiBM; /Bbi`B#miBQMX h?2`2 Bb ?Qr2p2` /B{+mHiv BM + ``vBM; i?Bb T`Q;` K iQ Bib 2M/ bBM+2 i?2 +QKTmi iBQM Q7 i?2 BMi2;` H
({x1 , . . . , xn })λ(dx1 ) . . . λ(dxn ) Bb BM ;2M2` H /B{+mHiX Wn n _2K `F eXRX9 LQi2 HbQ i? i r2 + MMQi mb2 M `#Bi` `v KmHiBTHB+ iBp2 7 +iQ` Kn bBM+2 r2 ? p2 iQ `2bT2+i i?2 +QM/BiBQM n≥0 pn = 1- i? i Bb Kn p n = 1 . n≥0
AM i?2 TTHB2/ HBi2` im`2- QM2 Q7i2M K F2b i?2 +?QB+2 Q7 SQBbbQM `272`2M+2 K2 bm`2X 1t KTH2 eXRX8, SQBbbQM _272`2M+2 J2 bm`2- h F2 RX h F2 Q = P ν iQ #2 i?2 /Bbi`B#miBQM Q7 SQBbbQM T`Q+2bb QM W rBi? i?2 }MBi2 MQM@ iQKB+ BMi2MbBiv K2 bm`2 ν Ui?2 +Q``2bTQM/BM; TQBMi T`Q+2bb Bb }MBi2 bBM+2 i?2 p2` ;2 MmK#2` Q7 TQBMib ν(W ) Bb }MBi2V M/ H2i : Mpf (W ) → [0, ∞] #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM bm+? i? i Eν [ (X)] = 1 . h?Bb +QM/BiBQM HHQrb mb iQ /2}M2 T`Q# #BHBiv P QM Mpf (W ) #v i?2 _ /QMĜ LBFQ/ɷK /2`Bp iBp2 dP (x) = (x) . dP ν h?Bb Bb T `iB+mH ` + b2 Q7 i?2 ;2M2` H +QMbi`m+iBQM #Qp2- rBi? πn = M/ λ(dx) = ν(dx)/ν(W )X ++Q`/BM; iQ h?2Q`2K eXRXR e−ν(W ) pn =
n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) n! Wn M/ jn (x1 , . . . , xn ) = Wn
n ({x1 , . . . , xn }) .
n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn )
e−ν(W ) ν(W )n n!
eXRX h>1 CLPaau .1LaAhu
kjj
AM T `iB+mH `- 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM ϕ : Mpf (W ) → RE [ϕ(X)] =
∞ e−ν(W ) n=0
n!
ϕ(x1 , . . . , xn ) n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) . UeXjV Wn
1t KTH2 eXRXe, > `/@+Q`2 SQBMi S`Q+2bb- h F2 RXj h?2 `272`2M+2 K2 bm`2 Bb b BM 1t KTH2 eXRX8X AM i?2 ? `/@+Q`2 KQ/2H i ` M;2 R > 0 n ({x1 , . . . , xn }) = αβ n Π1≤i2R} , r?2`2 β > 0 M/ α > 0 `2 +?Qb2M bm+? i? i (x) BMi2;` i2b iQ R rBi? `2bT2+i iQ P ν X AM i?2 + b2 i? i β = 1- i?2 T`Q# #BHBiv P Bb P ν +QM/BiBQM2/ #v i?2 2p2Mi i? i i?2`2 Bb MQ T B` Q7 TQBMib i Kmim H /Bbi M+2 ≤ RX * HHBM; i?Bb 2p2Mi F - r2 i?2M ? p2 P (·) = P ν (· | F ). PM2 UBM2{+B2MiV r v iQ b KTH2 P Bb iQ K F2 BM/2T2M/2Mi b KTHBM;b Q7 i?2 /Bbi`B@ #miBQM P ν mMiBH b KTH2 BM F Bb Q#i BM2/X h?2 BMi2;` H BMpQHp2/ BM i?2 2tT`2bbBQM 7Q` pn /Q2b MQi ? p2 +HQb2/ 7Q`KX amTTQb2 r2 +? M;2 α M/ β iQ α = a α M/ β = b β `2bT2+iBp2HvX h?2 /2MbBiB2b Q7 Mv Q`/2` `2 MQi Hi2`2/ #mi i?2 T`Q# #BHBiv Q7 ? pBM; n TQBMib BM i?2 +QM};m` iBQM #2+QK2b pn = abn pn X h?2 +QM/BiBQM E ν [ (x)] = 1 Bb i?2M pn = a bn p n = 1 . n≥0
n≥0
6BMBi2 J `F2/ SQBMi S`Q+2bb2b q2 MQr BMi`Q/m+2 K `FbX G2i G #2 #QmM/2/ K2 bm` #H2 bm#b2i Q7 Rm M/ H2i G #2 i?2 σ@}2H/ Q7 "Q`2H bm#b2ib Q7 GX }MBi2 K `F2/ TQBMi T`Q+2bb QM G rBi? K `Fb BM i?2 K2 bm` #H2 bT +2 (K, K) Bb ` M/QK 2H2K2Mi X ∈ Mpf (G × K) Q7 i?2 7Q`K {(Y1 , Z1 ), . . . , (Yn , Zn )} r?2M |X| = n- r?2`2 i?2 Zn Ƕb i F2 i?2B` p Hm2b BM KX h?Bb + b2 i?2`27Q`2 /Q2b MQi `2[mB`2 M / Ti iBQM Q7 i?2 i?2Q`v i i?Bb bi ;2bBM+2 X Bb }MBi2 +QM};m` iBQM QM E := G × KX 1t KTH2 eXRXd, h?2 SQBbbQM _272`2M+2 J2 bm`2- h F2 kX AM 1t KTH2 eXRX8- i?2 K2 bm`2 ν + M #2 +?Qb2M Q7 i?2 T`Q/m+i 7Q`K ν(dx) = ν(dy, dz) = σ(dy)QZ (dz) , r?2`2 QZ Bb T`Q# #BHBiv /Bbi`B#miBQM QM (K, K) M/ σ Bb }MBi2 MQM@ iQKB+ K2 bm`2 QM G U+QMb2[m2MiHv- ν Bb MQM@ iQKB+VX h?Bb BKTHB2b BM T `iB+mH ` i? i j
(ai` mbb- RNd8)X
kj9 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a mM/2` i?2 `272`2M+2 T`Q# #BHBiv Q- Y Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 σ M/ i? i i?2 K `Fb `2 BB/ rBi? +QKKQM /Bbi`B#miBQM QZ M/ BM/2T2M/2Mi Q7 i?2 # bB+ TQBMi T`Q+2bb YX 1t KTH2 eXRX3, h?2 S2M2i` #H2 aT?2`2b JQ/2H- h F2 RX AM i?Bb 2t KTH2G Bb K2 bm` #H2 bm#b2i Q7 Rm Q7 }MBi2 pQHmK2 U7Q` BMbi M+2 }MBi2 m@+m#2Vσ = λ m - K = {1, 2}- QZ ({1}) = α1 M/ QZ ({2}) = α2 X AM T `iB+mH `- Y Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ M/ i?2 TQBMi T`Q+2bb2b Y1 M/ Y2 QM E i? i +QmMi i?2 TQBMib Q7 Y rBi? K `Fb 1 M/ 2 `2bT2+iBp2Hv `2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? BMi2MbBiv `2bT2+iBp2 BMi2MbBiB2b λ1 := λα1 M/ λ2 := λα2 X AM i?2 T2M2i` #H2 bT?2`2b KQ/2H i ` M;2 R > 0 (x) = αβ |y| 1{d(y1 ,y2 )>R} , r?2`2 d(y1 , y2 ) Bb i?2 bK HH2bi /Bbi M+2 #2ir22M TQBMi Q7 y1 M/ TQBMi Q7 y2 r?2`2 β > 0 M/ α > 0 `2 +?Qb2M bm+? i? i (x) BMi2;` i2b iQ R rBi? `2bT2+i iQ P ν X AM i?2 + b2 i? i β = 1- i?2 T`Q# #BHBiv P Bb P ν +QM/BiBQM2/ #v i?2 2p2Mi i? i i?2`2 Bb MQ T B` Q7 TQBMib- QM2 BM Y1 i?2 Qi?2` BM Y2 - i Kmim H /Bbi M+2 ≤ RX * HHBM; i?Bb 2p2Mi F - r2 ? p2 P (·) = P ν (· | F ). ; BM- QM2 UBM2{+B2MiV r v iQ b KTH2 P Bb iQ K F2 BM/2T2M/2Mi b KTHBM;b Q7 i?2 /Bbi`B#miBQM P ν mMiBH b KTH2 BM F Bb Q#i BM2/X
h?2 1tpBbB#H2 *QM/BiBQM H AMi2MbBiv h?2 # bB+ /2}MBiBQM Bb i?2 7QHHQrBM;9 .2}MBiBQM eXRXN G2i X #2 }MBi2 TQBMi T`Q+2bb QM i?2 K2 bm` #H2 b2i W rBi? /Bbi`B#miBQM /2}M2/ #v T`Q# #BHBiv /2MbBiv (x) rBi? `2bT2+i iQ i?2 SQBbbQM T`Q# #BHBiv K2 bm`2 Q = P ν U1t KTH2 eXRX8V b BM UeXkVX h?2 7mM+iBQM λ : W × Mpf (W ) → [0, ∞) /2}M2/ #v λ(u, x) =
(x ∪ {u})
(x)
(u ∈ / x)
UeX9V
(= 0 B7 (x) = 0) Bb + HH2/ i?2 2tpBbB#H2 BMi2MbBiv Q7 i?2 }MBi2 TQBMi T`Q+2bb XX _2K `F eXRXRy 7Q`K H `2b2K#H M+2 rBi? i?2 MQiBQM Q7 biQ+? biB+ BMi2MbBiv 7Q` TQBMi T`Q+2bb QM i?2 HBM2 Bb i?2 7QHHQrBM;X _2im`MBM; KQK2Mi `BHv iQ i?2 MQi iBQM Q7 *? Ti2` 8- H2i N #2 bBKTH2 TQBMi T`Q+2bb QM i?2 BMi2`p H [0, T ] rBi? T`2@ /B+i #H2 FtN @BMi2MbBiv {λ(t, N )}t∈[0,T ] - bbmK2/ #QmM/2/ BM i?Bb 2t KTH2X q2 K v +QMbB/2` i? i i?2 mM/2`HvBM; T`Q# #BHBiv P Bb Q#i BM2/ 7`QK T`Q# #BHBiv P0 QM 9
a22 ?Qr2p2` a2+iBQM 3X8X
eXRX h>1 CLPaau .1LaAhu
kj8
i?2 b K2 T`Q# #BHBiv bT +2 i? i K F2b Q7 N M ?TT rBi? BMi2MbBiv 1 pB M #bQ@ Hmi2Hv +QMiBMmQmb +? M;2 Q7 T`Q# #BHBiv K2 bm`2 2K#Q/B2/ #v i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 L(T, N ) r?2`2 t L(t, N ) := Πn;Tn ∈(0,t] λ(Tn , N ) exp − (λ(s, N ) − 1) ds . 0
AM T `iB+mH `- B7 QM2 T`2i2M/b i? i i?2`2 Bb M //BiBQM H TQBMi i t- i?Bb 2tT`2bbBQM #2+QK2b L(t, N ∪ t) = λ(t, N )L(t, N ) , M/ i?2`27Q`2 λ(t, N ) =
L(t, N ∪ t) . L(t, N )
1t KTH2 eXRXRR, h?2 S2M2i` #H2 aT?2`2b JQ/2H- h F2 kX AM i?2 T2M2@ i` #H2 bT?2`2b KQ/2H i ` M;2 R > 0- i?2 2tpBbB#H2 BMi2MbBiv ;Bp2M #v UeX9V BM i?Bb + b2 i F2b i?2 7QHHQrBM; 7Q`K, B7 u = (v, 1)λ(u, x) = β1{d(v,y2 )>R} , r?2`2 d(v, y2 ) Bb i?2 bK HH2bi /Bbi M+2 #2ir22M v M/ TQBMi Q7 y2 - rBi? bBKBH ` `2bmHi 7Q` u = (v, 2)X S`QQ7X q2 mb2 i?2 MQi iBQM x = (y, z)- i?mb b2T ` iBM; i?2 K `Fb 7`QK i?2 TQBMibX h?Bb Bb bQK2r? i #mbBp2 MQi iBQM- #mi BMMQ+mQmb B7 r2 /QTi i?2 +QMp2MiBQM i? i- r?2M |x| = n- {x1 , . . . , xn } = {(y1 , z1 ), . . . , (yn , zn )} Ui?2 BM/2t Q7 K `F BM/B+ i2b i?2 TQBMi iQ r?B+? Bi Bb ii +?2/VX h?2`27Q`2- rBi? i?Bb MQi iBQM- (x) =
(y, z)X G2i v M/ ζ #2 TQBMib Q7 G M/ K `2bT2+iBp2HvX h?2 2tpBbB#H2 BMi2MbBiv i (v, ζ) Bb- #v /2}MBiBQMλ((v, ζ), (y, z)) =
(y ∪ {v}, z ∪ {ζ}) ,
(y, z)
λ((v, 1), (y, z)) =
(y ∪ {v}, z ∪ {1}) ,
(y, z)
M/ BM T `iB+mH `() .
* HH i?2 TQBMi T`Q+2bb y1 i?2 1@T`Q+2bb Q7 y- rBi? bBKBH ` K2 MBM; 7Q` i?2 2@ T`Q+2bb Q7 yX h?2`2 `2 irQ TQbbB#BHBiB2b +QM+2`MBM; i?2 /DmM+iBQM Q7 1@TQBMi i vX 1Bi?2` d(v, y2 ) > R- BM r?B+? + b2 i?2 /Bbi M+2 #2ir22M i?2 M2r 1@T`Q+2bb M/ y2 /Q2b MQi z2+i i?2 +QM/BiBQM Q7 b2T ` iBQM Q7 i?2b2 irQ TQBMi T`Q+2bb2b M/ i?2 MmK2` iQ` M/ /2MQKBM iQ` BM UV `2 i?2 b K2X AM i?2 Qi?2` + b2- i?Bb +QM/BiBQM Bb #`QF2M M/ i?2 MmK2` iQ` BM UV Bb MmHHX >2M+2 i?2 MMQmM+2/ `2bmHiX
kje *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a
eXk h?2 aT iB H aKQQi?BM; 7Q`KmH h?2 7QHHQrBM; `2bmHi 72 im`2b v2i MQi?2` p i ` Q7 * KT#2HHǶb 7Q`KmH - bKQQi?@ BM; 7Q`KmH Q7 i?2 b K2 ivT2 b i?2 QM2 7Q` TQBMi T`Q+2bb2b QM i?2 HBM2 Q7 i?2 T`2pBQmb +? Ti2`X h?2Q`2K eXkXR G2i Q = P ν #2 b BM 1t KTH2 eXRX8X h?2M- 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM g : W × Mpf (W ) → [0, ∞]
g(x, X\x) = E g(u, X) λ(u, X)ν(du) , UeX8V E W
x∈X
r?2`2 X\x := X\{x}X S`QQ7X 6`QK UeXjV
g(u, X)λ(u, X)ν(du) A := E W
∞ e−ν(W )
=
n!
n=0
g(u, {x1 , . . . , xn }) λ(u, {x1 , . . . , xn }) × · · · W n+1
· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn )ν(du)
∞ e−ν(W )
=
n!
n=0
g(u, {x1 , . . . , xn }) ({x1 , . . . , xn } ∪ {u}) W n+1
ν(dx1 ) . . . ν(dxn )ν(du) . qBi? i?2 +? M;2 Q7 p `B #H2b u → xi - x1 → x1 , . . . , xi−1 → xi−1 - xi → xi+1 , . . . , xn → xn+1 , g(u, {x1 , . . . , xn }) ({x1 , . . . , xn } ∪ {u}) ν(dx1 ) . . . ν(dxn )ν(du) = W n+1
g(xi , {x1 , . . . , xi−1 ,xi+1 , . . . , xn+1 }) ({x1 , . . . , xi−1 , xi+1 , . . . , xn+1 } ∪ {xi }) W n+1
ν(dx1 ) . . . ν(dxi−1 ) ν(dxi+1 ) . . . ν(dxn+1 ) ν(dxi ) . h?2`27Q`2 A=
∞ e−ν(W ) n=0
=
∞ n=1
n! e−ν(W ) n!
W n+1
n+1 1 g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn+1 }) × · · · n + 1 i=1
· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) ν(du) n
g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn }) × · · ·
W n i=1
· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) .
eXkX h>1 aShAG aJPPh>AL: 6P_JlG
kjd
g(x, X\x) = 0 B7 |X| = 0 g(x, X\x) B := E
LQr- bBM+2
x∈X
x∈X
=
∞ e−ν(W ) n=1
n!
n
g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn }) × · · ·
W n i=1
({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) ,
M/ i?2`27Q`2 A = BX
1t KTH2 eXkXk, h?2 aKQQi?BM; 6Q`KmH 7Q` SQBbbQM S`Q+2bb2bX A7 P ≡ P ν Q`- 2[mBp H2MiHv- (x) ≡ 1 M/ +QMb2[m2MiHv λ(u, x) ≡ 1- r2 ? p2 i?2 dzbKQQi?BM; 7Q`KmH Ǵ
E g(x, X\x) = E g(u, X) ν(du) . W
x∈X
_2K `F eXkXj M BMi2`T`2i iBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv bBKBH ` iQ i?2 BM}MBi2b@ BK H BMi2`T`2i iBQM Q7 i?2 biQ+? biB+ BMi2MbBiv 7Q` TQBMi T`Q+2bb2b QM i?2 HBM2 Bb p BH #H2X G2i B(u, ε) #2 i?2 +HQb2/ # HH +2Mi2`2/ i u ∈ Rm M/ rBi? ` /Bmb ε > 0X X G2i F #2 M `#Bi` `v b2i BM FB(u,ε) i?2 σ@}2H/ ;2M2` i2/ #v i?2 ` M/QK 2H2K2Mi X ∩ B(u, ε)X LQiB+2 i? i 1F (X) = 1F (X\v) B7 v ∈ B(u, ε)X h?2M E [1F (X)X(B(u, ε))] = E 1B(u,ε) (v)1F (X) v∈X
=E
1B(u,ε) (v)1F (X\v)
v∈X
1F (X)λ(v, X) ν(dv) B(u,ε)
λ(v, X) ν(dv) . = E 1F (X) =E
B(u,ε) X FB(u,ε)
r2 ? p2 T`Qp2/ i? i
X X =E . λ(v, X) ν(dv) | FB(u,ε) E X(B(u, ε)) | FB(u,ε)
aBM+2 F Bb `#Bi` `v BM
B(u,ε)
amTTQb2 i? i i?2 `272`2M+2 SQBbbQM T`Q+2bb Bb M ?TT rBi? mMBi BMi2MbBiv- i? i Bbν(dx) = dxX b ε Bb `#Bi` `BHv bK HH- i?Bb ;Bp2b +QMbBbi2M+v iQ i?2 7QHHQrBM; HQ+ H BMi2`T`2i iBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv, λ(u, X) = lim
1
u↓0 m (B(u, ε))
E [X(B(u, ε)) | X\{u}] .
UeXeV
>Qr2p2` iQ ;m ` Mi22 i?Bb- bQK2 i2+?MB+ H +QM/BiBQMb `2 M22/2/- 7Q` BMbi M+2 i? i u → λ(u, X) #2 HKQbi bm`2Hv #QmM/2/ M/ +QMiBMmQmb U1t2`+Bb2 eXdXjVX
kj3 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a
eXj 1tpBbB#BHBiv M/ S`2/B+i #BHBiv AM i?Bb b2+iBQM- r2 b? HH b22 ?Qr 7 ` i?2 7Q`K H M HQ;v #2ir22M i?2 i?2Q`B2b Q7 biQ+? biB+ BMi2MbBiv M/ Q7 2tpBbB#H2 BMi2MbBiv + M #2 Tmb?2/X .2}MBiBQM eXjXR biQ+? biB+ T`Q+2bb Q7 i?2 7Q`K {H(x, X)}x∈W r?2`2 H(x, X) = g(x, X\x) 7Q` bQK2 K2 bm` #H2 7mM+iBQM g : (W × Mp (W ), B(W ) ⊗ Mp (W ) → (R, B(R)) Bb + HH2/ M 2tpBbB#H2 T`Q+2bbX P#b2`pBM; i? i bBM+2 ν Bb /Bzmb2 K2 bm`2- i?2 `B;?i@? M/ bB/2 Q7 UeX8V + M #2 r`Bii2M b
g(u, X\u) λ(u, X)ν(du) .
E W
h?2`27Q`2 h?2Q`2K eXkXR + M #2 `2T?` b2/ BM i?2 7QHHQrBM; 7Q`K, h?2Q`2K eXjXk G2i Q = P ν #2 b BM 1t KTH2 eXRX8X h?2M- 7Q` Mv MQM@M2; iBp2 2tpBbB#H2 T`Q+2bb {H(x, X)}x∈X
H(x, X) = E H(u, X) λ(u, X)ν(du) . UeXdV E x∈X
W
aBKBH `Hv r2 K v `2TH +2 BM UeXdV λ(u, X) #v Bib dz2tpBbB#H2 p2`bBQMǴ λ(u, X\u) M/ T`QQ7 bBKBH ` iQ i? i Q7 h?2Q`2K 8XRXj3 ;Bp2b, h?2Q`2K eXjXj h?2 T`2/B+i #H2 p2`bBQM λ(u, X\u) Q7 S T M;2HQmǶb 2tpBbB#H2 BM@ i2MbBiv Bb bm+? i? i- P @ HKQbi@bm`2Hvλ(u, X\u) > 0 7Q` HH u ∈ X . _2K `F eXjX9 G2i (t, μ) → ϕ(t, μ) #2 K2 bm` #H2 MQM@M2; iBp2 K TTBM; 7`QK R × Mp (R) iQ R bm+? i? i B7 i?2 `2bi`B+iBQMb Q7 μ M/ μ iQ (−∞, t] `2 i?2 b K2i?2M ϕ(t, μ) = ϕ(t, μ )X AM T `iB+mH `- `2im`MBM; KQK2Mi `BHv iQ i?2 MQi iBQM Q7 *? Ti2` 8- i?2 T`Q+2bb {ϕ(t, N )}t∈R Bb FtN / Ti2/X G2iiBM; H(t, N ) := ϕ(t, N \t) r2 + HH {H(t, N )}t∈R M FtN @T`2/B+i #H2 T`Q+2bbX "v h?2Q`2K eXkXR- M/ H2iiBM; λ(t, N ) #2 i?2 2tpBbB#H2 BMi2MbBiv Q7 N
T
ϕ(t, N \t) N (dx) = E ϕ(t, N )λ(t, N ) dt . E 0
[0,T ]
aBM+2 ϕ(t, N ) M/ ϕ(t, N\t) /Bz2` QMHv QM b2i Q7 MmHH G2#2b;m2 K2 bm`2- i?2 T `B;?i@? M/ bB/2 2[m Hb E 0 ϕ(t, N \t)λ(t, N ) dt - M/ r2 Q#i BM E [0,T ]
H(t, N )N (dt) = E
T 0
H(t, N )λ(t, N ) dt .
eXjX 1soAaA"AGAhu L. S_1.A*h"AGAhu
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h?2Q`2K eXjX8 G2i Q = P ν #2 b BM 1t KTH2 eXRX8 M/ H2i g : W × Mpf (W ) → [0, ∞] #2 K2 bm` #H2 7mM+iBQM bm+? i? i E W |g(u, X)| λ(u, X)ν(du) < ∞ bQ i? i BM T `iB+mH ` 1A (x)g(x, X\x) − 1A (u)g(u, X) λ(u, X)ν(du) UeX3V MA (g) := W
x∈X
Bb r2HH /2}M2/ 7Q` HH K2 bm` #H2 b2ib A ⊆ W X G2i 7Q` Mv K2 bm` #H2 b2i C ⊆ W FC := σ(X ∩ C). h?2M- 7Q` HH K2 bm` #H2 b2ib A ⊆ W E [MA (g) | FA ] = 0 . S`QQ7X Ai bm{+2b iQ T`Qp2 i? i 7Q` HH #QmM/2/ K2 bm` #H2 7mM+iBQMb f : Mpf (W ) → [0, ∞] E MA (g)f (X ∩ A) = 0 , i? i Bb- α = β- r?2`2 α := E f (X ∩ A)
1A (x)g(x, X\x)
x∈X
M/
1A (u)g(u, X) λ(u, X)ν(du) . β := E f (X ∩ A) W
h FBM; BMiQ ++QmMi i?2 7 +i i? i B7 x ∈ A- i?2M f ((X\x) ∩ A) = f (X ∩ A)
r2 ? p2 i? i α=E
1A (x)f ((X\x) ∩ A))g(x, X\x) ,
x∈X
[m MiBiv 2[m H iQ β #v h?2Q`2K eXkXRX
_2K `F eXjXe h?2Q`2K eXjX8 b22Kb iQ #`BM; mb +HQb2 iQ i?2 MQiBQM Q7 K `iBM; H2X AM/22/- B7 A1 M/ A2 `2 K2 bm` #H2 bm#b2ib Q7 W bm+? i? i A1 ⊂ A2 - r2 ? p2 i? i MA2 (g) − MA1 (g) = MA2 −A1 (g) M/ i?2`27Q`2
E MA2 −A1 (g) | FA2 −A1 = 0 .
h?Bb Bb b 7 ` b r2 + M ;Q- M/ Bi Bb MQi +H2 ` b Q7 iQ/ v B7 i?Bb /2}MBiBQM Q7 K `iBM; H2 UBM/2t2/ #v b2ibV ? b Mv +? M+2 iQ #2 b T`Q/m+iBp2 b i? i Q7 K `iBM; H2 rBi? iBK2 BM/2tX
k9y *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a h?2 aT iB H q i M #2Ƕb h?2Q`2K Hi?Qm;? i?2 2tpBbB#H2 BMi2MbBiv i?2Q`v H +Fb BM Bib bT iB H 7Q`K HBMF rBi? K `@ iBM; H2 i?2Q`v Ur?B+? Bb i?2 b HB2Mi 72 im`2- i?2 Q`B;BM H ;Q H M/ i?2 K BM BMi2`2bi Q7 i?2 i?2Q`v Q7 biQ+? biB+ BMi2MbBivV- i?2`2 2tBbib +? ` +i2`Bx iBQM Q7 SQBbbQM T`Q+2bb2b `2KBMBb+2Mi Q7 q i M #2Ƕb i?2Q`2KX h?2Q`2K eXjXd G2i X #2 bBKTH2 }MBi2 TQBMi T`Q+2bb QM W X amTTQb2 i? i 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM g : W × Mpf (W ) → [0, ∞] E
g(x, X\x) = E
x∈X
g(u, X) ν(du) ,
UeXNV
W
r?2`2 ν Bb σ@}MBi2 /Bzmb2 K2 bm`2 QM EX h?2M X Bb SQBbbQM T`Q+2bb rBi? K2 M BMi2MbBiv K2 bm`2 νX S`QQ7X 6Q` 2 +? n ≥ 0- i?2 [m MiBiv E E 1A (x)1{X(A)=n+1} X(dx) rBHH #2 2p Hm@ i2/ BM irQ /Bz2`2Mi r vbX 6B`bi- Q#pBQmbHv- i?Bb [m MiBiv 2[m Hb (n + 1)P (X(A) = n + 1)X LQr
E 1A (x)1{X(A)=n+1} X(dx) = E 1A (x)1{(X\x)(A)=n} X(dx) , E
E
[m MiBiv r?B+? Bb- ++Q`/BM; iQ i?2 ?vTQi?2bBb- 2[m H iQ
1A (x)1{X(A)=n} ν(dx) = ν(A)P (X(A) = n) . E E
h?2`27Q`2- 7Q` HH n ≥ 0ν(A)P (X(A) = n) = (n + 1)P (X(A) = n + 1) , 7`QK r?B+? Bi 7QHHQrb i? i P (X(A) = n) = P (X(A) = 0)
ν(A)n n!
(n ≥ 0)
M/- #v MQ`K HBx iBQM- P (X(A) = 0) = e−ν(A) X h?2 `2bmHi i?2M 7QHHQrb 7`QK _ûMvBǶb pQB/ M+2 i?2Q`2K Uh?2Q`2K RXjXk8VX
eX9 6BMBi2 J `FQp SQBMi S`Q+2bb2b G2i MQr W #2 2M/Qr2/ rBi? bvKK2i`B+ M/ `2~2tBp2 `2H iBQM ∼ Ui? i Bb- 7Q` HH u- v BM q- u ∼ u M/ u ∼ v ⇔ v ∼ uVX A7 u ∼ v- u M/ v `2 + HH2/ M2B;?#Qm`bX MQM@2KTiv +QM};m` iBQM x ∈ Mpf (W ) Bb + HH2/ +HB[m2 UrBi? `2bT2+i iQ ∼V B7 u ∼ v 7Q` HH u, v ∈ xX AM T `iB+mH `- Mv bBM;H2iQM {u} Bb +HB[m2 bBM+2 u ∼ uX h?2 2KTiv +QM};m` iBQM ∅ Bb +HB[m2- #v +QMp2MiBQMX
eX9X 6ALAh1 J_EPo SPALh S_P*1aa1a
k9R
.2}MBiBQM eX9XR X Bb + HH2/ J `FQp TQBMi T`Q+2bb rBi? `2bT2+i iQ ∼ B7 7Q` HH x ∈ Mpf (W ) bm+? i? i (x) > 0U V (y) > 0 7Q` HH y ∈ Mpf (W ) bm+? i? i y ⊆ x- M/ U#V 7Q` HH u ∈ W - λ(u, x) /2T2M/b QMHv QM u M/ QM {x ∈ x ; x ∼ u}- i? i Bbλ(u, x) = λ(u, x ∪ {v}) r?2M2p2` v ∼ uX S`QT2`iv U V Bb + HH2/ i?2 ?2`2/Bi `v T`QT2`ivX S`QT2`iv U#V b vb i? i λ(u, x) /2@ T2M/b QMHv QM u M/ QM i?2 M2B;?#Qm`b Q7 u BM xX h?2 7QHHQrBM; Bb +? ` +i2`Bx iBQM Q7 i?2 J `FQp T`QT2`iv,8 h?2Q`2K eX9Xk G2i X #2 }MBi2 TQBMi T`Q+2bb /2}M2/ #v T`Q# #BHBiv /2MbBiv
(x) rBi? `2bT2+i iQ i?2 /Bbi`B#miBQM P ν Q7 SQBbbQM T`Q+2bb Q7 MQM@ iQKB+ }MBi2 BMi2MbBiv K2 bm`2 ν b BM 1t KTH2 eXRX8X 6Q` X iQ #2 J `FQp T`Q+2bb rBi? `2bT2+i iQ ∼- Bi Bb M2+2bb `v M/ bm{+B2Mi i? i
(x) = ϕ(y) UeXRyV y⊆x
7Q` bQK2 7mM+iBQM ϕ : Mpf (W ) → [0, ∞) bm+? i? i ϕ(y) = 1 r?2M2p2` y Bb MQi +HB[m2X S`QQ7X am{+B2M+v, amTTQb2 i? i UeXRyV ?QH/b 7Q` bQK2 ϕ : Mpf (W ) → [0, ∞) bm+? i? i ϕ(y) = 1 B7 y Bb MQi +HB[m2X q2 +?2+F U V M/ U#V Q7 .2}MBiBQM eX9XRX U V "v i?2 ?2`2/Bi `v T`QT2`iv-! (x) > 0 BKTHB2b i? i ϕ(y) > 0 7Q` HH y ⊆ x M/ BM T `iB+mH ` B7 z ⊆ x- (z) = y⊆z ϕ(y) > 0X U#V q`Bi2 7Q` u ∈ /x
! ϕ(y) ϕ(y ∪ {u})
(x ∪ {u}) y⊆x y⊆x ! = = ϕ(y ∪ {u}) . λ(u, x) =
(x) ϕ(y) y⊆x !
y⊆x
A7 y ⊆ x M/ y ∪ {u} Bb MQi +HB[m2- i?2M ϕ(y ∪ {u}) = 1 U M/ i?2`27Q`2 /Q2b MQi /2T2M/ QM Mvi?BM;VX A7 y ⊆ x M/ y ∪ {u} Bb +HB[m2- i?2M ϕ(y ∪ {u}) /2T2M/b QM u M/ QM HH i?2 TQBMib x ∈ y- r?B+? HH b iBb7v i?2 +QM/BiBQM x ∼ u bBM+2 y ∪ {u} Bb +HB[m2X L2+2bbBivX q2 bmTTQb2 i? i (x) Bb J `FQp /2MbBiv rBi? `2bT2+i iQ ∼X AM T `iB+mH `- 7Q` HH z ∈ Mpf (W )- HH v- w ∈ W bm+? i? i v ∼ w (z ∪ {v, w})
(z ∪ {w}) = ,
(z ∪ {v})
(z) #2+ mb2 i?Bb Bb 2[mBp H2Mi iQ λ(w, z ∪ {v}) = λ(w, z) , 8
(_BTH2v M/ E2HHv- RNdd)X
UeXRRV
k9k *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a r?B+? 2tT`2bb2b T`QT2`iv U#V Q7 .2}MBiBQM eX9XRX .2}M2 ϕ : Mpf (W ) → [0, ∞) #v BM/m+iBQM, ϕ(∅) = (∅) , ϕ(x) = 1 B7 x Bb MQi +HB[m2 ,
(x) ϕ(x) = ! ϕ(y)
B7 x Bb +HB[m2 ,
UeXRkV
y:y⊂x
rBi? i?2 +QMp2MiBQM i? i 0/0 = 0X h?2 7mM+iBQM ϕ Bb r2HH /2}M2/ #2+ mb2 B7 i?2 /2MQKBM iQ` BM UeXRkV Bb MmHH- i?2M (y) = 0 7Q` bQK2 y ⊂ x- M/ i?2`27Q`2- BM pB2r Q7 i?2 ?2`2/Bi `v T`QT2`iv- (x) = 0c i?Bb H2 /b BM UeXRkV iQ i?2 7Q`K 0/0 = 0X q2 b?Qr i? i (x) Bb Q7 i?2 7Q`K UeXRyV #v BM/m+iBQM QM i?2 MmK#2` Q7 TQBMib Q7 xX bbmK2 i? i UeXRyV ?QH/b 7Q` HH +QM};m` iBQMb rBi? n − 1 Q` 72r2` TQBMib M/ +QMbB/2` +QM};m` iBQM x bm+? i? i |x| = nX q2 T`Qp2 UeXRyV #v +QMbB/2`BM; i?2 /Bz2`2Mi + b2b QM2 #v QM2X UBV (x) > 0- x Bb +HB[m2X h?2M- #v UeXRkV
(x) = ϕ(x) ϕ(y) = ϕ(y) . y:y⊂x
y⊆x
UBBV (x) > 0- x Bb MQi +HB[m2X aBM+2 x Bb MQi +HB[m2- i?2`2 2tBbib z ∈ Mpf (W )u, w ∈ W bm+? i? i v ∼ w M/ x = z ∪ {u, w}X "v i?2 ?2`2/Bi `v T`QT2`iv- i?2 +QM/BiBQMb z ⊂ x- z ∪ {v} ⊂ x M/ (x) > 0 BKTHv i? i (z) > 0 M/ (z ∪ {v}) > 0X h?2`27Q`2
(z ∪ {u, w})
(z ∪ {v})
(z ∪ {v})
(z ∪ {w})
(z ∪ {v}) #v UeXRRV =
(z) ! ! ϕ(y ∪ {w}) ϕ(y) y⊆z y⊆z ! = ϕ(y) ϕ(y)
(x) =
=
y⊆z
ϕ(y ∪ {w})
y⊆z
y⊆z∪{v}
ϕ(y) =
y∈z∪{v}
+HB[m2b
ϕ(y) , y⊆x
bBM+2 +HB[m2 y ⊆ x + MMQi i i?2 b K2 iBK2 +QMi BM #Qi? v M/ wX UBBBV (x) = 0- x Bb !MQi +HB[m2X h?2M- b BM UBBV- x = z ∪ {v, ! w} r?2`2 v ∼ wX q2 rBHH b?Qr i? i ϕ(y) = 0 U M/ i?2`27Q`2 0 = (x) = ϕ(y)VX AM pB2r Q7 y:y⊂x
+QMi` /B+iBQM- bmTTQb2 i? i
y⊆x
ϕ(y) > 0 .
y:y⊂x
h?2M- #v i?2 BM/m+iBQM ?vTQi?2bBb QM ϕ- 7Q` HH +QM};m` iBQMb y ⊂ x- (y) > 0 M/ i?2M
eX9X 6ALAh1 J_EPo SPALh S_P*1aa1a 0=
k9j
(z ∪ {w})
(z ∪ {v, w})
(z ∪ {w}) =
(z ∪ {w}) > 0 ,
(z ∪ {v})
(z)
+QMi` /B+iBQMX ! UBpV (x) = 0- x Bb +HB[m2X h?2M- UeXRkV TTHB2b M/ vB2H/b ϕ(x) = 0X h?2`27Q`2 ϕ(y) = 0 M/ i?Bb 2[m Hb (x)X
y⊆x
_2+Q`/ BM T `iB+mH ` i?2 7QHHQrBM; 2tT`2bbBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv, λ(u, x) =
ϕ(y ∪ u) .
UeXRjV
y⊆x
1t KTH2 eX9Xj, 6BMBi2 SQBbbQM S`Q+2bb2bX h?2 `272`2M+2 K2 bm`2 Bb b BM 1t KTH2 eXRX8X h F2 ({x1 , . . . , xn }) = λn exp{(1 − λ)ν(W )}- BM r?B+? + b2 X Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 λνX q2 ? p2
(x) = e(1−λ)ν(W )
λ,
x∈x
M/ i?2`27Q`2- X Bb J `FQp TQBMi T`Q+2bb U7Q` Mv +?QB+2 Q7 ∼V- rBi? ϕ(∅) = e(1−λ)ν(W ) M/ ϕ({u}) = λ
(u ∈ W )
M/ ϕ(x) = 1 7Q` b2ib rBi? k Q` KQ`2 TQBMibX h?Bb Bb +QMbBbi2Mi rBi? i?2 B/2 i? i BM SQBbbQM T`Q+2bb i?2`2 Bb MQ BMi2` +iBQM #2ir22M TQBMibX
1t KTH2 eX9X9, h?2 > `/@+Q`2 SQBMi S`Q+2bb- h F2 kX h?2 `272`2M+2 K2 bm`2 Bb b BM 1t KTH2 eXRX8X AM i?2 ? `/@+Q`2 KQ/2H i ` M;2 R > 0 Q7 1t KTH2 eXRXe- u ∼ v ⇔ 'u − v' ≤ 2Rϕ(∅) = α ,
ϕ({u}) = β ,
ϕ({u, v}) = 1u−v>2R .
1t KTH2 eX9X8, h?2 S B`rBb2 AMi2` +iBQM JQ/2HX h?2 ? `/@+Q`2 KQ/2H #Qp2 Bb bT2+B H + b2 Q7 T B`rBb2 BMi2` +iBQM KQ/2Hb- 7Q` r?B+?
(x) = α
x∈x
β(x)
γ(u, v) ,
u,v∈x u∼v
r?2`2 α > 0- β : W → [0, ∞) M/ γ : W × W → [0, ∞)X
k99 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a
eX8 aT iB H "B`i?@ M/@.2 i? SQBMi S`Q+2bb2b h?Bb KQ/2He +QM+2`Mb i?2 2pQHmiBQM BM iBK2 Q7 TQBMi T`Q+2bb QM +QKT +i rBM/Qr W Q7 Rm i? i +QMp2`;2b BM /Bbi`B#miBQM iQ TQBMi T`Q+2bb rBi? bT2+B}2/ 2tpBbB#H2 BMi2MbBivX *QMbB/2` i?2 7QHHQrBM; biQ+? biB+ T`Q+2bb {X(t)}t≥0 rBi? p Hm2b BM Mpf (W )X Ai Bb +? ` +i2`Bx2/ #v #B`i? ` i2 7mM+iBQM b : Mpf (W ) × W → [0, ∞) M/ /2 i? ` i2 7mM+iBQM d : Mpf (W ) × W → [0, ∞)- r?2`2 Bi Bb bbmK2/ i? i 7Q` HH x ∈ Mpf (W ) B(x) := b(x, u)ν(du) < ∞, UeXR9V W
7Q` bQK2 }MBi2 MQM@ iQKB+ K2 bm`2 ν QM (W, W)X h?2 [m MiBiv B(x) Bb i?2 iQi H #B`i? ` i2 BM i?2 +QM};m` iBQM xX G2i D(x) := d(x, x) UeXR8V x∈x
#2 i?2 iQi H /2 i? ` i2 BM i?2 +QM};m` iBQM xX h?2 2pQHmiBQM Q7 {X(t)}t≥0 Bb b 7QH@ HQrbX G2i T0 = 0- T1 - T2 , . . . #2 i?2 i` MbBiBQM iBK2b- M/ H2i X0 = X(0), . . . , Xn = X(Tn ), . . . #2 i?2 p Hm2b i i?2 i` MbBiBQM iBK2bX AMBiB HBx2 rBi? X0 = x0 ∈ Mpf (W )X 6Q` ;Bp2M n ≥ 0- B7 Xn = x- i?2M Tn+1 − Tn Bb M 2tTQM2MiB H ` M/QK p `B #H2 rBi? K2 M 1 , D(x) + B(x) M/ BM/2T2M/2Mi Q7 T1 , . . . , Tn - X0 , . . . , Xn−1 X i iBK2 Tn+1 - i?2 i` MbBiBQM Bb • /2 i? rBi? T`Q# #BHBiv rBi? T`Q# #BHBiv
D(x) B(x)+D(x) d(x,x) c M/ D(x)
• #B`i? rBi? T`Q# #BHBiv
B(x) D(x)+B(x)
M/ i?2 TQBMi x iQ #2 2` b2/ Bb +?Qb2M BM x
M/ i?2 TQBMi u //2/ iQ x Bb +?Qb2M BM W
++Q`/BM; iQ i?2 T`Q# #BHBiv /Bbi`B#miBQM
b(x,u)ν(du) X B(x)
P7 +Qm`b2- i?Bb +QMbi`m+iBQM Bb BKTH2K2Mi #H2 QMHv mT iQ iBK2 T∞ = limn↑∞ Tn UMQi BM+Hm/2/VX 6Q` Mv iBK2 t ≥ T∞ - H2i X(t) = ∅ Ui?2 2KTiv TQBMi T`Q+2bbVX h?2 T`Q+2bb {X(t)}t≥0 - + HH2/ bT iB H #B`i?@ M/@/2 i? TQBMi T`Q+2bb- Bb DmKT J `FQp T`Q+2bb rBi? bi i2 bT +2 Mpf (W )X h?2Q`2K eX8XR G2i {λn }n≥0 M/ {μn }n≥1 #2 i?2 #B`i? M/ /2 i? T ` K2i2`b Q7 +QMiBMmQmb@iBK2 N@p Hm2/ #B`i?@ M/@/2 i? J `FQp +? BM {Y (t)}t≥0 X amTTQb2 i? i 7Q` HH x ∈ Mpf (W )B(x) ≤ λ|x| , D(x) ≥ μ|x| . UeXReV h?2M, U V B7 {Y (t)}t≥0 Bb `2;mH `- bQ Bb {X(t)}t≥0 - M/ U#V B7 {Y (t)}t≥0 Bb 2`;Q/B+- bQ Bb {X(t)}t≥0 X e
(S`2biQM- RNd8)X
eX8X aShAG "A_h>@[email protected]h> SPALh S_P*1aa1a
k98
U"v 2`;Q/B+Biv- Bi Bb K2 Mi i? i bi iBQM `v p2`bBQM Q7 i?2 bT iB H #B`i?@ M/@ /2 i? TQBMi T`Q+2bb rBi? i?2 `2[mB`2/ +? ` +i2`BbiB+b b M/ d 2tBbib M/ i? i Mv p2`bBQM Q7 Bi +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 bi iBQM `v /Bbi`B#miBQMX AM 7 +i- Bi rBHH #2 b?QrM i? i- mM/2` i?2 #Qp2 +QM/BiBQMb 7Q` i?2 #B`i?@ M/@/2 i? T ` K2i2`b+QMp2`;2M+2 Bb BM p `B iBQMXV 1t KTH2 eX8Xk, S`QQ7 Q7 G2KK eX8XRX _272` iQ a2+iBQM eX8 7Q` i?2 7` K2rQ`F M/ MQi iBQMb Q7 i?2 T`2b2Mi 2t KTH2X h?2 b2[m2M+2 {(Tn , Zn )}n≥0 r?2`2 Zn := X(Tn ) + M #2 pB2r2/ b TQBMi T`Q@ +2bb QM R+ rBi? K `Fb BM Mpf (W )- r?Qb2 FtX @HQ+ H +? ` +i2`BbiB+b (λ(t), Φ(t, dx)) `2 λ(t) = B(X(t)) + D(X(t)), Φ(t, F ) = Q(X(t−), F ) , r?2`2 B(x) Q(x, F ) := B(x) + D(x)
b(x, u)1F (x ∪ u)ν(du) W
+
D(x) d(x, x)1F (x\x) . B(x) + D(x) x∈x
h?2 TQBMi T`Q+2bb2b B M/ D +QmMiBM; i?2 iBK2b Q7 Q++m``2M+2 Q7 #B`i?b M/ /2 i?b ? p2 FtX @BMi2MbBiv {B(X(t))}t≥0 M/ {D(X(t))}t≥0 `2bT2+iBp2HvX q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 G2KK eX8XRX U V "v i?2 BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXeV- ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 rBi? BMi2MbBiv 1 + M #2 +QMbi`m+i2/ BM bm+? r v i? i 7Q` HH t ∈ [0, T∞ ) 1{0≤z≤B(X(s))} N (dz × ds) B(0, t] = (0,t]
M/
D(0, t] = (0,t]
6Q` HH t ∈ (0, T∞ )-
R
R
1{0≥z≥−D(X(s))} 1{X(s−)=∅} N (dz × ds) .
|X(t)| = |X(0)| + B(0, t] − D(0, t] .
*QMbi`m+i MQr {Y˜ (t)}t≥0 b 7QHHQrbX 6B`bi b2i Y˜ (0) = |X(0)|X h?2M- 7Q` t ∈ (0, T∞ ) 1{0≤z≤λY˜ (s−) } N (dz × ds) Y˜ (t) = Y˜ (0) + (0,t] R 1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) . − (0,t]
R
h?2 biQ+? biB+ T`Q+2bb {Y˜ (t)}t∈(0,T∞ ) /QKBM i2b {|X(t)|}t∈(0,T∞ ) U#Qi? T`Q+2bb2b ? p2 DmKTb Q7 K ;MBim/2 i KQbi 1- i?2v ? p2 i?2 b K2 BMBiB H p Hm2- M/ r?2M2p2` Y˜ (t) = |X(t)|- B(X(t)) ≤ λY˜ (t) , D(X(t)) ≥ μY˜ (t) VX AM T `iB+mH `- bBM+2 {Y˜ (t)}t≥0 Bb bbmK2/ `2;mH `- bQ Bb {|X(t)|}t≥0 M/ i?2`27Q`2 {X(t)}t≥0 X
k9e *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a U#V G2i X(0) #2 i?2 BMBiB H p Hm2 Q7 i?2 #B`i?@ M/@/2 i? TQBMi T`Q+2bb iQ #2 +QMbi`m+i2/X bbmK2 i? i i?2 T ` K2i2`b λn M/ μn `2 i?Qb2 Q7 M 2`;Q/B+ #B`i?@ M/@/2 i? T`Q+2bb QM i?2 TQbBiBp2 HBM2X *QMbi`m+i 7`QK ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 Q7 BMi2MbBiv 1 bm+? M 2`;Q/B+ #B`i?@ M/@/2 i? T`Q+2bb {Y˜ (t)}t≥0 #v Y˜ (t) = Y˜ (0) + 1{0≤z≤λY˜ (s−) } N (dz × ds) (0,t] R 1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) , − (0,t]
R
r?2`2 Y˜ (0) = |X(0)|X LQr bmTTQb2 i? i r2 ? p2 bm++2bb7mHHv +QMbi`m+i2/ i?2 bT iB H #B`i?@ M/@/2 i? T`Q+2bb mT iQ iBK2 t BM bm+? r v i? i 7Q` HH r ≤ t|X(r)| ≤ Y˜ (r) UMQi2 i? i i?Bb Bb i?2 bBim iBQM i iBK2 0VX GQQF MQr i i?2 }`bi TQBMi (s, z) Q7 N 7i2` t BM [−D(X(t)), +B(X(t))] × (t, t + a(t)]- r?2`2 t + a(t) Bb i?2 }`bi DmKT iBK2 Q7 {Y˜ (t)}t≥0 7i2` tX A7 Bi Bb TQBMi BM i?2 mTT2` ? H7 TH M2 Uz ≥ 0V- // iQ i?2 +m``2Mi bT iB H #B`i?@ M/@/2 i? T`Q+2bb TQBMi u +?Qb2M i ` M/QK ++Q`/BM; ν(du) iQ i?2 T`Q# #BHBiv /Bbi`B#miBQM b(X(t−),u) X A7 z < 0- `2KQp2 TQBMi 7`QK i?2 B(X(t−)) +m``2Mi TQBMi T`Q+2bb- +?QQbBM; i?2 TQBMi x ∈ X(t−) iQ #2 `2KQp2/ rBi? T`Q# #BHBiv d(X(t−),x) X i i?Bb TQBMi s i?2 /QKBM iBQM T`QT2`iv Y˜ (s) ≥ |X(s)| `2K BMb b iBb}2/X D(X(t−)) UAM 7 +i- i?2 QMHv T`Q#H2K +QmH/ `Bb2 7`QK i?2 + b2 r?2`2 |X(t)| = Y˜ (t) M/ s = t + a(t) M/ z ≥ 0- #mi i?2M t + a(t) Bb M2+2bb `BHv M mTr `/ DmKT Q7 i?2 Y˜ T`Q+2bb- #2+ mb2 i?2M D(X(s−)) ≤ λ|X(s−)| = λY˜ (s−) - M/ i?2`27Q`2 s Bb M2+2bb `BHv M mTr `/ DmKT Q7 i?2 Y˜ T`Q+2bbXV q2 + M MQr `2T`Q/m+2 i?2 #Qp2 +QMbi`m+iBQMi?Bb iBK2 rBi? s b bi `iBM; iBK2X h?2 /QKBM iBQM T`QT2`iv Dmbi T`Qp2/ rBHH 2Mi BH i? i irQ bT iB H #B`i?@ M/@ /2 i? T`Q+2bb2b {X1 (t)}t≥0 M/ {X2 (t)}t≥0 rBi? ;Bp2M BMBiB H bi i2b X1 (0) M/ X2 (0) + M #2 +QMbi`m+i2/ BM bm+? r v i? i i?2v +QmTH2 BM }MBi2 iBK2X hQ b22 i?Bb- T2`7Q`K 7Q` 2 +? Q7 i?2K i?2 #Qp2 +QMbi`m+iBQM rBi? i?2 b K2 N M/ H2i {Y˜1 (t)}t≥0 M/ {Y˜2 (t)}t≥0 #2 i?2 +Q``2bTQM/BM; Y˜ T`Q+2bbX _2+ HH i? i Y˜1 (0) = |X1 (0)| M/ Y˜2 (0) = |X2 (0)|X amTTQb2 i? i |X1 (0)| ≥ |X2 (0)|- M/ i?2`2@ 7Q`2 Y˜1 (0) ≥ Y˜2 (0)X 6`QK i?2 +QMbi`m+iBQM Q7 i?2 Y˜ T`Q+2bb2b M/ i?2 7 +i i? i r2 `2 mbBM; 7Q` i?Bb +QMbi`m+iBQM i?2 b K2 N - r2 b22 i? i 7Q` HH t ≥ 0- Y˜1 (t) ≥ Y˜2 (t)X AM T `iB+mH `- i?2b2 irQ T`Q+2bb2b rBHH +QmTH2 BM }MBi2 iBK2 U b bQQM b Y˜1 (t) = 0r?B+? ? TT2Mb 7Q` }MBi2 t #v i?2 2`;Q/B+Biv bbmKTiBQM 7Q` i?2 #B`i?@ M/@/2 i? T`Q+2bb rBi? T ` K2i2`b λn M/ μn V- M/ bQ rBHH i?2 #B`i?@ M/@/2 i? TQBMi T`Q@ +2bb2b X1 M/ X2 X q2 MQr T`Qp2 i?2 2tBbi2M+2 Q7 bi iBQM `v #B`i?@ M/@/2 i? TQBMi T`Q+2bb rBi? i?2 #Qp2 #B`i?@ M/@/2 i? +? ` +i2`BbiB+bX 6Q` i?Bb r2 }`bi +QMbi`m+i bi iBQM `v +QMiBMmQmb@iBK2 #B`i?@ M/@/2 i? J `FQp +? BM Y˜ QM i?2 2MiB`2 HBM2 R- M/ i?2M SQBbbQM T`Q+2bb N QM R2 bm+? i? i ˜ ˜ Y (t) = Y (0) + 1{0≤z≤λY˜ (s−) } N (dz × ds) (0,t] R 1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) . =− (0,t]
R
eX8X aShAG "A_h>@[email protected]h> SPALh S_P*1aa1a
k9d
G2i {Vn }n∈Z #2 i?2 b2[m2M+2 Q7 iBK2b bm+? i? i Y˜ (Vn ) = 0 M/ Y˜ (Vn −) > 0 UBM 7 +i- = 1VX *QMbi`m+i 7Q` 2 +? n ∈ Z- X(t), t ∈ [Vn , Vn+1 ) rBi? X(Vn ) = 0- b mbm HmbBM; i?2 TQBMi T`Q+2bb N X h?Bb T`Q+2bb Bb bi iBQM `v U 7Q`K H T`QQ7 Bb `2[mB`2/ BM 1t2`+Bb2 eXdXRVX AM pB2r Q7 i?2 Dmbi T`Qp2/ +QmTHBM; T`QT2`iv- i?2 bi iBQM `v /Bbi`B#miBQM Bb mMB[m2 M/ Mv i` MbB2Mi bQHmiBQM +QMp2`;2b BM p `B iBQM iQ i?2 bi iBQM `v /Bbi`B@ #miBQMX _2+ HH i? i M2+2bb `v M/ bm{+B2Mi +QM/BiBQM 7Q` `2;mH `Biv Q7 +QMiBMmQmb@ iBK2 #B`i?@ M/@/2 i? J `FQp +? BM rBi? TQbBiBp2 #B`i? M/ /2 i? T ` K2i2`b Bb
1 μn μn · · · μ1 = ∞, + + ··· + λn λn λn−1 λn · · · λ 0 n≥1 r?2`2 b M2+2bb `v M/ bm{+B2Mi +QM/BiBQM 7Q` 2`;Q/B+Biv Bb λ0 λ1 · · · λn−1 n≥1
μ1 μ2 · · · μn
< ∞.
h?2 #B`i?@ M/@/2 i? T`Q+2bb Bb J `FQp T`Q+2bb r?Qb2 bi i2 bT +2 Bb Mpf (W ) M/ Bi + M #2 b?QrM d i? i B7 i?2 7QHHQrBM; /2i BH2/ # H M+2 2[m iBQMb b(x, u) (x) = d(x ∪ {u}, u) (x ∪ {u})
UeXRdV
`2 b iBb}2/ 7Q` HH x, u bm+? i? i (x ∪ {u}) > 0 Ui?2M (x) > 0 #v i?2 ?2`2/Bi `v bbmKTiBQMV- Bi +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 }MBi2 TQBMi T`Q+2bb rBi? T`Q# #BHBiv /2MbBiv X h?2`2 `2 K Mv +?QB+2b 7Q` i?2 #B`i?@ M/@/2 i? ` i2b b iBb7vBM; i?2 /2i BH2/ # H M+2 2[m iBQMbX PM2 Q7 i?2K Bb d(x, x) ≡ 1 ,
(x ∪ {u}) = λ(u, x) . b(x, u) =
(x)
UeXR3V
1t KTH2 eX8Xj, h?2 SQBbbQM T`Q+2bbX G2i λ : E → [0, ∞) #2 K2 bm` #H2 MQM@M2; iBp2 7mM+iBQM bm+? i? i B= λ(u)ν(du) < ∞ . E
h?2 +?QB+2 d(x, x) ≡ 1- b(x, u) = λ(u) +Q``2bTQM/b iQ bT iB H #B`i?@ M/@/2 i? T`Q+2bb {Z(t)} r?Qb2 bi iBQM `v /Bbi`B#miBQM +Q``2bTQM/b iQ SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 λ(u)ν(du)X >2`2 B(x) = B- D(x) = n(x)- i?2 MmK#2` Q7 TQBMib BM i?2 +QM};m` iBQM xX h?2 T`Q+2bb Y (t) = n(Z(t)) +QmMib i?2 MmK#2` Q7 TQBMib Q7 i?2 bT iB H #B`i?@ M/@/2 i? T`Q+2bb i iBK2 tX AM i?Bb T `iB+mH ` + b2- r2 b22 i? i B7 i ;Bp2M iBK2 d
(S`2biQM- RNd8)X
k93 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a t- Y (t) = n- Bi bi vb i?2`2 7Q` M 2tTQM2MiB H iBK2 Q7 K2 M (B + n)−1 X i i?2 2M/ Q7 B i?Bb 2tTQM2MiB H iBK2- Bi K F2b i` MbBiBQM iQ n + 1 rBi? T`Q# #BHBiv B+n - iQ n − 1 n rBi? T`Q# #BHBiv B+n M/ i?2 T`Q+2bb Bb `2T2 i2/ bi `iBM; 7`QK i?2 M2r TQbBiBQM n ± 1- BM/2T2M/2MiHv Q7 i?2 T biX h?2`27Q`2 {Y (t)} Bb i?2 mbm H #B`i?@ M/@/2 i? T`Q+2bb rBi? T ` K2i2`b βn = B , δn = n. am+? T`Q+2bb Bb `2;mH ` DmKT J `FQp T`Q+2bb- B``2/m+B#H2 M/ 2`;Q/B+X h?2 bT iB H #B`i?@ M/@/2 i? T`Q+2bb {Z(t)}t∈R Bb `2;mH ` DmKT J `FQp T`Q+2bb UBib `2;mH `Biv 7QHHQrb 7`QK i? i Q7 {Y (t)}t∈R VX AM pB2r Q7 i?2 /2i BH2/ # H M+2 2[m iBQMb- Bi /KBib P λν Ui?2 SQBbbQM K2 bm`2 rBi? BMi2MbBiv K2 bm`2 λ(u)ν(du)V b BMp `B Mi T`Q# #BHBivX
eXe M Hi2`M iBp2 JQ/2H q2 ? p2 H`2 /v K2MiBQM2/ i?2Q`2iB+ H M/ T2`? Tb T` +iB+ H BM+QMp2MB2M+2 BM i?2 KQ/2HBM; Q7 }MBi2 TQBMi T`Q+2bb2b #v K2 Mb Q7 _ /QMĜLBFQ/ɷK /2`Bp iBp2bM K2Hv i?2 /B{+mHiv Q7 `2H iBM; i?2 /2MbBiB2b jn Un ≥ 1V iQ i?2 +QmMi /Bbi`B#miBQMX JQ`2 T`2+Bb2Hv- i?2 KQ/2H2` BMi2`2bi2/ QMHv BM i?2 BMi2` +iBQM #2ir22M i?2 TQBMib rBHH +QK2 mT rBi? i?2 /2MbBiB2b jn Un ≥ 1V M/ i?2M rBHH ;2M2` HHv #2 +QM7`QMi2/ rBi? i?2 TQbbB#Hv mM72 bB#H2 +QKTmi iBQM Q7 i?2 +QmMi /Bbi`B#miBQMX 6Q` BMbi M+2BM i?2 ? `/@+Q`2 KQ/2H- 2p2M i?2 K2 M K2 bm`2 Q7 i?2 TQBMi T`Q+2bb Bb /B{+mHi iQ Q#i BMX MQi?2` ivT2 Q7 KQ/2H Bb MQr T`2b2Mi2/- r?B+? /Q2b MQi pQB/ i?2 /B{+mHiB2b K2MiBQM2/- #mi +QMM2+ib iQ r2HH@bim/B2/ `2 X h?2 ;2M2` H B/2 Q7 i?2 ivT2 Q7 KQ/2HHBM; i? i Bb MQr T`QTQb2/ Bb iQ +QMbi`m+i SQBbbQM T`Q+2bb Q` Mv Qi?2` ivT2 Q7 bBKTH2 TQBMi T`Q+2bb r?B+? rBHH #2 i?2 bmTTQ`i Q7 i?2 TQBMi T`Q+2bb i? i QM2 rBb?2b iQ +QMbi`m+iX h?Bb K2 Mb i? i T`Q+2/m`2 /Bb+ `/BM; bQK2 TQBMib Q7 i?2 bmTTQ`i TQBMi T`Q+2bb Bb ;Bp2MX am+? T`Q+2/m`2 Bb ;Bp2M BM i2`Kb Q7 }MBi2 :B##bĜ J `FQp ` M/QK }2H/b- Q7 r?B+? i?2 K BM 72 im`2b Q7 i?2 2H2K2Mi `v i?2Q`v rBHH #2 `2+ HH2/ BM 72r HBM2bX LQi2 ?Qr2p2` i? i i?2 T?BHQbQT?v Bb bHB;?iHv /Bz2`2Mi 7`QK i?2 QM2 BM i?2 T`2pBQmb b2+iBQMb- M/ i?2 +?QB+2 rBHH /2T2M/ QM i?2 TTHB+ iBQMb +QM+2`M2/X G2i V #2 }MBi2 b2i- r?Qb2 2H2K2Mib `2 + HH2/ MQ/2b- Q` bBi2b- Q` p2`iB+2b ++Q`/BM; iQ i?2 +QMi2tiX h?2`2 Bb #BM `v bvKK2i`B+ `2H iBQM ∼ #2ir22M i?2 MQ/2b- M/ i?2 MQ/2b v1 M/ v2 bm+? i? i v1 ∼ v2 `2 + HH2/ M2B;?#Qm`bX 6Q` v ∈ V /2MQi2 #v Nv i?2 +QHH2+iBQM Q7 MQ/2b /Bz2`2Mi 7`QK v i? i `2 M2B;?#Qm`b Q7 vX +HB[m2 Bb bm#b2i Q7 V i? i Bb 2Bi?2` bBM;H2iQM Q` bm+? i? i Mv irQ /BbiBM+i 2H2K2Mib Q7 Bi `2 M2B;?#Qm`bX h?2`2 Bb }MBi2 bT +2 Λ + HH2/ i?2 T? b2 bT +2 M/ M 2H2K2Mi x = (x(v) ; v ∈ V ) ∈ ΛV Bb + HH2/ +QM};m` iBQMX h?2 MQi iBQM x(C)r?2`2 C ⊆ S- `2T`2b2Mib i?2 +QM};m` iBQM x `2bi`B+i2/ iQ C, x(C) := {x(v) ; v ∈ C}X ` M/QK 2H2K2Mi X = (X(v), v ∈ V )- r?2`2 X(v) ∈ Λ Bb i?2 ` M/QK K `FQ` T? b2- bbQ+B i2/ iQ i?2 bBi2 UQ` MQ/2V v- Bb + HH2/ ` M/QK }2H/ QM V rBi? T? b2b BM ΛX
eXeX L Gh1_LhAo1 JP.1G
k9N
.2}MBiBQM eXeXR :B##b TQi2MiB H QM ΛV `2H iBp2 iQ ∼ Bb +QHH2+iBQM {VC }C⊆V Q7 7mM+iBQMb VC : ΛV → R ∪ {+∞} bm+? i? i UBV VC ≡ 0 B7 C Bb MQi +HB[m2UBBV 7Q` HH x, x ∈ ΛV M/ HH C ⊆ V - x(C) = x (C) ⇒ VC (x) = VC (x )X AM bmKK `v, i?2 7mM+iBQM VC /2T2M/b QMHv QM i?2 T? b2b i i?2 bBi2b BMbB/2 i?2 bm#b2i C M/ Bb i?2 MmHH 7mM+iBQM B7 C Bb MQi +HB[m2X h?2 7mM+iBQM U : ΛV → R ∪ {+∞}- ?2M+27Q`i? + HH2/ i?2 2M2`;v 7mM+iBQM- Bb b B/ iQ /2`Bp2 7`QK i?2 TQi2MiB H {VC }C⊆V B7 U (x) =
VC (x) .
C
*QMbB/2` i?2 T`Q# #BHBiv /Bbi`B#miBQM π(x) =
1 −U (x) e Z
UeXRNV
QM i?2 +QM};m` iBQM bT +2 ΛV - r?2`2 Z Bb i?2 MQ`K HBxBM; +QMbi Mi- + HH2/ i?2 T `iBiBQM 7mM+iBQMX h?2 /Bbi`B#miBQM BM UeXRNV Bb + HH2/ :B##b /Bbi`B#miBQMX ` M/QK }2H/ X QM V rBi? T? b2b BM Λ Bb + HH2/ J `FQp ` M/QK }2H/ UK`7V rBi? `2bT2+i iQ ∼ B7 7Q` HH MQ/2b v ∈ V - i?2 ` M/QK p `B #H2b X(v) M/ X(V \(Nv ∪ {v})) `2 BM/2T2M/2Mi ;Bp2M X(Nv )X AM bvK#QHb, P (X(v) = x(v) | X(V \v) = x(V \v)) = P (X(v) = x(v) | X(Nv ) = x(Nv )) UeXkyV 7Q` HH x ∈ ΛV M/ HH v ∈ V X S`QT2`iv UeXkyV Bb +H2 `Hv Q7 i?2 J `FQp ivT2, i?2 /Bbi`B#miBQM Q7 i?2 T? b2 i bBi2 Bb /B`2+iHv BM~m2M+2/ QMHv #v i?2 T? b2b Q7 i?2 M2B;?#Qm`BM; bBi2bX LQi2 i? i Mv ` M/QK }2H/ Bb J `FQpB M rBi? `2bT2+i iQ i?2 i`BpB H `2H iBQM ∼ 7Q` r?B+? i?2 M2B;?#Qm`?QQ/ Q7 Mv bBi2 Bb i?2 r?QH2 b2i V X >Qr2p2`- i?2 BMi2`2biBM; J `FQp }2H/b U7`QK i?2 TQBMi Q7 pB2r Q7 KQ/2HHBM;- bBKmH iBQM M/ QTiBKBx iBQMV `2 i?Qb2 rBi? `2H iBp2Hv bK HH M2B;?#Qm`?QQ/bX h?2 HQ+ H +? ` +i2`BbiB+ Q7 i?2 K`7 i bBi2 v Bb i?2 7mM+iBQM π v : ΛV → [0, 1] /2}M2/ #v π v (x) := P (X(v) = x(v) | X(Nv ) = x(Nv )). PM2 HbQ r`Bi2b π v (x) = π(x(v) | x(Nv ))X M K`7 Bb b B/ iQ b iBb7v i?2 TQbBiBpBiv +QM/BiBQM B7 Bib T`Q# #BHBiv /Bbi`B#miBQM Bb bi`B+iHv TQbBiBp2X h?2Q`2K eXeXk hrQ /Bbi`B#miBQMb Q7 M K`7 rBi? }MBi2 +QM};m` iBQM bT +2 ΛV i? i b iBb7v i?2 TQbBiBpBiv +QM/BiBQM M/ ? p2 i?2 b K2 HQ+ H bT2+B}+ iBQM `2 B/2MiB+ HX
k8y *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a S`QQ7X 1MmK2` i2 V b {1, 2, . . . , K}X h?2`27Q`2 x = (x1 , . . . , xK−1 , xK ) ∈ ΛK X h?2 7QHHQrBM; B/2MiBiv π(x) =
K π(xi | x1 , . . . , xi−1 , yi+1 , . . . , yK ) π(y) π(yi | x1 , . . . , xi−1 , yi+1 , . . . , yK ) i=1
()
?QH/b 7Q` Mv x, y ∈ ΛK X 6Q` i?2 T`QQ7- QM2 +?2+Fb i? i π(x) =
π(xK | x1 , . . . , xK−1 ) π(x1 , . . . , xK−1 , yK ) π(yK | x1 , . . . , xK−1 )
#v bBKTHv `2r`BiBM; i?2 +QM/BiBQM H T`Q# #BHBiB2b i?2`2Q7 mbBM; " v2bǶb /2}MBiBQMX aBKBH `Hvπ(xK−1 | x1 , . . . , xK−2 , yK ) π(x1 , . . . , xK−2 , yK−1 , yK ) π(yK−1 | x1 , . . . , xK−2 , yK )
π(x1 , . . . , xK−1 , yK ) =
M/ bQ 7Q`i?X h?2 #Qp2 + H+mH iBQMb K F2 b2Mb2 #2+ mb2 Q7 i?2 TQbBiBpBiv +QM/BiBQMX G2i MQr π M/ π #2 T`Q# #BHBiv /Bbi`B#miBQMb rBi? i?2 b K2 HQ+ H bT2+B}+ iBQMb M/ b iBb7vBM; i?2 TQbBiBpBiv +QM/BiBQMX *?QQb2 Mv yX h?2 B/2MiBiv UV b?Qrb i? i 7Q` HH xπ (x) π (y) = , π(x) π(y) +QMbi Mi- M2+2bb `BHv 2[m H iQ 1 bBM+2 π M/ π `2 T`Q# #BHBiv /Bbi`B#miBQMbX :B##b /Bbi`B#miBQMb rBi? M 2M2`;v /2`BpBM; 7`QK :B##b TQi2MiB H `2H iBp2 iQ #BM `v bvKK2i`B+ `2H iBQM ∼ `2 /Bbi`B#miBQMb Q7 J `FQp }2H/b `2H iBp2 iQ i?2 b K2 #BM `v `2H iBQMX JQ`2 T`2+Bb2Hv, h?2Q`2K eXeXj A7 X Bb ` M/QK }2H/ QM V rBi? T? b2b BM Λ rBi? /Bbi`B#miBQM π Q7 i?2 7Q`K UeXRNV- r?2`2 i?2 2M2`;v U /2`Bp2b 7`QK :B##b TQi2MiB H {VC }C⊆V `2H iBp2 iQ ∼- i?2M X Bb J `FQpB M `2H iBp2 iQ ∼X JQ`2Qp2`- Bib HQ+ H bT2+B}+ iBQM Bb ;Bp2M #v i?2 7Q`KmH
e− C v VC (x) π (x) = , − C v VC (λ,x(V \v)) λ∈Λ e v
r?2`2 i?2 MQi iBQM i BMBM; bBi2 vX
Cv
UeXkRV
K2 Mb i? i i?2 bmK 2ti2M/b Qp2` i?2 b2ib C ⊆ V +QM@
S`QQ7X 6B`bi Q#b2`p2 i? i i?2 `B;?i@? M/ bB/2 Q7 UeXkRV /2T2M/b QM x QMHv i?`Qm;? x(v) M/ x(Nv )X AM/22/- VC (x) /2T2M/b QMHv QM (x(w), w ∈ C)- M/ 7Q` +HB[m2 C- B7 w ∈ C M/ v ∈ C- i?2M 2Bi?2` w = v Q` w ∼ vX h?2`27Q`2- B7 QM2 + M b?Qr i? i P (X(v) = x(v) | X(V \v) = x(V \v)) 2[m Hb i?2 `B;?i@? M/ bB/2 Q7 UeXkRV M/ BM T `iB+mH ` Bb 7mM+iBQM Q7 x(v) M/ x(Nv ) QMHv- i?2M Ub22 1t2`+Bb2 eXdXeVi?2 J `FQp T`QT2`iv UeXkyV M/ 2[m HBiv UeXkRV rBHH #2 T`Qp2/X "v /2}MBiBQM Q7 +QM/BiBQM H T`Q# #BHBiv-
eXeX L Gh1_LhAo1 JP.1G
k8R
P (X(v) = x(v) | X(V \v) = x(S\v)) = "mi π(x) =
π(x) . π(λ, x(V \v)) λ∈Λ
(†)
1 − C v VC (x)+C v VC (x) e , Z
M/ bBKBH `Hvπ(λ, x(S\v)) =
1 − C v VC (λ,x(S\v))−C v VC (λ,x(S\v)) e . Z
A7 C Bb +HB[m2 M/ v Bb MQi BM C- i?2M VC (λ, x(V \v)) = VC (x) M/ Bb i?2`27Q`2 BM/2T2M/2Mi Q7 ∈ ΛX h?2`27Q`2, λ - i?2 `B;?i@? M/ bB/2 Q7 U†V Bb 7QmM/- 7i2` 7 +iQ`BM; Qmi exp x(w) − Cv VC (x) - iQ #2 2[m H iQ i?2 `B;?i@? M/ bB/2 Q7 UeXkRVX h?2 HQ+ H 2M2`;v i bBi2 v Q7 +QM};m` iBQM x Bb VC (x). Uv (x) = Cv
qBi? i?Bb MQi iBQM- UeXkRV #2+QK2b π v (x) =
e−Uv (x) . −Uv (λ,x(V \v)) λ∈Λ e
h?2Q`2K eXeXj ? b M BKTQ`i Mi +QMp2`b2- i?2 > KK2`bH2vĜ*HBzQ`/ i?2Q`2Kr?B+? r2 b? HH MQi T`Qp2 ?2`2,3 h?2Q`2K eXeX9 G2i π #2 i?2 /Bbi`B#miBQM Q7 ` M/QK }2H/ i? i Bb J `FQpB M rBi? `2bT2+i iQ ∼ M/ bmTTQb2 i? i π > 0X h?2M π(x) =
1 −U (x) e Z
7Q` bQK2 2M2`;v 7mM+iBQM U /2`BpBM; 7`QK :B##b TQi2MiB H {VC }C⊆V bbQ+B i2/ rBi? ∼X h?2 Hi2`M iBp2 TT`Q +? iQ }MBi2 TQBMi T`Q+2bb KQ/2HHBM; +QMbBbib BM }`bi ;2M2` iBM; bBKTH2 TQBMi T`Q+2bb QM Rm Ui?2 dz7` K2ǴV rBi? }MBi2 K2 M K2 bm`2 ν QM Rd - M/ i?2M ` M/QKHv K `FBM; U b 2tTH BM2/ #Qp2V i?2 }MBi2 UbBM+2 ν Bb }MBi2 K2 bm`2V MmK#2` Q7 TQBMib #v 2Bi?2` 0 M/ 1X h?2 TQBMib K `F2/ rBi? 1 `2 `2i BM2/ M/ i?2 Qi?2`b `2 /2H2i2/X AM Qi?2` rQ`/b- r?2M i?2 `2 HBx iBQM Q7 i?2 SQBbbQM T`Q+2bb ? b n TQBMib v1 , . . . , vn - /2}M2 bBi2 bT +2 V := {v1 , . . . , vn } M/ T? b2 bT +2 Λ := {0, 1}X :Bp2M #BM `v bvKK2i`B+ `2H iBQM ∼ #2ir22M TQBMib BM Rm - i?2 `2H iBQM ∼ #2ir22M i?2 TQBMib Q7 V Bb i?2 QM2 BM?2`Bi2/ 7`QK BiX q?2M i?2 K `Fb 7Q`K J `FQpĜ:B##b }2H/ QM }MBi2 bBi2 bT +2- i?2 }MBi2 TQBMi T`Q+2bb + M #2 bBKmH i2/ #v K2 Mb Q7 JQMi2 * `HQ b KTHBM; H;Q`Bi?K 3
a22 7Q` BMbi M+2 ("`ûK m/- kyky)X
k8k *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a r?Qb2 ;Q H Bb iQ b KTH2 /Bbi`B#miBQM π QM E ⊆ ΛV X h?2 K2i?Q/ iQ +?B2p2 i?Bb Bb iQ +QMbi`m+i M T2`BQ/B+ B``2/m+B#H2 ?QKQ;2M2Qmb J `FQp +? BM {Xn }n≥0 rBi? bi i2 bT +2 E /KBiiBM; π b bi iBQM `v /Bbi`B#miBQMX "v 7mM/ K2Mi H `2bmHi Q7 J `FQp +? BM i?2Q`v- limn↑∞ P (Xn = x) = π(x) 7Q` HH x ∈ E- M/ i?2`27Q`2- dz7Q` bm{+B2MiHv H `;2 nǴ- i?2 /Bbi`B#miBQM Q7 Xn Bb +HQb2 iQ i?2 i `;2i /Bbi`B#miBQM π M/ i?2`27Q`2 Xn Bb M TT`QtBK i2 b KTH2 Q7 πX T? b2 bT +2 `B+?2` i? i {0, 1} HHQrb QM2 iQ +QMbB/2` BMi2` +iBQMb #2ir22M /Bz2`2Mi TQBMi T`Q+2bb2bX a22 7Q` BMbi M+2 1t2`+Bb2 eXdX3X 1t KTH2 eXeX8, .Bb+`2iBx2/ TQBMi T`Q+2bb KQ/2HX AM i?Bb KQ/2H- i?2 b2i Q7 p2`iB+2b V Bb }MBi2 TQ`iBQM Q7 i?2 ;`B/ Z2 M/ i?2 T? b2 bT +2 Bb Λ := {0, 1}X S? b2 1 i bBi2 v Bb iQ #2 BMi2`T`2i2/ b i?2 2tBbi2M+2 Q7 dzTQBMiǴ i i?Bb bBi2X aBM+2 i?2`2 `2 QMHv irQ TQbbB#H2 T? b2b- i?2 HQ+ H bT2+B}+ iBQM π v (x) := π(x(v) | x(Nv )) := P (X(v) = x(v) | X(Nv ) = x(Nv )) Bb 2MiB`2Hv bT2+B}2/ #v i?2 7mM+iBQM Q7 (v, x) α(v, x\v) := π(1 | x(Nv )) , r?B+? /2T2M/b QM x QMHv i?`Qm;? i?2 T? b2b Q7 bBi2b /Bz2`2Mi 7`QK v- M/ BM 7 +i i?`Qm;? T? b2b BM i?2 M2B;?#Qm`?QQ/ Nv Q7 bBi2 vX PM2 `2+Q;MBx2b ?2`2 M M HQ;v rBi? i?2 MQiBQM Q7 2tpBbB#H2 BMi2MbBivX
h?Bb M HQ;v Bb +QKTH2i2 BM i?2 BKTQ`i Mi + b2 i? i i?2 /2T2M/2M+2 QM x(Nv ) Bb i?`Qm;? i?2 bBi2b u ∈ Nv bm+? i? i x(u) = 1 M/ r?2`2 i?2 BMBiB H b2i V Q7 p2`iB+2b `2 TH +2/ QM i?2 bT +2 bQ b iQ 7Q`K ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 νX 1t KTH2 eXeXe, .Bb+`2iBx2/ #B`i?@ M/@/2 i? bBKmH iBQMX PM2 r v iQ b KTH2 i?2 /Bbi`B#miBQM Bb pB JQMi2 * `HQ bBKmH iBQM H;Q`Bi?KXN q2 +QMbi`m+i J `FQp +? BM QM i?2 +QM};m` iBQM bT +2 i? i HHQrb i` MbBiBQM 7`QK +QM};@ m` iBQM x iQ +QM};m` iBQM y QMHv B7 i?2 H ii2` Bb i?2 b K2 b x 2t+2Ti K v#2 i bBi2 vX q2 /2MQi2 bm+? +QM};m` iBQM #v y = ((v), x(V \{v})X h?2 +?QB+2 Q7 v Bb mMB7Q`K QM V M/ i?2 T? b2 Bb +?Qb2M ++Q`/BM; iQ i?2 T`Q# #BHBiv π(· | x(Nv )) QM {0, 1}X h?2`27Q`2- B7 x(v) = 0 M/ y(v) = 1- i?2`2 Bb dz#B`i?Ǵ i bBi2 v- r?2`2 b i?2 bBim iBQM x(v) = 1 M/ y(v) = 0 +Q``2bTQM/b iQ dz/2 i?Ǵ i bBi2 vX A7 i?2 +? BM Bb B``2/m+B#H2 U M/ i?2`27Q`2 TQbBiBp2 `2+m``2MiV M/ T2`BQ/B+ U M/ i?2M 2`;Q/B+V Bi +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 bi iBQM `v /Bbi`B#miBQM π U1t2`+Bb2 eXdXdVX
N AM i?2 bBim iBQM /2b+`B#2/ i i?2 2M/ Q7 i?2 T`2pBQmb 2t KTH2- Bi Bb bBKBH ` iQ S`2biQMǶb #B`i?@ M/@/2 i? H;Q`Bi?K Q7 a2+iBQM eX8X
eXdX 1s1_*Aa1a
eXd
k8j
1t2`+Bb2b
1t2`+Bb2 eXdXRX "B`i?@ M/@/2 i? S`Qp2 7Q`K HHv i?2 bi i2K2Mi BM i?2 H bi #27Q`2 H bi T ` ;` T? Q7 i?2 T`QQ7 Q7 h?2Q`2K eX8XRX 1t2`+Bb2 eXdXkX PM i?2 J `FQp T`QT2`iv Q7 TQBMi T`Q+2bb G2i W #2 #QmM/2/ K2 bm` #H2 bm#b2i Q7 Rm X G2i P #2 T`Q# #BHBiv K2 bm`2 QM dP Mpf (W ) /2}M2/ #v i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 dQ (x) = (x) rBi? `2bT2+i iQ f bQK2 T`Q# #BHBiv K2 bm`2 Q QM Mp (W )X amTTQb2 i? i mM/2` P - i?2 TQBMi T`Q+2bb X Bb J `FQp rBi? `2bT2+i iQ i?2 #BM `v bvKK2i`B+ `2H iBQM ∼X G2i C ⊆ W X a?Qr i? i i?2 +QM/BiBQM H /Bbi`B#miBQM Q7 X(C) ;Bp2M X(C) /2T2M/b QMHv QM X(∂(C)∩C) r?2`2 ∂(C) ∩ C := {u ∈ W ∩ C ; u ∼ v 7Q` bQK2 v ∈ C}X 1t2`+Bb2 eXdXjX GQ+ H BMi2`T`2i iBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv G2i X #2 }MBi2 TQBMi T`Q+2bb QM Rm +QMbi`m+i2/ b BM 1t KTH2 eXRX8 M/ bmTTQb2 i? i i?2 `272`2M+2 SQBbbQM T`Q+2bb Bb M ?TT rBi? mMBi BMi2MbBiv- i? i Bb- ν(du) = duX S`Qp2 i? i B7 u → λ(u, X) Bb HKQbi bm`2Hv #QmM/2/ M/ +QMiBMmQmbλ(u, X) = lim ε↓0
1 E [X(B(u, ε)) | X\{u}] .
m (B(u, ε))
1t2`+Bb2 eXdX9X J `F2/ T B`rBb2 BMi2` +iBQM T`Q+2bb *QMbB/2` i?2 7QHHQrBM; KQ/2H Q7 }MBi2 K `F2/ TQBMi T`Q+2bb rBi? # bB+ TQBMib BM +QKT +i b2i W ⊂ Rm M/ K `Fb BM i?2 b2i K = {1, . . . , M }X h?2 `272`2M+2 /Bbi`B#miBQM Bb i? i Q7 M ?TT QM W rBi? mMBi ` i2 M/ BB/ mMB7Q`KHv /Bbi`B#mi2/ K `Fb U`272` iQ 1t KTH2 eXRXd 7Q` i?2 7` K2rQ`F M/ MQi iBQMVX h?2 /2MbBiv rBi? `2bT2+i iQ i?Bb `272`2M+2 K2 bm`2 Bb
(x) = α
(u,k)∈x
βk
γkm (||u − v||) ,
(u,k),(v,m)∈x
r?2`2 α M/ i?2 βk `2 TQbBiBp2 `2 H MmK#2`b M/ r?2`2 γkm : [0, ∞) → [0, ∞) U+ HH2/ i?2 BMi2` +iBQM 7mM+iBQMbVX bbmK2 i? i γkm ≡ γmk X a?Qr i? i B7 7Q` HH k, m ∈ {1, . . . , M }- γk,m (r) = 1 r?2M r > rkm r?2`2 rkm > 0- M/ B7 7Q` bQK2 }MBi2 TQbBiBp2 MmK#2` R- supk,m rkm ≤ R- i?2M i?2 K `F2/ TQBMi T`Q+2bb X Bb J `FQp rBi? `2bT2+i iQ i?2 `2H iBQM ∼ /2}M2/ #v (u, k) ∼ (v, m) ⇐⇒ ||u − v|| ≤ R .
1t2`+Bb2 eXdX8X MQi?2` ? `/@+Q`2 TQBMi T`Q+2bb 6`QK M ?TT Q7 BMi2MbBiv 1 QM Rm - 2` b2 HH i?2 TQBMib i /Bbi M+2 ≤ R 7`QK MQi?2` TQBMi- r?2`2 R > 0X *QKTmi2 i?2 BMi2MbBiv K2 bm`2 Q7 i?2 `2K BMBM; TQBMi T`Q+2bbX
k89 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a 1t2`+Bb2 eXdXeX suw G2i X- Y - M/ Z #2 i?`22 /Bb+`2i2 ` M/QK p `B #H2b rBi? p Hm2b BM E- F - M/ G- `2bT2+iBp2HvX S`Qp2 i?2 7QHHQrBM;, A7 7Q` bQK2 7mM+iBQM g : E × F → [0, 1]P (X = x | Y = y, Z = z) = g(x, y) 7Q` HH x, y, z- i?2M P (X = x | Y = y) = g(x, y) 7Q` HH x, y- M/ X M/ Z `2 +QM/BiBQM HHv BM/2T2M/2Mi ;Bp2M Y X 1t2`+Bb2 eXdXdX .Bb+`2iBx2/ #B`i?@ M/@/2 i? bBKmH iBQM AM 1t KTH2 eXeXe ;Bp2 i?2 i` MbBiBQM T`Q# #BHBiB2b px,y Q7 i?2 JQMi2 * `HQ J `FQp +? BM i?2`2Q7 M/ b?Qr i? i 7Q` HH x, y- i?2 /2i BH2/ # H M+2 2[m iBQMb π(x)px,y = π(y)py,x `2 b iBb}2/X .2/m+2 7`QK i?Bb i? i π Bb BM/22/ i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?Bb J `FQp +? BMX :Bp2 i?2 +QM/BiBQMb QM i?2 2M2`;v 7mM+iBQM U i? i ;m ` Mi22 B``2/m+B#BHBiv M/ T2`BQ/B+BivX 1t2`+Bb2 eXdX3X S`QT2`Hv +QHQm`2/ ;` T?b *QMbB/2` +QM};m` iBQM bT +2 7Q` r?B+? i?2 T? b2 bT +2 Λ +QMbBbib Q7 }MBi2 MmK#2` Q7 dz+QHQm`bǴ H #2H2/ 7`QK 1 iQ qX h?2 K `FBM; Q7 i?2 bBi2b T2`KBib mb iQ /BbiBM;mBb? b2p2` H TQBMi T`Q+2bb2b- 7Q` BMbi M+2 i?2 /Bz2`2Mi ivT2b Q7 i`22b BM 7Q`2biX q2 /2b+`B#2 J `FQp +? BM {Xn }n≥0 i FBM; Bib p Hm2b BM i?2 bm#b2i E Q7 ΛV +QMbBbiBM; Q7 i?2 dzT`QT2`Hv +QHQm`2/Ǵ +QM};m` iBQMb- i? i Bb +QM};m` iBQMb x bm+? i? i x(v) = x(w) r?2M2p2` v ∼ wX q2 bi `i 7`QK T`QT2`Hv +QHQm`2/ +QM};m` iBQM X0 X amTTQb2 i iBK2 n i?2 bi i2 Bb xX q2 i?2M +?QQb2 mMB7Q`KHv i ` M/QK bBi2 v- M/ i?2M +?QQb2 mMB7Q`KHv i ` M/QK +QHQm` BM i?2 b2i Q7 +QHQm`b HHQr #H2 i v BM +QM};m` iBQM x- i? i BbAv (x) := {j ∈ {1, 2, . . . , q} ; j = x(w) 7Q` HH w , w ∼ v} . h?2 M2r bi i2 i iBK2 n + 1 Bb i?2M y- r?B+? Bb 2[m H iQ x 2t+2Ti 7Q` i?2 M2r +QHQm` j i bBi2 vX o2`B7v i? i i?Bb +? BM Bb B``2/m+B#H2 B7 i?2`2 `2 i H2 bi i?`22 +QHQm`br?B+? r2 ?2M+27Q`i? bbmK2X q? i Bb i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?Bb +? BM\ 1t2`+Bb2 eXdXNX 1tpBbB#H2 BMi2MbBiv S`Qp2 h?2Q`2K eXjXjX
*? Ti2` d S HK S`Q# #BHBiv QM i?2 GBM2 S HK i?2Q`v QM i?2 HBM2 HBMFb irQ ivT2b Q7 bi iBQM `Biv 7Q` K `F2/ TQBMi T`Q+2bb2b, iBK2@bi iBQM `Biv M/ 2p2Mi@bi iBQM `BivX h?2 M2ti irQ 2t KTH2b rBHH +H `B7v i?2b2 MQiBQMbX 1t KTH2 dXyXR, h?2 _2M2r H S`Q+2bbX AM i?Bb ivT2 Q7 TQBMi T`Q+2bb- QM2 /BbiBM;mBb?2b i?2 /2H v2/ iBK2@bi iBQM `v p2`bBQM 7`QK i?2 mM/2H v2/ p2`bBQM r?Qb2 /Bbi`B#miBQM Bb BMp `B Mi rBi? `2bT2+i iQ i?2 b?B7i i? i i` MbH i2b i?2 }`bi 2p2Mi iBK2 iQ i?2 Q`B;BMX q2 ? p2 b22M i? i i?2`2 2tBbib bBKTH2 `2H iBQM #2ir22M i?2 irQ p2`bBQMb- r?B+? `2 B/2MiB+ H 2t+2Ti 7Q` i?2 /Bbi`B#miBQM Q7 i?2 }`bi 2p2Mi iBK2X AM i?2 i2`KBMQHQ;v Q7 S HK i?2Q`v- i?2 mM/2H v2/ p2`bBQM Bb i?2 S HK p2`bBQM Q7 i?2 iBK2@bi iBQM `v p2`bBQMX
1t KTH2 dXyXk, h?2 *QMiBMmQmb@iBK2 J `FQp *? BMX *QMbB/2` M B``2@ /m+B#H2 TQbBiBp2 `2+m``2Mi +QMiBMmQmb@iBK2 ?K+ r?Qb2 BK#2//2/ +? BM Ui?2 +? BM Q#b2`p2/ i i?2 i` MbBiBQM iBK2bV Bb HbQ TQbBiBp2 `2+m``2MiX q?2M bm+? +? BM Bb UiBK2@Vbi iBQM `v- Bi Bb MQi i`m2 BM ;2M2` H i? i i?2 2K#2//2/ +? BM Bb bi iBQM `v2p2M r?2M Bi Bb bbmK2/ TQbBiBp2 `2+m``2MiX >Qr2p2`- i?2`2 Bb bBKTH2 `2H iBQM #2@ ir22M i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +QMiBMmQmb@iBK2 +? BM M/ i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 BK#2//2/ +? BMX h?2 +QMiBMmQmb@iBK2 +? BM bi `i2/ rBi? i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 BK#2//2/ +? BM Bb i?2 S HK p2`bBQM Q7 i?2 bi iBQM `v +QMiBMmQmb@iBK2 +? BMX JQ`2 ;2M2` HHv- S HK i?2Q`v Bb +QM+2`M2/ rBi? DQBMiHv bi iBQM `v biQ+? biB+ T`Q+2bb2b M/ TQBMi T`Q+2bb2b BM iBK2 M/ BM bT +2- M/ rBi? i?2 T`Q# #BHBbiB+ bBim iBQM i 2p2Mi iBK2b Q` TQBMi HQ+ iBQMbX h?2 ivT2 Q7 Bbbm2b i? i Bi //`2bb2b Bb 2bT2+B HHv `2H2p Mi BM [m2m2BM; i?2Q`v TTHB2/ iQ b2`pB+2 bvbi2Kb- 7Q` r?B+? i?2`2 `2 irQ /BbiBM+i TQBMib Q7 pB2r- i? i Q7 i?2 dzQT2` iQ`Ǵ- r?Q Bb BMi2`2bi2/ BM i?2 #2? pBQm` Q7 [m2m2 i `#Bi` `v iBK2b- M/ i? i Q7 i?2 dz+mbiQK2`Ǵ- r?Q Bb ;2M2` HHv BMi2`2bi2/ BM i?2 bBim iBQM i? i b?2 }M/b mTQM ``Bp HX AM KQ#BH2 +QKKmMB+ iBQMb +QMi2ti- r?2`2 TQBMi K v `2T`2b2Mi i?2 HQ+ iBQM Q7 KQ#BH2 T?QM2 Q` Q7 M Mi2MM `2H v- bBKBH ` /m HBiv 2tBbib #2ir22M i?2 bBim iBQM Q7 ;Bp2M mb2` M/ i?2 bBim iBQM i M `#Bi` `v TQBMiX
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_7
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Pm` }`bi bi2Tb +QM+2`M i?2 S HK i?2Q`v 7Q` TQBMi T`Q+2bb2b QM Rm - #mi r2 rBHH bQQM brBi+? iQ TQBMi T`Q+2bb2b QM i?2 HBM2- QMHv iQ `2im`M iQ ?B;?2`@/BK2MbBQM H bT +2b H i2` QM BM *? Ti2` 3X
dXR ai iBQM `v SQBMi S`Q+2bb2b h?2 Q`B;BM H 7` K2rQ`F Q7 i?2 KQ/2`M S HK T`Q# #BHBivR Bb i? i Q7 bi iBQM `v TQBMi T`Q+2bb2bX h?2 b2iiBM; Q7 K2 bm` #H2 ~Qrb M/ Q7 +QKT iB#BHBiv rBi? ~Qr BMi`Q@ /m+2/ BM i?2 +m``2Mi b2+iBQM ? b MmK#2` Q7 /p Mi ;2b i? i H `;2Hv +QKT2Mb i2 7Q` Bib `2H iBp2 #bi` +iM2bbX J2 bm` #H2 6HQrb q2 }`bi BMi`Q/m+2 i?2 2H2K2Mi `v +QM+2Ti Q7 i` MbH iBQM Q7 7mM+iBQMb M/ Q7 K2 @ bm`2b- M/ i?2M T`Q+22/ iQ i?2 KQ`2 #bi` +i MQiBQMb Q7 b?B7i M/ K2 bm` #H2 ~QrX h?2 H ii2` MQiBQM rBHH T`QpB/2 TQr2`7mH 7` K2rQ`F 7Q` i?2 /2p2HQTK2Mi Q7 i?2 S HK i?2Q`v BM i?Bb +? Ti2` M/ BM *? Ti2` 3X 1t KTH2 dXRXR, h` MbH i2/ 6mM+iBQMb M/ h` MbH i2/ J2 bm`2bX UBV G2i Ω #2 i?2 bT +2 Q7 +QMiBMmQmb 7mM+iBQMb ω : Rm → R M/ H2i F #2 i?2 σ@}2H/ QM Ω ;2M2` i2/ #v i?2 +QQ`/BM i2 7mM+iBQMb y → ω(y) Uy ∈ Rm VX .2}M2 K TTBM; τx : Ω → Ω #v τx (ω)(y) := ω(y + x) (x, y ∈ Rm , ω ∈ Ω) . Uh?Bb K TTBM; τx i` MbH i2b Mv 7mM+iBQM ω ∈ Ω #v −xXV UBBV G2i (Ω, F) = (M (Rm ), M(Rm ))- i?2 bT +2 Q7 HQ+ HHv }MBi2 K2 bm`2b QM (R , B(Rm ))X G2i θx := Sx - r?2`2 m
(Sx μ)(C) := μ(C + x)
(x ∈ Rm , μ ∈ M (Rm ), C ∈ B(Rm ) .
Uh?2 K TTBM; Sx i` MbH i2b Mv K2 bm`2 μ ∈ M (Rm ) #v −xXV
.2}MBiBQM dXRXk 7 KBHv {θx }x∈Rm Q7 K2 bm` #H2 K TTBM;b 7`QK i?2 K2 bm` #H2 bT +2 (Ω, F) BMiQ Bib2H7 Bb + HH2/ b?B7i QM (Ω, F) B7, U V
7Q` HH x ∈ Rm - θx Bb #BD2+iBp2- M/
U#V
7Q` HH x, y ∈ Rm - θx ◦ θy = θx+y X
h?2 b?B7i {θx }x∈Rm QM (Ω, F) Bb + HH2/ K2 bm` #H2 ~Qr B7 BM //BiBQM U+V
i?2 K TTBM; (x, ω) → θx (ω) Bb K2 bm` #H2 7`QK B(Rm ) ⊗ F iQ FX
AM T `iB+mH `- 7`QK U#V- θ0 Bb i?2 B/2MiBiv M/ θx −1 = θ−x 7Q` HH x ∈ Rm X R
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1t KTH2 dXRX9, J2 bm` #H2 6HQr QM aT +2 Q7 J2 bm`2bX h?2 b?B7i {Sx }x∈Rm Q7 1t KTH2 dXRXR UBBV Bb K2 bm` #H2 ~QrX hQ T`Qp2 i?Bb- Bi bm{+2b iQ b?Qr i? i i?2 K TTBM; (x, μ) → (Sx μ)(f ) := f (y − x)μ(dx) Rm
Bb K2 bm` #H2 r?2M2p2` f : R → R Bb MQM@M2; iBp2 +QMiBMmQmb 7mM+iBQM rBi? +QKT +i bmTTQ`iX h?Bb Bb i?2 + b2 bBM+2 (x, μ) → g(x, μ) := (Sx μ)(f ) Bb +QMiBMmQmb BM i?2 }`bi `;mK2Mi M/ K2 bm` #H2
BM i?2 b2+QM/X UAM/22/- g Bb i?2 HBKBi b n ↑ ∞ Q7 i?2 K2 bm` #H2 7mM+iBQMb k∈N 1Ck,n (x)g(xk,n , μ)- r?2`2 {xk,n }k∈N Bb M 1 1 m 2MmK2` iBQM Q7 i?2 ;`B/ n−1 Zm M/ Ck,n = xk,n + (− 2n , 2n ] XV m
1t KTH2 dXRX8, a?B7iBM; J `F2/ SQBMi S`Q+2bbX _272` iQ .2}MBiBQM RXRXky 7Q` i?2 b2iiBM; M/ MQi iBQMX AM i?2 + b2 r?2`2 (E, B(E)) = (Rm , B(Rm ))i?2 b?B7i Sx Ux ∈ Rm V Bb /2}M2/ #v Sx μ (C × L) := μ ((C + x) × L) (C ∈ B(Rm ), L ∈ K) . 6Q` Mv x ∈ Rm i?2 K TTBM; Sx i` MbH i2b i?2 # b2 TQBMi T`Q+2bb #v −x- i?2 K `Fb `2K BMBM; ii +?2/ iQ i?2B` # b2 TQBMi /m`BM; i?2 i` MbH iBQMX q2 MQr BMi`Q/m+2 i?2 7mM/ K2Mi H MQiBQM Q7 +QKT iB#BHBiv rBi? ~QrX .2}MBiBQM dXRXe G2i {θx }x∈Rm #2 K2 bm` #H2 ~Qr QM (Ω, F)X biQ+? biB+ T`Q+2bb {Z(x)}x∈Rm /2}M2/ QM (Ω, F) rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (K, K) Bb + HH2/ +QKT iB#H2 rBi? i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i, θx @+QKT iB#H2V B7 Z(x) = Z(0) ◦ θx , i? i BbZ(x, ω) = Z(0, θx (ω))
(ω ∈ Ω, x ∈ Rm ) .
` M/QK K2 bm`2 N QM Rm Bb + HH2/ +QKT iB#H2 rBi? i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i, θx @+QKT iB#H2V B7 N ◦ θx = Sx N , i? i BbN (θx (ω), C) = N (ω, C + x) (ω ∈ Ω, C ∈ B(Rm ), x ∈ Rm ) .
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1t KTH2 dXRX3, G2p2H *`QbbBM;bX AM 1t KTH2 dXRXR UBBV- rBi? m = 1- i?2 ` M/QK K2 bm`2 N `2+Q`/BM; i?2 +`QbbBM;b Q7 H2p2H a #v i?2 +QQ`/BM i2 T`Q+2bb Bb +QKT iB#H2 rBi? i?2 b?B7i i?2`2Q7X
1t KTH2 dXRXN, .Bb+QMiBMmBiv J2 bm`2X G2i Ω #2 i?2 bT +2 Q7 TB2+2rBb2 +QMbi Mi +Q`HQH U+QMiBMmQmb QM i?2 `B;?i- rBi? HBKBib QM i?2 H27iV 7mM+iBQMb ω : R → R- M/ H2i F #2 i?2 σ@}2H/ QM Ω ;2M2` i2/ #v i?2 +QQ`/BM i2 7mM+iBQMb X(y) : Ω → R Uy ∈ RV /2}M2/ #v X(y, ω) := ω(y)X .2}M2 7Q` 2 +? x ∈ R i?2 K TTBM; θx : Ω → Ω #v θx (ω)(y) := ω(y + x), θx i` MbH i2b b KTH2 7mM+iBQM #v −xX h?2 ` M/QK K2 bm`2 N +QmMiBM; i?2 /Bb+QMiBMmBiB2b Q7 i?2 +QQ`/BM i2 T`Q+2bb Bb +QKT iB#H2 rBi? i?2 #Qp2 b?B7iX
1t KTH2 dXRXRy, J `F a2H2+iBQMX AM 1t KTH2 dXRX8- r2 K F2 i?2 7QHHQrBM; +? M;2 Q7 MQi iBQM, p (E) , F := M p (E) , θx := Sx . Ω := M h?2 TQBMi T`Q+2bb NL /2}M2/ #v NL (C) := μ (C × L)
(L ∈ K)
Bb θx @+QKT iB#H2X
.2}MBiBQM dXRXRR G2i {θx }x∈Rm #2 K2 bm` #H2 ~Qr M/ H2i N #2 bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM Rm +QKT iB#H2 rBi? i?Bb ~Qr- rBi? b2[m2M+2 Q7 TQBMib {Xn }n∈N X G2i {Zn }n∈N #2 b2[m2M+2 Q7 ` M/QK p `B #H2b rBi? p Hm2b BM bQK2 K2 bm` #H2 bT +2 (K, K)X h?2 K `F2/ TQBMi T`Q+2bb (N, Z) Bb b B/ iQ #2 +QKT iB#H2 rBi? i?2 ~Qr NZ (θx (ω), C × L) = NZ (ω, (C + x) × L) (C ∈ B(Rm ), L ∈ K, x ∈ Rm ) . h?2 b2[m2M+2 {Zn }n∈Z Bb i?2M + HH2/ K `F b2[m2M+2 Q7 N +QKT iB#H2 rBi? i?2 ~QrX h?2 K `F Zn Bb ii +?2/ iQ i?2 TQBMi Xn - BM i?2 b2Mb2 i? i Bi dz7QHHQrb Bib TQBMiǴ /m`BM; i?2 i` MbH iBQMb T`QpQF2/ #v i?2 #bi` +i b?B7i θx X q2 b v, ((N, Z), θx ) Bb K `F2/ TQBMi T`Q+2bb +QKT iB#H2 rBi? i?2 ~QrX
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rQ`/ #Qmi MQi iBQM 6B`bi- `2+ HH i? i #v /2}MBiBQM Q7 +QKT iB#BHBiv- i?2 TQBMi T`Q+2bb N Bb b B/ iQ #2 +QKT iB#H2 rBi? i?2 ~Qr {θx }x∈Rm B7 N ◦ θx := Sx N r?2`2 Sx Bb i?2 i` MbH iBQM QT2` iQ` +iBM; QM K2 bm`2b U1t KTH2 dXRXR- UBBVV- i? i Bb(Sx N )(ω) = N (ω) − x := {y − x ; y ∈ N (ω)} . AM bmKK `v- r2 ? p2 i?`22 MQi iBQMb 7Q` i?2 b K2 Q#D2+i N ◦ θx , Sx N , N − x . h?2 H ii2` Bb MQi iQ #2 +QM7mb2/ rBi? N − εx - r?B+? `2T`2b2Mib N \{x} B7 x ∈ N M/ N B7 x ∈ / NX ai iBQM `v 6` K2rQ`Fb G2i (Ω, F, P ) #2 T`Q# #BHBiv bT +2 M/ H2i {θx }x∈Rm #2 K2 bm` #H2 ~Qr QM (Ω, F)X .2}MBiBQM dXRXRk h?2 T`Q# #BHBiv P Bb + HH2/ BMp `B Mi rBi? `2bT2+i iQ i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i- θx @BMp `B MiV B7 7Q` HH x ∈ Rm P ◦ θx−1 = P . PM2 i?2M b vb, (P, θx ) Bb bi iBQM `v 7` K2rQ`F QM Rm X 1t KTH2 dXRXRj, ai iBQM `v SQBMi S`Q+2bbX G2i (P, θx ) #2 bi iBQM `v 7` K2rQ`F QM Rm X A7 i?2 TQBMi T`Q+2bb N QM Rm Bb +QKT iB#H2 rBi? i?2 b?B7i- Bi Bb bi iBQM `vX AM/22/- H2iiBM; A = {ω; N (ω, C1 ) = k1 , . . . , N (ω, Cm ) = km }, r?2`2 C1 , . . . , Cm ∈ B(Rm ) M/ k1 , . . . , km ∈ N- r2 ? p2- #v /2}MBiBQMθx−1 (A) := {ω; θx (ω) ∈ A} = {ω; N (θx (ω), C1 ) = k1 , . . . , N (θx (ω), Cm ) = km } = {ω; N (ω, C1 + x) = k1 , . . . , N (ω, Cm + x) = km }. h?2`27Q`2- bBM+2 P ◦ θx −1 = P P (N (C1 ) = k1 , . . . , N (Cm ) = km ) = P (N (C1 + x) = k1 , . . . , N (Cm + x) = km ) .
AM i?2 bBim iBQM Q7 i?2 T`2pBQmb 2t KTH2- QM2 b vb- 7Q` b?Q`i, (N, θx , P ) Bb bi iBQM `v TQBMi T`Q+2bbX A7 BM .2}MBiBQM dXRXRR- i?2 mM/2`HvBM; T`Q# #BHBiv P Bb BMp `B Mi rBi? `2bT2+i iQ i?2 ~Qr- r2 b v, ((N, Z), θx , P ) Bb bi iBQM `v K `F2/ TQBMi T`Q+2bbX
key
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1t KTH2 dXRXR9, ai iBQM `v aiQ+? biB+ S`Q+2bbX G2i (P, θx ) #2 bi iBQM@ `v 7` K2rQ`F QM Rm X "v i?2 b K2 `;mK2Mi b BM i?2 2t KTH2 #Qp2- biQ+? biB+ T`Q+2bb {Z(x)}x∈Rm rBi? p Hm2b BM (K, K) i? i Bb θx @+QKT iB#H2 Bb bi`B+iHv bi iBQM@ `vX 6Q` b?Q`i, (Z, θx , P ) Bb bi iBQM `v biQ+? biB+ T`Q+2bbX U>2`2 Z bi M/b 7Q` {Z(x)}x∈Rm XV h?2 M2ti `2bmHi b vb i? i bi iBQM `v TQBMi T`Q+2bb ? b 2Bi?2` MQ TQBMib i HH Q` M BM}MBiv Q7 TQBMib #Qi? BM R+ M/ R− X h?2Q`2K dXRXR8 G2i (N, θt , P ) #2 bi iBQM `v TQBMi T`Q+2bb QM RX h?2M P ({N (R) = 0} ∪ {N ((0, ∞)) = N ((−∞, 0)) = +∞}) = 1. S`QQ7X h?2 b2i Ht := {N ((t, ∞)) = 0} BM+`2 b2b rBi? t M/ H :=
∞ 4 n=0
H−n = {N (R) = 0} ⊆ {N ((0, ∞)) < ∞} =
∞
Hn := G.
n=0
AM //BiBQM θt Hs = Hs−t 7Q` HH s, t ∈ R- bQ i? i P (Hn ) /Q2b MQi /2T2M/ QM n ∈ ZX >2M+2 P (G) = limn↑∞ P (Hn ) = limn↓−∞ P (Hn ) = P (H)- r?B+?- iQ;2i?2` s := rBi? H ⊆ G- BKTHB2b i? i P (G − H) = 0X bBKBH ` `;mK2Mi # b2/ QM H {N ((−∞, s)) = 0} H2 /b iQ P (G − H) = 0- r?2`2 G = {N ((−∞, 0)) < ∞}X h?2`27Q`2 P ((G ∪ G ) − H) = 0X bBKBH ` T`QQ7 b?Qrb i? i B7 (N, θx , P ) Bb bi iBQM `v TQBMi T`Q+2bb QM Rm i?2M rBi? T`Q# #BHBiv 1- 2Bi?2` N (Rm ) = 0 Q` ∞X h?2 MQiBQMb BMi`Q/m+2/ 7Q` +QMiBMmQmb T ` K2i2` ? p2 +QmMi2`T `ib BM i?2 /Bb+`2i2 T ` K2i2` + b2X G2i (Ω, F, P 0 ) #2 T`Q# #BHBiv bT +2 M/ H2i θ #2 #BD2+iBp2 K2 bm` #H2 K T 7`QK Ω iQ Bib2H7 rBi? K2 bm` #H2 BMp2`b2X h?2 7 KBHv {θn }n∈Z Bb + HH2/ /Bb+`2i2 b?B7i UQ` ~QrV QM (Ω, F) M/ biQ+? biB+ T`Q+2bb {Zn }n∈Z QM (Ω, F) Bb b B/ iQ #2 +QKT iB#H2 rBi? i?2 b?B7i U7Q` b?Q`i, θ@+QKT iB#H2V B7 7Q` HH n ∈ Z M/ HH ω ∈ Ω- Bi ?QH/b i? i Zn = Z0 ◦θ- i? i Bb- Zn (ω) = Z0 (θn (ω))X amTTQb2 i? i P 0 Bb θ@BMp `B Mi T`Q# #BHBiv QM (Ω, F)- i? i Bb- P 0 ◦ θ = P 0 X h?2M (P 0 , θ) Bb + HH2/ bi iBQM `v 7` K2rQ`FX aBKBH `Hv iQ i?2 +QMiBMmQmb T ` K2i2` + b2- θ@+QKT iB#H2 biQ+? biB+ T`Q+2bb Bb bi iBQM `v B7 P 0 Bb θ@BMp `B MiX 1`;Q/B+Biv AM i?Bb bm#b2+iBQM- i?2 bi M/ `/ `2bmHib +QM+2`MBM; 2`;Q/B+Biv rBHH #2 `2pB2r2/X G2i (P, θx ) #2 bi iBQM `v 7` K2rQ`FX _2+ HH i?2 7QHHQrBM; /2}MBiBQMb, .2}MBiBQM dXRXRe M 2p2Mi A ∈ F Bb + HH2/ bi`B+iHv θx @BMp `B Mi B7 A = θx−1 A 7Q` HH x ∈ Rm X Ai Bb + HH2/ θx @BMp `B Mi B7 P (A # θx−1 A) = 0 7Q` HH x ∈ Rm X .2}MBiBQM dXRXRd h?2 ~Qr {θx }x∈Rm Bb + HH2/ P @2`;Q/B+ B7 HH θx @BMp `B Mi 2p2Mib `2 i`BpB HX AM Qi?2` rQ`/b, (P, θx ) Bb 2`;Q/B+X
dXRX ahhAPL_u SPALh S_P*1aa1a
keR
.2}MBiBQM dXRXR3 MQM@/2+`2 bBM; b2[m2M+2 {An }n≥1 Q7 #QmM/2/ +QMp2t bm#b2ib Q7 Rm Bb + HH2/ +QMp2t p2` ;BM; b2[m2M+2 B7 lim sup{r ; B(0; r) ⊂ An } = ∞ ,
n↑∞
r?2`2 B(0; r) Bb i?2 +HQb2/ # HH Q7 ` /Bmb r +2Mi2`2/ QM 0X h?2 7mM/ K2Mi H `2bmHi Q7 2`;Q/B+ i?2Q`v Bb "B`F?QzǶb TQBMirBb2 2`;Q/B+ i?2Q@ `2K, h?2Q`2K dXRXRN G2i (P, θx ) #2 2`;Q/B+X h?2M- 7Q` Mv P @BMi2;` #H2 7mM+iBQM f : (Ω, F) → (R, B(R) 1 f ◦ θx dx = E [f ] . lim m n↑∞ (An ) A n M 2[mBp H2Mi +? ` +i2`Bx iBQM Q7 2`;Q/B+Biv Bb i?2 7QHHQrBM;, 1 P (A ∩ θx B) = P (A)P (B) (A, B ∈ F ) . lim a↑∞ (2a)m [−a,+a]m
()
AM 7 +i- Bi bm{+2b i? i UV ?QH/b 7Q` HH A, B BM M H;2#` A ;2M2` iBM; F 7Q` {θx }x∈Rm iQ #2 P @2`;Q/B+X .2}MBiBQM dXRXky h?2 ~Qr {θx }x∈Rm Bb + HH2/ P @KBtBM; B7 lim P (A ∩ θx B) = P (A)P (B) (A, B ∈ F ) .
|x|↑∞
()
AM Qi?2` rQ`/bc (P, θx ) Bb KBtBM;X AM 7 +i- Bi bm{+2b i? i UV ?QH/b 7Q` HH A, B BM M H;2#` A ;2M2` iBM; F 7Q` {θx }x∈Rm iQ #2 P @KBtBM;X *H2 `Hv- #v /QKBM i2/ +QMp2`;2M+2 BM UV- KBtBM; ~Qr Bb 2`;Q/B+X .2}MBiBQM dXRXkR G2i N #2 bi iBQM `v ` M/QK K2 bm`2 QM Rm X Ai Bb + HH2/ 2`;Q/B+ U`2bTX- KBtBM;V B7 (PN , Sx ) Bb 2`;Q/B+ U`2bTX- KBtBM;V- r?2`2 PN Bb i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 N M/ {Sx }x∈Rm Bb i?2 b?B7i Q7 M (Rm )X q2 K v ?2M+27Q`i? bmTTQb2 i? i N Bb i?2 + MQMB+ H ` M/QK K2 bm`2 QM M (Rm )X h?2 7QHHQrBM; +? ` +i2`Bx iBQMb Q7 2`;Q/B+Biv M/ KBtBM; Bb H27i b 1t2`+Bb2 dX3XjX G2i LN #2 i?2 G TH +2 i` Mb7Q`K Q7 N X
kek
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
h?2Q`2K dXRXkk U V h?2 bi iBQM `v ` M/QK K2 bm`2 N Bb 2`;Q/B+ B7 M/ QMHv B7 1 lim LN (f + Sx g) dx = LN (f )LN (f ) a↑∞ (2a)m [−a,+a]m 7Q` Mv MQM@M2; iBp2 #QmM/2/ K2 bm` #H2 7mM+iBQMb f, g : Rm → R rBi? #QmM/2/ bmTTQ`ibX U#V h?2 bi iBQM `v ` M/QK K2 bm`2 N Bb KBtBM; B7 M/ QMHv B7 lim LN (f + Sx g) = LN (f )LN (f )
|x|↑∞
7Q` Mv MQM@M2; iBp2 #QmM/2/ K2 bm` #H2 7mM+iBQMb f, g : Rm → R rBi? #QmM/2/ bmTTQ`ibX 1t KTH2 dXRXkj, h?2 >QKQ;2M2Qmb SQBbbQM S`Q+2bb Bb JBtBM;X h?Bb Bb M BKK2/B i2 +Q`QHH `v Q7 h?2Q`2K dXRXkkX
dXk 6B`bi GQQF i S HK S`Q# #BHBiv G2i mb `2+ HH MQi iBQMX A7 N Bb TQBMi T`Q+2bb rBi? TQBMi b2[m2M+2 {Xn }n≥1 - N − Xn `2T`2b2Mib i?2 TQBMi T`Q+2bb Q#i BMBM; #v i` MbH iBM; N #v −Xn X AM T `iB+mH ` i?2 TQBMi Xn #2+QK2b i?2 Q`B;BMX h?2 TQBMi T`Q+2bb N − εXn Bb i?2 Q`B;BM H N rBi?Qmi i?2 TQBMi Xn X h?2`2 `2 b2p2` H TT`Q +?2b iQ i?2 i?2Q`v Q7 S HK T`Q# #BHBivX h?2 QM2 /QTi2/ BM i?Bb +? Ti2` #2;BMb rBi? M2r HQQF i * KT#2HHǶb 7Q`KmH 7Q` bi @ iBQM `v K `F2/ TQBMi T`Q+2bb2bX h?2 S HK .Bbi`B#miBQM Q7 i?2 J `Fb G2i N #2 bBKTH2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2 b2[m2M+2 Q7 ` M/QK p `B #H2b rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (K, K)X 1 +? Zn Bb +QMbB/2`2/ b K `F Q7 i?2 +Q``2bTQM/BM; TQBMi Zn X _2+ HH i? i i?2 TQBMi T`Q+2bb M/ Bib b2[m2M+2 Q7 K `Fb `2 `272``2/ iQ b dzi?2 K `F2/ TQBMi T`Q+2bb (N, Z)ǴX Ai + M HbQ #2 `2T`2b2Mi2/ #v TQBMi T`Q+2bb NZ QM Rm × K /2}M2/ #v 1D (Xn , Zn ) (D ∈ B(Rm ) ⊗ K) . NZ (D) := n∈N
q2 bbmK2 i? i i?2 K `F2/ TQBMi T`Q+2bb (N, Z) Bb bi iBQM `v- i? i Bb- 7Q` HH x ∈ Rm - i?2 ` M/QK K2 bm`2 Sx (NZ ) /2}M2/ #v Sx (NZ )(D) := 1D (Xn + x, Zn ) n∈N
? b i?2 b K2 /Bbi`B#miBQM b NZ X h?2 BMi2MbBiv Q7 i?2 Ubi iBQM `vV TQBMi T`Q+2bb N Bb bbmK2/ #2 TQbBiBp2 M/ }MBi2,
dXkX 6A_ah GPPE h SGJ S_P""AGAhu
kej
0 < λ := E[N ((0, 1]m )] < ∞ . G2i i?2 σ@}MBi2 K2 bm`2 νZ QM (Rm × K) #2 /2}M2/ #v (D ∈ B(Rm ) ⊗ K) .
νZ (D) := E [NZ (D)]
_2+ HH i?2 7QHHQrBM; MQi iBQM, 7Q` Mv C ⊆ Rm - Mv x ∈ Rm C + x := {y + x ; y ∈ C} . "v bi iBQM `BivνZ ((C + x) × L) = νZ (C × L) (x ∈ Rm , C ∈ B(Rm ), L ∈ K) , M/ i?2`27Q`2- 7Q` }t2/ L- i?2 K2 bm`2 C → νZ (C × L) Bb BMp `B Mi mM/2` i` Mb@ H iBQM bQ i? i- #v h?2Q`2K XRX3- Bi Bb KmHiBTH2 Q7 i?2 G2#2b;m2 K2 bm`2 m QM (Rm , B(Rm )), νZ (C × L) = γ(L) m (C) , 7Q` bQK2 γ(L)X h?2 K TTBM; L → γ(L) Bb K2 bm`2 QM (K, K) i? i Bb }MBi2 bBM+2- rBi? L = K- γ(K) = λ- i?2 }MBi2 TQbBiBp2 BMi2MbBiv Q7 N X AM T `iB+mH `Q0N := λ−1 γ Bb T`Q# #BHBiv K2 bm`2 QM (K, K)- M/ νZ (C × L) = λ Q0N (L) m (C) . h?2`27Q`2 Q0N (L)
=
E
1C (Xn )1L (Zn ) . λ m (C)
n∈N
UdXRV
.2}MBiBQM dXkXR h?2 T`Q# #BHBiv Q0N QM (K, K) /2}M2/ #v UdXRV Bb i?2 S HK /Bb@ i`B#miBQM Q7 i?2 K `FbX h?2Q`2K dXkXk G2i f : Rm × K → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX h?2M
E f (x, z)NZ (dx × dz) = λ f (x, z)Q0N (dz) dx . UdXkV Rm ×K
Rm
K
S`QQ7X 6Q`KmH UdXkV Bb i`m2 7Q` f (x, z) := 1C (x) 1L (z), r?2`2 C ∈ B(Rm ) M/ L ∈ K- bBM+2 Bi i?2M `2/m+2b iQ UdXRVX h?2 ;2M2` H + b2 ; BM 7QHHQrb #v i?2 mbm H KQMQiQM2 +H bb `;mK2Mi # b2/ QM .vMFBMǶb h?2Q`2K XRXeX 6Q`KmH UdXkV #Qp2 Bb i?2 S HKĜ* KT#2HH 7Q`KmH 7Q` bi iBQM `v K `F2/ TQBMi T`Q+2bb2bX
ke9
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
h?2 S HK S`Q# #BHBiv hQ T bb 7`QK i?2 S HK /Bbi`B#miBQM Q7 K `Fb iQ S HK T`Q# #BHBiv- Bi Bb +QMp2MB2Mi iQ /Q i?Bb BM i2`Kb Q7 K2 bm` #H2 ~Qrb QM #bi` +i T`Q# #BHBiv bT +2bX q2 `272` iQ a2+iBQM dXR 7Q` i?2 # bB+ /2}MBiBQMbX G2i ((N, Z), θx , P ) #2 bi iBQM `v K `F2/ TQBMi T`Q+2bb QM Rm bm+? i? i N Bb bBKTH2 M/ rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX AM 7 +i- i?2 rQ`F M22/2/ 7Q` i?2 /2}MBiBQM Q7 S HK T`Q# #BHBiv Bb H`2 /v /QM2X Ai bm{+2b iQ +?QQb2 7Q` K `F bT +2 i?2 b KTH2 bT +2 Ω Bib2H7X 6Q` 2 +? n ∈ Z- θXn Bb ` M/QK 2H2K2Mi i FBM; Bib p Hm2b BM i?2 K2 bm` #H2 bT +2 (Ω, F)X h F2 BM UdXRV (K, K) = (Ω, F) M/ Zn = θXn (ω)X .2MQi2 BM i?Bb + b2 Q0N #v PN0 X AM T `iB+mH `- PN0 Bb T`Q# #BHBiv QM (Ω, F)X 6Q`KmH UdXRV i?2M `2 /b 7Q` HH C ∈ B(Rm ) Q7 }MBi2 TQbBiBp2 G2#2b;m2 K2 bm`2 E n∈N 1C (Xn )1A ◦ θXn 0 . UdXjV PN (A) = λ m (C) .2}MBiBQM dXkXj h?2 T`Q# #BHBiv PN0 /2}M2/ #v UdXjV Bb + HH2/ i?2 S HK T`Q# @ #BHBiv bbQ+B i2/ rBi? P UQ`- KQ`2 T`2+Bb2Hv- rBi? (N, θx , P )VX h?2Q`2K dXkX9 G2i v : Rm ×Ω → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX h?2M
(v(x) ◦ θx ) N (dx) = λ E0N v(x) dx . UdX9V E Rm
Rm
h?Bb 2[m HBiv Bb Q7i2M ;Bp2M BM i?2 bHB;?iHv KQ`2 2tTHB+Bi 7Q`K, v(x, θx ω) N (ω, dx) P (dω) = λ v(x, ω) dx PN0 (dω) . Ω
Rm
Ω
Rm
UdX8V
MQi?2` 2tT`2bbBQM 72 im`BM; 2tTHB+BiHv i?2 TQBMib Q7 i?2 TQBMi T`Q+2bb QM i?2 `B;?i@ ? M/ bB/2 Bb v(Xn , θXn ) = λ v(x, ω) dx PN0 (dω) . UdXeV E n∈N
Ω
Rm
S`QQ7X 6Q`KmH UdXjV Bb bT2+B H + b2 Q7 i?2 MMQmM+2/ 2[m HBiv 7Q` i?2 +?QB+2 v(x, ω) = 1C (x)1A (ω) , 7`QK r?B+? i?2 ;2M2` H + b2 7QHHQrb #v i?2 mbm H KQMQiQM2 +H bb `;mK2Mi # b2/ QM .vMFBMǶb h?2Q`2K XRXeX 6Q`KmH UdX9V Bb Q7i2M + HH2/ i?2 * KT#2HHĜJ2+F2 7Q`KmH #2+ mb2 Bi Bb bQ@ T?BbiB+ i2/ p i ` Q7 * KT#2HHǶb 7Q`KmH Xk Ai Bb bQK2iBK2b mb2/ BM i?2 Hi2`M iBp2 2[mBp H2Mi 7Q`K
E v(x) N (dx) = λE0N (v(x) ◦ θ−x ) dx . UdXdV Rm
k
(_vHH@L `/x2rbFB- RNeR)X
Rm
dXkX 6A_ah GPPE h SGJ S_P""AGAhu
ke8
_2K `F dXkX8 h?2 Tm`TQb2 Q7 i?Bb `2K `F Bb iQ K F2 6Q`KmH dX9 HQQF HBiiH2 H2bb #bi` +iX G2i h : Rm × M (Rm ) → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX h FBM; v(x, ω) := h(x, N (ω)) BM UdX9V- r2 ? p2
0 E h(Xn , N − Xn ) = λEN h(x, N ) dx . Rm
n∈N
aT2+B HBxBM; i?Bb iQ h(x, N ) = g(x)1Γ (N )- r?2`2 Γ ∈ M(Rm )- r2 Q#i BM E g(Xn )1Γ (N − Xn ) = λPN0 (N ∈ Γ) g(x) dx . Rm
n∈N
qBi? g(x) = 1C (x) r?2`2 C ∈ B(Rm ) Bb #QmM/2/ E 1C (Xn )1Γ (N − Xn ) = λ m (C)PN0 (N ∈ Γ) .
UdX3V
n∈N
b2i Γ ∈ M(Rm ) `2T`2b2Mib T`QT2`iv i? i K2 bm`2 μ ∈ M(Rm ) K v TQbb2bb Q` MQi- M/ N −Xn ∈ Γ K2 Mb i? i i?2 TQBMi T`Q+2bb b22M #v M Q#b2`p2` TH +2/ QM i?2 TQBMi Xn Ui? i Bb- T`2+Bb2Hv- N − Xn V b iBb}2b i?Bb T`QT2`ivX 6Q` BMbi M+2- rBi? Γ = {μ ; μ(B(0, a)\{0}) = 0}- r?2`2 B(x, a) Bb i?2 QT2M # HH Q7 ` /Bmb a ≥ 0 +2Mi2`2/ BM x- {N − Xn ∈
Γ} = {(N − Xn )(B(0, a)\{0}) = 0}- i? i Bb {N (B(Xn , a)\{Xn }) = 0}X h?2 bmK n∈N 1C (Xn )1Γ (N −Xn ) +QmMib i?2 TQBMib Q7 N HvBM; BM C M/ r?Qb2 M2 `2bi M2B;?#Qm` Bb i /Bbi M+2 ≥ aX qBi? ;2M2` H Γ ∈ M(Rm )- i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb NΓ (C) := E 1C (Xn )1Γ (N − Xn ) (C ∈ B(Rm )) UdXNV n∈N
Bb- BM pB2r Q7 UdX3V- λΓ = λPN0 (N ∈ Γ)X 1t KTH2 dXkXe, ǭ`2M2r HǮ 7Q`KmH X G2i (N, θt , P ) #2 bi iBQM `v bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R rBi? }MBi2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 bbQ+B i2/ S HK T`Q# #BHBivX h?2M- 7Q` HH a ≥ 0 a 2 E N (0, a] = (2R(t) − 1)λ dt , 0
r?2`2 R(t) := E0N [N (0, t]]
(t ≥ 0) .
Ub r2 b? HH bQQM b22- B7 N Bb `2M2r H TQBMi T`Q+2bb- R Bb i?2 `2M2r H 7mM+iBQMXV S`QQ7X "v i?2 T`Q/m+i 7Q`KmH Q7 aiB2HiD2b + H+mHmb N (0, a]2 = 2 N ((0, t]) N (dt) − N (0, a] . (0,a]
h?2`27Q`2
kee
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 E N (0, a]2 = 2E
R+
N (0, t]1(0,a] (t) N (dt) − λa
AM pB2r Q7 UdXdV rBi? v(t) := N (0, t]1(0,a] (t) U M/ i?2`27Q`2 v(t) ◦ θ−t = N (−t, 0]1(0,a] (t)V
0 E N (0, t]1(0,a] N (dt) = EN N (−t, 0]1(0,a] (t)λ dt R+ R+ a E0N [N (−t, 0]] 1(0,a] (t)λ dt = R(t)λ dt . = R+
0
h?2Q`2K dXkXd lM/2` i?2 S HK T`Q# #BHBiv- i?2`2 Bb TQBMi i 0 Ui?2 Q`B;BM Q7 Rm V- i? i BbPN0 (N ({0}) = 1) = 1 . S`QQ7X qBi? g(x) = 1C (x) M/ Γ = {μ ; μ({0}) = 1}- 1[MX UdX3V #2+QK2b- bBM+2 N − Xn ? b Hr vb QM2 TQBMi 2t +iHv i i?2 Q`B;BM E 1C (Xn ) = λ m (C)PN0 (N ({0}) = 1) . n∈N
aBM+2 i?2 H27i@? M/ bB/2 Bb λ m (C)- i?2 `2bmHi Bb T`Qp2/X
1t KTH2 dXkX3, amT2`TQbBiBQM Q7 BM/2T2M/2Mi bi iBQM `v TQBMi T`Q+2bb2bX _2+ HH i? i Sx Bb- 7Q` Mv x ∈ Rm - i?2 i` MbH iBQM #v x TTHB2/ iQ K2 bm`2b μ ∈ M (Rm ), Sx (μ)(C) = μ(C + x) . G2i P #2 T`Q# #BHBiv K2 bm`2 QM (M (Rm ), M(Rm )) bm+? i? i P ◦ Sx = P 7Q` HH x ∈ Rm X h FBM; N 2[m H iQ Φ- i?2 B/2MiBiv K T Q7 M (Rm )- r2 Q#@ i BM bi iBQM `v ` M/QK (Φ, Sx , P)- r?B+? Bb b B/ iQ #2 BM + MQMB+ H 7Q`KX G2i (Mi - Mi - Sx (i) , Φi ) (1 ≤ i ≤ k) #2 `2THB+ b Q7 (M (Rm ), M(Rm ), Sx , Φ) M/ H2i Pi #2 T`Q# #BHBiv QM (Mi , Mi ) r?B+? Bb Sx (i) @BMp `B Mi 7Q` HH x ∈ Rm X amTTQb2 i? i 7Q` HH i U1 ≤ i ≤ kV- Φi Bb Pi @ HKQbi bm`2Hv bBKTH2 TQBMi T`Q+2b rBi? }MBi2 M/ TQbBiBp2 BMi2MbBiv λi X .2}M2 i?2 T`Q/m+i bT +2 k k k Mi , ⊗i=1 Mi , ⊗i=1 Pi (Ω, F, P ) = i=1
M/- 7Q` 2 +? x ∈ Rm - /2}M2 θx := ⊗ki=1 Sx (i) - rBi? i?2 K2 MBM; i? i θx (ω) = (Sx (i) μi ; 1 ≤ i ≤ k)- r?2`2 ω = (μi ; 1 ≤ i ≤ k). .2}M2
dXkX 6A_ah GPPE h SGJ S_P""AGAhu Ni (ω) := μi M/ N (ω) :=
ked
k
μi .
i=1
h?2M (N, θx , P ) Bb bi iBQM `v TQBMi T`Q+2bb- i?2 bmT2`TQbBiBQM Q7 i?2 bi iBQM `v TQBMi T`Q+2bb2b (Ni , θx , P ) (1 ≤ i ≤ k). q2 b? HH /2MQi2 #v Pi0 i?2 S HK T`Q# #BHBiv bbQ+B i2/ iQ (Φi , Pi )X Ai rBHH #2 T`Qp2/ #2HQr i? i PN0 =
k λi i−1 ⊗j=1 Pj ⊗ Pi0 ⊗ ⊗kj=i+1 Pj , λ i=1
UdXRyV
r?2`2 λ = ki=1 λi X Uh?2 BMi2`T`2i iBQM Q7 UdXRyV Bb i?2 7QHHQrBM;X qBi? T`Q# #BHBiv λi i?2 TQBMi i i?2 Q`B;BM BM i?2 S HK p2`bBQM +QK2b 7`QK i?2 i@i? TQBMi T`Q+2bbλ M/ i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 i@i? T`Q+2bb Bb i?2M Bib S HK T`Q# #BHBivr?2`2 b i?2 Qi?2` T`Q+2bb2b F22T i?2B` bi iBQM `v /Bbi`B#miBQMbX HH i?2 k TQBMi T`Q+2bb2b `2K BM BM/2T2M/2MiXV ! "v /2}MBiBQM- 7Q` A = ki=1 Ai - r?2`2 Ai ∈ Mi 1 E λ ...
(1A ◦ θx )N (dx)
PN0 (A) = 1 = λ
M1
(0,1]m
Mk
k k
(0,1]m j=1
k 1 = ... λ M1 Mk j=1
1Ai ◦
Φj (dx)P1 (dμ1 ) . . . Pk (dμk )
Sx(i)
i=1
k
(0,1]m
1Ai ◦
Sx(i)
Φj (dx) P1 (dμ1 ) . . . Pk (dμk ).
i=1
"mi U6m#BMB M/ i?2 /2}MBiBQM Q7 S HK T`Q# #BHBiv Pj0 V 1 λj
k
... M1
(0,1]m i=1
Mk
(1Ai ◦
P1 (dμ1 ) . . . Pk (dμk )
Sx(i) ) Φj (dx)
= Pj0 (Aj )
k
Pi (Ai ),
i=1, i=j
(i)
r?2`2 r2 ? p2 i F2M BMiQ ++QmMi i?2 Sx @BMp `B M+2 Q7 Pi X h?2`27Q`2 PN0
k i=1
Ai
⎫ ⎪ ⎪ ⎬
⎧ ⎪
=
k ⎪ ⎨ i=1
λi 0 Pj (Aj ) , Pi (Ai ) ⎪ ⎪ λ ⎪ ⎪ 1≤j≤k ⎭ ⎩ j=i
r?B+? BKTHB2b UdXRyV- #v h?2Q`2K XRXdX
ke3
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
h?2 * b2 Q7 JQiBQM AMp `B Mi S`Q# #BHBiv G2i (Ω, F) = (Mp (Rm ), Mp (Rm ) M/ H2i N /2MQi2 i?2 + MQMB+ H TQBMi T`Q+2bbX amTTQb2 i? i N ? b HQ+ HHv }MBi2 K2 M K2 bm`2 M/ Bb KQiBQM@BMp `B Mi- i? i Bb7Q` Mv /BbTH +2K2Mi τ - τ N ∈ A M/ N `2 B/2MiB+ HHv /Bbi`B#mi2/X Ai Bb BM T `iB+mH ` i` MbH iBQM BMp `B Mi- M/ i?2 S HK /Bbi`B#miBQM PN0 Bb r2HH /2}M2/X HbQ- i?2 K2 M K2 bm`2 Q7 N Bb Q7 i?2 7Q`K λ m 7Q` }MBi2 MQM@M2; iBp2 λ- bbmK2/ TQbBiBp2 UN Bb MQi 2KTivVX h?2Q`2K dXkXN 6Q` Mv `Qi iBQM r Q7 Rm `QmM/ i?2 Q`B;BM 0PN0 (A) = PN0 (rA)
(A ∈ F ) .
UAM Qi?2` rQ`/b- KQiBQM@BMp `B M+2 Q7 N mM/2` P BKTHB2b `Qi iBQM BMp `B M+2 Q7 PN0 XV S`QQ7X .2MQiBM; #v B(0; r) i?2 +HQb2/ # HH Q7 +2Mi2` 0 M/ ` /Bmb R m 0 λ (B(0; r))PN (A) = 1A (θx ω) N (ω, dx) P (dω) Ω B(0;r) 1A (f (x, ω) N (ω, dx) P (dω) = Ω
B(0;r)
U_2+ HH _2K `F dXRXjV 1A (f (x, r−1 ω) N (r−1 ω, dx) P (dω) = Ω
B(0;r)
U#v `Qi iBQM H BMp `B M+2 Q7 N mM/2` P ) 1A (f (r−1 y, r−1 ω) N (r−1 ω, r−1 dy) P (dω) = Ω
B(0;r)
U+? M;2 Q7 p `B #H2 x = r−1 y) 1rA (f (y, ω) N (ω, dy) P (dω) = Ω
B(0;r)
= λ m (B(0; r))PN0 (rA) . h?BMMBM; M/ *QM/BiBQMBM; G2i (N, θx , P ) #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM Rm rBi? }MBi2 TQbBiBp2 BMi2MbBivX 6Q` U ∈ F - /2}M2 i?2 TQBMi T`Q+2bb NU #v NU (ω, C) := 1U (θx (ω))N (ω, dx), C ∈ B(Rm ). UdXRRV C
am+? TQBMi T`Q+2bb Bb i?2`27Q`2 Q#i BM2/ #v i?BMMBM; Q7 N - TQBMi x ∈ N (ω) #2BM; `2i BM2/ B7 M/ QMHv B7 θx (ω) ∈ U X 1t KTH2 dXkXRy, J `F b2H2+iBQMX G2i (N, Z, θx , P ) #2 bi iBQM `v K `F2/ TQBMi T`Q+2bbX h F2 U = {Z0 ∈ L} 7Q` bQK2 L ∈ KX h?2M- bBM+2 Z0 (θXn (ω)) = Zn (ω)-
dXkX 6A_ah GPPE h SGJ S_P""AGAhu NU (C) =
keN
1L (Zn )1C (Xn ).
n∈N
h?2 TQBMi T`Q+2bb NU Bb Q#i BM2/ #v i?BMMBM; N - QMHv `2i BMBM; i?2 TQBMib Q7 Xn rBi? K `F Zn 7 HHBM; BM LX (NU , θx , P ) Bb Q#pBQmbHv bi iBQM `v TQBMi T`Q+2bb M/ Bi ? b }MBi2 BMi2MbBiv UbBM+2 NU ≤ N VX A7 i?2 BMi2MbBiv Q7 λU Q7 NU Bb TQbBiBp2- Bib S HK T`Q# #BHBiv Bb ;Bp2M #v 1 0 PNU (A) = E (1A ◦ θx )NU (dx) , λU (0,1]m
r?2`2
(1U ◦ θx ) N (dx) = λPN0 (U ) .
λU = E (0,1]m
AM //BiBQM- r2 ? p2
(1A ◦ θx ) NU (dx) = E E (0,1]m
(1A ◦ θx )(1U ◦ θx ) N (dx) = λPN0 (A ∩ U ) . (0,1]m
h?2`27Q`2
PN0 (A ∩ U ) = PN0 (A | U ) . PN0 (U ) LQi2 i? i i?2 b2[m2M+2 Q7 K `Fb +QmH/ i F2 Bib p Hm2b BM M (Rm )- 7Q` BMbi M+2Zn = N − Xn X _2+ HH i? i N − Xn = SXn (N )X h FBM; U := {ω ; , N (ω) ∈ Γ}r?2`2 Γ ∈ M(Rm )- r2 b22 i? i BM i?Bb + b2 NU ≡ NΓ - r?2`2 i?2 H ii2` Bb /2}M2/ #v UdXNVX PN0 U (A) =
1t KTH2 dXkXRR, amT2`TQbBiBQM Q7 bi iBQM `v TQBMi T`Q+2bb2bX h?Bb 2t@ KTH2 ;2M2` HBx2b 1t KTH2 dXkX3X G2i Ni (1 ≤ i ≤ k) #2 TQBMi T`Q+2bb2b QM Rm - HH +QKT iB#H2 rBi? i?2 ~Qr {θx }x∈Rm - rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λi (1 ≤ i ≤ k) `2bT2+iBp2Hv- #mi MQi M2+2bb `BHv BM/2T2M/2MiX * HH N i?2B` bmT2`TQbBiBQM- bbmK2/ bBKTH2X 6`QK " v2bǶb `mH2, PN0 (A) =
k
PN0 (Ni ({0}) = 1) PN0 (A | Ni ({0}) = 1) .
i=1
"mi PN0
1 λi 1 . (Ni ({0}) = 1) = E 1(0,1]m (Xn )1{Ni ({Xn })=1} = E [Ni (0, 1]] = λ λ λ n∈N
G2i U = {Ni ({0}) = 1}X aBM+2 r2 ? p2 NU = Ni UrBi? i?2 MQi iBQM Q7 UdXRRVV- r2 Q#i BM PN0 (A | Ni ({0}) = 1) = PN0 i (A) . h?2`27Q`2 PN0 (A) =
k λi i=1
λ
PN0 i (A) .
kdy
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
dXj S HK h?2Q`v QM i?2 GBM2, " bB+ 6Q`KmH b 6`QK MQr QM BM i?Bb +? Ti2`- ii2MiBQM rBHH #2 `2bi`B+i2/ iQ bBKTH2 bi iBQM `v TQBMi T`Q+2bb2b QM i?2 `2 H HBM2X h?2 MQi iBQM t BMbi2 / Q7 x rBHH 2KT? bBx2 i?2 7 +i i? i QM2 Bb rQ`FBM; QM i?2 `2 H HBM2X h?2 S HK T`Q# #BHBiv PN0 bbQ+B i2/ rBi? i?2 bi iBQM `v TQBMi T`Q+2bb (N, θt , P ) ? b- b r2 b r- Bib K bb +QM+2Mi` i2/ QM Ω0 := {T0 = 0}X _2+ HH i? i i?2 b2[m2M+2 Q7 TQBMib {Tn }n∈Z Bb /2}M2/ BM bm+? r v i? i Bi Bb bi`B+iHv BM+`2 bBM; M/ bm+? i? i T0 ≤ 0 < T1 X T0
T1
T2
t
T3
T−1 ◦θt
T0 ◦θt
0
T1 ◦θt
0
T−2 ◦θt
N
N ◦θt = St N
h?2 K TTBM; θ := θT1 , /2}M2/ 7`QK Ω0 BMiQ Ω0 - Bb #BD2+iBQM- rBi? BMp2`b2 θ−1 = θT−1 X HbQ- QM Ω0 θTn := θn 7Q` HH n ∈ ZXj h?Bb Bb MQi i`m2 QM Ω U7Q` BMbi M+2- i?2 BMp2`b2 Q7 θT1 Bb MQi θT−1 c 1t2`+Bb2 dX3X9VX h?2 T`QT2`iv BM i?2 M2ti i?2Q`2K Bb `272``2/ iQ b i?2 dz2p2Mi@iBK2 bi iBQM `BivǴX h?2Q`2K dXjXR PN0 Bb θ@BMp `B Mi. S`QQ7X 6B`bi Q#b2`p2 i? i 7Q` HH A ∈ F 1{θ−1 (A)◦θTn } = 1{θTn ∈θ−1 (A)} = 1{θTn+1 ∈A} = 1A (θTn+1 ) . LQr- rBi? A ∈ F M/ C = (0, t]- 7Q`KmH UdXjV ;Bp2b, & & & & & & 0 1 & 0 −1 &PN (A) − PN (θ (A))& ≤ E & (1A ◦ θTn − 1θ−1 (A) ◦ θTn )1(0,t] (Tn )&& & λt & n∈Z & & & 1 && 2 & = E & (1A ◦ θTn − 1A ◦ θTn+1 )1(0,t] (Tn )& ≤ . & λt λt & n∈Z G2iiBM; t → ∞- r2 Q#i BM PN0 (A) = PN0 (θ−1 (A))X
AM T `iB+mH `- B7 {Z(t)}t∈R Bb θt @+QKT iB#H2 M/ i?2`27Q`2 bi iBQM `v mM/2` P i?2 b2[m2M+2 {Z(Tn )}n∈Z Bb- mM/2` PN0 - bi iBQM `v b2[m2M+2X j Ai Bb T2`? Tb rQ`i?r?BH2 iQ 2KT? bBx2 i?2 7 +i i? i θTn Bb dzi?2 b?B7i i? i KQp2b i?2 n@i? TQBMi Q7 TQBMi T`Q+2bb iQ i?2 Q`B;BMǴX
dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa
kdR
1t KTH2 dXjXk, S HKėE?BM+?BM 2[m iBQMbX G2i 7Q` k ∈ N M/ t ≥ 0ϕk (t) := PN0 (N (0, t] = k) . q2 ? p2 i?2 S HKĜE?BM+?BM 2[m iBQMb,9
t
P (N (0, t] > k) = λ
ϕk (s) ds. 0
hQ T`Qp2 i?Bb- Q#b2`p2 i? i 1{N (s,t]=k} N (ds),
1N (0,t]>k = (0,t]
M/ /2/m+2 7`QK i?Bb M/ N (s, t] = N (0, t − s] ◦ θs i? i 1(0,t] (s) 1{N (0,t−s]=k} ◦ θs N (ds). 1{N (0,t]>k} = R
"v i?2 J2+F2 7Q`KmH - i?2 2tT2+i iBQM Q7 i?2 `B;?i@? M/ bB/2 rBi? `2bT2+i iQ P Bb 2[m H iQ t λ 1(0,t] (s)PN0 (N (0, t − s] = k) ds = λ PN0 (N (0, t − s] = k) ds R 0 t = λ PN0 (N (0, s] = k) ds. 0
h?2Q`2K dXjXj G2i (N, θt , P ) #2 bi iBQM `v TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i? i P (N (R) = 0) = 0X h?2M λE0N [T1 ] = 1 . S`QQ7X 6`QK i?2 S HKĜE?BM+?BM 2[m iBQM rBi? k = 0 t ϕ0 (s) ds . P (N (0, t] = 0) = 1 − λ 0
"mi ϕ0 (s) = PN0 (N (0, s] = 0) = P (T1 > s)- M/ i?2`27Q`2 0 = P (N (R) = 0) = lim P (N (0, t] = 0) t↑+∞ ∞ PN0 (T1 > s) ds = 1 − λE0N [T1 ] . =1−λ 0
9
(S HK- RN9j) 7Q` k = 0- (E?BM+?BM- RNey) 7Q` k ≥ 1X
kdk
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
AMp2`bBQM 6Q`KmH b >Qr iQ T bb 7`QK i?2 S HK T`Q# #BHBiv iQ i?2 bi iBQM `v /Bbi`B#miBQM\ h?2 7Q`KmH b /QBM; i?Bb `2 + HH2/ BMp2`bBQM 7Q`KmH bX h?2Q`2K dXjX9 G2i (N, P, θt ) #2 bi iBQM `v bBKTH2 TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i? i P (N (R) = 0) = 0X 6Q` Mv MQM@M2; iBp2 ` M/QK p `B #H2 f
T1
E [f ] = E0N
(f ◦ θs )ds .
0
T`QQ7 rBHH #2 ;Bp2M pB i?2 7QHHQrBM; +QMb2`p iBQM T`BM+BTH2- r?B+? b vb i? i dzi?2 bKQQi? p `B iBQM Q7 bi iBQM `v biQ+? biB+ T`Q+2bb Bb # H M+2/ #v i?2 p `B iBQM /m2 iQ DmKTbǴX JQ`2 T`2+Bb2Hv, h?2Q`2K dXjX8 U8 V G2i (N, P, θt ) #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞X G2i {Y (t)}t∈R #2 `2 H@p Hm2/ biQ+? biB+ T`Q+2bb- `B;?i@ +QMiBMmQmb rBi? H27i@? M/ HBKBib- M/ H2i {Y (t)}t∈R #2 `2 H@p Hm2/ biQ+? biB+ T`Q+2bb bm+? i? i 1 Y (1) − Y (0) = Y (s)ds + (Y (s) − Y (s−))N (ds). UdXRkV 0
(0,1]
amTTQb2 i? i i?2 T`Q+2bb2b Y M/ Y `2 θt @+QKT iB#H2X amTTQb2 KQ`2Qp2` i? i E[|Y (0)|] < ∞ M/ E0N [|Y (0) − Y (0−)|] < ∞ .
UdXRjV
E[Y (0)] + λE0N [Y (0) − Y (0−)] = 0 .
UdXR9V
h?2M
6Q` i?2 T`QQ7- r2 b? HH M22/ i?2 7QHHQrBM; bBKTH2 H2KK , G2KK dXjXe G2i (P 0 , θ) #2 bi iBQM `v 7` K2rQ`FX G2i Z #2 MQM@M2; iBp2- P 0 @ XbX }MBi2 ` M/QK p `B #H2 bm+? i? i Z − Z ◦ θ ∈ L1 (P 0 )X h?2M E0 [Z − Z ◦ θ] = 0X S`QQ7X 6Q` Mv C > 0 |Z ∧C −(Z ∧C)◦θ| ≤ |Z −Z ◦θ|X h?2M- #v i?2 θ@BMp `B M+2 Q7 P 0 - E[Z ∧C −(Z ∧C)◦θ] = 0 M/ i?2 +QM+HmbBQM 7QHHQrb #v /QKBM i2/ +QMp2`;2M+2H2iiBM; C ↑ ∞ BM i?2 H bi 2[m HBivX ULQi2 i? i i?2 /2HB+ i2 TQBMi ?2`2 Bb i? i r2 /Q MQi bbmK2 i? i Z Bb BMi2;` #H2XV S`QQ7X P#b2`p2 i? i 1 E[ |Y (s)| ds] = E[|Y (0)|] M/ 0 E[ |Y (s) − Y (s−)| N (ds)] = λE0N [|Y (0) − Y (0−)|] . (0,1] 8
(JBv x r - RNN9)X
dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa
kdj
h?2`27Q`2- +QM/BiBQM UdXRjV ;m ` Mi22b i? i Y (1) − Y (0) Bb S@BMi2;` #H2 M/ #v G2KK dXjXe- E [Y (1) − Y (0] = 0X 1[m iBM; i?Bb iQ i?2 2tT2+i iBQM Q7 i?2 `B;?i@ 1 ? M/ bB/2 Q7 UdXRkVr2 Q#i BM i?2 MMQmM+2/ `2bmHi bBM+2 E[ Y (s) ds] = E[Y (0)] 0 0 M/ E[ (0,1] (Y (s) − Y (s−)) N (ds)] = λEN [Y (0) − Y (0−)]X _2K `F dXjXd G2i mb K F2 KQ`2 T`2+Bb2 i?2 BMi2`T`2i iBQM Q7 i?2 +QMb2`p iBQM T`BM+BTH2, E[Y (0)] Bb i?2 p2` ;2 ` i2 Q7 BM+`2 b2 UT2` mMBi iBK2V /m2 iQ i?2 bKQQi? 2pQHmiBQM Q7 i?2 biQ+? biB+ T`Q+2bb Y #2ir22M DmKTb- r?2`2 b λE0N [Y (0) − Y (0−)] Bb i?2 p2` ;2 ` i2 Q7 BM+`2 b2 /m2 iQ i?2 DmKTb U i /Bb+QMiBMmBiv iBK2bVX h?2`27Q`2 1[MX UdXR9V bi i2b i? i i?2 iQi H p2` ;2 ` i2 Q7 BM+`2 b2 Bb MmHH- r?B+? Bb 2tT2+i2/ bBM+2 i?2 T`Q+2bb Bb bi iBQM `vX AM i?Bb b2Mb2- UdXR9V Bb +QMb2`p iBQM 2[m iBQMX q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K dXjX9X S`QQ7X Ai Bb 2MQm;? iQ T`Qp2 i?2 `2bmHi 7Q` #QmM/2/ f X G2i T+ (t) Y (t) := (f ◦ θs ) ds, t
r?2`2 T+ (t) = inf{Tn ; Tn > t}VX h?Bb T`Q+2bb b iBb}2b i?2 +QM/BiBQMb Q7 h?2Q`2K T dXjX8 rBi? Y (0) = −f M/ Y (0) − Y (0−) = 0 1 (f ◦ θs ) ds Umb2 i?2 7 +i i? i E0N [T1 ] = λ−1 < ∞VX "v i?2 θ@BMp `B M+2 Q7 P E [f ] = E0N
Tn+1
(f ◦ θs ) ds ,
UdXR8V
Tn
M/ BM T `iB+mH `- 7Q` f = 1A +∞ P (A) = PN0 (Tn < s ≤ Tn+1 , θs ∈ A) ds . −∞
UdXReV
M AMi2`T`2i iBQM Q7 i?2 AMp2`bBQM 6Q`KmH h?2 BMp2`bBQM 7Q`KmH `2+2Bp2b M BMi2`2biBM; BMi2`T`2i iBQM r?2M r`Bii2M b
T1 1 (f ◦ θt ) dt = E0N [λT1 (f ◦ θV )] , E[f ] = E0N λT1 T1 0 r?2`2 V Bb ` M/QK p `B #H2 r?B+?- dz+QM/BiBQM HHv QM 2p2`vi?BM; 2Hb2Ǵ- Bb mMB@ 7Q`KHv /Bbi`B#mi2/ QM [0, T1 ] U7Q` i?2 #Qp2 iQ K F2 b2Mb2- r2 Kmbi Q7 +Qm`b2 2M@ H `;2 i?2 T`Q# #BHBiv bT +2VX h?Bb BMi2`T`2i iBQM T`QpB/2b M 2tTHB+Bi +QMbi`m+iBQM Q7 P 7`QK PN0 - b 7QHHQrbX 6B`bi +QMbi`m+i i?2 T`Q# #BHBiv P0 #v dP0 = (λT1 ) dPN0 . aBM+2 PN0 (T0 = 0) = 1 M/ P0 Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ PN0 P0 (T0 = 0) = 1X h?2 bi iBQM `v T`Q# #BHBiv P Bb i?2M Q#i BM2/ #v TH +BM; i?2 Q`B;BM i ` M/QK BM i?2 BMi2`p H [0, T1 ]- i? i Bb-
kd9
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 E[f ] = E0 [f ◦ θV ].
1t KTH2 dXjX3, aBKmH iBQM H;Q`Bi?KX M TTHB+ iBQM Q7 i?2 #Qp2 `2bmHi Bb iQ i?2 2t +i b KTHBM; Q7 i?2 bi iBQM `v T`Q# #BHBiv P r?2M b KTH2 Q7 i?2 S HK T`Q# #BHBiv PN0 Bb FMQrMX h?Bb + M #2 /QM2 BM i?2 + b2 i? i PN0 (T1 ≤ a) = 1 7Q` bQK2 a < ∞Xe 6Q` i?Bb- 2t KBM2 bm++2bbBp2Hv i?2 BMi2`@2p2Mi iBK2b Sn := Tn − Tn−1 Un ≥ 1V Q7 i?2 PN0 b KTH2 U7Q` r?B+? r2 `2+ HH i? i T0 = 0V mMiBH i?2 2p2Mi Un ≤ Sn Bb `2 HBx2/- r?2`2 {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv a /Bbi`B#mi2/ QM [0, 1]X G2i M #2 i?2 ` M/QK BM/2t r?2M Bi Bb `2 HBx2/X h?2M i?2 TQBMi T`Q+2bb N 7i2` TM −1 Bb `2 HBx iBQM Q7 P0 X A7 U Bb MQi?2` ` M/QK p `B #H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] M/ BM/2T2M/2Mi Q7 2p2`vi?BM; 2Hb2- i?2 TQBMi T`Q+2bb N 7i2` TM −1 + U SM Bb b KTH2 Q7 P X h?2 /2i BHb Q7 i?2 T`QQ7 `2 `2[mB`2/ BM 1t2`+Bb2 dX3XdX
1t KTH2 dXjXN, J2 M p Hm2 7Q`KmH bX G2i (N, θt , P ) #2 bi iBQM `v bBKTH2 TQBMi T`Q+2bb QM R rBi? }MBi2 TQbBiBp2 BMi2MbBiv M/ bbQ+B i2/ S HK T`Q# #BHBiv PN0 X G2i {Zt }t∈R #2 θt @+QKT iB#H2 biQ+? biB+ T`Q+2bb rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (K, K)X h?2 7QHHQrBM; 7Q`KmH b `2 irQ T `iB+mH ` 7Q`Kb Q7 i?2 BMp2`bBQM 7Q`KmH M/ Q7 i?2 /2}MBiBQM Q7 S HK T`Q# #BHBivX 6Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : (K, K) → (R, B)
E [g(Z0 )] = M/
E0N g(Z0 ) =
E0N
T1 0
g(Zt )dt
E0N [T1 ]
UdXRdV
E
g(ZTn )1{Tn ∈(0,1]} . E n∈Z 1{Tn ∈(0,1]} n∈Z
UdXR3V
" +Fr `/ M/ 6Q`r `/ _2+m``2M+2 hBK2b h?2Q`2K dXjXRy G2i (N, P, θt ) #2 bi iBQM `v bBKTH2 TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i? i P (N (R) = 0) = 0- M/ rBi? bbQ+B i2/ S HK T`Q# #BHBiv PN0 X G2i F0 #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 T1 mM/2` PN0 , F0 (x) = PN0 (T1 ≤ x). h?2M- 7Q` HH v, w ∈ R+ P (T1 > v, −T0 > w) = λ
∞ v+w
e
(h?Q`BbbQM- kyyy)- b2+iBQM eX9- *? Ti2` 3X
(1 − F0 (u))du .
UdXRNV
dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa
kd8
S`QQ7X G2i u, v ∈ R+ X qBi? A := {T1 > v, −T0 > w} BM UdXReV r`Bii2M 7Q` n = 0 M/ i FBM; BMiQ ++QmMi i? i PN0 @ XbX- −T0 ◦ θt = t M/ T1 ◦ θt = T1 − t 7Q` HH t ∈ [0, T1 )- r2 Q#i BM P (T1 > v, −T0 > w) =
λE0N
= λE0N
T1
1{t>v} 1{T1 −t>w} dt (T1 − (v + w))+ . 0
h?2 `2bmHi i?2M 7QHHQrb 7`QK i?2 7QHHQrBM; +QKTmi iBQM Up HB/ 7Q` Mv MQM@M2; iBp2 ` M/QK p `B #H2 X M/ MQM@M2; iBp2 MmK#2` aV, ∞ E[(X − a)+ ] = P ((X − a)+ > x) dx 0 ∞ ∞ = P (X − a > x) dx = P (X > x) dx . 0
a
λ
h FBM; v = 0 M/ w = 0- r2 Q#i BM i? i P (T1 > 0, −T0 > 0) = 1 bBM+2 ∞ (1 − F0 (u))du = λE0N [T1 ] = 1X AM T `iB+mH ` 0
h?2Q`2K dXjXRR lM/2` P - i?2`2 Bb MQ TQBMi Q7 N i i?2 Q`B;BM Q7 iBK2X h FBM; MQr v = 0- r2 Q#i BM
∞
P (−T0 > w) = λ
(1 − F0 (u)) du.
w
aBKBH `Hv-
∞
P (T1 > v) = λ
(1 − F0 (u)) du.
v
h?mb −T0 M/ T1 `2 B/2MiB+ HHv /Bbi`B#mi2/ mM/2` P U#mi MQi BM/2T2M/2Mi BM ;2M2` HVX h?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM QM R+ x F (x) = λ (1 − F0 (u))du 0
Bb i?2 2t+2bb /Bbi`B#miBQM Q7 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F0 X h?2 1t+? M;2 6Q`KmH G2i (N, θt , P ) M/ (N , θt , P ) #2 irQ bi iBQM `v TQBMi T`Q+2bb2b rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λ M/ λ `2bT2+iBp2HvX LQi2 i? i N M/ N `2 DQBMiHv bi iBQM `v- BM i?2 b2Mb2 i? i i?2B` bi iBQM `Biv Bb `2H iBp2 iQ i?2 b K2 [m /`mTH2 (Ω, F, P, θt )X h?2 7QHHQrBM; `2bmHi HBMFb i?2B` S HK /Bbi`B#miBQMb,d d
(L2p2m- RNe3 M/ RNdd)X
kde
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
h?2Q`2K dXjXRk λE0N [f ]
=λ
0 EEN
(0,T1 ]
(f ◦ θt )N (dt)
UdXkyV
7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb f : (Ω, F) → (R, B(R))X S`QQ7X "v i?2 KQMQiQM2 +QMp2`;2M+2 i?2Q`2K- r2 K v bbmK2 f #QmM/2/ Ub v #v RVX qBi? bm+? f - bbQ+B i2 i?2 7mM+iBQM g := (f ◦ θt )N (dt). (T0 ,T1 ]
6Q` HH t ∈ R+ - r2 ? p2 (f ◦ θs ) N (ds) = (0,t]
(g ◦ θs ) N (ds) + R(t), (0,t]
r?2`2- /2MQiBM; #v T+ (t) i?2 }`bi TQBMi Q7 N bi`B+iHv 7i2` t (f ◦ θs )N (ds) − (f ◦ θs )N (ds). R(t) = (0)] (0,T+
(t)] (t,T+
6Q` HH > 0- H2i f := f 1{N (T0 ,T1 ]≤−1}
M/ g :=
(T0 ,T1 ]
(f ◦ θt ) N (dt) .
6Q` i?2b2 7mM+iBQMb- 2 +? i2`K BM R(t) Bb #QmM/2/ #v - M/ i?2`27Q`2 i?2 2tT2+i @ iBQMb `2 }MBi2X JQ`2Qp2`- #v θt @BMp `B M+2 Q7 P - i?2v ? p2 i?2 b K2 2tT2+i iBQMbbQ i? i E[R(t)] = 0X q2 i?2`27Q`2 ? p2
E (f ◦ θs ) N (ds) = E (g ◦ θs ) N (ds) . (0,t]
(0,t]
h?Bb T`Qp2b UdXkyV rBi? f = f , g = g X G2iiBM; ;Q iQ BM}MBiv vB2H/b i?2 MMQmM+2/ 2[m HBivX
dX9 6`QK S HK S`Q# #BHBiv iQ ai iBQM `v S`Q# #BHBiv h?Bb b2+iBQM3 b?Qrb ?Qr iQ +QMbi`m+i P 7`QK PN0 X G2i {θt }t∈R #2 K2 bm` #H2 ~Qr QM (Ω, F)- M/ H2i N #2 TQBMi T`Q+2bb +QKT iB#H2 rBi? i?Bb ~QrX G2i P 0 #2 T`Q# #BHBiv K2 bm`2 QM (Ω, F) i? i Bb U V +QM+2Mi` i2/ QM Ω0 = {T0 = 0}- i? i Bb- bm+? i? i P 0 (Ω0 ) = 1- M/ U#V θTn @BMp `B Mi- i? i Bb- bm+? i? i P 0 (θTn ∈ .) = P 0 (.) 7Q` HH n ∈ ZX 3
(_vHH@L `/x2rbFB- RNeR)- (aHBpMv F- RNek)X
dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu
kdd
h?2Q`2K dX9XR bbmK2 KQ`2Qp2` i? i i?2 7QHHQrBM; i?`22 T`QT2`iB2b `2 b iBb}2/, 0 < E0 [T1 ] < ∞, P 0 [T1 > 0] = 1, E0 [N (0, t0 ]] < ∞ 7Q` bQK2 t0 > 0 .
UBV UBBV UBBBV
UdXkRV
h?2M P 0 Bb i?2 S HK T`Q# #BHBiv PN0 bbQ+B i2/ rBi? (N, P ) r?2`2 P Bb i?2 θt @ BMp `B Mi T`Q# #BHBiv ;Bp2M #v
T1 1 0 P (A) = 0 E (1A ◦ θt ) dt , (A ∈ B(R)) . UdXkkV E [T1 ] 0 S`QQ7X Ai Kmbi #2 }`bi p2`B}2/ i? i P Bb θt @BMp `B Mi 7Q` HH t ∈ R- M/ i?2M i? i PN0 = P 0 X PM Ω0 - 7Q` HH j ∈ Z
T1 ◦θTj T0 ◦θTj
Tj+1
(1A ◦ θs ◦ θTj ) ds =
(1A ◦ θs ) ds,
Tj
M/ i?2`27Q`2- bBM+2 P 0 Bb θTn @BMp `B MiP (A) =
1 1 E0 n E0 [T1 ]
Tn
(1A ◦ θs )ds .
0
HbQ- 7Q` HH t ∈ RP (θt (A)) =
1 1 E0 n E0 [T1 ]
Tn 0
Tn −t 1 1 0 E (1θt (A) ◦ θs )ds = (1 ◦ θ )ds . A s n E0 [T1 ] −t
h?2`27Q`2 |P (A) − P (θt (A))| ≤
0 & & Tn 1 1 2t & 0& E . (1 ◦ θ ) ds − (1 ◦ θ ) ds &≤ & A s A s n E0 [T1 ] nE0 [T1 ] −t Tn −t
G2iiBM; n ↑ ∞- Bi 7QHHQrb i? i P (A) = P (θt (A))X q2 MQr T`Qp2 i? i i?2 BMi2MbBiv Q7 N Bb (E0 [T1 ])−1 X 6`QK UdXkkV- r2 Q#i BM T1
1 E[f ] = 0 E0 (f ◦ θt ) dt , E [T1 ] 0 7Q` HH MQM@M2; iBp2 f X AM T `iB+mH `- 7Q` HH ε > 0
T1 E[N (0, ε]] 1 1 0 N (t, t + ε] dt . = 0 E ε E [T1 ] ε 0 "v bbmKTiBQM UBBV1 ε→0 ε
T1
N (t, t + ε] dt = 1 .
lim
0
h?2`27Q`2- B7 r2 + M BMi2`+? M;2 HBKBi M/ 2tT2+i iBQM-
kd3
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 1 lim E0 ε→0 ε
bQ i? i lim
ε→0
T1
N (t, t + ε] dt = 1 ,
0
E[N (0, ε]] 1 = 0 . ε E [T1 ]
h?2 BMi2`+? M;2 Bb DmbiB}2/ #v /QKBM i2/ +QMp2`;2M+2 BM pB2r Q7 i?2 #QmM/ 1 T1 N ((t, t + ε])dt ≤ 1 + N ((T1 , T1 + ε]) ε 0 M/ Q7 bbmKTiBQM UBBBVX aBM+2 E[Nε(0,ε]] = λ- r2 Q#i BM λ = E01[T1 ] M/ i?2`27Q`2- BM pB2r Q7 UBV- 0 < λ < ∞X h?2`27Q`2- r2 K v /2}M2 i?2 S HK T`Q# #BHBiv PN0 bbQ+B i2/ rBi? (N, θt , P )X q2 b? HH +QM+Hm/2 #v T`QpBM; i? i P 0 = PN0 X "v i?2 BMp2`bBQM 7Q`KmH - 7Q` HH A ∈ F
T1 T1 0 0 (1θ−1 A ◦ θt )dt = λEN 1A dt = λE0N [T1 1A ] , () P (θT0 ∈ A) = λEN 0
T0
0
bBM+2 QM Ω - θT0 ◦ θt Bb i?2 B/2MiBiv 7Q` HH t ∈ (0, T1 ]X 6`QK i?2 /2}MBiBQM Q7 P BM i2`Kb Q7 P 0 , 1 P (θT0 ∈ A) = 0 E0 [T1 1A ]. E [T1 ] 0
h?2`27Q`2- 7Q` HH A ∈ F - E0 [T1 1A ] = E0N [T1 1A ]X 6`QK i?Bb- r2 Q#i BM i? i 7Q` HH MQM@M2; iBp2 ` M/QK p `B #H2b f - E0 [T1 f ] = E0N [T1 f ]X aBM+2 T1 > 0 #v +QMbi`m+iBQMr2 + M i F2 f = T1−1 1A iQ Q#i BM P 0 (A) = PN0 (A)X _2K `F dX9Xk q2 b? HH bBM;H2 Qmi 2[m HBiv UV #2+ mb2 Bi b?Qrb v2i MQi?2` +QM@ M2+iBQM #2ir22M i?2 bi iBQM `v T`Q# #BHBiv M/ i?2 S HK T`Q# #BHBiv, 7Q` HH A ∈ F P (θT−1 A) = λE0N [T1 1A ] 0
(A ∈ F) .
1t KTH2 dX9Xj, ai iBQM `Bx iBQM Q7 _2M2r H S`Q+2bbX h?2 `2bmHi #2@ HQr- r?B+? ? b H`2 /v #22M T`Qp2/ BM i?2 +? Ti2` QM `2M2r H i?2Q`v- rBHH #2 `2pBb@ Bi2/ BM i?2 HB;?i Q7 S HK i?2Q`vX amTTQb2 i? i mM/2` PN0 - i?2 BMi2`@2p2Mi b2[m2M+2 {Sn }n∈Z /2}M2/ #v Sn = Tn+1 − Tn Bb BB/ rBi? }MBi2 K2 MX h?2M (N, PN0 ) Bb M mM/2H v2/ `2M2r H T`Q+2bb M/ (N, P ) bi iBQM `v `2M2r H T`Q+2bbX h?2 2tBbi2M+2 Q7 P Bb ;` Mi2/ #v i?2 BMp2`b2 +QMbi`m+iBQMX q2 MQr b?Qr i? i, U V h?2 /Bbi`B#miBQM Q7 i?2 b2[m2M+2 S ∗ = {Sn }n∈Z\{0} Bb i?2 b K2 mM/2` P M/ PN0 X U#V S0 M/ S ∗ `2 P @BM/2T2M/2MiX S`QQ7X U V Ai bm{+2b iQ b?Qr i? i E [g(S ∗ )] = E0N [g(S ∗ )]
dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu
kdN
7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : (RZ , (B(R)⊗Z ) → (R, B(R))X "v i?2 BMp2`bBQM 7Q`KmH T1
∗ 0 ∗ E [g(S )] = λEN g(S (θu ))du . 0
"mi B7 u Bb BM [0, T1 )- i?2M S ∗ (θu ) = S ∗ - bQ i? i
T1
E [g(S ∗ )] = λE0N
g(S ∗ )du = λE0N [T1 g(S ∗ )]
0
= λE0N [T1 ] E0N [g(S ∗ )] = E0N [g(S ∗ )] , r?2`2 r2 ? p2 mb2/ i?2 BM/2T2M/2M+2 Q7 T1 = S0 M/ S ∗ mM/2` PN0 X U#V aBKBH ` +QMbB/2` iBQMb ;Bp2 E [f (S0 )g(S ∗ )] = λE0N [f (S0 )T1 g(S ∗ )] = λE0N [f (S0 )T1 ] E0N [g(S ∗ )] = λE0N [f (S0 )T1 ] E [g(S ∗ )] = E [f (S0 )] E [g(S ∗ )] .
_2K `F dX9X9 h?2 7Q`KmH E[f (S0 )] = E0N [λS0 f (S0 )] Bb i`m2 7Q` ;2M2` H bi iBQM `v TQBMi T`Q+2bbX Ai bi i2b i? i QM i?2 σ@}2H/ ;2M2` i2/ #v S0 - P Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ PN0 - rBi? i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 λS0 X AM T `iB+mH ` P (−T0 + T1 ≤ x) = λyF0 (dy), [0,x]
r?2`2 F0 (x) = PN0 (T1 ≤ x)X 1t KTH2 dX9X8, ai iBQM `Bx iBQM Q7 b2KB@J `FQp T`Q+2bb2bX h?2 T`2pB@ Qmb 2t KTH2 + M #2 ;2M2` HBx2/ iQ b2KB@J `FQp T`Q+2bb2b QM /2MmK2` #H2 bi i2 bT +2 E Ub22 a2+iBQM 9Xe 7Q` /2}MBiBQMVX am+? biQ+? biB+ T`Q+2bb Bb +QMbi`m+i2/ b 7QHHQrbX G2i P = {pij }i,j∈E - #2 biQ+? biB+ K i`Bt QM E- bbmK2/ B``2/m+B#H2 M/ TQbBiBp2 `2+m``2Mi rBi? bi iBQM `v /Bbi`B#miBQM πX 6Q` 2 +? i, j ∈ E- H2i Gij #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 bQK2 bi`B+iHv TQbBiBp2 M/ T`QT2` ` M/QK p `B #H2 rBi? K2 M ∞ ∞ mij = tGij (dt) = (1 − Gij (t))dt < ∞ . 0
0
_2+ HH i i?Bb bi ;2 i? i B7 U Bb ` M/QK p `B #H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] , G−1 ij (U ) Bb ` M/QK p `B #H2 rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Gij U?2`2 G−1 ij Bb i?2 BMp2`b2 Q7 Gij VX
k3y
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
G2i {Xn }n∈Z - #2 bi iBQM `v J `FQp +? BM rBi? i` MbBiBQM K i`Bt P- /2}M2/ QM bQK2 T`Q# #BHBiv bT +2 rBi? T`Q# #BHBiv P 0 - M/ H2i {Un }n∈Z - #2 b2[m2M+2 Q7 BB/ ` M/QK p `B #H2b- /2}M2/ QM i?2 b K2 bT +2 M/ mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X bbmK2 KQ`2Qp2` i? i i?2 b2[m2M+2b {Un }n∈Z M/ {Xn }n∈Z `2 BM/2T2M/2Mi mM/2` P 0 X .2}M2 Sn = G−1 Xn Xn+1 (Un ). AM T `iB+mH `- +QM/BiBQM HHv QM Xn = i M/ Xn+1 = j- Sn ? b i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Gij X JQ`2Qp2`- +QM/BiBQM HHv QM i?2 r?QH2 b2[m2M+2 {Xn }n∈Z i?2 b2[m2M+2 {Sn }n∈Z 7Q`Kb M BM/2T2M/2Mi 7 KBHv Q7 ` M/QK p `B #H2bX q2 + M MQr /2}M2 TQBMi T`Q+2bb N #v T0 = 0 , Tn+1 − Tn = Sn
(n ∈ Z)
M/ i?2 b2KB@J `FQp T`Q+2bb {X(t)}t∈R #v X(t) = Xn , if Tn ≤ t < Tn+1
(t ∈ R) .
LQi2 i?2 #b2M+2 Q7 2tTHQbBQM Ui? i Bb- limn→∞ Tn = ∞ M/ limn→−∞ Tn = −∞ HKQbi bm`2HvV bBM+2 {Xn }n∈Z Bb `2+m``2Mi ?K+X Ai rBHH #2 bbmK2/ i? i E0 [T1 ] = π(i) pij mij < ∞ i∈E
M/ i? i
j∈E
E0 [N (0, t]] < ∞,
(t ∈ R+ ).
G2i (Ω, F) #2 i?2 + MQMB+ H bT +2 Q7 +QMiBMmQmb 7mM+iBQMb rBi? HBKBib QM i?2 H27i i FBM; i?2B` p Hm2b BM EX G2i {θt }t∈R #2 i?2 mbm H i` MbH iBQM ~Qr +iBM; QM 7mM+@ iBQMb- M/ H2i P 0 = P 0 ◦ X −1 X PM i?Bb + MQMB+ H 7` K2rQ`F- N Bb θt @+QKT iB#H2 K `F2/ TQBMi T`Q+2bb- i?2 K `F Q7 Tn #2BM; (Xn , Un ) rBi? Xn = X(Tn )X G2i Sn = GXn ,Xn+1 (Un )X lM/2` i?2 #Qp2 bbmKTiBQMb- HH i?2 +QM/BiBQMb Q7 i?2 BM@ p2`b2 +QMbi`m+iBQM `2 b iBb}2/- bQ i? i i?2`2 2tBbib θt @BMp `B Mi T`Q# #BHBiv P QM (Ω, F) 7Q` r?B+? N Bb bi iBQM `v TQBMi T`Q+2bb M/ bm+? i? i PN0 = P 0 X h?2 T`Q# #BHBbiB+ bi`m+im`2 Q7 {X(t)}t∈R - mM/2` i?2 bi iBQM `v T`Q# #BHBiv P Bb i?2 7QHHQrBM;, U V *QM/BiBQM HHv QM X1 = j- i?2 b2[m2M+2 S1 , X2 , S2 , X3 , S3 , . . . ? b i?2 b K2 /Bbi`B#miBQM mM/2` P Q` PN0 X U#V *QM/BiBQM HHv QM X0 = i- i?2 b2[m2M+2 X−1 , S−1 , X−2 , S−2 , . . . ? b i?2 b K2 /Bbi`B#miBQM mM/2` P M/ PN0 X U+V *QM/BiBQM HHv QM X0 = i, X1 = j, −T0 > x, T1 > y- i?2 b2[m2M+2b BM U V M/ U#V `2 BM/2T2M/2MiX U/V JQ`2Qp2` P (X0 = i , X1 = j , −T0 > x , T1 > y)
∞
= λπ(i)pij
(1 − Gij (t)) dt.
x+y
UdXkjV
dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu
k3R
S`QQ7X h?2 T`QQ7 Q7 U VĜU+V Bb bBKBH ` iQ i? i BM 1t KTH2 dX9Xj 7Q` `2M2r H T`Q@ +2bb2bX 6Q` BMbi M+2- 7Q` n ≥ 1 T1
0 1{Xn0 θs =i , Xn+10 θs =j} ds P (Xn = i, Xn+1 = j) = λEN 0
T1 0 = λEN 1{Xn =i , Xn+1 =j} ds , 0
bBM+2 s ∈ (0, T1 ) BKTHB2b Xn0 θs = Xn 7Q` HH n ∈ ZX h?2`27Q`2 P (Xn = i , Xn+1 = j) = λE0N T1 1{Xn =i} 1{Xn+1 =j} = λE0N G−1 X0 X1 (U0 )1{Xn =i} 1{Xn+1 =j} = λE0N G−1 (U0 )1{X =i} pij X0 X1 n 0 = λEN T1 1{Xn =i} pij , r?2`2 r2 ? p2 mb2/ i?2 ?vTQi?2bBb n ≥ 1- ;m ` Mi22BM; i? i Xn+1 Bb +QM/BiBQM HHv BM/2T2M/2Mi Q7 X0 , X1 , U0 ;Bp2M Xn = iX aBKBH `Hv UbmKKBM; mT i?2 H bi 2[m HBiv BM jV P (Xn = i) = λE0N T1 1{Xn =i} M/ i?2`27Q`2- 7Q` n ≥ 1P (Xn+1 = j|Xn = i) = pij . JQ`2 ;2M2` HHv- Bi + M #2 b?QrM rBi? i?2 b K2 ivT2 Q7 + H+mH iBQMb i? i {Xn }n∈Z Bb mM/2` P J `FQp +? BM rBi? i` MbBiBQM K i`Bt PX HbQ- ; BM rBi? i?2 b K2 T`QQ7- mM/2` P - U1 , U2 , . . . `2 BB/ ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] M/ BM/2T2M/2Mi Q7 X1 , X2 , . . .- M/ i?Bb T`Qp2b U VX b 7Q` U#V- Bi bm{+2b iQ `2p2`b2 iBK2- M/ iQ Q#b2`p2 i? i mM/2` PN0 - i?2 b2@ [m2M+2 {X−n }n∈Z Bb HbQ bi iBQM `v M/ 2`;Q/B+ J `FQp +? BM- i?Bb iBK2 rBi? i?2 i` MbBiBQM K i`Bt Q = {qij }i,j∈E ;Bp2M #v qij = pji
π(j) . π(i)
h?2 T`QQ7 Q7 U+V Bb bBKBH `X Ai `2K BMb iQ ;Bp2 i?2 DQBMi H r Q7 X0 , X1 , T0 , T1 mM/2` P X "v i?2 BMp2`bBQM 7Q`KmH - i?2 H27i@? M/ bB/2 Q7 UdXkjV 2[m Hb
T1 1{X0 ◦θs =i , X1 ◦θs =j} 1{−T0 ◦θs >x , T1 ◦θs >y} ds . λE0N 0
"mi B7 s ∈ (0, T1 ) , X0 ◦ θs = X0 , X1 ◦ θs = X1 , −T0 ◦ θs = s , T1 ◦ θs = T1 − sX h?2`27Q`2 i?2 H bi [m MiBiv 2[m Hb
T1 0 λEN 1{X0 =i , X1 =j} 1{s>x} 1{T1 −s>y} ds 0 ∞
= λπ(i)pij E0N E0N 1{T1 >y+s} ds | X0 = i, X1 = j x ∞ (1 − Gij (t)) dt . = λπ(i)pij x+y
k3k
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
_2+ HHBM; mij =
∞ 0
(1 − Gij (t))dt M/ b2iiBM; x = y = 0 BM UdXkjV ;Bp2b,
P (X0 = i , X1 = j) =
π(i)p m
ij ij k∈E l∈E π(k)pkl mkl
M/ i?2`27Q`2 P (−T0 > x , T1 > y | X0 = i , X1 = j) =
1 mij
(1 − Gij (t))dt. x+y
1t KTH2 dX9Xe, h?2 +Qp `B M+2 Q7 rbb biQ+? biB+ T`Q+2bb 7mM+@ iBQM Q7 b2KB@J `FQp T`Q+2bbX *QMbB/2` i?2 b2KB@J `FQp bi iBQM `v T`Q@ +2bb {X(t)}t∈R /2b+`B#2/ #Qp2X G2i f : E → R #2 bQK2 7mM+iBQM bm+? i? i i?2 T`Q+2bb {Y (t)}t∈R - r?2`2 Y (t) := f (X(t))- Bb Q7 i?2 b2+QM/ Q`/2` U M/ i?2`27Q`2 rbbV rBi? +Qp `B M+2 7mM+iBQM CY X q2 ? p2- 7Q` τ ≥ 0, CY (τ ) = E [f (X(0))f (X(τ ))] = E f (X(0))2 1{τ 1) = o(t) M/ U.Q#`mb?BMǶb 2biBK i2V P (N ((0, t]) > 0) = λt + o(t). S`QQ7X G2i PN0 #2 i?2 bbQ+B i2/ S HK T`Q# #BHBivX h?2 BMp2`bBQM 7Q`KmH UdXReVrBi? n = −1- ;Bp2b ∞ P (N ((0, t]) > 1) = P (T2 ≤ t) = λ PN0 (u < −T−1 , T2 ◦ θ−u ≤ t) du. 0
"mi QM Ω0 - u < −T−1 BKTHB2b T2 ◦ θ−u = T1 + uX h?2`27Q`2 t PN0 (u < −T−1 , T1 ≤ t − u) du P (N ((0, t]) > 1) = λ 0 t ≤λ PN0 (T1 ≤ t) du = λtPN0 (T1 ≤ t). 0
aBM+2
PN0 (T1
> 0) = 1-
limt↓0 PN0 (T1
≤ t) = 0X
6Q` .Q#`mb?BMǶb 2biBK i2- r2 mb2 ; BM i?2 BMp2`bBQM 7Q`KmH UdXReV- i?Bb iBK2 rBi? n = −1X aBM+2 T1 ◦ θ−u = u QM Ω0 ∩ {0 < u ≤ −T−1 } ∞ PN0 (u < −T−1 , T1 ◦ θ−u ≤ t) du P (N ((0, t]) > 0) = P (T1 ≤ t) = λ 0 t = λ PN0 (u < −T−1 ) du 0 t = λt − λ PN0 (−T−1 ≤ u) du = λt + o(t) . 0
q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K dX8XRX S`QQ7X q2 ? p2 iQ b?Qr i? i P (θT1 ∈ A | T1 ≤ t) =
P (T1 ≤ t, θT1 ∈ A) λt P (T1 ≤ t) λt
i2M/b iQ PN0 (A) b t ↓ 0X h FBM; BMiQ ++QmMi .Q#`mb?BMǶb 2biBK i2lim t→0
λt = 1, P (T1 ≤ t)
Bi bm{+2b iQ b?Qr i? i- mMB7Q`KHv BM A& 1 && P (T1 ≤ t, θT1 ∈ A) − PN0 (A)& → 0 . λt
dX8X GP*G ALh1_S_1hhAPL P6 SGJ S_P""AGAhu
k38
h?2 BMp2`bBQM 7Q`KmH UdXReV UrBi? n = −1V ;Bp2b ∞ P (T1 ≤ t, θT1 ∈ A) = λ PN0 (u < −T−1 , T1 ◦ θ−u ≤ t, θT1 ◦ θ−u ∈ A) du 0 t = λ PN0 (u < −T−1 , A) du, 0
bBM+2 θT1 ◦ θ−u Bb i?2 B/2MiBiv M/ T1 ◦ θ−u = u QM Ω0 ∩ {0 < u < −T−1 }X h?2`27Q`2`2r`BiBM; t λtPN0 (A) = λ PN0 (A) du , 0
r2 Q#i BM & 1 && 1 t 0 P (T1 ≤ t, θT1 ∈ A) − PN0 (A)& = P (u ≥ −T−1 , A) du λt t 0 N 1 t 0 ≤ P (u ≥ −T−1 ) du t 0 N t 1 ≤ P 0 (t ≥ −T−1 ) du = PN0 (t ≥ −T−1 ) . t 0 N "mi bBM+2 PN0 (−T−1 > 0)- PN0 (t ≥ −T−1 ) → 0 b t → 0X
1t KTH2 dX8Xj, ;2M2` H bBim iBQMX G2i {X(t)}t∈R #2 θt @+QKT iB#H2 +Q`HQH biQ+? biB+ T`Q+2bb rBi? p Hm2b BM Rm M/ H2i f : Rm → R #2 #QmM/2/ +QMiBMmQmb 7mM+iBQMX h FBM; Z := f (X(0−)) BM UdXk8V- r2 ? p2 i? i lim E [f (X(T1 −)) | N (0, h] ≥ 1] = E0N [f (X(0−)] .
h→0
.2}MBM; T+ (t) := t + inf {h > 0; N (t, t + h] = 1} , r2 ? p2 2[mBp H2MiHv lim E [f (X(T+ (t)−) | N (t, t + h] ≥ 1] = E0N [f (X(0−)] .
h→0
aBM+2 i?2 biQ+? biB+ T`Q+2bb {Z(t)}t∈R Bb +Q`HQHlim E [f (X(t) | N (t, t + h] ≥ 1] = E0N [f (X(0−)] .
h→0
_2K `F dX8X9 ivTB+ H 2t KTH2 Q7 TTHB+ iBQM + M #2 7QmM/ BM [m2m2BM; i?2Q`vr?2M QM2 +QKTmi2b i?2 H r Q7 i?2 MmK#2` Q7 +mbiQK2`b BM bi iBQM `v bvbi2K ;Bp2M i? i bQK2 2p2Mi U/2T `im`2 Q` ``Bp HV Q++m``2/X a22 a2+iBQM RRX9X
k3e
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
dXe h?2 *`Qbb@2`;Q/B+ h?2Q`2K h?Bb b2+iBQM ;Bp2b MQi?2` BMi2`T`2i iBQM Q7 S HK T`Q# #BHBiv BM i?2 2`;Q/B+ + b2X q2 }`bi `2+ HH 72r 7 +ib M/ /2}MBiBQMb Q7 2`;Q/B+ i?2Q`vX 1`;Q/B+Biv M/ JBtBM; h?Bb bm#b2+iBQM +QMi BMb `2pB2r Q7 i?2 # bB+ `2bmHib Q7 2`;Q/B+ i?2Q`v mb2/ BM i?Bb b2+iBQMXN q2 bi `i rBi? /Bb+`2i2@iBK2 ~QrbX G2i (P 0 , θ) #2 bi iBQM `v 7` K2rQ`F QM (Ω, F)X .2}MBiBQM dXeXR M 2p2Mi A ∈ F Bb b B/ iQ #2 bi`B+iHv BMp `B Mi B7 θ−1 A = A M/ BMp `B Mi B7 P 0 (A # θ−1 A) = 0- r?2`2 # /2MQi2b i?2 bvKK2i`B+ H /Bz2`2M+2X .2}MBiBQM dXeXk h?2 /Bb+`2i2 ~Qr {θn }n∈Z Bb + HH2/ P 0 @2`;Q/B+ B7 HH θ@BMp `B Mi 2p2Mib `2 i`BpB H UQ7 T`Q# #BHBiv 2Bi?2` y Q` RVX q2 b? HH b v, (P 0 , θ) Bb 2`;Q/B+X P#b2`p2 i? i 7Q` HH θ@BMp `B Mi 2p2Mib A- i?2 2p2Mi B = lim supn θ−n A Bb bi`B+iHv θ@BMp `B Mi M/ bm+? i? i P 0 (A) = P 0 (B)X h?2`27Q`2- 7Q` HH BMp `B Mi 2p2Mib- i?2`2 2tBbib bi`B+iHv BMp `B Mi 2p2Mi rBi? i?2 b K2 T`Q# #BHBivX AM T `iB+@ mH `- i?2 ~Qr Bb 2`;Q/B+ B7 M/ QMHv B7 HH bi`B+iHv θ@BMp `B Mi 2p2Mib `2 Q7 T`Q# #BHBiv 2Bi?2` y Q` RX h?2 +QMiBMmQmb T ` K2i2` b2i Bb- 7`QK MQr QM BM i?Bb b2+iBQM- RX q2 `2T2 i BM i?Bb b2iiBM; i?2 /2}MBiBQMb Dmbi ;Bp2M 7Q` /Bb+`2i2@iBK2 ~QrbX G2i (P, θt ) #2 bi iBQM `v 7` K2rQ`F QM (Ω, F)- {θt }t∈R Bb UK2 bm` #H2V ~Qr QM (Ω, F)X .2}MBiBQM dXeXj M 2p2Mi A ∈ F Bb b B/ iQ #2 bi`B+iHv θt @BMp `B Mi B7 A = θt−1 A 7Q` HH t ∈ R- θt @BMp `B Mi B7 P (A # θt−1 A) = 0 7Q` HH t ∈ RX .2}MBiBQM dXeX9 h?2 ~Qr {θt }t∈R Bb + HH2/ P @2`;Q/B+ B7 HH θt @BMp `B Mi 2p2Mib `2 Q7 T`Q# #BHBiv 2Bi?2` y Q` RX q2 b? HH b v, (P, θt ) Bb 2`;Q/B+X .2}MBiBQM dXeX8 h?2 /Bb+`2i2 ~Qr {θn }n∈Z Bb + HH2/ P 0 @KBtBM; B7 7Q` HH 2p2Mib A, B ∈ Flim P 0 (A ∩ θ−n B) = P 0 (A)P 0 (B). UdXkeV n
q2 b? HH b v, (P 0 , θ) Bb KBtBM;X LQi2 i? i UdXkeV Bb 2[mBp H2Mi iQ lim P 0 (θ−n B|A) = P 0 (B). n
h?mb- KBtBM; Bb T`QT2`iv Q7 dz7Q`;2i7mHM2bb Q7 i?2 BMBiB H +QM/BiBQMbǴX N
a22 ("BHHBM;bH2v- RNe8) 7Q` i?2 T`QQ7bX
dXeX h>1 *_Paa@1_:P.A* h>1P_1J
k3d
.2}MBiBQM dXeXe h?2 +QMiBMmQmb ~Qr {θt }t∈R Bb + HH2/ P @KBtBM; B7 7Q` HH 2p2Mib A M/ Blim P (A ∩ θ−t B) = P (A)P 0 (B). t↑∞
q2 b? HH b v, (P, θt ) Bb KBtBM;X A7 A Bb bi`B+iHv BMp `B Mi 7Q` i?2 KBtBM; ~Qr θ- i?2M P 0 (A) = P 0 (A)2 - bQ i? i P (A) Bb y Q` RX aBKBH `Hv 7Q` KBtBM; +QMiBMmQmb ~QrX h?2`27Q`2, 0
h?2Q`2K dXeXd KBtBM; ~Qr Bb M 2`;Q/B+ ~QrX JBtBM; ~Qrb `2 2`;Q/B+- M/ Bi Bb 2 bv iQ +?2+F i? i QM i?2 T`Q/m+i Q7 irQ T`Q# #BHBiv bT +2b- 2 +? 2M/Qr2/ rBi? KBtBM; b?B7i- i?2 T`Q/m+i b?B7i Bb KBtBM; M/ ?2M+2 2`;Q/B+X h?2Q`2K dXeX3 (P 0 , θ) Bb 2`;Q/B+ B7 M/ QMHv B7 i?2`2 2tBbib MQ /2+QKTQbBiBQM P 0 = α1 P10 + α2 P20 ,
α1 + α2 = 1, α1 > 0, α2 > 0,
UdXkdV
r?2`2 P10 M/ P20 `2 /BbiBM+i θ@BMp `B Mi T`Q# #BHBiB2bX >2`2 Bb i?2 +QMiBMmQmb@iBK2 p2`bBQM Q7 h?2Q`2K dXeX3, h?2Q`2K dXeXN (P, θt ) Bb 2`;Q/B+ B7 M/ QMHv B7 i?2`2 2tBbib MQ /2+QKTQbBiBQM P = β1 P1 + β2 P2 ,
β1 + β2 = 1, β1 > 0,
β2 > 0,
UdXk3V
r?2`2 P1 M/ P2 `2 /BbiBM+i θt @BMp `B Mi T`Q# #BHBiB2b 7Q` HH tX aBKmHi M2Qmb 1`;Q/B+Biv Q7 i?2 S HK M/ ai iBQM `v o2`bBQMb G2i (N, θt , P ) #2 bBKTH2 HQ+ HHv }MBi2 bi iBQM `v TQBMi T`Q+2bb QM R rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX G2i PN0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? (N, θt , P )X G2i θ := θT1 X h?2Q`2K dXeXRy U V G2i A ∈ F #2 M 2p2Mi BMp `B Mi rBi? `2bT2+i iQ i?2 ~Qr {θt }t∈R X h?2M P (A) = 1 B7 M/ QMHv B7 PN0 (A) = 1. U#V G2i A ∈ F #2 θ@BMp `B Mi 2p2MiX h?2M PN0 (A) = 1 B7 M/ QMHv B7 P (A) = 1X S`QQ7X U V amTTQb2 i? i PN0 (A) = 1X "v i?2 BMp2`bBQM 7Q`KmH ∞ P (A) = λ PN0 (u < T1 , θ−u A) du 0 ∞ = λ PN0 (u < T1 , A) du (θt @BMp `B M+2 Q7 A) 0 ∞ = λ PN0 (u < T1 ) du (PN0 (A) = 1) = λE00 [T1 ] = 1 . 0
k33
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
*QMp2`b2Hv- bmTTQb2 i? i P (A) = 1X q2 ? p2- ; BM #v i?2 BMp2`bBQM 7Q`KmH ∞ PN0 (u < T1 , A) du , 1 = P (A) = λ 0
∞
M/ i?2`27Q`2- bBM+2 1 = λE0N [T1 ] = λ 0 PN0 (u < T1 ) du ∞ PN0 (u < T1 , A) du = λE0N [T1 1A ] . 0=λ 0
"mi T1 < ∞ XbX bBM+2 E0N [T1 ] = λ−1 < ∞X h?2`27Q`2 E0N [T1 1A ] = 0 BKTHB2b PN0 (A) = 0X U#V A7 P (A) = 1- i?2M 7`QK i?2 /2}MBiBQM Q7 S HK T`Q# #BHBiv 1 0 PN (A) = E 1θ−n A 1{0 0) = P20 (T1 > 0) = 1 bBM+2 PN0 (T1 > 0) = 1X AM T `iB+mH `- E01 [T1 ] M/ E02 [T1 ] `2 TQbBiBp2X h?2 BMp2`bBQM 7Q`KmH ;Bp2b T1
1 E01 [T1 ] 0 P (A) = α1 0 E 1A ◦ θt dt EN [T1 ] E01 [T1 ] 1 0 T1
1 E0 [T1 ] 0 + α2 02 E 1 ◦ θ dt , A t EN [T1 ] E02 [T1 ] 2 0
dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu
k3N
i? i Bb P = β1 P1 + β2 P2 , E0 [T ] α1 E01 [T11 ] N
E0 [T ]
M/ β2 = α2 E02 [T11 ] ∈ (0, 1) `2 bm+? i? i β1 +β2 = 1X JQ`2Qp2`N T P1 M/ P2 - /2}M2/ #v Pi (A) = E01[T1 ] E0i 0 1 1A ◦ θt dt Ui = 1, 2V `2 θt @BMp `B Mi i 7Q` HH t ∈ RX h?2`27Q`2- #v h?2Q`2K dXeXN- P + MMQi #2 2`;Q/B+- +QMi` /B+iBQMX r?2`2 β1 =
h?2 T`QQ7 Q7 i?2 +QMp2`b2 T `i 7QHHQrb i?2 b K2 HBM2 Q7 `;mK2Mi U1t2`+Bb2 dX3XRRVX h?2Q`2K dXeXRk G2i (N, θt , P ) #2 bi iBQM `v TQBMi T`Q+2bb rBi? }MBi2 TQbBiBp2 BMi2MbBiv λ- M/ bmTTQb2 i? i (P, θt ) Bb 2`;Q/B+X G2i PN0 #2 i?2 bbQ+B i2/ S HK T`Q# #BHBivX G2i f #2 BM L1 (P )X h?2M 1 T ↑∞ T
T
f ◦ θt dt = E[f ],
PN0 @ XbX
UdXkNV
1 f ◦ θTk = E0N [f ], n↑∞ n k=1
P @ XbX
UdXjyV
lim
M/
0
n
lim
S`QQ7X aBM+2 (P, θt ) Bb 2`;Q/B+- i?2 T`Q# #BHBiv Q7 2p2Mi , A :=
1 T →∞ T
T
f ◦ θt dt = E[f ]
lim
-
0
Bb 1X JQ`2Qp2` A Bb θt @BMp `B MiX h?2`27Q`2 #v G2KK dXeXRy- PN0 (A) = 1X h?Bb T`Qp2b UdXkNVX aBKBH `Hv- bBM+2 (PN0 , θ) Bb 2`;Q/B+- 7Q` HH f ∈ L1 (PN0 )- i?2 2p2Mi , B :=
1 f ◦ θTk = E0N [f ] n↑∞ n k=1 n
lim
? b T`Q# #BHBiv 1X Ai Bb KQ`2Qp2` θ@BMp `B Mi- M/ i?2`27Q`2 P (B) = 1 #v G2KK dXeXRyX h?Bb T`Qp2b UdXjyVX _2K `F dXeXRj h?Bb `2bmHi Bb + HH2/ i?2 +`Qbb@2`;Q/B+ i?2Q`2K 7Q` Q#pBQmb `2 @ bQMbX Ai Bb 2bb2MiB H 7Q` [m2m2BM; i?2Q`v BM i? i Bi T`QpB/2b M 2 bv BMi2`T`2i iBQM Q7 i?2 +`Qbb@7Q`KmH b- i? i Bb- 7Q`KmH b HBMFBM; PN0 @K2 Mb M/ P @K2 Mb Q7 QT2` iBQM H [m MiBiB2bX
dXd
S HK S`Q# #BHBiv M/ aiQ+? biB+ AMi2MbBiv
_Qm;?Hv bT2 FBM;- S HK T`Q# #BHBiv /2b+`B#2b r? i ? TT2Mb dzr?2M i?2`2 Bb TQBMi i iBK2 tǴX aiQ+? biB+ BMi2MbBiv ;Bp2b i?2 2tT2+i iBQM Q7 dzb22BM; TQBMi i iBK2 tǴ FMQrBM; i?2 T bi ?BbiQ`v Q7 i?2 TQBMi T`Q+2bbX Ai Bb M im` H iQ 2tT2+i i?2b2 TQBMib Q7 pB2r iQ #2 +QMM2+i2/X
kNy
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
.2}MBiBQM dXdXR G2i {θt }t∈R #2 K2 bm` #H2 ~Qr QM (Ω, F)X ?BbiQ`v {Ft }t∈R Bb b B/ iQ #2 θt @+QKT iB#H2 B7 θt Fs = Fs−t 7Q` HH s, t ∈ RX aiQ+? biB+ BMi2MbBiv T`QpB/2b /2i BH2/ /2b+`BTiBQM Q7 i?2 /vM KB+b Q7 TQBMi T`Q+2bb- M/ r?2M QM2 FMQrb i?2 mM/2`HvBM; T?vbB+b Q7 i?2 ;2M2` iBQM Q7 2p2MibQM2 rBHH BM KQbi + b2b #2 #H2 iQ T`QpB/2 i?2 biQ+? biB+ BMi2MbBiv rBi? `2bT2+i iQ ?BbiQ`v i? i bmKK `Bx2b i?2 BM7Q`K iBQM p BH #H2 i Mv BMbi MiX Ai Bb i?2`27Q`2 Q7 i?2Q`2iB+ H BKTQ`i M+2 iQ FMQr r?2i?2` Q` MQi i?2 biQ+? biB+ BMi2MbBiv Bb i?2 b K2 mM/2` i?2 bi iBQM `v T`Q# #BHBiv M/ i?2 S HK T`Q# #BHBivX Ai im`Mb Qmi i? irBi? bQK2 T`QpBbBQMb iQ #2 bi i2/ bQQM- i?Bb Bb i?2 + b2c i?2`27Q`2 i?2 bi iBQM `v T`Q# #BHBiv M/ i?2 S HK T`Q# #BHBiv /2b+`B#2 i?2 b K2 /vM KB+bX JQ`2 T`2+Bb2Hv, h?2Q`2K dXdXk G2i {Ft }t∈R #2 ?BbiQ`v Q7 HQ+ HHv }MBi2 bBKTH2 TQBMi T`Q+2bb N - #Qi? θt @+QKT iB#H2X G2i P #2 θt @BMp `B Mi T`Q# #BHBivX amTTQb2 i? i N ? b }MBi2 TQbBiBp2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? (N, θt , P )X amTTQb2 i? i N /KBib θt @+QKT iB#H2 (P, Ft )@BMi2MbBiv {λ(t)}t∈R X PM R+ - N /KBib i?2 (PN0 , Ft )@BMi2MbBiv {λ(t)}t∈R X _2K `F dXdXj LQi2 i?2 //2/ T`2+BbBQM BM i?2 i2`KBMQHQ;v, i?2 /2T2M/2M+2 Q7 biQ+? biB+ BMi2MbBiv QM i?2 mM/2`HvBM; T`Q# #BHBiv ? b #22M 2KT? bBx2/X h?Bb /2T2M/2M+2 Bb +im HHv i?2 K BM +QM+2`M Q7 i?2 i?2Q`2KX S`QQ7XRy q2 ? p2 iQ T`Qp2 i? i 7Q` HH (a, b] ⊂ R+ , A ∈ Fa , b
λ(t)dt . E0N [1A N (a, b]] = E0N 1A a
"v /2}MBiBQM Q7 S HK T`Q# #BHBiv- i?2 H27i@? M/ bB/2 Q7 i?2 #Qp2 2[m HBiv Bb 1 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ] . λ n∈Z P#b2`p2 i? i- B7 n ≥ 1, Tn Bb M Ft @biQTTBM; iBK2, BM/22/ {Tn ≤ t} = {N (0, t] ≥ n} ∈ Ft X U6Q` n ≤ 0, Tn Bb MQi M Ft @biQTTBM; iBK2XV HbQ- bBM+2 a Bb MQM@ M2; iBp2- 1(0,1] (Tn ) Bb Fa+Tn @K2 bm` #H2X "2+ mb2 {Ft }t∈R Bb θt @+QKT iB#H2 M/ A Bb BM Fa - i?2 ` M/QK p `B #H2 1A ◦ θTn Bb Fa+Tn @K2 bm` #H2X h?2 T`Q+2bb 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t) Bb i?2`27Q`2 Ft @ / Ti2/X "2BM; BM //BiBQM H27i@ +QMiBMmQmb- Bi Bb Ft @T`2/B+i #H2X h?2`27Q`2- 7Q` n ≥ 1, a ≥ 0 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ]
= E 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t)N (dt) R
= E 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t)λ(t)dt R
b+Tn λ(t)dt . = E 1(0,1] (Tn )(1A ◦ θTn ) a+Tn Ry
("`ûK m/- RN3N)X
dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu
kNR
aBM+2 {λ(t)}t∈R Bb θt @+QKT iB#H2 b b+Tn λ(t)dt = λ(t)dt ◦ θTn , a+Tn
a
M/ i?2`27Q`2 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ]
b λ(t)dt ◦ θTn . = E 1(0,1] (Tn )(1A ◦ θTn ) a
amKKBM; mT i?2 H bi 2[m HBiv rBi? `2bT2+i iQ n = 1, 2, . . .- r2 Q#i BM i?2 MMQmM+2/ `2bmHiX G2i {Ft }t∈R #2 θt @+QKT iB#H2 ?BbiQ`vX A7 BM //BiBQM Mv Ft @T`2/B+i #H2 T`Q+2bb {H(t)}t∈R ? b i?2 7Q`K H(t, ω) = v(t, θt (ω)), UdXjRV r?2`2 (t, ω) → v(t, ω) Bb B(R) ⊗ 8 F K2 bm` #H2 M/ 7Q` HH t ∈ R, ω → v(t, ω) Bb F0− @K2 bm` #H2 Ur?2`2 F0− = s aX N,Z @K2 bm` #H2- i?2M S`QQ7 Q7 (β), Bi bm{+2b iQ b?Qr i? i B7 W : Ω → R Bb F0− N,Z (t, ω) → W (θt (ω)) Bb P(Ft )@K2 bm` #H2X Ai Bb BM im`M 2MQm;? iQ b?Qr i?Bb 7Q` HH ` M/QK p `B #H2b W Q7 i?2 7Q`K W = ϕ(N ([a, b) × L)), a ≤ b ≤ 0, L ∈ KX AM i?Bb + b2 {W ◦ θt } Bb 7mM+iBQM Q7 H27i@+QMiBMmQmb FtN,Z @ / Ti2/ T`Q+2bb- M/ i?2`27Q`2 Bi Bb FtN,Z @T`2/B+i #H2X
kNk
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G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb M/ H2i {Ft }t∈R #2 ?BbiQ`v Q7 N - #Qi? θt @+QKT iB#H2X amTTQb2 i? i i?2 BMi2MbBiv λ Q7 N Bb }MBi2 M/ TQbBiBp2 M/ H2i PN0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? (N, P, θt )X amTTQb2 KQ`2Qp2` i? i {Ft }t∈R ? b T`2/B+i #H2 bi`m+im`2 / Ti2/ iQ {θt }t∈R X h?2 # bB+ `2bmHi Bb S T M;2HQmǶb i?2Q`2K, h?2Q`2K dXdX8 lM/2` i?2 #Qp2 +QM/BiBQMb- N /KBib (P, Ft )@BMi2MbBiv {λ(t)}t∈R B7 M/ QMHv B7 PN0 P on F0− , UdXjjV M/ BM i? i + b2 {λ(t)}t∈R + M #2 +?Qb2M Q7 i?2 7Q`K λ(t) = (μ ◦ θt )λ ,
UdXj9V
r?2`2 μ Bb i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 QM F0− Q7 PN0 rBi? `2bT2+i iQ P , μ=
dPN0 | dP
F0−
.
UdXj8V
h?2 7QHHQrBM; Bb +QKT +i 2tT`2bbBQM Q7 i?2 #Qp2 i?2Q`2K, E[f (Z(0))λ(0)] = λE0N [f (Z(0))] ,
UdXjeV
r?2`2 {Z(t)}t∈R Bb θt @+QKT iB#H2 biQ+? biB+ T`Q+2bb bm+? i? i Z(0) Bb F0− @ K2 bm` #H2 U M/ i?2`27Q`2 {Z(t)}t∈R Bb Ft @T`2/B+i #H2 /m2 iQ i?2 bbmKTiBQM i? i i?2 T`2/B+i #H2 bi`m+im`2 Q7 {Ft }t∈R Bb / Ti2/ iQ {θt }t∈R V- M/ f Bb MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM 7`QK i?2 bi i2 bT +2 Q7 {Z(t)}t∈R BMiQ RX S`QQ7XRR amTTQb2 i? i UdXjjV ?QH/b- M/ /2}M2 μ #v UdXj8VX G2i MQr {H(t)}t∈R #2 MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bbX aBM+2 {Ft }t∈R ? b T`2/B+i #H2 bi`m+@ im`2 / Ti2/ iQ {θt }t∈R , H(t, ω) = v(t, θt (ω)) 7Q` bQK2 7mM+iBQM v i? i Bb B ⊗ F @ K2 bm` #H2 M/ bm+? i? i 7Q` 2 +? t ∈ R, ω → v(t, ω) Bb F0− @K2 bm` #H2X 6`QK i?2 H ii2` T`QT2`iv Q7 v M/ i?2 /2}MBiBQM Q7 μE0N [v(t)] = E[μv(t)],
M/ i?2`27Q`2
R
E0N [v(t)] dt =
aBM+2 P Bb θt @BMp `B Mi
E[μv(t)] dt =
R
E[μv(t)] dt. R
R
E[(μ ◦ θt )(v(t) ◦ θt )] dt .
PM i?2 Qi?2` ? M/- #v i?2 J2+F2 7Q`KmH
1 1 E0N [v(t)] dt = E (v(t) ◦ θt )N (dt) = E H(t)N (dt) . λ λ R R R RR h?2 7QHHQrBM; T`QQ7 Q7 S T M;2HQmǶb Q`B;BM H BM i2`Kb Q7 biQ+? biB+ BMi2MbBiv Bb i F2M 7`QK ("`ûK m/- RN3N)- r?Qb2 Tm`TQb2 r b iQ `2p2 H Bib HBMF rBi? i?2 T bi T`QT2`iv Q7 [m2m2BM; i?2Q`vX a22 a2+iBQM RRX9X
dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu
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*QK#BMBM; HH i?2 #Qp2- r2 Q#i BM
E H(t)N (dt) = E (H(t)λμ ◦ θt ) dt , R
R
r?2`2 {H(t)}t∈R Bb Mv `#Bi` `v MQM@M2; iBp2 Ft @T`2/B+i #H2 T`Q+2bbX h?2`27Q`2#v h?2Q`2K X9Xj- {λμ ◦ θt }t∈R Bb i?2 Ft @BMi2MbBiv Q7 N X "27Q`2 im`MBM; iQ i?2 T`QQ7 Q7 i?2 +QMp2`b2 T `i- r2 `2K `F i? i {λμ ◦ θt }t∈R Bb M Ft @T`2/B+i #H2 T`Q+2bb #2+ mb2 μ Bb F0− @K2 bm` #H2 M/ {Ft }t∈R Bb bbmK2/ iQ ? p2 T`2/B+i #H2 bi`m+im`2 / Ti2/ iQ {θt }t∈R X AM Q`/2` iQ T`Qp2 i?2 +QMp2`b2 T `i- r2 bi `i rBi? i?2 ?vTQi?2bBb i? i N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X q2 FMQr i? i bm+? M BMi2MbBiv + M #2 bbmK2/ Ft @ T`2/B+i #H2 Uh?2Q`2K 8XRX9yV- i? i Bb- BM pB2r Q7 i?2 bbmKTiBQM QM i?2 T`2/B+i #H2 bi`m+im`2 Q7 {Ft }t∈R λ(t, ω) = λ μ(t, θt (ω)), r?2`2 μ Bb B ⊗ F @K2 bm` #H2 M/ 7Q` 2 +? t ∈ R, ω → μ (t, ω) Bb F0− @K2 bm` #H2X G2i MQr v #2 MQM@M2; iBp2 F0− @K2 bm` #H2 ` M/QK p `B #H2X "v i?2 bbmKTiBQM QM i?2 T`2/B+i #H2 bi`m+im`2 Q7 {Ft }t∈R - i?2 biQ+? biB+ T`Q+2bb {v ◦ θt }t∈R Bb Ft @ T`2/B+i #H2X h?2`27Q`2- #v h?2Q`2K X9Xj- 7Q` HH (a, b] ⊂ R
E (v ◦ θt )(λ μ(t) ◦ θt ) dt = E (v ◦ θt )N (dt) . (a,b]
(a,b]
"v i?2 θt @BMp `B M+2 Q7 P - i?2 H27i@? M/ bB/2 Q7 i?Bb 0 2[m HBiv Bb 2[m H iQ λE[v (a,b] μ (t) dt]- r?2`2 b i?2 `B;?i@? M/ bB/2 Bb λ(b − a)EN [v]X h?2`27Q`2- 7Q` HH MQM@M2; iBp2 ` M/QK p `B #H2b v i? i `2 F0− @K2 bm` #H2
b 1 μ (t) dt . E0N [v] = E v b−a a h?Bb b?Qrb i? i PN0 P QM F0− UrBi? μ = /2`Bp iBp2
0 dPN | VX dP F0−
1 b−a
b a
μ (t) dt b i?2 _ /QMĜLBFQ/ɷK
q2 `2K `F i? i i?2 T`QQ7 BM i?2 }`bi T `i + M #2 mb2/ iQ b?Qr i? i {λμ ◦ θt }t∈R Bb T`2/B+i #H2 Ft @BMi2MbBiv Q7 N X AM T `iB+mH `- i?Bb BMi2MbBiv Bb θt @+QKT iB#H2 Ur2 /B/ MQi bi `i rBi? i?Bb b ?vTQi?2bBbVX h?2 7QHHQrBM; Bb M BKK2/B i2 +QMb2[m2M+2 Q7 h?2Q`2K dXdX8 M/ Q7 q i M #2Ƕb +? ` +i2`Bx iBQM Q7 SQBbbQM T`Q+2bb2bX *Q`QHH `v dXdXe lM/2` i?2 +QM/BiBQMb bi i2/ BM i?2 T ` ;` T? T`2+2/BM; h?2Q`2K dXdX8- i?2 TQBMi T`Q+2bb N Bb ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb B7 M/ QMHv P = PN0 QM F0− X aBM+2 i?2 ``Qr Q7 iBK2 + M #2 BMp2`i2/ rBi?Qmi +? M;BM; i?2 /Bbi`B#miBQM Q7 M ?TT-
kN9
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
*Q`QHH `v dXdXd A7 N Bb M ?TT- i?2M i?2 PN0 @/Bbi`B#miBQM Q7 N − ε0 M/ i?2 P @/Bbi`B#miBQM Q7 N `2 i?2 b K2X S`QQ7X h?Bb 7QHHQrb 7`QK h?2Q`2K dXdXe rBi? Ft ≡ FtN Ut ∈ RV- M/ Q#b2`pBM; i? i F0− = σ{N (C) ; C ∈ B(R) , C ⊂ (−∞, 0)}X AM Qi?2` rQ`/b- B7 N Bb mM/2` P M ?TT Q7 BMi2MbBiv λ- mM/2` i?2 S HK /Bbi`B#miBQMN Bb SQBbbQM T`Q+2bb iQ r?B+? ? b #22M //2/ TQBMi i 0X M 2H2K2Mi `v 7Q`K Q7 i?Bb `2bmHi TT2 `b BM 1t2`+Bb2 jX3XkX
dX3 1t2`+Bb2b 1t2`+Bb2 dX3XRX J2 bm` #BHBiv Q7 i?2 b?B7i a?Qr i? i- 7Q` }t2/ x ∈ Rm - i?2 K TTBM;b θx Q7 1t KTH2 dXRXR `2 K2 bm` #H2X 1t2`+Bb2 dX3XkX θt @+QKT iB#BHBiv G2i {θx }t∈R #2 K2 bm` #H2 ~Qr M/ H2i N #2 bBKTH2 θt @+QKT iB#H2 TQBMi T`Q+2bb QM RX G2i {Tn }n∈Z #2 Bib b2[m2M+2 Q7 TQBMibX S`Qp2 i?2 7QHHQrBM;, {Zn }n∈Z Bb θx @ +QKT iB#H2 T`Q+2bb Q7 K `Fb Q7 N B7 M/ QMHv B7 i?2`2 2tBbib biQ+? biB+ T`Q+2bb {Z(t)}t∈R +QKT iB#H2 rBi? i?2 ~Qr M/ bm+? i? i Z(Tn ) = Zn 7Q` HH n ∈ NX 1t2`+Bb2 dX3XjX S`QQ7 Q7 h?2Q`2K dXRXkk S`Qp2 h?2Q`2K dXRXkkX 1t2`+Bb2 dX3X9X θT1 _272` iQ i?2 /Bb+mbbBQM #27Q`2 h?2Q`2K dXjXRX :Bp2 bBKTH2 2t KTH2 b?QrBM; i? i BM ;2M2` H- QM Ω- i?2 BMp2`b2 Q7 θT1 Bb MQi θT−1 X 1t2`+Bb2 dX3X8X 6B`bi TQbBiBp2 2p2Mi iBK2 Q7 bmT2`TQbBiBQM h F2 Rm = R BM i?2 1t KTH2 dXkX3X G2i Fi M/ Fi0 #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 }`bi TQBMi Q7 μi mM/2` Pi M/ Pi0 `2bT2+iBp2HvX S`Qp2 i? i- mM/2` i?2 S HK T`Q# #BHBiv PN0 - i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM G0 Q7 i?2 }`bi TQBMi bi`B+iHv iQ i?2 `B;?i Q7 i?2 Q`B;BM Q7 i?2 bmT2`TQbBiBQM T`Q+2bb N Bb ⎛ ⎞ G0 (x) = 1 −
k ⎜ λi ⎟ ⎜ (1 − Fi0 (x)) (1 − Fj (x))⎟ ⎝λ ⎠. i=1
1≤j≤k j=i
1t2`+Bb2 dX3XeX JQiBQM@BMp `B Mi bi iBQM `v TQBMi T`Q+2bb2bX G2i P #2 T`Q# #BHBiv QM (Mp (Rm ), Mp (Rm )) i? i K F2b Q7 i?2 +QQ`/BM i2 T`Q+2bb N KQiBQM@BMp `B Mi bi iBQM `v TQBMi T`Q+2bb Q7 }MBi2 MQM@MmHH BMi2MbBivX JQiBQM BMp `B M+2 K2 Mb i? i 7Q` Mv /Bbi M+2@T`2b2`pBM; i` Mb7Q`K iBQM τ Q7 Rm - M/ Mv Γ ∈ M(Rm )- P (τ N ∈ Γ) = P (N ∈ Γ)X a?Qr i? i i?2 S HK T`Q# #BHBiv P0N Bb BMp `B Mi mM/2` `Qi iBQM Q7 +2Mi2` 0- i? i Bb 7Q` Mv bm+? `Qi iBQM r M/ Mv Γ ∈ Mp (Rm )- P0N (rN ∈ Γ) = P0N (N ∈ Γ)X
dX3X 1s1_*Aa1a
kN8
1t2`+Bb2 dX3XdX P @b KTH2 7`QK PN0 @b KTH2 S`QpB/2 i?2 /2i BHb BM 1t KTH2 dXjX3X 1t2`+Bb2 dX3X3X 62HH2`ǰb #mb T ` /QtX lM/2` i?2 +QM/BiBQMb T`2p BHBM; BM 1t KTH2 dX9Xj rBi? i?2 //BiBQM H +QM/BiBQM i? i E0N [S0 2 ] < ∞- T`Qp2 i? i E[S0 ] =
E0N [S0 2 ] , E0N [S0 ]
r?2`2 S0 := T1 − T0 X .2/m+2 7`QK i?Bb i? i 7Q` S0 iQ ? p2 i?2 b K2 /Bbi`B#miBQM mM/2` P M/ mM/2` PN0 Bi Bb M2+2bb `v M/ bm{+B2Mi i? i i?2 BMi2``2M2r H iBK2b #2 /2i2`KBMBbiB+X 1t2`+Bb2 dX3XNX h?2 `2HB #BHBiv TQBMi T`Q+2bb *QMbB/2` MQM@/2H v2/ `2M2r H T`Q+2bb QM R UMQi Dmbi R+ V 7Q` r?B+? i?2 BMi2``2@ M2r H b2[m2M+2 {Sn }n∈Z Bb Q7 i?2 7Q`K Sn = Un + Vn - r?2`2 {Un }n∈Z M/ {Vn }n∈Z `2 BM/2T2M/2Mi BB/ b2[m2M+2b Q7 ` M/QK p `B #H2b rBi? }MBi2 K2 MX .2b+`B#2 i?2 iBK2@bi iBQM `v p2`bBQM Q7 i?2 TQBMi T`Q+2bb r?Qb2 2p2Mi iBK2b `2 U0 , U0 + V0 , U0 + V0 + U1 , U0 + V0 + U1 + V1 , . . .
1t2`+Bb2 dX3XRyX +QMiBMmQmb@iBK2 ?K+ M/ Bib 2K#2//2/ +? BM *QMbB/2` +QMiBMmQmb@iBK2 ?K+ {X(t)}T ∈R rBi? bi i2 bT +2 E i? i Bb TQbBiBp2 `2@ +m``2Mi M/ bi iBQM `vX :Bp2 M 2tT`2bbBQM BM i2`Kb Q7 i?2 BM}MBi2bBK H ;2M2` iQ` Q M/ Q7 i?2 bi iBQM `v /Bbi`B#miBQM π Q7 i?2 p2` ;2 BMi2MbBiv Q7 i?2 bi iBQM `v TQBMi T`Q+2bb +QmMiBM; i?2 i` MbBiBQM iBK2b Q7 i?2 +? BMX bbmKBM; i?Bb p2` ;2 BMi2M@ bBiv }MBi2- ;Bp2 i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 /Bb+`2i2@iBK2 ?K+ {X(Tn −)}n∈N r?2`2 Tn Bb i?2 n@i? i` MbBiBQM iBK2 Q7 i?2 +? BMX 1t2`+Bb2 dX3XRRX 1`;Q/B+Biv :Bp2 i?2 /2i BHb Q7 i?2 T`QQ7 Q7 h?2Q`2K dXeXRR- i? i Bb- T`Qp2 i? i B7 (PN0 , θ) Bb 2`;Q/B+- i?2M (P, θt ) Bb 2`;Q/B+X 1t2`+Bb2 dX3XRkX 1`;Q/B+Biv M/ i?2 bi iBQM `v `2M2r H T`Q+2bb Ab bi iBQM `v `2M2r H T`Q+2bb 2`;Q/B+\ CmbiB7v vQm` Mbr2`X 1t2`+Bb2 dX3XRjX JBtBM; M/ MQ KBtBM; *QMbB/2` i?2 TQBMi T`Q+2bb rBi? 2[mBbT +2/ TQBMib Ub v, Tn+1 − Tn = 1 7Q` HH n ∈ ZVX *H2 `Hv- 7Q` i?2 S HK p2`bBQM- (PN0 , θT1 ) Bb KBtBM;X a?Qr i? i i?2 bi iBQM `v p2`bBQM (P, θt ) Bb MQi KBtBM;X
kNe
*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1
1t2`+Bb2 dX3XR9X o `B iBQM /Bbi M+2 #2ir22M i?2 bi iBQM `v M/ mM@ /2H v2/ `2M2r H T`Q+2bb2b a?Qr i? i r?2M i?2 bBKTH2 HQ+ HHv }MBi2 N Bb mM/2` PN0 M mM/2H v2/ `2M2r H TQBMi T`Q+2bb rBi? M BMi2`@`2M2r H +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F /KBiiBM; /2MbBiv f rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2dV (P, PN0 ) = dV (F, G) , r?2`2 G Bb +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM rBi? /2MbBiv g(x) = λ(1 − F (x)) Ui?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 `2bB/m H BMi2`@`2M2r H /Bbi`B#miBQMVX U>2`2 P M/ PN0 `2 `2bi`B+i2/ iQ R+ XV 1t2`+Bb2 dX3XR8X E0N
λ λ(0)
=1
h?2 b2iiBM; Bb i? i Q7 h?2Q`2K dXdX8X a?Qr i? i E0N /B+i #H2 p2`bBQM Q7 i?2 biQ+? biB+ BMi2MbBivVX
λ λ(0)
= 1 UmbBM; i?2 T`2@
1t2`+Bb2 dX3XReX .Bbi M+2 BM p `B iBQM #2ir22M i?2 S HK M/ bi iBQM@ `v p2`bBQMb AM i?2 bBim iBQM Q7 h?2Q`2K dXdX8- b?Qr i? i i?2 /Bbi M+2 #2ir22M i?2 p `B iBQM `2bi`B+iBQMb Q7 P M/ PN0 iQ F0− Bb 2[m H iQ 12 E |λ(0)−λ| X S`Qp2 i? i Bi Bb 2[m H iQ λ |λ(0)−λ| 1 0 UmbBM; i?2 T`2/B+i #H2 p2`bBQM Q7 i?2 biQ+? biB+ BMi2MbBivVX E 2 N λ(0) 1t2`+Bb2 dX3XRdX J2+F2ǰb +? ` +i2`Bx iBQM Q7 SQBbbQM T`Q+2bb G2i {θt }t∈R #2 K2 bm` #H2 ~Qr QM RX G2i N #2 TQBMi T`Q+2bb QM R M/ H2i {Ft }t∈R #2 ?BbiQ`v Q7 N - #Qi? θt @+QKT iB#H2X amTTQb2 i? i N ? b }MBi2 TQbBiBp2 p2` ;2 BMi2MbBiv λ- M/ H2i PN0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? (N, θt , P )X S`Qp2 U V pB S T M;2HQmǶb i?2Q`2K M/ U#V pB q i M #2Ƕb i?2Q`2K- i? i M2+2b@ b `v M/ bm{+B2Mi +QM/BiBQM 7Q` N iQ #2 ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb Ui? i Bb- bm+? i? i- 7Q` HH (a, b] ⊂ R- N (a, b] Bb SQBbbQM p `B #H2 rBi? K2 M λ × (b − a) M/ Bb P @BM/2T2M/2Mi Q7 Fa V Bb i? i P ≡ PN0 QM F0− X 1t2`+Bb2 dX3XR3X a2H2+i2/ i` MbBiBQMb Q7 bi iBQM `v ?K+ G2i (P, θt ) #2 bi iBQM `v 7` K2rQ`F M/ H2i {X(t)}t∈R #2 θt @+QKT iB#H2 biQ+? b@ iB+ T`Q+2bb rBi? p Hm2b BM i?2 /2MmK2` #H2 bT +2 E i? i Bb KQ`2Qp2` M 2`;Q/B+ +QMiBMmQmb@iBK2 ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` Q = {qij }i,j∈E M/ bi iBQM `v /Bbi`B#miBQM πX G2i N (C) #2 i?2 MmK#2` Q7 /Bb+QMiBMmBiB2b Q7 i?Bb +? BM BM C ⊆ EX h?Bb /2}M2b bi iBQM `v T`Q+2bb (N, θt , P )X G2i 7Q` H ∈ E × E − /B ;(E × E)- NH #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 H@ i` MbBiBQMb Q7 i?2 +? BM- i? i Bb, NH (C) :=
1H (Xs− , Xs )N (ds) . C
UdXjdV
dX3X 1s1_*Aa1a
kNd
h?2M (NH , θt , P ) Bb bi iBQM `v TQBMi T`Q+2bbX AM T `iB+mH `- 7Q` H = E × E − /B ;(E ×E)- r2 ? p2 NH = N - i?2 # bB+ TQBMi T`Q+2bb Q7 i?2 +? BM- r?B+? +QmMib HH i` MbBiBQMbX :Bp2 i?2 2tT`2bbBQMb Q7 i?2 [m MiBiB2b PN0 H (X0− = k) M/ PN0 H (X0 = k) BM i2`Kb Q7 i?2 BM}MBi2bBK H ;2M2` iQ` M/ Q7 i?2 bi iBQM `v /Bbi`B#miBQM πX
*? Ti2` 3 S HK S`Q# #BHBiv BM aT +2 h?2 S HK i?2Q`v BM bT +2 Bb # b2/ QM J2+F2Ƕb K2 bm`2R M/ HbQ TTHB2b iQ MQM@ bi iBQM `v TQBMi T`Q+2bb2bX Ai `2i BMb bQK2 72 im`2b Q7 i?2 i?2Q`v `2bi`B+i2/ iQ i?2 HBM2X h?2 KBbbBM; `2bmHi Bb i?2 dz2p2Mi@iBK2 bi iBQM `BivǴ, Bi Bb MQi i`m2 BM ;2M2` H i? i ;Bp2M M 2MmK2` iBQM {Xn }n∈N Q7 i?2 TQBMib Q7 N - i?2 /Bbi`B#miBQM Q7 N M/ i? i Q7 N − Xn rBi? `2bT2+i iQ i?2 S HK T`Q# #BHBiv `2 i?2 b K2X
3XR
h?2 oQ`QMQB *2HH M/ i?2 AMp2`bBQM 6Q`KmH
h?2Q`2K dXjX9 ;Bp2b 7Q` bi iBQM `v TQBMi T`Q+2bb2b QM i?2 HBM2 M BMp2`bBQM 7Q`KmH K FBM; Bi TQbbB#H2 iQ +QKTmi2 2tT2+i iBQMb rBi? `2bT2+i iQ i?2 bi iBQM `v T`Q# @ #BHBiv BM i2`Kb Q7 2tT2+i iBQMb rBi? `2bT2+i iQ i?2 S HK T`Q# #BHBivX h?2 M HQ;Qmb BMp2`bBQM 7Q`KmH BM i?2 bT iB H + b2 Bb BM i2`Kb Q7 oQ`QMQś +2HHb- MQiBQM i? i r2 MQr BMi`Q/m+2X .2}MBiBQM 3XRXR G2i N #2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X h?2 UQT2MV oQ`QMQB +2HH Q7 N i x ∈ Rm Bb i?2 b2i Vx (N ) := {y ∈ Rm ; ||y − x|| < inf {||y − Xn || ; Xn = x}} . n∈N
R
(J2+F2- RNek)- +QKKQMHv `272``2/ iQ b i?2 * KT#2HHĜJ2+F2 K2 bm`2X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_8
kNN
jyy
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
1t KTH2 3XRXk, h?2 oQ`QMQB +2HH QM i?2 HBM2X A7 N Bb TQBMi T`Q+2bb QM R- V0 (N ) = ( 12 T0 , 12 T1 ) P @ XbX- M/ V0 (N ) = ( 12 T−1 , 12 T1 ) PN0 @ XbX
h?2Q`2K 3XRXj G2i N #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX 6Q` HH MQM@M2; iBp2 7mM+iBQMb f : Ω → R
E [f ] = λE0N (f ◦ θx ) 1V0 (N ) (x) dx . Rm
S`QQ7X G2i h(x, ω) := 1{N (ω,B(0,||x||))=0} , r?2`2 B(0, ||x||) Bb i?2 QT2M # HH Q7 ` /Bmb ||x|| +2Mi2`2/ i 0X "v G2KK 3XRX9 #2HQr- i?2`2 Bb HKQbi bm`2Hv MQ T B` Q7 TQBMib Q7 N i i?2 b K2 /Bbi M+2 7`QK ;Bp2M a ∈ Rm M/ r2 K v i?2`27Q`2 bb2`i i? i HKQbi bm`2Hv h(x, ω) N (ω, dx) = 1 . Rm
UAM/22/- i?2 #Qp2 BMi2;` H +QmMib i?2 TQBMib Xn bm+? i? i i?2`2 Bb MQ TQBMi Q7 N BMbB/2 i?2 QT2M # HH Q7 +2Mi2` 0 M/ ` /Bmb ||Xn ||- M/ i?2`2 Bb QM2 M/ QMHv QM2 bm+? TQBMi- M K2Hv i?2 TQBMi +HQb2bi iQ 0XV LQr
E [f ] = E f h(x) N (dx) = E f h(x) N (dx) m Rm R
=E (f ◦ θ−x ◦ θx ) (h(x) ◦ θ−x ◦ θx ) N (dx) Rm
= λE0N (f ◦ θ−x ) (h(x) ◦ θ−x ) dx . Rm
h?2`27Q`2- Q#b2`pBM; i? i h(x) = h(−x) M/ i? i mM/2` i?2 S HK T`Q# #BHBivx ∈ Rm #2HQM;b iQ i?2 oQ`QMQś +2HH V0 (N ) B7 M/ QMHv B7 i?2`2 Bb MQ TQBMi Q7 N i /Bbi M+2 < ||x|| 7`QK x E [f ] = λE0N
Rm
(f ◦ θx ) (h(x) ◦ θx ) dx = λE0N
Rm
(f ◦ θx ) 1V0 (N ) (x) dx .
G2KK 3XRX9 6Q` bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM Rm - i?2`2 HKQbi bm`2Hv /Q2b MQi 2tBbi T B` Q7 TQBMib i i?2 b K2 /Bbi M+2 7`QK i?2 Q`B;BMX S`QQ7X .2MQi2 #v F i?2 2p2Mi i? i i?2`2 2tBbi i H2 bi irQ TQBMib i i?2 b K2 /Bbi M+2 7`QK i?2 Q`B;BMX q2 ? p2
3XRX h>1 oP_PLPA *1GG L. h>1 ALo1_aAPL 6P_JlG P (F ) ≤ E
R
jyR
m
1{N ({y ; y=x,||y||=||x||})≥1} N (dx)
1{(N −x)({y ; y =0,||y +x||=||x||})≥1} N (dx) Rm
= λE0N 1{N ({y ; y =0,||y +x||=||x||})≥1} dx m R
0 ≤ λEN 1{||y +x||=||x||} N (dy ) dx Rm Rm \{0}
1{||y +x||=||x||} dx N (dy ) ≤ λE0N Rm \{0} Rm
m ({x ; ||y + x|| = ||x||}) N (dy ) = 0 . = λE0N
=E
Rm \{0}
_2K `F 3XRX8 h?Bb H2KK HHQrb mb iQ /2}M2 HKQbi@bm`2Hv mM K#B;mQmbHv i?2 TQBMi R1 +HQb2bi 7`QK i?2 Q`B;BMX G2i N #2 i?2 +QQ`/BM i2 T`Q+2bb Q7 i?2 + MQMB+ H K2 bm` #H2 bT +2 Q7 TQBMi K2 bm`2b (Mp (Rm ), Mp (Rm ))X h?2 BMp2`bBQM 7Q`KmH Q7 h?2Q`2K 3XRXj i?2M i F2b i?2 7QHHQrBM; 7Q`K, 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM g : M (Rm ) → R
0 g(N − x) 1V0 (N ) (x) dx . E [g(N )] = λEN Rm
1t KTH2 3XRXe, h?2 QM2@/BK2MbBQM H + b2X AM i?2 mMB/BK2MbBQM H + b2 M/ mM/2` i?2 S HK T`Q# #BHBiv- i?2 oQ`QMQś +2HH i 0 = T0 Bb i?2 BMi2`p H 12 T−1 , 12 T1 M/ i?2`27Q`2 i?2 BMp2`bBQM 7Q`KmH `2 /b 0 E [f ] = λEN f ◦ θt dt . [ 12 T−1 , 12 T1 ]
h?2 1t+? M;2 6Q`KmH 6Q` bT iB H TQBMi T`Q+2bb2b- i?2`2 2tBbib M 2t+? M;2 7Q`KmH bBKBH ` iQ i?2 QM2 BM i?2 QM2@/BK2MbBQM H + b2X _2+ HH i?2 MQi iBQM ∂C 7Q` i?2 #QmM/ `v rBi? `2bT2+i iQ i?2 1m+HB/2 M iQTQHQ;v Q7 b2i C ⊂ Rm X h?2Q`2K 3XRXd G2i (P, θx ) #2 bi iBQM `v 7` K2rQ`F M/ H2i N M/ N #2 irQ θx @+QKT iB#H2 U M/ i?2`27Q`2V DQBMiHv bi iBQM `v TQBMi T`Q+2bb2b QM Rm rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λ M/ λ `2bT2+iBp2HvX 6Q` Mv MQM@M2; iBp2 7mM+iBQM f 7`QK (Mp (E) × Mp (E), Mp (E) ⊗ Mp (E)) iQ (R, B(R))- mM/2` i?2 +QM/BiBQM i? i E0N [N (∂V0 (N )] = 0
0 0 λEN [f (N, N )] = λ EN f (N − x, N − x)1V0 (N )(x) N (dx) . Rm
jyk
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
S`QQ7X lbBM; i?2 +H bbB+ H KQMQiQM2 +QMp2`;2M+2 `;mK2Mi- r2 K v bbmK2 f Bb #QmM/2/- b v #v 1X "v J2+F2Ƕb 7Q`KmH ,
E f (x, N − x, N − x) N (dx) = λ E0N [f (x, N, N )] dx . U3XRV Rm
Rm
"v /2}MBiBQM Q7 S HK T`Q# #BHBiv M/ U3XRV
0 f (N − x, N − x) N (dx) λEN [f (N, N )] = E [0,1]m
f (N − x, N − x)1Vy (N ) (x) N (dx) N (dy) + A − B , E [0,1]m
r?2`2
Rm
A := E Rm \[0,1]m
M/
[0,1]m
f (N − x, N − x)1Vy
(N )
(x) N (dx)
B := E [0,1]m
N (dy)
Rm \[0,1]m
f (N − x, N − x)1Vy
(N )
(x)N (dx)
N (dy)
`2 bm+? i? i A − B = 0- b rBHH #2 T`Qp2/ H i2`X h?2 T`QQ7 Q7 i?2 i?2Q`2K Bb i?2M ;Bp2M BM i?2 7QHHQrBM; + H+mH iBQMb,
f (N − x, N − x)1Vy (N ) (x) N (dx) N (dy) E [0,1]m Rm
f (N − x, N − x)1Vy−y (N −y) (x − y) N (dx) N (dy) =E [0,1]m Rm
f (N − x − y, N − x − y)1V0 (N −y) (x) (N − y)(dx) N (dy) =E [0,1]m Rm
f (N − x, N − x)1V0 (N )(x) N (dx) . = λ E0N Rm
Ai `2K BMb iQ b?Qr i? i A − B = 0X q2 }`bi MQi2 i? i
1Vy (N ) (x) N (dy) N (dx) = E [N ([0, 1]m )] < ∞ , A≤E [0,1]m
Rm \[0,1]m
bBM+2 Rm \[0,1]m 1Vy (N ) (x) N (dy) ≤ Rm 1Vy (N ) (x) N (dy) = 1X h?2`27Q`2- BM Q`/2` iQ T`Qp2 i? i A − B = 0- Bi bm{+2b iQ b?Qr i? i A = B- r?B+? r2 MQr T`Q+22/ iQ /QX h?2 bT +2 Rm #2BM; i?2 +QmMi #H2 mMBQM ∪r ([0, 1]m + r)- r2 ? p2 i? i
E f (N − x, N − x)1Vy (N ) (x) N (dx) N (dy) A=
r=0
M/
[0,1]m +r
[0,1]m
3XkX h>1 GP*G ALh1_S_1hhAPL B=
E r=0
[0,1]m
[0,1]m +r
jyj
f (N − x, N − x)1Vy (N ) (x) N (dx) N (dy) .
6BM HHv- 7Q` 2 +? r = 0
E f (N − x, N − x)1Vy (N ) (x) N (dx) N (dx) [0,1]m +r [0,1]m
f (N − x, N − x)1Vy+r (N ) (x) N (dx) (N − r)(dy) =E [0,1]m [0,1]m
f (N − x, N − x)1Vy (N −r) (x − r) N (dx) (N − r)(dy) =E [0,1]m [0,1]m
f (N − x − r, N − x − r)1Vy (N −r) (x) (N − r)(dx) (N − r)(dy) =E [0,1]m [0,1]m
f (N − x, N − x)1Vy (N ) (x) N (dx) N (dy) . =E [0,1]m
[0,1]m +r
3Xk
h?2 GQ+ H AMi2`T`2i iBQM
h?2Q`2K dX8XR Bb MQr ;2M2` HBx2/ iQ TQBMi T`Q+2bb2b QM Rm X q2 bi `i rBi? i?2 ;2M2` HBx iBQM Q7 i?2 EQ`QHvmFĜ.Q#`mb?BM 2biBK i2b Q7 h?2Q`2K dX8XkX h?2Q`2K 3XkXR G2i {Cn }n≥1 #2 MQM@BM+`2 bBM; b2[m2M+2 Q7 +QKT +i b2ib BM Rm bm+? i? i ∩n≥1 Cn = ∅X h?2M lim
P (N (Cn ) = 1) =λ
m (Cn )
lim
P (N (Cn ) > 1) = 0.
m (Cn )
n↑∞
M/ n↑∞
S`QQ7X G2i C ∈ B(Rm )X TTHv 7Q`KmH UdXdV rBi? v(x, ω) = N (ω, C)1{N (ω,C)≥2} iQ Q#i BM
0 E N (C)1{N (C)≥2} = E 1{N (C)≥2} N (dx) = λEN 1{N (C−x)≥2} dx . () C
C
P#b2`p2 i? i 7Q` HH x ∈ C- N (C − x) ≤ N (C − C) Ur?2`2 C − C := {x − y ; x ∈ C, y ∈ C}VX h?2`27Q`2- #v UV E N (C)1{N (C)≥2} ≤ λ m (C)PN0 (N (C − C) ≥ 2) . qBi? C = Cn b BM i?2 bi i2K2Mi Q7 i?2 i?2Q`2K- r2 ? p2 i? i {N (Cn −Cn ) ≥ 2} ↓ ∅ M/ i?2`27Q`2- #v /QKBM i2/ +QMp2`;2M+2- bBM+2 E [N (Cn )] ≤ E [N (C1 )] < ∞-
jy9
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1 ( m (Cn ))−1 E N (Cn )1{N (Cn )≥2} → 0 .
AM T `iB+mH ` LQr
( m (Cn ))−1 E 1{N (Cn )≥2} → 0 . E [N (Cn )] = E N (Cn )1{N (Cn )≥2} + P (N (Cn ) = 1)
M/ i?2`27Q`2
E N (Cn )1{N (Cn )≥2} E [N (Cn )] − P (N (Cn ) = 1) = → 0.
m (Cn )
m (Cn )
AM T `iB+mH `- bBM+2 E [N (Cn )] = λ m (Cn )lim ( m (Cn ))−1 P (N (Cn ) = 1) = λ .
n↑∞
G2i R1 /2MQi2 i?2 HKQbi bm`2Hv mMB[m2 Ub22 _2K `F 3XRX8V TQBMi Q7 N +HQb2bi iQ i?2 Q`B;BMX h?2Q`2K 3XkXk G2i {Cn }n≥1 #2 /2+`2 bBM; b2[m2M+2 Q7 +HQb2/ bT?2`2b +2Mi2`2/ i i?2 Q`B;BM M/ bm+? i? i ∩n Cn = ∅X G2i f : Mp (Rm ) → R #2 #QmM/2/ K2 bm` #H2 7mM+iBQMX h?2M lim E [f (SR1 (N )) | N (Cn ) > 0] = E0N [f (N )] .
n↑∞
S`QQ7X q`Bi2 & & &E [f (SR1 (N )) | N (Cn ) > 0] − E0N [f (N )]& & & & & E f (S (N ))1 R1 {N (Cn )>0} & & 0 − EN [f (N )]& =& & & P (N (Cn ) > 0) & & & E f (S (N ))1 & m 0 R1 {N (Cn )>0} − λ (Cn )EN [f (N )] & & ≤& & & & λ m (Cn ) & E f (SR1 (N ))1{N (Cn )>0} && −1 + λ − m (Cn )P (N (Cn ) > 0)−1 & . m
(Cn ) "v h?2Q`2K 3XkXR- i?2 KQ/mHmb BM i?2 H bi i2`K Q7 i?2 `B;?i@? M/ bB/2 i2M/b iQ 0 b n ↑ ∞ r?BH2 E f (SR1 (N ))1{N (Cn )>0} P (N (Cn ) > 0) ≤ (sup f ) → λ(sup f ) < ∞ .
m (Cn )
m (Cn ) h?2`27Q`2- i?2 H bi i2`K Q7 i?2 `B;?i@? M/ bB/2 i2M/b iQ 0X aBM+2 #v /2}MBiBQM Q7 S HK T`Q# #BHBiv m 0 f (SXi (N )1{Xi ∈Cn } , λ (Cn )EN [f (N )] = E i∈N
3XjX 1_:P.A*Ahu
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Bi Dmbi `2K BMb iQ T`Qp2 i? i & & & & 1 & & E f (S (N )1 − f (S (N )1 & R1 Xi {N (Cn )>0} {Xi ∈Cn } & → 0 . & &
m (Cn ) i∈N
h?Bb [m MiBiv Bb MmHH B7 N (Cn ) = 0X A7 N (Cn ) > 0- R1 Bb QM2 Q7 i?2 Xi Ƕb BM Cn h?2`27Q`2 i?Bb [m MiBiv Bb #QmM/2/ #v (sup f )
P (N (Cn ) ≥ 2) → 0.
m (Cn )
L2 `2bi L2B;?#Qm` M/ 1KTiv aT +2 .Bbi M+2b b BM i?2 + b2 Q7 TQBMi T`Q+2bb2b QM i?2 HBM2- i?2 HQ+ H BMi2`T`2i iBQM i2HHb mb i? i 7Q` bi iBQM `v TQBMi T`Q+2bb i?2 S HK /Bbi`B#miBQM /2b+`B#2b i?2 bBim iBQM i dzivTB+ HǴ TQBMiX lM/2` i?2 S HK /Bbi`B#miBQM- i?2 Q`B;BM `2T`2b2Mib bm+? ivTB+ H TQBMiX 6Q` BMbi M+2 i?2 [m MiBiv G(r) := PN0 (N (B(0, r) > 1) r?2`2 B(0, r) Bb i?2 +HQb2/ # HH Q7 ` /Bmb r +2Mi2`2/ i i?2 Q`B;BM Bb BMi2`T`2i2/ b i?2 T`Q# #BHBiv i? i i?2`2 Bb MQ TQBMi i /Bbi M+2 ≤ r 7`QK ivTB+ H TQBMi Q7 i?2 TQBMi T`Q+2bbX PM i?2 Qi?2` ? M/- i?2 [m MiBiv F (r) := P (N (B(0, r) > 0) Bb i?2 T`Q# #BHBiv i? i i?2`2 Bb MQ TQBMi Q7 i?2 bi iBQM `v TQBMi T`Q+2bb i /Bbi M+2 ≤ r 7`QK M `#Bi` `v HQ+ iBQMX h?2 o M GB2b?QmiĜ" //2H2v +Hmbi2`BM; BM/2t J(r) :=
1 − G(r) 1 − F (r)
+QKT `2b M2 `2bi@M2B;?#Qm` M/ 2KTiv bT +2 /Bbi M+2bX dzAi i F2b p Hm2b H2bb i? M 1 r?2M2p2` i?2 2KTiv bT +2b i2M/ iQ #2 H `;2` i? M i?2 /Bbi M+2 #2ir22M M2 `2bi M2B;?#Qm` T B`b- i?mb BM/B+ iBM; +Hmbi2`BM;- r?2`2 b p Hm2b 2t+22/BM; 1 bm;;2bi KQ`2 `2;mH ` T ii2`MXǴk AM T `iB+mH `- 7Q` ?QKQ;2M2Qmb SQBbbQM T`Q+2bb- J(r) = 1 7Q` HH r > 0X
3Xj
1`;Q/B+Biv
q2 MQr ;Bp2 T `iB H M HQ;m2 Q7 UdXjyV BM h?2Q`2K dXeXRkX G2i (P, θx , N ) #2 bi iBQM `v 2`;Q/B+ TQBMi T`Q+2bb QM Rm rBi? }MBi2 TQbBiBp2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 bbQ+B i2/ S HK T`Q# #BHBivX k
6`QK (o M GB2b?Qmi- kyyy)- TX jdc (o M GB2b?Qmi M/ " //2H2v- RNNe)X
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_2+ HH i?2 /2}MBiBQM Q7 +QMp2t p2` ;BM; b2[m2M+2 U.2}MBiBQM dXRXR3VX MQM@ /2+`2 bBM; b2[m2M+2 {An }n≥1 Q7 #QmM/2/ +QMp2t bm#b2ib Q7 Rm Bb + HH2/ +QMp2t p2` ;BM; b2[m2M+2 B7 lim sup{r ; B(0; r) ⊂ An } = ∞ .
n↑∞
h?2Q`2K 3XjXR G2i {An }n≥1 #2 +QMp2t p2` ;BM; b2[m2M+2 Ub22 .2}MBiBQM dXRXR3VX A7 f Bb PN0 @BMi2;` #H2 ` M/QK p `B #H2 1 f ◦ θx N (dx) = λE0N [f ] P @ XbX lim m n (An ) A n AM T `iB+mH `lim
n↑∞
N (An ) = λ.
m (An )
S`QQ7X ai2T RX f ≡ 1 Ui?2 T `iB+mH ` + b2 BM i?2 i?2Q`2KVX 6Q` Mv K2 bm` #H2 b2i A Q7 Rm M/ Mv ε > 0- /2}M2 i?2 ε@ TT`QtBK iBQMb Q7 A 7`QK #Qp2 M/ 7`QK #2HQr `2bT2+iBp2Hv #v Aε := ∪x∈A B(x, ε) M/ Aε := {x ∈ A ; B(x, ε) ⊂ A} . G2i MQr gε : Rm → R+ #2 MQM@M2; iBp2 +QMiBMmQmb 7mM+iBQM BMi2;` iBM; iQ 1 Qp2` Rm M/ p MBb?BM; QmibB/2 B(0, ε) M/ H2i hε := gε (x) N (dx) . Rm
AM T `iB+mH `- #v * KT#2HHǶb 7Q`KmH E[hε ] = λ
gε (x) dx = λ . Rm
P#b2`p2 7Q` 7mim`2 mb2 i? i
hε ◦ θt =
Rm
gε (x − t) N (dx) .
()
qBi? A ≡ An - r2 ? p2 i? i gε (y − t) dt = 1
(y ∈ A)
gε (y − t) dt = 0
(y ∈ A) .
Aε
M/
Aε
h?2`27Q`2
gε (y − t) dt ≤ 1A (y) ≤ Aε
gε (y − t) dt . Aε
()
3XjX 1_:P.A*Ahu
jyd
AMi2;` iBM; rBi? `2bT2+i iQ N gε (y − t) dt N (dy) ≤ N (A) ≤ Rm
Aε
aBM+2 r? i2p2` A ∈ B(Rm ) gε (y − t) dt N (dy) = Rm
A
A
Aε
Rm
gε (y − t) dt
( )
Rm
N (dy) .
gε (y − t) N (dy)
hε ◦ θt dt ,
dt = A
i?2 #QmM/b BM U V #2+QK2- K FBM; mb2 Q7 UV hε ◦ θt dt ≤ N (A) ≤ hε ◦ θt dt . Aε
Aε
_2im`MBM; iQ i?2 7mHH MQi iBQM A = An lim sup n
N (An )
m ((An )ε ) 1 ≤ lim sup m m
(An )
(An ) m ((An )ε ) n
hε ◦ θt dt . (An )ε
"v 2`;Q/B+Biv- i?2 b2+QM/ 7 +iQ` Q7 i?2 `B;?i@? M/ bB/2 +QMp2`;2b iQ λX h?2 }`bi 7 +iQ` +QMp2`;2b iQ 1 U?2`2 ; BM r2 mb2 i?2 /2}MBiBQM Q7 +QMp2t p2` ;BM; b2@ [m2M+2VX h?2`27Q`2 N (An ) lim sup m ≤ λ. n (An ) aBKBH ` `;mK2Mib vB2H/ lim inf n
N (An )
m ((An )ε ) 1 ≥ lim inf m m n
(An )
(An ) m ((An )ε )
M/ i?2M lim inf n
hε ◦ θt dt (An )ε
N (An ) ≥ λ.
m (An )
ai2T kX f BMi2;` #H2X h?2 `;mK2Mib 7QHHQr +HQb2Hv i?Qb2 Q7 ai2T RX h?Bb iBK2- H2i gε (x)(f ◦ θx ) N (dx) . hε := Rm
AM T `iB+mH ` UJ2+F2Ƕb 7Q`KmH V E0N [gε (x)f ] dx = λE0N [f ] E[hε ] = λ Rm
Rm
P#b2`p2 7Q` 7mim`2 mb2 i? i gε (x)(f ◦ θx ◦ θt ) (N ◦ θt )(dx) = hε ◦ θ t = Rm
gε (x) dx = λE0N [f ] .
Rm
gε (x − t)(f ◦ θx ) N (dx) .
JmHiBTHvBM; #Qi? bB/2b Q7 UV #v f ◦ θy M/ BMi2;` iBM; rBi? `2bT2+i iQ N -
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Rm
gε (y − t) dt (f ◦ θy ) N (dy) ≤ (f ◦ θy ) N (dy) Aε A gε (y − t) dt (f ◦ θy ) N (dy) . ≤ Rm
Aε
aBM+2 7Q` Mv A ∈ B(Rm ) gε (y − t) dt (f ◦ θy ) N (dy) Rm A = gε (y − t)(f ◦ θy ) N (dy) dt = hε ◦ θt dt , A
r2 ? p2 i? i
Rm
hε ◦ θt dt ≤ Aε
h?2`27Q`2 lim sup n
An
A
(f ◦ θy ) N (dy)
m (An )
(f ◦ θy ) N (dy) ≤
hε ◦ θt dt .
(†)
Aε
A
m ((An )ε ) 1 hε ◦ θt dt
m (An ) m ((An )ε ) (An )ε n
m ((An )ε ) 1 ≤ lim sup m lim sup m hε ◦ θt dt ε
(An ) n n ((An ) ) (An )ε ≤ lim sup
= λE0N [f ] . aBKBH `Hv- bi `iBM; 7`QK i?2 `B;?i@? M/ BM2[m HBiv BM U†V (f ◦ θy ) N (dy) ≥ λE0N [f ] . lim inf An m n
(An ) :2M2` HBx2/ _2M2r H h?2Q`2K G2i (N, θx , P ) #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM Rm Q7 i?2 b2+QM/ Q`/2`- Q7 BMi2MbBiv λ- M/ H2i PN0 #2 i?2 bbQ+B i2/ S HK T`Q# #BHBivX G2i σ(C) := E0N [N (C)] UC ∈ B(Rm )VX .2}MBiBQM 3XjXk h?2 T`Q# #BHBiv P Bb + HH2/ N @KBtBM; B7 7Q` HH G2#2b;m2 BMi2@ ;` #H2 7mM+iBQMb a, b : Rm → R p MBb?BM; QmibB/2 #QmM/2/ b2i
a(x)N (ω, dx) b(x)N (θx ω, dx) lim E ||x||↑∞ m Rm R
=E a(x)N (ω, dx) E b(x)N (ω, dx) m m R R = λ2 a(x) dx b(x) dx . Rm
Rm
_2K `F 3XjXj A7 P Bb KBtBM;- Bi Bb 7Q`iBQ`B N @KBtBM;X
3XjX 1_:P.A*Ahu
jyN
h?2Q`2K 3XjX9 Uj V A7 P Bb N @KBtBM;- i?2M 7Q` HH b : Rm → R p MBb?BM; QmibB/2 #QmM/2/ b2i 2 lim b(x)Sx σN (dx) = λ b(x) dx . ||x||↑∞
Rm
Rm
Uh?2 #Qp2 `2bmHi b vb- BM Qi?2` i2`Kb- i? i i?2 K2 bm`2 Sx σ +QMp2`;2b p ;m2Hv b ||x|| ↑ ∞ iQ λ2 iBK2b i?2 G2#2b;m2 K2 bm`2 QM Rm XV S`QQ7X q2 b? HH mb2 J2+F2Ƕb 7Q`KmH BM i?2 7Q`K,
f (−x, ω, θx ω) PN0 (dω) dx ,
f (x, θx ω, ω)N (ω, dx)P (dω) = λ Ω
Rm
Rm
r?B+? ;Bp2b- rBi? i?2 MQi iBQM N (ω, a) :=
Rm
Rm
a(x)N (ω, dx) M/ i?2 HBF2
a(x)N (ω, dx) Ω
Ω
b(x)N (θx ω, dx) P (dω) R
m
a(x)N (θx ω, b)N (ω, dx) P (dω)
= Ω Rm
=λ a(−x)N (θx+t ω, b) dx PN0 (dω) m Ω R = N (θx ω, b ∗ a) PN0 (dω) = (b ∗ a)(x)Sx σ(dx) , Rm
Ω
r?2`2 a(x) := a(−x)X h?2`27Q`2
Rm
(b ∗ a)(x)Sx σ(dx) → λ
2
a(x) dx Rm
b(x) dx , Rm
7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrb #v bi M/ `/ `;mK2Mi mbBM; b2[m2M+2 Q7 7mM+iBQMb an : R → R+ bm+? i? i Rm an (x) dx = 1 M/ b ∗ an → bX
_2K `F 3XjX8 aBM+2 bi iBQM `v `2M2r H i?2Q`2K rBi? MQM@H iiB+2 /Bbi`B#miBQM Bb P @KBtBM;- i?2 +H bbB+ H `2M2r H i?2Q`2K + M #2 b22M b T `iB+mH ` + b2 Q7 i?2 #Qp2 `2bmHiX >Qr2p2`- i?Bb Bb bQK2r? i KBbH2 /BM; bi i2K2Mi- bBM+2 i?2 T`QQ7 i? i bi iBQM `v MQM@H iiB+2 `2M2r H TQBMi T`Q+2bb Bb KBtBM; Bb MQi i HH 2 bv- M/ Bi ? b H`2 /v #22M MQi2/9 i? i i?2 T`QQ7 Q7 i?2 KBtBM; T`QT2`iv 7Q` `2M2r H T`Q+2bb2b 7QHHQrb `;mK2Mib i? i `2 p2`v +HQb2 iQ i?Qb2 mb2/ BM i?2 T`QQ7 Q7 "H +Fr2HHǶb i?2Q`2KX
j 9
ii`B#mi2/ iQ L2p2m BM (.2H bM2`B2- RNdd)X (. H2v M/ o2`2@CQM2b- RN33)- TX 93NX
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3X9 h?2 J2+F2 J2 bm`2 G2i E #2 M HX+X/X#X bT +2 M/ H2i B(E) #2 i?2 bbQ+B i2/ "Q`2H σ@}2H/X G2i N #2 ` M/QK K2 bm`2 QM E rBi? HQ+ HHv }MBi2 K2 M K2 bm`2 νX G2KK 3X9XR h?2 b2i 7mM+iBQM C : B(E) × F → R+ /2}M2/ QM i?2 `2+i M;H2b A × B UA ∈ B(E), F ∈ F V #v C(A × F ) := E[N (A)1F ]
U3XkV
+ M #2 mMB[m2Hv 2ti2M/2/ iQ σ@}MBi2 K2 bm`2 C : B(E) ⊗ F → [0, ∞]X S`QQ7X h?2 +H bb Q7 b2ib S := {A × F ; A ∈ B(E), F ∈ F} 7Q`Kb b2KB@ H;2#` X h?2 7mM+iBQM C : S → [0, ∞] /2}M2/ #v U3XkV Bb +H2 `Hv +QmMi #Hv //BiBp2X A7 Bi + M #2 b?QrM iQ #2 σ@}MBi2- i?2 `2bmHi 7QHHQrb 7`QK * ` i?ûQ/Q`vǶb 2ti2MbBQM i?2Q`2K Uh?2Q`2K XRXRkV bBM+2 B(E) ⊗ F = σ(S)X hQ T`Qp2 σ@}MBi2M2bb- +QMbB/2` +QmMi #H2 +Qp2`BM; Q7 E #v `2H iBp2Hv +QKT +i b2ib Am Um ∈ NV M/ H2i Fmn := {ω; N (ω, Am ) ≤ n} Um, n ∈ NVX h?2 7 KBHv {Am × Fm,n ; m, n ∈ N} Bb +Qp2`BM; Q7 E × F X AM/22/- 7Q` Mv (x, ω) ∈ E × F - i?2`2 2tBbib M Am bm+? i? i x ∈ Am - M/ 7Q` bm+? }t2/ Am - i?2`2 2tBbib M n ∈ N bm+? i? i N (Am ) ≤ n- bBM+2 N (Am ) < ∞X h?2`27Q`2 (x, ω) ∈ Am × Fmn X JQ`2Qp2` C(Am × Fmn ) = E[N (Am )1Fmn ] ≤ nE[1Fmn ] ≤ n < ∞ r?B+? T`Qp2b σ@}MBi2M2bbX .2}MBiBQM 3X9Xk h?2 K2 bm`2 C /2}M2/ BM G2KK 3X9XR Bb + HH2/ i?2 J2+F2 K2 bm`2 Q7 i?2 ` M/QK K2 bm`2 N X h?2Q`2K 3X9Xj 6Q` HH MQM@M2; iBp2 K2 bm` #H2 g : E × Ω → [0, ∞] g(x, ω)N (ω, dx)P (dω) = g(x, ω) C(dx × dω) . Ω
E
E
U3XjV
Ω
S`QQ7X "v i?2 mbm H KQMQiQM2 +H bb `;mK2Mi- Bi bm{+2b iQ T`Qp2 i?2 H ii2` 2[m H@ Biv 7Q` 7mM+iBQMb Q7 i?2 ivT2 g(x, ω) = 1B (x, ω) (B ∈ B(E) ⊗ F) . AM/22/ i?2 +QHH2+iBQM D Q7 b2ib B ∈ B(E) ⊗ F 7Q` r?B+? 2[m HBiv U3XjV Bb i`m2 7Q` g = 1B Bb .vMFBM bvbi2K M/ +QMi BMb i?2 π@bvbi2K S = {A × F ; A ∈ B(E)F ∈ F} #v /2}MBiBQM Q7 i?2 K2 bm`2 CX h?2`27Q`2 D ⊇ σ(S) = B(E) ⊗ F X h?2 `2bmHi Bb i?2M Q#i BM2/ 7`QK i?2`2- #v i FBM; i?2 mbm H `Qmi2 7`QK BM/B+ iQ` 7mM+iBQMb iQ bBKTH2 7mM+iBQMb M/ i?2M iQ MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMbX *Q`QHH `v 3X9X9 G2i g : E × Ω → R #2 K2 bm` #H2 7mM+iBQM bm+? i? i 2Bi?2` QM2 Q7 i?2 irQ 7QHHQrBM; +QM/BiBQMb Bb b iBb}2/, |g(x, ω)|N (ω, dx)P (dω) < ∞ Q` |g(x, ω)|C(dx × dω) < ∞ . Ω
E
E
Ω
h?2M i?2 Qi?2` +QM/BiBQM Bb b iBb}2/ M/ 7Q`KmH U3XjV TTHB2bX
3X9X h>1 J1*E1 J1al_1
jRR
h?2 +QM/BiBQMb T`2p BHBM; #Qp2 `2 K BMi BM2/X 6Bt F ∈ F- M/ /2MQi2 #v CF i?2 K2 bm`2 QM (E, B(E)) /2}M2/ #v CF (A) = C(A × F )X *H2 `Hv CF ν- M/ F i?2`27Q`2- i?2`2 2tBbib _ /QMĜLBFQ/ɷK /2`Bp iBp2 dC /2}M2/ 7Q` ν@ HKQbi HH dν x ∈ E, dCF C(A × F ) = (x) ν(dx) . U3X9V A dν G2i dCF Px (F ) := (x) . dν h?Bb ν@ HKQbi@2p2`vr?2`2 /2}M2/ 7mM+iBQM i?2`27Q`2 b iBb}2b C(A × F ) = Px (F ) ν(dx) . U3X8V A
AM T `iB+mH `-
ν(A) = C(A × Ω) =
Px (Ω) ν(dx) A
M/ i?2`27Q`2 Px (Ω) = 1- i? i Bb- Px Bb ν@ X2X T`Q# #BHBiv K2 bm`2X .2}MBiBQM 3X9X8 h?2 T`Q# #BHBiv Px bQ /2}M2/ Uν@ X2XV Bb + HH2/ i?2 S HK T`Q# @ #BHBiv Q7 N i x ∈ EX KQ`2 +QKTH2i2 MQi iBQM rBHH bQK2iBK2b #2 mb2/ BM Q`/2` iQ bT2+B7v i?2 TQBMi T`Q+2bb BMpQHp2/, PN,x X q2 rQmH/ HBF2 iQ #2 #H2 iQ +?QQb2 i?2 T`Q# #BHBiB2b Px Ux ∈ EV BM bm+? r v i? i 7Q` ν@ HKQbi HH x ∈ E F → Px (F ) Bb T`Q# #BHBiv K2 bm`2 QM (Ω, F) M/ 7Q` }t2/ F
x → Px (F ) Bb B(E)@K2 bm` #H2 .
h?2 2tBbi2M+2 Q7 bm+? `2;mH ` p2`bBQM Bb ;m ` Mi22/ mM/2` +2`i BM +QM/BiBQMb- 7Q` BMbi M+2- #v i?2 K2 bm`2 /BbBMi2;` iBQM i?2Q`2K Uh?2Q`2K XRXRjV- B7 i?2 bT +2 Ω Bb +QKTH2i2 b2T ` #H2 K2i`B+ bT +2 M/ F Bb i?2 "Q`2H }2H/ bbQ+B i2/ iQ i?2 iQTQHQ;v BM/m+2/ #v Bib K2i`B+X h?Bb Bb i?2 + b2 BM T `iB+mH ` B7 (Ω, F) ≡ (M (E), M(E)) #2+ mb2 i?2`2 2tBbib /Bbi M+2 QM M (E) K FBM; Q7 Bi +XbXKXbX h?2 T`Q# #BHBiv F2`M2H 7`QK E × Ω iQ (Ω, F)) /2}M2/ #v (x, A) → Px (A) Bb + HH2/ i?2 S HK F2`M2HX 6`QK MQr- i?2 2tBbi2M+2 Q7 `2;mH ` p2`bBQM Bb bbmK2/X AM pB2r Q7 U3X8V- i?2 7QHHQrBM; `2bmHi Bb Dmbi `2r`BiBM; Q7 7Q`KmH U3XjVX h?2Q`2K 3X9Xe 6Q` HH MQM@M2; iBp2 K2 bm` #H2 g : E × Ω → [0, ∞] g(x, ω)N (ω, dx)P (dω) = g(x, ω)Px (dω) ν(dx) . Ω
E
E
U3XeV
Ω
6Q`KmH U3XeV HbQ ?QH/b i`m2 7Q` Mv K2 bm` #H2 g : E × Ω → R bm+? i? i 2Bi?2` QM2 Q7 i?2 7QHHQrBM; BM2[m HBiB2b Bb i`m2, |g(x, ω)|N (ω, dx)P (dω) < ∞ Q` |g(x, ω)|Px (dω) ν(dx) < ∞ Ω
E
E
Ω
Ui?2 Qi?2` QM2 i?2M #2BM; miQK iB+ HHv b iBb}2/VX
jRk
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
.B`2+iBQM H .2`Bp iBp2 Q7 i?2 G TH +2 h` Mb7Q`K q2 MQr BMi`Q/m+2 mb27mH iQQH 7Q` i?2 /2i2`KBM iBQM Q7 i?2 G TH +2 i` Mb7Q`K mM/2` i?2 S HK T`Q# #BHBiv i ;Bp2M TQBMi xX .2}MBiBQM 3X9Xd G2i N #2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM E rBi? BMi2MbBiv K2 @ bm`2 ν M/ G TH +2 7mM+iBQM H LN X G2i ϕ M/ f #2 irQ MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb 7`QK E iQ RX h?2 [m MiBiv ∂LN LN (ϕ + εf ) − LN (ϕ) (ϕ, f ) := lim ε↓0 ∂f ε Bb + HH2/- B7 Bi 2tBbib- i?2 /B`2+iBQM H /2`Bp iBp2 i ϕ Q7 i?2 G TH +2 i` Mb7Q`K BM i?2 /B`2+iBQM f X h?2Q`2K 3X9X3 A7 f ∈ L1E (ν)- i?2 /2`Bp iBp2 i ϕ Q7 i?2 G TH +2 i` Mb7Q`K BM i?2 /B`2+iBQM f 2tBbib M/ Bb ;Bp2M #v i?2 7Q`KmH ∂LN (ϕ, f ) = − f (x)Lx (ϕ)ν(dx) , U3XdV ∂f E exp − ϕ(y) N (ω, dy) Px (dω)
r?2`2 Lx (ϕ) :=
Ω
E
Bb i?2 G TH +2 /Bbi`B#miBQM Q7 N mM/2` i?2 T`Q# #BHBiv K2 bm`2 Px X S`QQ7X .Bz2`2MiB iBQM rBi? `2bT2+i iQ ε i i?2 p Hm2 ε = 0 Q7
LN (ϕ + εf ) = E exp − (ϕ(x) + εf (x)) N (dx) E
= E exp − ϕ(x) N (dx) exp −ε f (x) N (dx) E
E
;Bp2b
∂LN (ϕ, f ) = E exp − ϕ(y) N (dy) f (x) N (dx) ∂f E E
=E f (x) exp − ϕ(y) N (dy) N (dx) . E
E
Ai bm{+2b MQr iQ TTHv 7Q`KmH U3XeV rBi? g(x, ω) := f (x) exp − ϕ(y) N (ω, dy) E
iQ Q#i BM ∂LN (ϕ, f ) = ∂f
exp − ϕ(y) N (ω, dy) Px (dω) ν(dx) ,
f (x) E
r?B+? Bb i?2 MMQmM+2/ `2bmHiX
Ω
E
3X9X h>1 J1*E1 J1al_1
jRj
1t KTH2 3X9XN, S HK F2`M2H Q7 SQBbbQM T`Q+2bbX A7 N Bb mM/2` P SQBbbQM T`Q+2bb Q7 HQ+ HHv }MBi2 BMi2MbBiv K2 bm`2 ν −ϕ(x) LN (ϕ) = exp e − 1 ν(dx) E
M/ i?2`27Q`2
e−ϕ(x)−εf (x) − 1 ν(dx)
LN (ϕ + εf (x)) = exp
E
7`QK r?B+? Bi 7QHHQrb i? i 1 −ϕ(x) 1 −εf (x) LN (ϕ + εf ) − LN (ϕ) = LN (ϕ) exp (e − 1) ν(dx) e ε ε E M/ i?2`27Q`2- H2iiBM; ε → 0 ∂LN (ϕ, f ) = − f (x)LN (ϕ)e−ϕ(x) ν(dx) . ∂f E h?2`27Q`2- 7`QK U3XdV f (x)Lx (ϕ)ν(dx) = f (x)LN (ϕ)e−ϕ(x) ν(dx) . E
E
h?Bb #2BM; i`m2 7Q` HH #QmM/2/ f Lx (ϕ) = LN (ϕ)e−ϕ(x)
(ν@ X2X) .
−ϕ(x)
Bb i?2 G TH +2 7mM+iBQM H Q7 i?2 bBKTH2 /2i2`KBMBbiB+ TQBMi T`Q+2bb "mi ϕ → e rBi? bBM;H2 TQBMi x- M/ i?2`27Q`2 ϕ → LN (ϕ)e−ϕ(x) Bb i?2 G TH +2 i` Mb7Q`K Q7 SQBbbQM TQBMi T`Q+2bb Q7 BMi2MbBiv K2 bm`2 ν iQ r?B+? QM2 //b TQBMi i xX h?Bb b?Qrb i? i- ν@ HKQbi@2p2`vr?2`2- N ? b mM/2` Px i?2 b K2 /Bbi`B#miBQM b N + εx mM/2` P X 1t KTH2 3X9XRy, G2i X1 , . . . , Xk #2 BB/ ` M/QK 2H2K2Mib Q7 (E, B(E)) M/ H2i N :=
k
εX j .
j=1
h?2 G TH +2 i` Mb7Q`K Q7 i?Bb TQBMi T`Q+2bb Bb k LN (ϕ) = E e−ϕ(X1 ) M/ M 2H2K2Mi `v +QKTmi iBQM ;Bp2b k−1 ∂LN (ϕ, f ) = −E e−ϕ(X1 ) f (x)e−ϕ(x) kP (X1 ∈ dx) . ∂f E h?2`27Q`2 k−1 −ϕ(x) Lx (ϕ) = E e−ϕ(X1 ) e , r?B+? Bb i?2 G TH +2 i` Mb7Q`K Q7 TQBMi T`Q+2bb Nx :=
k−1 j=1
ε X j + εx .
jR9
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h?2 ai iBQM `v * b2 h?2Q`2K 3X9XRR G2i N #2- mM/2` P - bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM Rm Q7 }MBi2 BMi2MbBivX h?2M Px (N ∈ Γ) = PN0 (N + x ∈ Γ) U3X3V 7Q` HKQbi HH x rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 QM Rm X
S`QQ7X >2`2 ν(dx) = λ dxX TTHvBM; U3XeV rBi? g(x, ω) := 1C (x)1Γ (N (ω) + x) (C ∈ B(Rm ) , Γ ∈ M(Rm ) , r2 Q#i BM 7Q` i?2 `B;?i@? M/ bB/2 λ C Px (N + x ∈ Γ) dx M/ 7Q` i?2 H27i@? M/ bB/2mbBM; J2+F2Ƕb 7Q`KmH 7Q` bi iBQM `v TQBMi T`Q+2bb2b E
h?2`27Q`2
PN0 (N + x ∈ Γ) dx .
1C (Xn )1Γ (N (ω) + Xn ) = λ C
n∈N
Px (N ∈ Γ) dx = C
PN0 (N + x ∈ Γ) dx . C
aBM+2 C Bb `#Bi` `v BM B(Rm )- i?2 `2bmHi 7QHHQrbX Ai b vb i? i 7Q` HKQbi HH x UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2V- i?2 /Bbi`B#miBQM Q7 N mM/2` Px Bb i?2 BK ;2 Q7 i?2 /Bbi`B#miBQM Q7 N mM/2` PN0 #v i?2 K TTBM; N → N + xX *Q`QHH `v 3X9XRk A7 N Bb mM/2` P bBKTH2 bi iBQM `v TQBMi T`Q+2bb QM E Q7 }MBi2 BMi2MbBiv- Px (N ({x} = 1) = 1 7Q` HKQbi HH x rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 QM EX S`QQ7X h F2 Γ := {μ ; μ({x}) = 1} BM U3X3VX AM T `iB+mH `- {N ∈ Γ} ≡ {N ({x}) = 1} M/ {N + x ∈ Γ} ≡ {N ({0}) = 1}X h?2`27Q`2 U3X3V `2 /b 7Q` i?Bb +?QB+2 Q7 ΓPx (N ({x}) = 1) = PN0 (N ({0}) = 1) , i? i Bb- #v h?2Q`2K dXkXd- Px (N ({x}) = 1) = 1X
h?2Q`2K 3X9XRR Bb bi i2/ BM i2`Kb Q7 i?2 /Bbi`B#miBQM Q7 N X >2`2 Bb i?2 #bi` +i 7Q`K Q7 Bi BM i?2 θx @7Q`K HBbK Ui?Bb Bb KQ`2 ;2M2` H b Bi HHQrb mb iQ +QMbB/2` ` M/QK 2H2K2Mib Qi?2` i? M i?2 TQBMi T`Q+2bb N Bib2H7VX h?2Q`2K 3X9XRj G2i (N, θx , P ) #2 bi iBQM `v TQBMi T`Q+2bb QM Rm X h?2M- 7Q` HH F ∈ F - 7Q` HKQbi HH UrX`XiX G2#2b;m2 K2 bm`2V x ∈ Rm Px (F ) = PN0 (θx (F )) .
()
3X9X h>1 J1*E1 J1al_1
jR8
−1 S`QQ7X q2 T`Qp2 i?2 2[mBp H2Mi 7Q`K, Px (F ) = PN0 (θ−x (F )) 7Q` HKQbi HH UrX`XiX m G2#2b;m2 K2 bm`2V x ∈ R X 6Q` i?Bb- r2 TTHv J2+F2Ƕb 7Q`KmH UdX9V rBi? v(x, ω) := 1A (x)1F (θx (ω)) iQ Q#i BM −1 λ PN0 θ−x (F ) dx = 1A (x)1F ((θx ◦ θ−x )(ω)) N (ω, dx) P (dω) A Ω E = 1A (x)1F (ω) N (ω, dx) P (dω) = C(A × F ) .
−1 θ−x (F )
h?2`27Q`2 λ A PN0 B(Rm )- i?2 `2bmHi 7QHHQrbX
Ω
E
dx = λ
A
Px (F ) dxX h?Bb #2BM; i`m2 7Q` HH A ∈
h?2 SQBbbQM * b2, aHBpMv FǶb i?2Q`2K h?2 7QHHQrBM; `2bmHi ;2M2` HBx2b *Q`QHH `v dXdXdX h?2Q`2K 3X9XR9 G2i P #2 T`Q# #BHBiv i? i K F2b Q7 i?2 TQBMi T`Q+2bb N QM Rm bBKTH2 SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2 bm`2 νX h?2M- 7Q` HH Γ ∈ M(Rm )Px (N ∈ Γ) = P (N + εx ∈ Γ) ,
dν@ X2X
U3XNV
S`QQ7X aBM+2 #Qi? Px (N ∈ ·) M/ P (N + εx ∈ ·) `2 T`Q# #BHBiv /Bbi`B#miBQMb Q7 bBKTH2 TQBMi T`Q+2bb2b- Bi bm{+2b #v i?2 JƺM+?Ĝ_ûMvB i?2Q`2K Uh?2Q`2K RXjXk8V iQ T`Qp2 i? i i?2B` `2bT2+iBp2 pQB/ M+2 T`Q# #BHBiv 7mM+iBQMb vx M/ v +QBM+B/2X 6Q` HH A- B ∈ B(Rm ) C(A × {N (B) = 0}) = Px (N (B) = 0) ν(dx) = vx (B) ν(dx) . A
A
"v i?2 BM/2T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b C(A × {N (B) = 0}) = E N (A)1{N (B)=0} = E N (A\B)1{N (B)=0} v(B) ν(dx) . = ν(A\B) v(B) = A\B
h?2`27Q`2-
vx (B) ν(dx) =
v(B) ν(dx) .
A
"mi
A\B
v(B) ν(dx) =
A\B
v(B)1{x∈B} ν(dx) /
A
P (N (B) = 0)1{x∈B} ν(dx) = /
= A
P ((N + εx )(B) = 0) ν(dx) . A
h?2`27Q`2 7Q` HH A- B ∈ B(Rm ) Px (N (B) = 0) ν(dx) = P ((N + εx )(B) = 0) ν(dx) A
A
jRe
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
M/ i?Bb BKTHB2b Px (N (B) = 0) = P ((N + εx )(B) = 0)- ν@ X2X
AM i?2 bi iBQM `v + b2- i FBM; BMiQ ++QmMi U3XNV- r2 ? p2 i? i 7Q` HKQbi HH x UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2VPN0 (N + x ∈ Γ) = P (N + εx ∈ Γ) . dz?QKQ;2M2Qmb SQBbbQM T`Q+2bb Bb i?2 b K2 mM/2` i?2 S HK T`Q# #BHBiv 2t+2Ti 7Q` TQBMi //2/ i i?2 Q`B;BMǴX JQ`2 T`2+Bb2Hv, h?2Q`2K 3X9XR8 U8 V G2i P #2 T`Q# #BHBiv i? i K F2b Q7 i?2 TQBMi T`Q+2bb N ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λX h?2M- 7Q` HH Γ ∈ Mp (Rm )PN0 (N ∈ Γ) = P (N + ε0 ∈ Γ) . S`QQ7X *QK#BM2 h?2Q`2K 3X9XR9 M/ 1[MX U3X3V iQ Q#i BM i? i 7Q` i H2 bi QM2 a UBM 7 +i 7Q` HKQbi HH aV PN0 (N + a ∈ Γ) = P (N + εa ∈ Γ) . _2TH +2 N #v N := N + aX aBM+2 N M/ N `2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? i?2 b K2 BMi2MbBiv- P (N + εa ∈ Γ) = P (N + a + εa ∈ Γ) = P (N + ε0 ∈ Γ)X HbQ PN0 (N ∈ Γ) = PN0 (N ∈ Γ) b QM2 + M +?2+F 7`QK i?2 /2}MBiBQM Q7 i?2 S HK T`Q# #BHBivX 1t KTH2 3X9XRe, h?2 J iö`M ? `/@+Q`2 KQ/2HXe G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv λX G2i {Xn }n∈N #2 Bib b2[m2M+2 Q7 TQBMibX h?2 Q#i BM2/ #v i?BMMBM; N BM bm+? r v i? i HH J iû`M KQ/2H Bb TQBMi T`Q+2bb N T B`b Q7 TQBMib Q7 i?2 i?BMM2/ TQBMi T`Q+2bb `2 i H2 bi i /Bbi M+2 R > 0 T `iX h?2 T`Q#H2K Bb, ?Qr iQ +?QQb2 i?2 TQBMiUbV iQ 2` b2 T B` Q7 TQBMib i Kmim H /Bbi M+2 ≤ R Ur2 /Q MQi r Mi i?2 2 bv bQHmiBQM i? i +QMbBbib BM 2` bBM; #Qi? TQBMibV\ h?2 B/2 BM i?Bb KQ/2H Bb iQ BMi`Q/m+2 dzT`BQ`Biv K `FbǴX JQ`2 T`2+Bb2HvH2i {Un }n∈N #2 M BB/ b2[m2M+2 Q7 `2 H ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM i?2 BMi2`p H [0, 1]- BM/2T2M/2Mi Q7 N X TQBMi Xn Q7 N Bb `2i BM2/ b TQBMi Q7 N B7 M/ QMHv B7 Un < Uk 7Q` HH k = n bm+? i? i Xk ∈ B(Xn ; R) , r?2`2 B(x, ; R) /2MQi2b i?2 +HQb2/ # HH Q7 +2Mi2` x M/ ` /Bmb RX P#pBQmbHv- N Bb bi iBQM `v TQBMi T`Q+2bbX AM Q`/2` iQ +QKTmi2 Bib BMi2MbBiv K2 bm`2 ν- }`bi +QM/BiBQM rBi? `2bT2+i iQ N , 6Q` HH C ∈ B(Rm )
N ν(C) := E . 1C (x) N (dx) = E E 1C (x) N (dx) | F Rn
AM pB2r Q7 i?2 BM/2T2M/2M+2 Q7 N M/ i?2 Un Ƕb, 8 e
(aHBpMv F- RNek)X (J iû`M- RNey)- (J iû`M- RN3e)X
Rn
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E
(dx) | F N 1C (x) N Rn N =E 1C (Xn )1(Un < Uk 7Q` HH k = n bm+? i? i Xk ∈ B(Xn ; r)) | F n∈N
=E
1C (Xn ) P (Un < Uk 7Q` HH k = n bm+? i? i Xk ∈ B(Xn ; R) | F ) . N
n∈N
LQr P (Un < Uk 7Q` HH k = n bm+? i? i Xk ∈ B(Xn ; R) | F N ) min Uk | F N = P Un < k=n ; Xk ∈B(Xn ;R) 1 = P min Uk > u | F N du
k=n ; Xk ∈B(Xn ;R)
0 1
(1 − u)N (B(Xn ;R))−1 du .
= 0
h?2`27Q`2 ν(C) = E
1C (x)g(x, N ) N (dx) , Rm
1 r?2`2 g(x, N ) := 0 (1 − u)N (B(x;R))−1 du M/ #v J2+F2Ƕb 7Q`KmH M/ i?2 7 +i i? i BM i?2 SQBbbQMB M + b2 Px (N ∈ ·) = P (N + εx ∈ ·)
1C (x)g(x, N ) N (dx) = λ 1C (x)E [ g(x, N + εx )] dx E Rm Rm 1
=λ 1C (x)E (1 − u)N (B(x;R)) du dx 0 Rm 1 =λ 1C (x) E uN (B(x;R)) du dx. Rm
0
aBM+2 N (B(x; R)) Bb SQBbbQM p `B #H2 rBi? K2 M λVm Rm - r?2`2 Vm := Bb i?2 pQHmK2 Q7 i?2 mMBi m@/BK2MbBQM H bT?2`2 m E uN (B(x;R)) = e−λ(1−u)Vm R M/ i?2`27Q`2
1
E u
N (B(x;R))
1
du =
0
0 1 =
e−λ(1−u)Vm R du m
1 − e−λV R . λV m Rm m
e−λuVm R du = m
0
r?2`2 6BM HHv ν(C) = m (C)λ−λV R = 1−e λ . m V Rm m
m
m
m
π2 Γ(1+ m 2 )
jR3
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
h?Bb BMi2MbBiv Bb M BM+`2 bBM; 7mM+iBQM Q7 λ M/ Bib HBKBi b λ ↑ ∞ Ub im` i2/ KQ/2HV Bb V m1Rm X 1t KTH2 3X9XRd, +Qp2` ;2 T`Q#H2K BM +QKKmMB+ iBQMbXd G2i N #2 mM/2` T`Q# #BHBiv P bi iBQM `v ?TT QM R2 Q7 BMi2MbBiv λ rBi? TQBMi b2[m2M+2 {Xn }n∈N X *QMbB/2` i?2 7QHHQrBM; dz`2+2TiBQM +2HHǴ3 bbQ+B i2/ rBi? TQBMi Xn , C(Xn , N ) :=
F D(|y − Xn |)
n >T y∈R ; W + k∈N ; k=n Fk D(|y − Xk |) 2
,
r?2`2 W ≥ 0- T > 0- D : R+ → R+ Bb MQM@M2; iBp2 /2+ v 7mM+iBQM- i? i Bb- 7mM+iBQM i? i /2+`2 b2b iQ 0 b u → ∞- M/ bm+? i? i 7Q` HH a > 0 ∞ uD(u) du < ∞ , U3XRyV a
7Q` BMbi M+2 D(u) = u−β
(β > 2) ,
U3XRRV
M/ {Fn }n∈N Bb M BB/ b2[m2M+2 Q7 2tTQM2MiB H ` M/QK p `B #H2b rBi? K2 M μ−1 X *QM/BiBQM U3XRyV ;m ` Mi22b i? i i?2 /2MQKBM iQ` BM i?2 /2}MBiBQM Q7 i?2 `2+2TiBQM
+2HH Bb }MBi2X AM/22/- i?2 UH `;2`V bmK k∈N Fk D(|y − Xk |) + M #2 b2T ` i2/ BM bmK +Q``2bTQM/BM; iQ TQBMib BM i?2 # HH B(0; a)- a > 0- r?B+? `2 }MBi2 BM MmK#2` M/ /Bz2`2Mi 7`QK 0- M/ i?2 `2bi Q7 i?2 bmK r?Qb2 2tT2+i iBQM Bb }MBi2 b i?2 7QHHQrBM; +QKTmi iBQM b?Qrb, E Fk D(|y − Xk |)1B(0;a) (Xk ) = E [F0 ] 1B(0;a) (x)D(|y − x|)λ dx
k∈N
= E [F0 ]
R2
1B(0;a) (x)D(|x|)λ dx ∞ = E [F0 ] 2π uD(u)λ du < ∞ . R2
a
h?2 +2HH C(Xn , N ) Bb i?2 b2i Q7 TQBMib rBi? dz;QQ/Ǵ `2+2TiBQM 7`QK i?2 Mi2MM Ui?2 2KBii2`V HQ+ i2/ i Xn - i? i Bb- 7Q` r?B+? i?2 dzbB;M H@iQ@BMi2`72`2M+2@MQBb2@ ` iBQǴ UbBM`V Bb H `;2` i? M i?2 i?`2b?QH/ T X h?2 bBM` Bb i?2 ` iBQ Q7 i?2 bB;M H `2+2Bp2/ Fn D(|y − Xn |) 7`QK i?2 bi iBQM iQ i?2 dzMQBb2Ǵ- +QMbBbiBM; Q7 bQK2 dz`2@ +2TiBQM 2H2+i`QMB+ MQBb2Ǵ W M/ Q7 i?2 dzBMi2`72`2M+2 MQBb2Ǵ 7`QK Qi?2` bi iBQMb
k∈N ; k=n Fk D(|y − Xk |)X G2i PN0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? P M/ N X h?2 [m MiBiv PN0 (x ∈ C(0, N )) K2 bm`2b i?2 [m HBiv Q7 `2+2TiBQM Q7 ;Bp2M `2+2TiQ` HQ+ i2/ i /Bbi M+2 |x| 7`QK bQK2 ;Bp2M Mi2MM X "v aHBpMv FǶb i?2Q`2K d
(" ++2HHB- "ƈ bx+xvbxvM M/ JɃ?H2i? H2`- kyyj)X h?Bb KQ/2H r b BMi`Q/m+2/ BM +QKKmMB+ iBQMb +QMi2ti #v (" ++2HHB M/ "ƈ bx+xvbxvMkyyR)X 3
3X9X h>1 J1*E1 J1al_1
jRN
PN0 (x ∈ C(0, N )) = P
W+
F0 D(|x|) >T k∈N Fk D(|x − Xk |)
= P (F0 > D(|x|)−1 T (W + I)) ,
r?2`2 I = k∈N Fk D(|x − Xk |)X LQr- bBM+2 F0 Bb M 2tTQM2MiB H ` M/QK p `B #H2 rBi? K2 M μ−1 M/ BM/2T2M/2Mi Q7 i?2 Fk Ƕb rBi? k ≥ 1 −1 P (F0 > D(|x|)−1 T (W + I)) = E e−μD(|x|) T (W +I) −1 −1 = e−μD(|x|) T W E e−μD(|x|) T I . aBM+2 N Bb M ?TT E e−γI = E e−γ k Fk D(|x−Xk |) −γF D(|x−y|) E e − 1 λ dy = exp 2 R −γF D(|y|) = exp E e − 1 λ dy 2 R = exp (LF (γD(|y|)) − 1) λ dy , R2
r?2`2 LF Bb i?2 G TH +2 i` Mb7Q`K Q7 F0 X S bbBM; iQ TQH ` +QQ`/BM i2b
−γI
E e
∞
= exp −2πλ
u (LF (γD(u)) − 1) du
.
0
h?2`27Q`2- rBi? γ = μD(|x|)−1 T PN0 (x ∈ C(0, N )) = e−μD(|x|)
−1 T W
exp −2πλ
∞
u (LF (D(|x|)−1 T D(u)) − 1) du
0
1p Hm iBM; i?2 G TH +2 i` Mb7Q`K LF PN0 (x
∈ C(0, N )) = e
−μD(|x|)−1 T W
∞
exp −2πλ 0
u du 1 + D(|x|)/(T D(u))
.
qBi? i?2 /2+ v KQ/2H U3XRRV M/ i?2 ?vTQi?2bBb W = 0- i?2 +QKTmi iBQMb H2 / iQ PN0 (x ∈ C(0, N )) = exp −λ|x|2 T 2/β K(β) , r?2`2 K(β) =
2πΓ(2/β)Γ(1 − 2/β) 2π 2 = . β β sin(2π/β)
.
jky
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
S HK E2`M2H Q7 SQBbbQM *Hmbi2` S`Q+2bb *QMbB/2` i?2 +Hmbi2` TQBMi T`Q+2bb Q7 1t KTH2 RX8Xk rBi? i?2 7QHHQrBM; bT2+B HBx @ iBQM, i?2 ;2`K T`Q+2bb N0 Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ0 X h?2Q`2K 3X9XR3 UN V lM/2` i?2 S HK K2 bm`2 PN0 - i?2 /Bbi`B#miBQM Q7 i?2 +Hmbi2` TQBMi T`Q+2bb N Bb i? i Q7 i?2 bi iBQM `v +Hmbi2` TQBMi T`Q+2bb iQ r?B+? QM2 ? b //2/ +Hmbi2` i 0 r?Qb2 /Bbi`B#miBQM Bb i?2 dzS HKǴ F2`M2H πZ0 i 0 Q7 i?2 ;2M2`B+ +Hmbi2` Z,
1 πZ0 (Γ) = E 1 (Z − y) Z(dy) (Γ ∈ Mp (Rm )) . Γ νZ (Rm ) m R "27Q`2 i?2 T`QQ7- r2 b? HH ;Bp2 KQ`2 /2i BHb +QM+2`MBM; πZ0 X 6B`biHv- MQi2 i? i Bb MQi 2[m H iQ PZ0 - i?2 S HK F2`M2H i 0 Q7 ZX h?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 MmK#2` T Q7 TQBMib Q7 Z mM/2` i?Bb T`Q# #BHBiv Bb
πZ0
πZ0 (T = n) =
1 nP (Z(Rm ) = n) E[T ]
(n ≥ 0) .
UP#b2`p2 i? i πZ0 (T = 0) = 0- b 2tT2+i2/ 7`QK dzS HKǴ F2`M2HXV hQ T`Qp2 i?BbBi bm{+2b iQ TTHv i?2 /2}MBiBQM Q7 πZ0 rBi? Γ := {μ ; μ(Rm ) = n},
1 m E 1 Z(dy) πZ0 (Z(Rm ) = n) = {(Z−y)(R )=n} νZ (Rm ) m R
1 m )=n} Z(dy) = E 1 {(Z)(R νZ (Rm ) Rm 1 1 = E 1{(Z)(Rm )=n} Z(Rm )) = E 1{(Z)(Rm )=n} n . m m νZ (R ) νZ (R ) .2MQi2 #v x1 , . . . , xn i?2 TQBMib Q7 Z r?2M T = nX q`Bi2 ∞ n 1 0 πZ (Γ) = E 1T =n E 1Γ (Z − xi ) | T = n E[T ] n=0 i=1 n ∞ 1 1 n 1T =n E 1Γ (Z − xi ) | T = n E = E[T ] n i=1 n=0 n ∞ n 1 P (T = n)E 1Γ (Z − xi ) | T = n = E[T ] n i=1 n=0 n ∞ 1 0 πZ (T = n)E 1Γ (Z − xi ) | T = n . = n i=1 n=0 h?Bb + M #2 BMi2`T`2i2/ b 7QHHQrbX AM Q`/2` iQ ;2M2` i2 b KTH2 Q7 Z mM/2` T`Q# #BHBiv πZ0 - +?QQb2 n rBi? T`Q# #BHBiv πZ0 (T = n)- i?2M +`2 i2 i?2 n TQBMib Q7 Z mM/2` i?2 T`Q# #BHBiv P (Z ∈ · | T = n)X h?2M +?QQb2 QM2 Q7 i?2b2 TQBMib i ` M/QK- b v U - M/ +QMbi`m+i i?2 /2bB`2/ b KTH2 b {x1 − U, x2 − U, . . . , xn − U 'X h?2 7Q`i?+QKBM; T`QQ7 Q7 h?2Q`2K 3X9XR3 K F2b mb2 Q7 i?2 7QHHQrBM; i2+?MB+ H H2KK X N
(K# `ixmKB M- RNee)X
3X9X h>1 J1*E1 J1al_1
jkR
G2KK 3X9XRN G2i LZ,x #2 i?2 G TH +2 7mM+iBQM H Q7 i?2 HQ+ HHv }MBi2 TQBMi T`Q+2bb Z mM/2` i?2 S HK F2`M2H PZ,x X .2MQi2 #v νZ Bib BMi2MbBiv K2 bm`2- M/ #v LZ,x i?2 G TH +2 i` Mb7Q`K Q7 Z mM/2` PZ,x - i? i Bb LZ,x (ϕ) := e− Rm ϕ(t) Z(dt) P (dω) . Ω
h?2M- 7Q` HH b ∈ Rm - HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb ϕ : Rm → R M/ HH B ∈ B(Rm )νZ @ X2X LZ,x (ϕ) = LZ+b,x−b (ϕ(· − b)) , S`QQ7XRy qBi? i?2 +? M;2 Q7 p `B #H2b x → x + b LZ+b,x−b (ϕ(· − b)) 1B (x)νZ (dx) Rm = LZ+b,x−b+b (ϕ(· − b)) 1B (x − b) νZ+b (dx) m R = LZ+b,x (ϕ(· − b)) 1B+b (x) νZ+b (dx) m R = e Rm ϕ(t−b) (Z(ω)+b)(dt) PZ+b,x P (dω)1B+b (x) νZ+b (dx) . Rm
Ω
AM pB2r Q7 U3XeV- i?2 H bi i2`K Q7 i?2 #Qp2 b2[m2M+2 Q7 2[m HBiB2b Bb 2[m H iQ
e− Rm ϕ(t−b) (Z+b)(dt) 1B+b (x) (Z + b)(dx) E Rm
− Rm ϕ(t) Z(dt) =E e 1B (x) Z(dx) m R = LZ,x (ϕ) 1B (x) νZ (dx) . Rm
h?2`27Q`2- 7Q` HH B ∈ B(Rm ) LZ+b,x−b (ϕ(· − b))νZ (dx) = LZ,x (ϕ) νZ (dx) , B
B
r?B+? BKTHB2b i?2 MMQmM+2/ `2bmHiX q2 `2 MQr `2 /v 7Q` i?2 T`QQ7 Q7 h?2Q`2K 3X9XR3 S`QQ7X _2+ HH 7Q`KmH URXkyV, LN (ϕ) = exp Rm
E e− Rm ϕ(y+x) Z(dy) − 1 λ0 dx .
1H2K2Mi `v +QKTmi iBQMb vB2H/ Ry
h?Bb T`QQ7 r b +QKKmMB+ i2/ iQ K2 #v " `i2F "ƈ bx+xvbxvMX
jkk −
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1 ∂LN (ϕ, f ) ∂f
= LN (ϕ)
f (y + x)Z(dy)
E Rm
ϕ(y + x)Z(dy)
exp
Rm
TTHvBM; 7Q`KmH U3XeV rBi?
g(y, ω) := f (y + x) exp −
;Bp2b
E
λ0 dx .
Rm
f (y + x)Z(dy) Rm =
ϕ(y + x) Z(ω, dy) Rm
exp −
ϕ(y + x)Z(dy) Rm
f (y + x) LZ+y,y+x (ϕ)νZ (dy) .
Rm
h?2`27Q`2 ∂LN − (ϕ, f ) = LN (ϕ) f (y + x) LZ+x,y+x (ϕ)νZ (dy) λ0 dx ∂f m Rm R = LN (ϕ) f (y + x) LZ+x,y+x (ϕ)λ0 dx νZ (dy) . Rm
Rm
qBi? i?2 +? M;2 Q7 p `B #H2b x → u − y- M/ mbBM; i?2 2tT`2bbBQM λ = λ0 νZ (Rm ) 7Q` i?2 BMi2MbBiv Q7 N - r2 Q#i BM BM pB2r Q7 G2KK 3X9XRN ∂LN − (ϕ, f ) = LN (ϕ) f (u) LZ+u−y,u (ϕ)λ0 du νZ (dy) ∂f m Rm
R 1 = LN (ϕ) f (u) LZ+u−y,u (ϕ) νZ (dy) λ du ν (Rm ) Rm m Z
R 1 = LN (ϕ) f (u) L (ϕ(· − y + u)) ν (dy) λ du Z,y Z ν (Rm ) Rm m Z
R 1 = LN (ϕ) f (u) EZy eϕ(t−y+u)Z(dt) νZ (dy) λ du . m) ν (R m m Z R R "v 7Q`KmH U3XeV ; BM−
∂LN (ϕ, f ) = LN (ϕ) ∂f
f (u) Rm
1 E νZ (Rm )
e−
Rm
ϕ(t−y+u)Z(dt)
Z(dy) λ du ,
Rm
7`QK r?B+? r2 /2/m+2- b BM 1t KTH2 3X9XN- i? i 7Q` G2#2b;m2 HKQbi HH u ∈ Rm
1 − Rm ϕ(t−y+u)Z(dt) LN,u (ϕ) = LN (ϕ) E e Z(dy) . νZ (Rm ) Rm AM T `iB+mH `- i?2`2 2tBbib M a ∈ Rm bm+? i? i i?2 #Qp2 Bb i`m2 7Q` u = a- M/ #v bi iBQM `Biv
1 − Rm ϕ(t−y)Z(dt) LN,0 (ϕ) = LN (ϕ) E e Z(dy) . νZ (Rm ) Rm
3X9X h>1 J1*E1 J1al_1
jkj
>B;?2` P`/2` J2+F2 J2 bm`2b G2i E #2 M HX+X/X#X bT +2 rBi? "Q`2H σ@}2H/ B(E)X G2i (Ω, F) := (Mp (E), Mp (E))X G2i N #2 TQBMi T`Q+2bb QM E /2}M2/ QM (Ω, F)X Ai Bb bbmK2/ iQ #2 M m@i? Q`/2` bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM E- rBi? m@i? KQK2Mi K2 bm`2b Mm X aBKBH `Hv iQ i?2 + b2 m = 1- r2 + M b?Qr i? i i?2 b2i 7mM+iBQM Cm (A × F ) := E N (m) (A)1F (A ∈ B(E)⊗m, F ∈ F ) mMB[m2Hv /2}M2b σ@}MBi2 K2 bm`2 Cm QM (E m × Ω, B(E)⊗m ⊗ F)- + HH2/ i?2 m@i? Q`/2` J2+F2 K2 bm`2X .2MQi2 #v Cm,F i?2 K2 bm`2 F → Cm (· × F )X aBKBH `Hv iQ i?2 mMBp `B i2 + b2rBi? i?2 b K2 T`QQ7- QM2 b?Qrb i?2 2tBbi2M+2 Q7 _ /QMĜLBFQ/ɷK /2`Bp iBp2 dCm,F Px1 ,...,xm (F ) = (x1 , . . . , xm ) dMm /2}M2/ 7Q` Mm @ HKQbi HH (x1 , . . . , xm ) ∈ E m X Ai b iBb}2b Px1 ,...,xm (F ) Mm (dx1 × · · · × dxm ) U3XRkV C(A × F ) = A
M/ BM T `iB+mH `-
Mm (A) = C(A × Ω) =
Px1 ,...,xm (Ω)Mm (dx1 × · · · × dxm ) . A
h?2`27Q`2 Px1 ,...,xm (Ω) = 1- Mm @ X2X h?2 T`Q# #BHBiv Px1 ,...,xm bQ /2}M2/ UMm @ X2XV Bb + HH2/ i?2 m@i? Q`/2` S HK T`Q# #BHBiv Q7 N i (x1 , . . . , xm ) ∈ E m X h?2 T`Q# #BHBiB2b Px1 ,...,xm U(x1 , . . . , xm ) ∈ E m V + M #2 +?Qb2M BM bm+? r v i? i 7Q` Mm @ HKQbi HH (x1 , . . . , xm ) ∈ E m F → Px1 ,...,xm (F ) Bb T`Q# #BHBiv K2 bm`2 QM (Ω, F) M/ 7Q` }t2/ F ∈ F (x1 , . . . , xm ) → Px1 ,...,xm (F ) Bb B(E)@K2 bm` #H2 . h?2 T`Q# #BHBiv F2`M2H 7`QK E m × Ω iQ (Ω, F)) /2}M2/ #v ((x1 , . . . , xm ), A) → Px1 ,...,xm (A) Bb + HH2/ i?2 m@i? Q`/2` S HK F2`M2H Q7 N X h?2Q`2K 3X9Xky 6Q` HH MQM@M2; iBp2 K2 bm` #H2 g : E m × Ω → [0, ∞] g(x1 , . . . , xm , ω)N (m) (ω, dx1 × · · · × dxm )P (dω) Ω En = g(x1 , . . . , xm , ω)Px1 ,...,xm (dω) Mm (dx1 × · · · × dxm ) . En
Ω
U3XRjV h?2 T`QQ7 Q7 i?2 2ti2MbBQM Q7 h?2Q`2K 3X9XR9 #2HQr 7QHHQrb 7`QK i?2 b K2 `;mK2MibX h?2Q`2K 3X9XkR G2i P #2 T`Q# #BHBiv i? i K F2b Q7 i?2 TQBMi T`Q+2bb N QM Rm bBKTH2 SQBbbQM T`Q+2bb rBi? m@i? Q`/2` KQK2Mi K2 bm`2 Mm X h?2M- 7Q` HH Γ ∈ M(Rm )Px1 ,...,xm (N ∈ Γ) = P (N + εx1 + · · · + εxm ∈ Γ) ,
Mm @ X2X
U3XR9V
jk9
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
3X8 h?2 _2/m+2/ J2+F2 J2 bm`2 AM i?Bb b2+iBQM- r2 rQ`F QM i?2 + MQMB+ H K2 bm` #H2 bT +2 (Mp (E), Mp (E)) QM r?B+? Bb ;Bp2M T`Q# #BHBiv K2 bm`2 P X .2MQi2 #v N i?2 +QQ`/BM i2 K TTBM;, N (μ) := μX bbmK2 i? i N Bb bBKTH2 rBi? σ@}MBi2 BMi2MbBiv K2 bm`2 νX 6Q` Mv b2ib A ∈ B(E) M/ Γ ∈ Mp (E)- H2i
C ! (A × Γ) := E 1Γ (N − εx ) N (dx) . U3XR8V A
_2+ HH i? i i?2 MQi iBQM μ − εa - r?2`2 μ ∈ Mp (E) Bb bBKTH2 M/ a ∈ E- `2T`2b2Mib i?2 K2 bm`2 μ B7 μ({a}) = 0 M/ i?2 K2 bm`2 μ KBMmb i?2 mMBi K bb i a Qi?2`rBb2X h?2`27Q`2- B7 {Xn }n∈N /2MQi2b i?2 TQBMi b2[m2M+2 Q7 N ! 1A (Xn )1Γ (N \{Xn }) . () C (A × Γ) := E n∈N
b BM i?2 T`QQ7 Q7 G2KK 3X9XR- Bi + M #2 b?QrM i? i i?2 b2i 7mM+iBQM C ! /2}M2/ #v U3XR8V 7Q` b2ib Q7 i?2 7Q`K A × Γ- r?2`2 A ∈ B(E) M/ Γ ∈ Mp (E)+ M #2 mMB[m2Hv 2ti2M/2/ iQ σ@}MBi2 K2 bm`2 C ! : B(E) ⊗ Mp (E) → [0, ∞]X h?2 K2 bm`2 C ! : B(E) ⊗ Mp (E) → [0, ∞] bQ /2}M2/ Bb + HH2/ i?2 U}`bi@Q`/2`V `2/m+2/ J2+F2 K2 bm`2X qBi? T`QQ7 bBKBH ` iQ i? i Q7 h?2Q`2K 3X9Xj- r2 ? p2 i?2 `2/m+2/ J2+F2 7Q`KmH , h?2Q`2K 3X8XR 6Q` HH MQM@M2; iBp2 K2 bm` #H2 g : E × Mp (E) → [0, ∞] g(x, μ − εx ) N (μ, dx) P (dμ) = g(x, μ) C ! (dx × dμ) . Mp (E)
E
E×Mp (E)
U3XReV AM i?2 MQi iBQM Q7 UV- i?2 H27i@? M/ bB/2 Q7 1[MX U3XReV Bb g(Xn , N \{Xn }) . E n∈N
Ai 7QHHQrb 7`QK h?2Q`2K 3X8XR i? i 7Q` HH K2 bm` #H2 g : E × Mp (E) → R bm+? i? i 2Bi?2` QM2 Q7 i?2 irQ 7QHHQrBM; +QM/BiBQMb, |g(x, μ − εx )| N (μ, dx) P (dμ) < ∞ Mp (E)
Q`
E
|g(x, μ)|C ! (dx × dμ) < ∞ , E×Mp (E)
Bb b iBb}2/- i?2M i?2 Qi?2` Bb HbQ b iBb}2/- M/ i?2 7Q`KmH BM h?2Q`2K 3X8XR Bb i`m2X
3X8X h>1 _1.l*1. J1*E1 J1al_1
jk8
6Q` 2 +? Γ ∈ Mp (E)- /2MQi2 #v CΓ! i?2 K2 bm`2 QM B(E) /2}M2/ #v CΓ! (A) = C (A × Γ)X b 7Q` i?2 J2+F2 K2 bm`2 Ua2+iBQM 3X9V- r2 + M bb2`i i?2 2tBbi2M+2 Q7 i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2!
Px! (Γ) :=
dCΓ! (x) dν
U3XRdV
/2}M2/ 7Q` ν@ HKQbi HH x ∈ EX Ai b iBb}2b ! Px! (Γ)ν(dx) C (A × Γ) =
U3XR3V
A
M/ Px! (Mp (E)) = 1- ν@ X2X .2}MBiBQM 3X8Xk h?2 T`Q# #BHBiv bQ /2}M2/ ν@ X2X Bb + HH2/ i?2 `2/m+2/ S HK T`Q# #BHBiv i x ∈ EX aBKBH `Hv iQ r? i r b /QM2 BM a2+iBQM 3X9 7Q` J2+F2Ƕb K2 bm`2- r2 + M +?QQb2 7Q` HH Γ p2`bBQM Q7 i?2 +QM/BiBQM H T`Q# #BHBiv Px (Γ) bm+? i? i ν@ HKQbi HH x ∈ E Px! (·) Bb T`Q# #BHBiv K2 bm`2 QM (Mp (E), Mp (E)) M/ 7Q` }t2/ Γ
x → Px! (Γ) Bb B(E)@K2 bm` #H2 .
h?2 `2/m+2/ J2+F2 7Q`KmH Q7 h?2Q`2K 3X8XR MQr `2 /b ! g(x, μ − εx ) μ(dx) P (dμ) = g(x, μ) Px (dμ) ν(dx) , Mp (E)
E
E
Mp (E)
7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM g : E × Mp (E) → RX
U3XRNV
h?2Q`2K 3X8Xj h?2 `2/m+2/ J2+F2 K2 bm`2 Px! M/ i?2 MQM@`2/m+2/ J2+F2 K2 @ bm`2 Px `2 HBMF2/ #v i?2 7QHHQrBM; `2H iBQM, 7Q` HH Γ ∈ Mp (E)Px (Γ) = Px! ({μ ; μ + εx ∈ Γ})
ν@ X2X
U3XkyV
S`QQ7X TTHv 6Q`KmH U3XRNV rBi? g(x, μ) := 1C (x)1Γ (μ + εx )- r?2`2 C ∈ B(E)X h?2 H27i@? M/ bB/2 2[m Hb 1C (x)1Γ (μ) μ(dx) P (dμ) = Px (Γ) ν(dx) , Mp (E)
E
C
r?2`2 b i?2 `B;?i@? M/ bB/2 2[m Hb 1Γ (μ + εx ) Px! (dμ) C
Px! ({μ ; μ + εx ∈ Γ}) ν(dx) .
ν(dx) =
Mp (E)
b Bi Bb i`m2 Q7 HH C ∈ B(E)- Bi BKTHB2b U3XkyVX
C
jke
*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1
h?2Q`2K 3X8X9 6Q` i?2 T`Q# #BHBiv P iQ K F2 Q7 i?2 TQBMi T`Q+2bb N bBKTH2 SQBbbQM T`Q+2bb rBi? σ@}MBi2 MQM@ iQKB+ BMi2MbBiv K2 bm`2 ν- Bi Bb M2+2bb `v M/ bm{+B2Mi i? i 7Q` HH Γ ∈ Mp (Rm )Px! (N ∈ Γ) = P (N ∈ Γ)
dν@ X2X
U3XkRV
S`QQ7X h?2 T`QQ7 Bb bBKBH ` iQ i? i Q7 h?2Q`2K eXjXdX L2+2bbBiv, *QK#BM2 h?2@ Q`2Kb 3X8Xj M/ 3X9XR9X 6Q` i?2 bm{+B2M+v- Bi bm{+2b iQ T`Qp2 i? i 7Q` 2p2`v #QmM/2/ "Q`2H b2i A ⊂ Rm - N (A) Bb SQBbbQM p `B #H2 rBi? K2 M ν(A)X qBi? Γ := {μ ; μ(A) = n} Un ≥ 0V- i?2 H27i@? M/ bB/2 Q7 U3XkyV Bb C (A × Γ) = E !
1A (Xn )1{N (A)=n+1} = (n + 1)P (N (A) = n + 1) .
n∈N
h?2 `B;?i@? M/ bB/2 Q7 U3XR3V Bb- #v ?vTQi?2bBb U3XkRV P (N ∈ Γ)ν(dx) = ν(A)P (N ∈ Γ) = ν(A)P (N (A) = n) . A
h?2`27Q`2 (n + 1)P (N (A) = n + 1) = ν(A)P (N (A) = n) 7Q` HH n ≥ 0- 7`QK r?B+? r2 ? p2 ν(A)n P (N (A) = n) = P (N (A) = 0) . n! LQ`K HBx iBQM ;Bp2b P (N (A) = 0) = e−ν(A) X
*Q`QHH `v 3X8X8 G2i P #2 T`Q# #BHBiv i? i K F2b Q7 i?2 TQBMi T`Q+2bb N M ?TT rBi? BMi2MbBiv λX h?2M- 7Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM f : Rm × Mp (Rm ) → R E Rm
f (x, N − εx )N (dx) = λ
E [f (x, N )] dx. Rm
1t KTH2 3X8Xe, _@BbQH i2/ SQBMib BM SQBbbQM S`Q+2bbX G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm Q7 BMi2MbBiv λ > 0X TQBMi Xn ∈ N Bb + HH2/ R@BbQH i2/ B7 i?2`2 Bb MQ Qi?2` TQBMi Q7 N i /Bbi M+2 7`QK Xn H2bb i? M Q` 2[m H iQ R > 0X AM Qi?2` i2`Kb- i?2 TQBMi T`Q+2bb N1 Q7 R@BbQH i2/ TQBMib Q7 N Bb /2}M2/ #v N1 (A) := 1N (B(Xn ;R))=1 (A ∈ B(Rm )) . n∈N
q2 +QKTmi2 Bib BMi2MbBiv K2 bm`2 ν1 ,
3X8X h>1 _1.l*1. J1*E1 J1al_1 ν1 (A) = E
jkd
1A (Xn )1N (B(Xn ;R))=1
n∈N
R
1A (x)1μ(B(x;R))=1 N (dx)
=E m
1A (x)1μ(B(x;R))=1 N (dx) m R =λ 1A (x) 1μ(B(x;R))=1 Px (dμ) dx Rm M (Rm ) 1A (x)P ((N + εx )(B(x; R)) = 0) dx =λ =E
Rm
= λ m (A)e−λ
m (B(0;R))
.
1t KTH2 3X8Xd, Jmim H L2 `2bi L2B;?#Qm`bXRR hrQ TQBMib Q7 bBKTH2 TQBMi T`Q+2bb N QM Rm `2 + HH2/ Kmim H M2 `2bi M2B;?#Qm`b UKMMV B7 2 +? QM2 Bb i?2 +HQb2bi TQBMi Q7 i?2 Qi?2`X h?2 T B`b Q7 Kmim H M2 `2bi M2B;?#Qm`b `2 Q#pBQmbHv /BbDQBMiX TQBMi Q7 N Bb + HH2/ BbQH i2/ UrBi? `2bT2+i iQ i?2 KMM T`QT2`ivV B7 Bi /Q2b MQi #2HQM; iQ M KMM T B`X
q2 MQr bmTTQb2 i? i N Bb mM/2` P M ?TT QM R2 Q7 BMi2MbBiv λX .2MQi2 #v x ↔ y i?2 bvKK2i`B+ `2H iBQM U/2T2M/BM; QM N V b vBM; i? i i?2`2 Bb MQ TQBMi Q7 N BM i?2 mMBQM Q7 QT2M # HHb B(x, |y − x|) ∪ B(y, |y − x|)X AM T `iB+mH `- 7Q` k = n- Xk ↔ Xn K2 Mb i? i (Xk , Xn ) Bb T B` Q7 Kmim H M2 `2bi M2B;?#Qm`bX q2 b22F M 2tT`2bbBQM 7Q` E g(Xn , Xk )1Xn ↔Xk = E g(Xn , y)1Xn ↔y (N − εXn )(dy) . n
q`Bi2
n
k;k=n
Rm
RR
Rm
g(Xn , y)1Xn ↔y (N − εXn )(dy) := f (Xn , N − εXn ) .
(:BQp MB/Bb- Hp `2x *Q`` H2b M/ .2+`2mb27QM/- kyR9)X
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"v *Q`QHH `v 3X8X8 E
f (Xn , N − εXn = λ
E [f (x, N )] dx . R2
n
q2 + M r`Bi2 f (x, N ) =
R2
g(x, y)1x↔y (N − εy )(dy)
M/ i?2`27Q`2- #v *Q`QHH `v 3X8X8 ; BM
E [f (x, N )] = E g(x, y)1x↔y dy = λ R2
R2
g(x, y)E [1x↔y ] dy .
LQr 2
E [1x↔y ] = P (x ↔ y) = P (N (B(x, |y − x|) ∪ B(y, |y − x|) = 0) = e−Cλ(y−x) , √
r?2`2 C = 43 + 2π3 X 6BM HHv E g(Xn , Xk )1Xn ↔Xk = n
k;k=n
R2
2
g(x, y)e−Cλ(y−x) λ2 dx dy . R2
>B;?2` P`/2` _2/m+2/ J2+F2 J2 bm`2 q2 rQ`F QM i?2 + MQMB+ H K2 bm` #H2 bT +2 (Mp (E), Mp (E)) QM r?B+? Bb ;Bp2M T`Q# #BHBiv K2 bm`2 P X .2MQi2 #v N i?2 +QQ`/BM i2 TTHB+ iBQM, N (μ) := μX bbmK2 i? i N Bb bBKTH2 m@i? Q`/2` TQBMi T`Q+2bb rBi? σ@}MBi2 m@i? KQK2Mi ! K2 bm`2 Mm X 6Q` Mv b2ib A ∈ B(E)⊗m M/ Γ ∈ Mp (E)- H2i
! (m)! Cm (A × Γ) := E 1Γ (N − εx1 − · · · − εxm ) N (dx1 × · · · × dxm ) . U3XkkV A ! b BM i?2 + b2 r?2`2 m = 1- Bi + M #2 b?QrM i? i i?2 b2i 7mM+iBQM Cm /2}M2/ ⊗m #v U3XkkV 7Q` b2ib Q7 i?2 7Q`K A × Γ- r?2`2 A ∈ B(E) M/ Γ ∈ Mp (E)- + M ! #2 mMB[m2Hv 2ti2M/2/ iQ σ@}MBi2 K2 bm`2 Cm : B(E)⊗m ⊗ Mp (E) → [0, ∞]X ! ⊗m h?2 K2 bm`2 Cm : B(E) ⊗ Mp (E) → [0, ∞] bQ /2}M2/ Bb + HH2/ i?2 m@i?@Q`/2` `2/m+2/ J2+F2 K2 bm`2X qBi? T`QQ7 bBKBH ` iQ i? i Q7 h?2Q`2K 3X8XR- r2 ? p2 i?2 `2/m+2/ m@i?@Q`/2` J2+F2 7Q`KmH ,
h?2Q`2K 3X8X3 6Q` HH MQM@M2; iBp2 K2 bm` #H2 g : E m × Mp (E) → [0, ∞] (m)! g(x, μ − εx1 − · · · − εxm ) N (μ, dx1 × · · · × dxm ) P (dμ) Mp (E) E ! g(x1 , . . . , xm , μ) Cm (dx1 × · · · × dxm × dμ) . = E m ×Mp (E)
3X8X h>1 _1.l*1. J1*E1 J1al_1
jkN
! 6Q` 2 +? Γ ∈ Mp (E)- /2MQi2 #v Cm,Γ i?2 K2 bm`2 QM B(E) /2}M2/ #v CΓ! (A) = C (A×Γ)X b BM i?2 + b2 m = 1- r2 + M bb2`i i?2 2tBbi2M+2 Q7 i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2!
Px! 1 ,...,xm (Γ) :=
! dCm,Γ (x1 , . . . , xm ) dM!m
U3XkjV
! @ HKQbi HH x1 , . . . , xm ∈ EX Ai b iBb}2b /2}M2/ 7Q` Mm ! (A × Γ) = Px! 1 ,...,xm (Γ)M!m (dx1 × · · · × dxm ) Cm
U3Xk9V
A
M/ Px! 1 ,...,xm (Mp (E)) = 1- M!m @ X2X .2}MBiBQM 3X8XN h?2 T`Q# #BHBiv Px! 1 ,...,xm QM (Mp (E), Mp (E)) bQ /2}M2/ M!m @ X2X Bb + HH2/ i?2 `2/m+2/ m@i? Q`/2` S HK T`Q# #BHBiv i (x1 , . . . , xm ) ∈ E m X aBKBH `Hv iQ i?2 + b2 m = 1- r2 + M +?QQb2 7Q` HH Γ p2`bBQM Q7 i?2 T`Q# #BHBiv ! @ HKQbi HH (x1 , . . . , xm ) ∈ E m Px! 1 ,...,xm (Γ) bm+? i? i 7Q` Mm Px! 1 ,...,xm (·) Bb T`Q# #BHBiv K2 bm`2 QM (Mp (E), Mp (E)) M/ 7Q` }t2/ Γ
x → Px! 1 ,...,xm (Γ) Bb B(E)⊗m @K2 bm` #H2 .
h?2 `2/m+2/ J2+F2 7Q`KmH Q7 h?2Q`2K 3X8XR MQr `2 /b g(x1 , . . . , xm , μ − εx1 − · · · − εxm ) μ(dx) P (dμ) Mp (E) E = E
Mp (E)
g(x1 , . . . , xm , μ) Px! 1 ,...,xm (dμ)
! Mm (dx1 × · · · × dxm ) ,
7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : E × Mp (E) → RX h?2 `2bmHi 2ti2M/BM; h?2Q`2K 3X8Xj Bb, h?2Q`2K 3X8XRy h?2 `2/m+2/ S HK K2 bm`2 Px! 1 ,...,xm M/ i?2 MQM@`2/m+2/ S HK K2 bm`2 Px1 ,...,xm `2 HBMF2/ #v i?2 7QHHQrBM; `2H iBQM, 7Q` HH Γ ∈ Mp (E)Px1 ,...,xm (Γ) = Px! 1 ,...,xm ({μ ; μ + εx1 + · · · + εxm ∈ Γ})
! Mm @ X2X
U3Xk8V
aHBpMv FǶb i?2Q`2K Uh?2Q`2K 3X8X9V 2ti2M/b iQ, h?2Q`2K 3X8XRR 6Q` i?2 T`Q# #BHBiv P iQ K F2 Q7 i?2 m@i? Q`/2` TQBMi T`Q+2bb N bBKTH2 SQBbbQM T`Q+2bb rBi? MQM@ iQKB+ BMi2MbBiv K2 bm`2 ν- Bi Bb M2+2bb `v M/ bm{+B2Mi i? i 7Q` HH Γ ∈ Mp (Rm )Px! 1 ,...,xm (N ∈ Γ) = P (N ∈ Γ)
dν@ X2X
_2K `F 3X8XRk h?2Q`2K 3X8XRR 7Q`K HBx2b i?2 `2bmHi Q7 1t2`+Bb2 jX3XkX
U3XkeV
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1tpBbB#H2 AMi2MbBiv, :HBKTb2 i i?2 :2M2` H * b2 h?Bb iQTB+ r b H`2 /v i`2 i2/ BM *? Ti2` e 7Q` }MBi2 TQBMi T`Q+2bb2bX h?2 ;2M2` H + b2 Bb KQ`2 BMpQHp2/ M/ r2 b? HH QMHv T`QpB/2 ?2m`BbiB+b- bbmKBM; i? i i?2 `2/m+2/ J2+F2 K2 bm`2 C ! Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ m ⊗ P M/ μ)X h?2M /2MQiBM; i?2 +Q``2bTQM/BM; _ /QMĜLBFQ/ɷK /2`Bp iBp2 #v λ(x, μ)P (dμ) dx g(x, μ − εx ) μ(dx) P (dμ) = g(x, μ)λ(x, Rm
Rm
Mp (Rm )
Q`- BM i?2 Qi?2` MQi iBQM g(x, N − εx ) = E x∈N
Mp (Rm )
N )g(x, N ) dx . E λ(x, Rm
AM T `iB+mH `- rBi? g(x, μ) = 1B (x) UB ∈ Rm V N ) dx , E λ(x, ν(B) = B
r?B+? BKTHB2b i? i ν ? b /2MbBiv UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2V ;Bp2M #v N) . λ(x) := E λ(x, h?2 `2/m+2/ J2+F2 7Q`KmH i?2`27Q`2 `2 /b dx , g(x, N − εx ) = E!x [g(x, N )] λ(x) E Rm
x∈N
M/ i?2`27Q`2 7Q` HKQbi HH UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2V x ∈ Rm N) λ(x, , U3XkdV E!x [g(x, N )] = E g(x, N ) λ(x) r?B+? b vb i? i 7Q` HKQbi HH x ∈ Rm λ(x, N ) :=
N) λ(x, λ(x)
U+ HH2/ i?2 2tpBbB#H2 BMi2MbBiv i TQBMi x Q7 i?2 TQBMi T`Q+2bb N V Bb i?2 _ /QMĜ LBFQ/ɷK /2`Bp iBp2 Q7 Px! rBi? `2bT2+i iQ P X h?2 b K2 ?2m`BbiB+ `;mK2Mi b i?2 QM2 7Q` }MBi2 TQBMi T`Q+2bb2b b?Qrb i? i λ(x, N ) dx = P (N (dx) > 0 | N (· ∩ dx)) , λ(x) i?2 +QM/BiBQM H T`Q# #BHBiv i? i i?2`2 Bb TQBMi Q7 N BM M BM}MBi2bBK H M2B;?#Qm`@ ?QQ/ dx Q7 x- ;Bp2M i?2 HQ+ iBQM Q7 HH TQBMib Q7 N QmibB/2 i?Bb M2B;?#Qm`?QQ/X _2K `F 3X8XRj _2+ HH i?2 HBMF rBi? i?2 K `iBM; H2 TT`Q +? Q#b2`p2/ BM *? T@ i2` eX a22 HbQ 1t2`+Bb2b 3XeX3 M/ 3XeXNX
3XeX 1s1_*Aa1a
3Xe
jjR
1t2`+Bb2b
1t2`+Bb2 3XeXRX MQi?2` 7Q`K Q7 i?2 BMp2`bBQM 7Q`KmH AM i?2 mMB/BK2MbBQM H + b2- b?Qr ?Qr iQ T bb 7`QK i?2 BMp2`bBQM 7Q`KmH Q7 1t@ KTH2 3XRXe iQ i?2 BMp2`bBQM 7Q`KmH BM h?2Q`2K dXjX9X 1t2`+Bb2 3XeXkX p2` ;2 pQHmK2 Q7 i?2 oQ`QMQŖ +2HH pb BMi2MbBivX G2i N #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb rBi? BMi2MbBiv 0 < λ < ∞X a?Qr i? i mM/2` i?2 S HK T`Q# #BHBiv- i?2 2tT2+i2/ pQHmK2 Q7 i?2 oQ`QMQś +2HH V0 Bb 2[m H iQ i?2 BMp2`b2 BMi2MbBivX 1t2`+Bb2 3XeXjX 62HH2`ǰb T ` /Qt G2i N #2 bBKTH2 bi iBQM `v TQBMi T`Q+2bb rBi? BMi2MbBiv 0 < λ < ∞X G2i W (0, N ) #2 i?2 oQ`QMQB +2HH +2Mi2`2/ i TQBMi Q7 N +Qp2`BM; 0 Ui?Bb +2HH Bb mMB[m2- #v G2KK 3XRX9VX a?Qr i? i 7Q` Mv MQM@M2; iBp2 7mM+iBQM g Q7 W0 (N )E [g(W (0, N ))] = λE0N [g(V0 (N ))pQH(V0 (N ))] M/ BM T `iB+mH ` E pQH(W (0, N ))−1 = E0N [pQH(V0 (N ))]−1 . a?Qr i? i E [pQH(W (0, N ))] ≥ E0N [pQH(V0 (N ))] . Uh?Bb Bb i?2 bT iB H p2`bBQM Q7 62HH2`Ƕb T ` /Qt- Q` #mb T ` /Qt XV 1t2`+Bb2 3XeX9X S HK /Bbi`B#miBQM Q7 i?2 BMi2MbBiv Q7 KBt2/ SQBbbQM T`Q+2bb G2i N #2 ?QKQ;2M2Qmb *Qt T`Q+2bb Q7 BMi2;` #H2 +QM/BiBQM H BMi2MbBiv ΛX S`Qp2 i? i E Λ1{Λ≤λ} Px (Λ ≤ λ) = . E [Λ] 1t2`+Bb2 3XeX8X .2MQi2 #v Nx1 ,...,xn M/ Nx! 1 ,...,xn TQBMi T`Q+2bb2b QM Rnm rBi? i?2 /Bbi`B#miBQMb Px1 ,...,xn M/ Px! 1 ,...,xn `2bT2+iBp2HvX bbmK2 MNn+m iQ #2 σ@}MBi2X 6Q` A ∈ B(E)n M/ B ∈ B(E)m MNn+m (A × B) = MNx!m,...,xn (dx1 , . . . , dxm )MNn (dx1 , . . . , dxn ) . A
B
1
! (A × B)X >BMi, *QKTmi2 Cn+m
1t2`+Bb2 3XeXeX LQM@R@BbQH i2/ TQBMib Q7 SQBbbQM T`Q+2bb AM 1t KTH2 3X8Xe- b?Qr i? i i?2 iQi H MmK#2` Q7 MQM@R@BbQH i2/ TQBMib Bb HKQbi@ bm`2Hv BM}MBi2X U"2r `2, MQM@M2; iBp2 }MBi2 ` M/QK p `B #H2 K v ? p2 M BM}MBi2 K2 MXV
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1t2`+Bb2 3XeXdX >B;?2bi /2MbBiv Q7 R@BbQH i2/ TQBMib Q7 SQBbbQM T`Q@ +2bb AM 1t KTH2 3X8Xe- }M/ i?2 BMi2MbBiv λ Q7 i?2 ?TT i? i K tBKBx2b i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb Q7 R@BbQH i2/ TQBMibX *QKT `2 i?Bb rBi? i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb Q7 R@BbQH i2/ TQBMib BM i?2 b im` i2/ J iû`M KQ/2H Q7 1t KTH2 3X9XReX 1t2`+Bb2 3XeX3X S HK S`Q# #BHBiv M/ biQ+? biB+ BMi2MbBiv G2i N #2 HQ+ HHv }MBi2 bBKTH2 TQBMi T`Q+2bb QM i?2 HBM2 rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X G2i A ∈ Fa X S`Qp2 i? i 7Q` G2#2b;m2 HKQbi HH t ≥ aPt (F ) =
E [1F λ(t)] , λ(t)
r?2`2 λ(t) := E [λ(t)]X 1t2`+Bb2 3XeXNX J `iBM; H2b BM/2t2/ #v b2ib a?Qr i? i mM/2` TT`QT`B i2 +QM/BiBQMb Ub22 h?2Q`2K eXjX8V g(x, μ − εx )μ(dx) − E!x [g(x, μ)] ν(dx) MA (g) := A
Bb bm+? i? i
A
E [MA (g) | FA ] = 0 ,
r?2`2 7Q` Mv b2i C ⊆ B(E)- FC := σ(N ∩ C)X aT2+B HBx2 iQ i?2 + b2 r?2`2 N Bb M ?TT Q7 BMi2MbBiv λX
*? Ti2` N h?2 SQr2` aT2+i` H J2 bm`2 h?2 MQiBQM Q7 TQr2` bT2+i` H K2 bm`2 7Q` rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb2bR Bb M im` H 2ti2MbBQM Q7 i?2 M HQ;Qmb MQiBQM 7Q` rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb2b- i?2 "Q+?M2` bT2+i` H K2 bm`2X AM i?Bb +? Ti2`- T `iB+mH ` ii2MiBQM rBHH #2 T B/ iQ i?2 TQr2` bT2+i` Q7 +QKTH2t bB;M Hb # b2/ QM TQBMi T`Q+2bb2b bm+? b i?Qb2 `BbBM; BM +QKKmMB+ iBQMb BM i?2 T`2b2M+2 Q7 `2~2+iBQMb 7Q` BMbi M+2Xk h?2 K BM `2bmHi rBi? `2bT2+i iQ i?Bb ;Q H Bb bBKTH2 mMBp2`b H 7Q`KmH 7`QK r?B+? MmK#2` Q7 bT2+B}+ TQr2` bT2+i` + M #2 Q#i BM2/ i HBiiH2 2tT2Mb2X
NXR
h?2 *Qp `B M+2 J2 bm`2
aBM+2 i?2 TQr2` bT2+i` H i?2Q`v Q7 TQBMi T`Q+2bb2b BKBi i2b i?2 +H bbB+ H i?2Q`v +QM+2`MBM; rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb2b- r2 b? HH i F2 bQK2 iBK2 iQ `2pB2r i?2 # bB+ `2bmHib Q7 i?2 H ii2` i?2Q`vX qB/2@a2Mb2 ai iBQM `v aiQ+? biB+ S`Q+2bb2b 72r `2bmHib +QM+2`MBM; i?2 bT2+i` H M HvbBb Q7 rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb2b rBHH #2 `2+ HH2/ BM pB2r Q7 +QKT `BbQM rBi? i?2 M HQ;Qmb QM2b 7Q` rB/2@ b2Mb2 bi iBQM `v TQBMi T`Q+2bb2bX h?2B` T`QQ7b `2 QKBii2/ bBM+2 i?2v + M #2 B;MQ`2/ 7Q` i?2 /2p2HQTK2Mi Q7 i?2 i?2Q`v Q7 i?Bb +? Ti2` U2t+2Ti 7Q` h?2Q`2K RyX8XR BM i?2 M2ti +? Ti2`VX rB/2@b2Mb2 bi iBQM `v UrbbV biQ+? biB+ T`Q+2bb {X (t)}t∈Rm Bb- #v /2}MBiBQM- b2+QM/@Q`/2` biQ+? biB+ T`Q+2bb Ui? i Bb- E |X(t)|2 < ∞ 7Q` HH t ∈ Rm V bm+? i? i E [X(t)] = μ UBM/2T2M/2Mi Q7 tV M/ +Qp (X(t + τ ), X(t)) = CX (τ ) UBM/2T2M/2Mi Q7 tVX h?2 7mM+iBQM CX Bb + HH2/ i?2 +Qp `B M+2 7mM+iBQMX h?2Q`2K NXRXR 6Q` HH ϕ ∈ L1C (Rm )- i?2 BMi2;` H Rm ϕ(t)X(t) dt Bb r2HH /2}M2/ M/ BM L1C (Rm ) ∩ L2C (Rm )X HbQ- 7Q` HH 7mM+iBQMb ϕ, ψ ∈ L1C (Rm ) +Qp ϕ(t)X(t) dt, ψ(s)X(s) ds Rm Rm = ϕ(t)ψ ∗ (t + s) dt CX (s) ds . UNXRV Rm
Rm
R
(" `iH2ii- RNej)X k (_B/QH}- RNNN)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_9
jjj
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*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 Uj V _2+ HH i?2 /2}MBiBQM Q7 i?2 6Qm`B2` i` Mb7Q`K ϕ Q7 7mM+iBQM ϕ ∈ L1C (Rm ), ϕ(t)e−2iπν,t h(t) dt . ϕ(ν) := Rm
h?2Q`2K NXRXk lM/2` i?2 //BiBQM H bbmKTiBQM Q7 +QMiBMmBiv BM i?2 [m /` iB+ K2 M U2[mBp H2Mi iQ +QMiBMmBiv Q7 i?2 +Qp `B M+2 7mM+iBQMV i?2`2 2tBbib U"Q+?M2`Ƕb i?2Q`2KV mMB[m2 }MBi2 K2 bm`2 μ QM Rm - + HH2/ i?2 TQr2` bT2+i` H K2 bm`2 Q7 i?Bb rbb biQ+? biB+ T`Q+2bb- bm+? i? i CX (τ ) = e2iπν,τ μX (dν) (τ ∈ Rm ) . Rm
HbQ- 7Q` HH ϕ ∈ L1C (Rm ) rBi? 6Qm`B2` i` Mb7Q`K ϕ 2
2 ϕ(t)X(t) dt |ϕ(ν)| μ(dν) . =E E Rm
Rm
UNXkV
h?2Q`2K NXRXj G2i {X (t)}t∈Rm #2 rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb +QM@ iBMmQmb BM i?2 [m /` iB+ K2 M rBi? TQr2` bT2+i` H K2 bm`2 μX X G2i h(t) Ut ∈ Rm V #2 /2i2`KBMBbiB+ 7mM+iBQM BMi2;` #H2 rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 QM R M/ /KBiiBM; i?2 6Qm`B2` i` Mb7Q`K hX h?2M- i?2 biQ+? biB+ T`Q+2bb Y (t) := h(t − s)X(s) ds (t ∈ Rm ) Rm
Bb r2HH@/2}M2/ rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb +QMiBMmQmb BM i?2 [m /` iB+ K2 M M/ rBi? TQr2` bT2+i` H K2 bm`2 & &2 & & μY (dν) = &h(ν)& μX (dν) .
h?2Q`2K NXRX9 G2i {X(t)}t∈R #2 +2Mi2`2/ rbb biQ+? biB+ T`Q+2bb- +QMiBM@ mQmb BM [m /` iB+ K2 M- M/ H2i μ #2 Bib TQr2` bT2+i` H K2 bm`2X h?2`2 2t@ Bbib mMB[m2 UKQ`2 T`2+BbBQM #2HQr i?2 i?2Q`2KV +2Mi2`2/ biQ+? biB+ T`Q+2bb {x(ν)}ν∈R rBi? mM+Q``2H i2/ BM+`2K2Mib M/ rBi? bi`m+im` H K2 bm`2 μ Ui? i BbE [|x(ν1 ) − x(ν2 )|2 ] = μ((ν1 , ν2 ])V- bm+? i? i P @ XbX X(t) = e2iπνt dx(ν) (t ∈ R) , UNXjV R
r?2`2 i?2 BMi2;` H Q7 i?2 `B;?i@? M/ bB/2 Bb .QQ# BMi2;` HX h?2 /2+QKTQbBiBQM UNXjV Bb i?2 *` Kû`ĜE?BM+?BM /2+QKTQbBiBQMX q2 MQr im`M iQ i?2 bT2+i` H i?2Q`v Q7 TQBMi T`Q+2bb2bX j 6Q` T`QQ7 Q7 i?2 #Qp2 `2bmHi M/ Q7 i?2 M2ti QM2b- b22 7Q` BMbi M+2 ("`ûK m/- kyR9- `2bTX kyky)- *? TX j- `2bTX RkX
NXRX h>1 *Po_AL*1 J1al_1
jj8
aTBF2 h` BMb TQBMi T`Q+2bb N QM R rBi? b2[m2M+2 Q7 2p2Mi iBK2b {Tn }n∈Z + M #2 `2T`2b2Mi2/ bvK#QHB+ HHv #v i?2 dz` M/QK .B` + +QK#Ǵ X(t) := δ(t − Tn ), UNX9V n∈Z
i?2 bmK i?2`2BM 2ti2M/BM;- ++Q`/BM; iQ i?2 mbm H +QMp2MiBQM- iQ HH 2p2Mi iBK2b i }MBi2 /Bbi M+2X h?2 `B;?i@? M/ bB/2 Bb +2`i BMHv MQi #QM }/2 biQ+? biB+ T`Q+2bb bBM+2 i?2 /2Hi 7mM+iBQM Bb MQi 7mM+iBQM BM i?2 Q`/BM `v b2Mb2- #mi /Bbi`B#miBQMX AM T `iB+mH `- QM2 + MMQi /2}M2 7Q` i?2 ` M/QK .B` + +QK# bbQ+B i2/ rBi? bi iBQM `v TQBMi T`Q+2bb TQr2` bT2+i` H K2 bm`2 b BM i?2 + b2 Q7 rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb2bX h?2 *Qp `B M+2 J2 bm`2 Q7 SQBMi S`Q+2bb _2+ HH 72r /2}MBiBQMb M/ 7 +ib 7`QK a2+iBQM RXkX TQBMi T`Q+2bb N QM Rm Bb + HH2/ b2+QM/@Q`/2` TQBMi T`Q+2bb B7 7Q` HH +QKT +i b2ib C ⊂ Rm E N (C)2 < ∞ . h?Bb BKTHB2b BM T `iB+mH ` i? i i?2 BMi2MbBiv K2 bm`2 ν Bb HQ+ HHv }MBi2X h?2 b2+QM/ KQK2Mi K2 bm`2 M2 Bb HQ+ HHv }MBi2 K2 bm`2 QM Rm × Rm , B(Rm ) ⊗ B(Rm ) i? i Bb r2HH M/ mMB[m2Hv /2}M2/ #v Bib p Hm2b QM i?2 K2 bm` #H2 `2+i M;H2b, M2 (A × B) = E [N (A) N (B)]
(A, B ∈ B(Rm )) .
6Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb g : Rm × Rm → R g(t, s) N (dt)N (ds) = g(t, s) M2 (dt × ds) , E Rm
Rm
Rm
Rm
7Q`KmH r?B+? 2ti2M/b iQ HH K2 bm` #H2 7mM+iBQMb g : Rm × Rm → C bm+? i? i |g(t, s)| M2 (dt × ds) < ∞ . Rm
Rm
AM T `iB+mH `- B7 ϕ, ψ ∈ L2N (M2 ) ∗ = ϕ(t) N (dt) ψ(t)N (dt) E Rm
Rm
Rm
Rm
ϕ(t)ψ ∗ (s)M2 (dt × ds) . UNX8V
.2}MBiBQM NXRX8 .2}M2 L2N (M2 ) iQ #2 i?2 +QHH2+iBQM Q7 K2 bm` #H2 7mM+iBQMb ϕ : R → C bm+? i? i |ϕ(t)ϕ(s)| M2 (dt × ds) < ∞ Rm
Q`- 2[mBp H2MiHv-
Rm
E N (|ϕ|)2 < ∞ .
UNXeV
jje
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
h?2Q`2K NXRXe L2N (M2 ) Bb p2+iQ` bT +2 i? i +QMi BMb HH #QmM/2/ 7mM+iBQMb rBi? +QKT +i bmTTQ`iX AM 7 +i- r2 ? p2 i?2 7QHHQrBM; BM+HmbBQM, L2N (M2 ) ⊆ L1C (ν) ∩ L2C (ν) .
UNXdV
S`QQ7X AM2[m HBiv UNXeV BKTHB2b BM T `iB+mH ` i? i E [N (|ϕ|)] < ∞ M/ i?2`27Q`2 U* KT#2HHǶb 7Q`KmH V ϕ ∈ L1C (ν)X aBM+2 2 2 |ϕ(Xn )| ≤ |ϕ(Xn )| , n∈N 2 B7 ϕ ∈ LN2(M2 )- i?2M E |ϕ(t)| ν(dt) < ∞X Rm
n∈N
Rm
|ϕ(t)|2 N (dt) < ∞- i? i Bb- #v * KT#2HHǶb 7Q`KmH
.2}MBiBQM NXRXd rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb Bb- #v /2}MBiBQM- b2+QM/@ Q`/2` TQBMi T`Q+2bb N bm+? i? i 7Q` HH b2ib C ∈ B(Rm ) M/ HH t ∈ Rm E [N (C + t)] = E [N (C)] , M/ bm+? i? i 7Q` HH #QmM/2/ A, B ∈ B(Rm ) M/ HH t ∈ Rm E [N (A + t)N (B + t)] = E [N (A)N (B)] . h?2 H bi `2[mB`2K2Mi BKTHB2b BM T `iB+mH ` i? i 7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb ϕ, ψ : Rm → R- i?2 [m MiBiv
E ϕ(τ + t) N (dt) ψ(τ + t) N (dt) R
R
Bb BM/2T2M/2Mi Q7 τ ∈ RX U1t2`+Bb2 NX8XjXV G2i N #2 rB/2@b2Mb2 bi iBQM `v bBKTH2 M/ HQ+ HHv }MBi2 TQBMi T`Q+2bb QM Rm X AM T `iB+mH `- Bib BMi2MbBiv K2 bm`2 ν Bb BMp `B Mi mM/2` i` MbH iBQM M/ i?2`27Q`2 Uh?2Q`2K XRX3V ν (C) = λ m (C) U m Bb i?2 G2#2b;m2 K2 bm`2 QM Rm V 7Q` bQK2 λ ∈ R+ - + HH2/ i?2 BMi2MbBivX AM T `iB+mH ` LpC (ν) = LpC (Rm ) M/ i?2`27Q`2- #v h?2Q`2K NXRXeL2N (M2 ) ⊆ L1C (Rm ) ∩ L2C (Rm ) .
UNX3V
"v bi iBQM `Biv ; BM- 7Q` HH b2ib A, B ∈ B(Rm ) M/ HH t ∈ Rm M2 ((A + t) × (B + t)) = M2 (A × B) . Ai 7QHHQrb 7`QK i?Bb M/ h?2Q`2K XRXR9- i? i 7Q` HH ϕ, ψ ∈ L2N (M2 ) ϕ (t) ψ ∗ (s) M2 (dt × ds) = ϕ (t) ψ ∗ (s + t) dt σ (ds) Rm
Rm
7Q` bQK2 HQ+ HHv }MBi2 K2 bm`2 σX
Rm
Rm
UNXNV
NXRX h>1 *Po_AL*1 J1al_1
jjd
h?2Q`2K NXRX3 6Q` HH ϕ, ψ ∈ L2N (M2 ) +Qp
ϕ (t) N (dt) , ψ (s) N (ds) m Rm R = ϕ (t) ψ ∗ (t + s) dt ΓN (ds) Rm
Rm
UNXRyV
r?2`2 i?2 HQ+ HHv }MBi2 K2 bm`2 ΓN := σ − λ2 m
UNXRRV
Bb + HH2/ i?2 +Qp `B M+2 K2 bm`2 Q7 i?2 bi iBQM `v b2+QM/@Q`/2` TQBMi T`Q+2bb N X
S`QQ7X ai `iBM; 7`QK UNXNV Bi bm{+2b iQ Q#b2`p2 i? i 7Q` ϕ, ψ ∈ L1C (Rm ) ∗ E [N (ϕ)] E [N (ψ)] = λ ϕ (t) dt λ ψ (s) ds m Rm R = λ2 ϕ (t) ψ ∗ (t + s) dt ds . ∗
Rm
Rm
LQi2 i?2 7Q`K H `2b2K#H M+2 Q7 1[Mb UNXRV M/ UNXRyVX 1t KTH2 NXRXN, h?2 *Qp `B M+2 J2 bm`2 Q7 i?2 >QKQ;2M2Qmb SQBbbQM S`Q+2bbX G2i N #2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM i?2 HBM2 rBi? BMi2MbBiv λ > 0X "v i?2 +Qp `B M+2 7Q`KmH UjX9V +Qp (N (ϕ), N (ψ)) = λ
ϕ (t) ψ ∗ (t) dt .
R
i 0- r2 ? p2 7Q` i?2 `B;?i@? M/ bB/2 Q7 i?2 H bi .2MQiBM; #v ε0 i?2 .B` + K2 bm`2 /BbTH v i?2 2tT`2bbBQM λ R R ϕ (t) ψ ∗ (t + s) dt ε0 (ds) M/ i?2`27Q`2- +QKT `BM; rBi? i?2 `B;?i@? M/ bB/2 Q7 UNXRVΓN = λε0 .
1t KTH2 NXRXRy, h?2 *Qp `B M+2 J2 bm`2 Q7 i?2 _2M2r H S`Q+2bbX G2i N #2 bi iBQM `v `2M2r H TQBMi T`Q+2bb rBi? `2M2r H 7mM+iBQM R Ui?2 b K2 MQi iBQM rBHH #2 mb2/ 7Q` i?2 bbQ+B i2/ `2M2r H K2 bm`2VX 6Q` Mv K2 bm` #H2 MQM@M2; iBp2 7mM+iBQMb ϕ, ψ : R+ → R-
jj3
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
ϕ(t) N (dt) ψ(t) N (dt) R R =E ϕ(Tn ) ψ(Tn + (Tn+m − Tn ))
E
m≥n
n∈Z
+E
=
E ϕ(Tn )
ψ(Tn )
+
ϕ(Tn + (Tn+m − Tn ))
m>1
n∈Z
ψ(Tn + (Tn+m − Tn ))
m≥1
n∈Z
E ψ(Tn )
ϕ(Tn + (Tn+m − Tn ))
.
m>1
n∈Z
* HH A M/ B i?2 bmKb QM i?2 `B;?i@? M/ bB/2X aBM+2 {(Tn+m − Tn )}m≥1 Bb M mM@ /2H v2/ `2M2r H T`Q+2bb BM/2T2M/2Mi Q7 Tn M/ rBi? i?2 b K2 BMi2`p H /Bbi`B#miBQM b i?2 Q`B;BM H `2M2r H T`Q+2bb- TTHvBM; * KT#2HHǶb 7Q`KmH irB+2 ;Bp2b A= E ϕ(Tn )E ψ(Tn + (Tn+m − Tn )) | Tn n∈Z
=
E ϕ(Tn )
ψ(t) R(dt + Tn ) R
n∈Z
=E
m≥1
ϕ(Tn )
n∈Z
ψ(t) R(dt + Tn )
R
ϕ(s)
=E R
aBKBH `Hv
R ϕ(s)
=E
ψ(t) R(dt + s) λ ds
ψ(s + t) R(dt) λ ds .
R
[s,+∞)
ψ(s)
B=E R
R
ϕ(s + t) R(dt) λ ds . (s,+∞)
6BM HHv
ϕ(t) N (dt) ψ(t) N (dt) =E E R
ϕ(s) Rm
ψ(s + t) R(dt) λ ds , R
7`QK r?B+? Bi 7QHHQrb i? i σ = λ R M/ i?2 +Qp `B M+2 K2 bm`2 Bb BM i?Bb + b2, ΓN (dt) = λ(R(dt) − λ dt) .
UNXRkV
Ua22 HbQ 1t KTH2 NXkXN 7Q` M Hi2`M iBp2 T`QQ7XV h?2 `2M2r H K2 bm`2 Q7 i?2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ Bb i?2 .B` + K2 bm`2 ε0 THmb λ iBK2b i?2 G2#2b;m2 K2 bm`2X h?2`27Q`2 ΓN (dt) = λ ε0 (dt)- b H`2 /v T`Qp2/ BM 1t KTH2 NXRXNX
NXRX h>1 *Po_AL*1 J1al_1
jjN
_BTH2vǶb 6mM+iBQM _2+ HH i?2 /2}MBiBQM Q7 i?2 7 +iQ`B H KQK2Mi K2 bm`2 M2! Q7 b2+QM/@Q`/2` TQBMi T`Q+2bb QM Rm , ! 1A (Xj )1B (Xk )1j=k (A, B ∈ B(Rm )) . M2 (A × B) := E j
k
6Q` Mv MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM f : Rm × Rm → R- Bi ?QH/b i? i f (Xj , Xk )1j=k = f (t, s) M2! (dt × ds) . E j
Rm
k
Rm
A7 N Bb KQ`2Qp2` rB/2@b2Mb2 bi iBQM `v rBi? p2` ;2 BMi2MbBiv λ- `;mK2Mib bBKBH ` iQ i?Qb2 H2 /BM; iQ i?2 /2}MBiBQM Q7 i?2 +Qp `B M+2 K2 bm`2 b?Qr i?2 2tBbi2M+2 Q7 σ@}MBi2 K2 bm`2 K QM Rm bm+? i? i f (Xj , Xk )1j=k = λ f (t, t + s) dt K(ds) . E j
Rm
k
Rm
TTHvBM; i?Bb 7Q`KmH iQ f (t, s) := 1A (t)1B (s − t) UA, B ∈ B(Rm )V ;Bp2b 1A (Xj )1B (Xk − Xj )1j=k = λ1A (t)1B (u) dx K(du) = λ m (A)K(B) , E j
k
i? i BbK(B) =
E[
n
1A (Xn )N ((B + Xn )\Xn )] . E [N (A)]
UNXRjV
.2}MBiBQM NXRXRR U9 V h?2 _BTH2v 7mM+iBQM Bb i?2 7mM+iBQM K : R+ → R+ /2}M2/ #v 1 K(r) := K(B(0, r) . λ aT2+B HBxBM; UNXRjV iQ B = B(0; r)- r2 Q#i BM i?2 2tT`2bbBQM
E [ n 1A (Xn )N ((B(0; r) + Xn )\Xn )] K(r) = E [N (A)]
E [ n 1A (Xn (N (B(0; r) + Xn ) − 1)] . = E [N (A)] h?2 [m MiBiv λK(r) `2T`2b2Mib 7Q` Mv TQBMi Xn i?2 p2` ;2 MmK#2` Q7 TQBMib Xj bm+? i? i 0 < d(Xj , Xk ) ≤ rX 6`QK J2+F2Ƕb 7Q`KmH , K(r) =
E0N [N (B(0; r) − 1] . λ
_BTH2vǶb 7mM+iBQM Bb M BKTQ`i Mi bi iBbiB+ H iQQH M/ Bib 2biBK i2b `2 mb27mH BM /2i2`KBMBM; i?2 M im`2 Q7 TQBMi T`Q+2bbX h?2 [m MiBiv 9
(_BTH2v- RNde)X
j9y
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
K(r) :=
j
k
1W (Xj )1W (Xk )1d(Xj ,Xk )≤r 1k=j
,
λW r?2`2 W Bb M Q#b2`p iBQM rBM/Qr M/ λW :=
N (W ) ,
m (W )
Bb mb2/ b M 2biBK iQ` Q7 K(r) i? i + M #2 mb2/ iQ i2bi M ?TT ?vTQi?2bBb- b i?2 M2ti 2t KTH2 bm;;2bibX8 1t KTH2 NXRXRk, _BTH2v 6mM+iBQM Q7 ?TTX A7 N Bb M ?TT- i?2M K(r) = νm rm , r?2`2 νm Bb i?2 m@pQHmK2 Q7 i?2 m@/BK2MbBQM H bT?2`2 Q7 ` /Bmb 1 U1t2`+Bb2 NX8XkXV
NXk h?2 " `iH2ii aT2+i` H J2 bm`2 h?2 2ti2MbBQM Q7 i?2 MQiBQM Q7 TQr2` bT2+i` H K2 bm`2 7Q` bi iBQM `v TQBMi T`Q@ +2bb2b i? i Bb bmBi #H2 7Q` bi iBQM `v TQBMi T`Q+2bb2b Bb i?2 bQ@+ HH2/ " `iH2ii bT2+i` H K2 bm`2X .2}MBiBQM NXkXR G2i N #2 bBKTH2 rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λX K2 bm`2 μN QM Rm Bb + HH2/ i?2 TQr2` bT2+i` H K2 bm`2 Q7 N B7 Bi Bb i?2 mMB[m2 HQ+ HHv }MBi2 K2 bm`2 μN bm+? i? i ϕ (t) N (dt) = |ϕ (ν)|2 μN (dν) UNXR9V o ` Rm
Rm
7Q` HH ϕ ∈ BN - r?2`2 BN ⊆ L2N (M 2 ) Bb p2+iQ` bT +2 Q7 7mM+iBQMb + HH2/ i?2 i2bi 7mM+iBQM bT +2- 7mM+iBQM i?2`2BM #2BM; + HH2/ i2bi 7mM+iBQMX AM T `iB+mH `,
o ` ULUUy-h)V =
R
sin πνT πν
2 μN (dν) .
UNXR8V
h?Bb BM 7 +i +? ` +i2`Bx2b i?2 TQr2` bT2+i` H K2 bm`2Xe _2K `F NXkXk LQi2 i?2 7Q`K H `2b2K#H M+2 Q7 1[Mb UNXkV M/ UNXR9VX "v TQH `Bx iBQM Q7 UNXR9V- 7Q` HH ϕ, ψ ∈ BN +Qp (N (ϕ) , N (ψ)) = ϕ(ν)ψ ∗ (ν)μN (dν). Rm
UNXReV
8 6Q` i?2 bi iBbiB+ H TTHB+ iBQMb Q7 i?2 i?2Q`v Q7 TQBMi T`Q+2bb2b- b22 (*?Bm- aiQv M- E2M/ HH M/ J2+F2- kyRj)X e (. H2v- RNdR)X
NXkX h>1 "_hG1hh aS1*h_G J1al_1
j9R
_2K `F NXkXj h?2 2ti2Mi Q7 i?2 i2bi 7mM+iBQM bT +2 BN Bb ;Bp2M BM 2 +? bBim @ iBQMXd Ai K v MQi #2 i?2 H `;2bi QM2- #mi Bi b?QmH/ BM Mv + b2 +QMi BM +H bb Q7 7mM+iBQMb `B+? 2MQm;? iQ ;m ` Mi22 mMB[m2M2bb Q7 i?2 K2 bm`2 μN X "v i?Bb- i?2 7QHHQrBM; Bb K2 MiX A7 i?2 HQ+ HHv }MBi2 K2 bm`2b μ1 M/ μ2 `2 bm+? i? i |ϕ(ν)|2 μ1 (dν) = |ϕ(ν)|2 μ2 (dν) Rm
Rm
7Q` HH ϕ ∈ BN - i?2M μ1 ≡ μ2 X h?2Q`2K NXkXRy rBHH i2HH mb i? i BN ⊇ B- i?2 b2i Q7 +QMiBMmQmb 7mM+iBQMb bm+? i? i #Qi? f M/ Bib 6Qm`B2` i` Mb7Q`K f `2 O 1/ |t|2 b |t| → ∞X _2K `F NXkX9 LQi2 i? i BN ⊆ L1C (Rm ) bBM+2- b r2 Q#b2`p2/ 2 `HB2`- L2N (M 2 ) ⊆ L1C (Rm )X AM T `iB+mH `- i?2 6Qm`B2` i` Mb7Q`K Q7 Mv ϕ ∈ BN Bb r2HH /2}M2/X h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 " `iH2ii bT2+i` H K2 bm`2 Bb T`Qp2/ BM h?2Q`2K NXkXRy #2HQr- r?2`2 Bi rBHH #2 b?QrM i? i BN +QMi BMb i H2 bi i?2 7mM+iBQMb i? i `2- iQ;2i?2` rBi? i?2B` 6Qm`B2` i` Mb7Q`K- O 1/ |t|2 b |t| → ∞X _2K `F NXkX8 PM2 bQK2iBK2b T`272`b iQ bi i2 i?2 `2bmHib BM i2`Kb Q7 dzTmHb iBQMbǴ BMbi2 / Q7 dz7`2[m2M+B2bǴX h?2 6Qm`B2` i` Mb7Q`K ϕ Q7 ϕ i?2M #2BM; /2}M2/ #v ϕ(t)e−iωt dt , ϕ(ω) := R
M 2H2K2Mi `v +? M;2 Q7 p `B #H2b vB2H/b i?2 7Q`KmH
o `
ϕ (t) N (dt) R
= R
|ϕ(ω)|2 fN (ω) dω ,
r?2`2 i?2 dzω@bT2+i` H /2MbBivǴ Bb fN (ω) =
1 ω f . 2π N 2π
h?2 /2}MBiBQM Q7 i?2 " `iH2ii bT2+i` H K2 bm`2 BKK2/B i2Hv vB2H/b i?2 bT2+i` H K2 bm`2 Q7 i?2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb, 1t KTH2 NXkXe, SQBbbQM AKTmHbBp2 q?Bi2 LQBb2X Ai r b b?QrM BM 1t K@ TH2 NXRXRy i? i i?2 +Qp `B M+2 7mM+iBQM Q7 SQBbbQM T`Q+2bb Bb λ iBK2b i?2 .B` + K2 bm`2 i i?2 Q`B;BM M/ i?2`27Q`2 Bib bT2+i` H K2 bm`2 Bb λ iBK2b i?2 G2#2b;m2 K2 bm`2- i? i Bb- Bi /KBib i?2 TQr2` bT2+i` H /2MbBiv fN (ν) ≡ λ.
d AM i?2 i?2Q`2iB+ H HBi2` im`2- i?2 +?QB+2 Bb mbm HHv b2i Q7 p2`v bKQQi? 7mM+iBQMb- r?B+? K v #2 iQQ `2bi`B+iBp2 7Q` TTHB+ iBQMbX
j9k
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
6Q` i?Bb `2 bQM i?2 ` M/QK .B` + +QK# UNX9V +Q``2bTQM/BM; iQ SQBbbQM T`Q+2bb Bb bQK2iBK2b + HH2/ i?2 SQBbbQM BKTmHbBp2 r?Bi2 MQBb2X3 1t KTH2 NXkXd, a?Qi LQBb2X G2i N #2 rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb QM Rm X 6Q` HH h ∈ BN M/ HH u, v ∈ Rm - H2i ϕ(t) := h(u − t), ψ(t) = h(v − t). "v UNXReV & &2 & & h(u − t)N (dt), h(v − t)N (dt) = +Qp &h(ν)& e2iπν(v−u) μN (dν). R
R
R
AM i?2 mMBp `B i2 + b2 Um = 1V- i?2 TQBMi T`Q+2bb Bb `2T`2b2Mi2/ #v Bib b2[m2M+2 Q7 2p2Mi iBK2b {Tn }n∈Z - M/ X(t) = h(t − Tn ) n∈Z
Bb i?2 biQ+? biB+ T`Q+2bb Q#i BM2/ #v T bbBM; i?2 .B` + +QK# n δ(t − Tn ) i?`Qm;? i?2 }Hi2`X h?2 TQr2` bT2+i` H K2 bm`2 Q7 i?2 QmiTmi {X(t)}t∈R Bb & &2 & & UNXRdV μX (dν) = &h(ν)& μN (dν) , r?B+? BM/22/ +Q``2bTQM/b iQ i?2 }Hi2`BM; 7Q`KmH Q7 h?2Q`2K NXRXj 7Q` #QM }/2
rbb bB;M Hb B7 r2 bbBKBH i2 i?2 .B` + +QK# δ(t − Tn ) iQ #QM }/2 rbb bB;M H Ur?B+? Q7 +Qm`b2 Bi Bb MQiV rBi? bT2+i` H K2 bm`2 μN X h?2 bT2+i` H K2 bm`2 μN Bb ?Qr2p2` MQi }MBi2 K2 bm`2 b Bi rQmH/ #2 7Q` Q`/BM `v rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb2bX
1tBbi2M+2 M/ lMB[m2M2bb h?2 T`QQ7 Q7 2tBbi2M+2 Q7 i?2 " `iH2ii bT2+i` H K2 bm`2 rBHH #2 /QM2 BM i?2 mMB@ p `B i2 + b2 7Q` i?2 b F2 Q7 HB;?i2` MQi iBQM- #mi Bi `2K BMb i?2 b K2 BM i?2 ;2M2` H + b2X G2i N #2 bBKTH2 HQ+ HHv }MBi2 rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb QM R M/ H2i σ #2 i?2 HQ+ HHv }MBi2 K2 bm`2 bm+? i? i 7Q` HH 7mM+iBQMb ϕ, ψ : R → R BM L2N (M2 ) ϕ(t)ψ(s + t) dt
E [N (ϕ)N (ψ)] = R
σ(ds) .
R
G2KK NXkX3 6Q` HH +QKT +i b2ib K ⊂ Rsup σ (K + t) < ∞ .
UNXR3V
t∈R
JQ`2Qp2`- B7 f : R → R Bb MQM@M2; iBp2 7mM+iBQM
i? i Bb MQM@BM+`2 bBM; QM [1, ∞)-MQM@/2+`2 bBM; QM (−∞, 0] M/ bm+? i? i n∈Z f (n) < ∞- r2 ? p2 i? i supt∈R R f (t + s) σ (ds) < ∞X 3 AM S?vbB+b- QM2 mbm HHv + HHb dzr?Bi2 MQBb2Ǵ ` M/QK 7mM+iBQM rBi? ~ i bT2+i`mK Qp2` H `;2 ` M;2 Q7 7`2[m2M+B2bX
NXkX h>1 "_hG1hh aS1*h_G J1al_1
j9j
S`QQ7X a2H2+i ϕ : R → R- MQM@M2; iBp2 7mM+iBQM rBi? +QKT +i bmTTQ`i bm+? i? i ϕ ∗ ϕˇ ≥ 1K Ur?2`2 ϕ(t) ˇ = ϕ(−t)VX ˇ h?2M- τt /2MQiBM; i?2 b?B7i QT2` iQ` s → s − t ˇ (s − t) σ (ds) σ (K + t) ≤ (ϕ ∗ ϕ) R = ((ϕ ◦ τt ) ∗ ϕ) ˇ (s) σ (ds) = E [N (ϕ ◦ τt ) N (ϕ)] . R
h?2`27Q`2- #v a+?r `xǶb BM2[m HBiv σ (K + t) ≤ E N (ϕ)2 , }MBi2 [m MiBiv bBM+2 ϕ Bb #QmM/2/ rBi? +QKT +i bmTTQ`i M/ N Bb b2+QM/@Q`/2` TQBMi T`Q+2bbX LQr- 7Q` f b BM i?2 bi i2K2Mi Q7 i?2 H2KK f (n) σ ([n, n + 1) − t) + f (n) σ ([n − 1, n) − t) sup f (t + s) σ (ds) ≤ t∈R
R
n≥0
≤
n≤−1
f (n)
sup σ ([l, l + 1) − t)
≤C
l∈Z,t∈R
n∈Z
f (n) ,
n∈Z
r?2`2 C < ∞- #v TTHB+ iBQM Q7 UNXR3V iQ i?2 +QKT +i b2i [0, 1]X
1t KTH2 NXkXN, *Qp `B M+2 J2 bm`2 M/ S HK S`Q# #BHBivX q2 MQr ;Bp2 MQi?2` T`QQ7- # b2/ QM J2+F2Ƕb 7Q`KmH - Q7 h?2Q`2K XRXR9 M/ G2KK NXkX3 7Q` i?2 +Qp `B M+2 K2 bm`2 Q7 b2+QM/@Q`/2` TQBMi T`Q+2bb BM i?2 bi`B+iHv bi iBQM `v + b2X G2i (Ω, F, P) #2 T`Q# #BHBiv bT +2 2M/Qr2/ rBi? K2 bm` #H2 ~Qr {θt }t∈Rm - M/ bmTTQb2 i? i P Bb θt @BMp `B Mi 7Q` HH t ∈ Rm X G2i N #2 bBKTH2 TQBMi T`Q+2bb /2}M2/ QM i?Bb T`Q# #BHBiv bT +2- M/ θt @+QKT iB#H2X Ai Bb BM T `iB+mH ` bi iBQM `v- rBi? BMi2MbBiv λ bbmK2/ }MBi2 M/ TQbBiBp2X *QMbB/2` i?2 K2 bm`2 σ QM (Rm , B(Rm )) /2}M2/ #v σ (C) = λE0N [N (C)]
UNXRNV
7Q` HH C ∈ B(Rm )- r?2`2 E0N /2MQi2b BMi2;` iBQM rBi? `2bT2+i iQ i?2 S HK T`Q#@ #BHBiv PN0 X h?2 K2 bm`2 σ Bb HQ+ HHv }MBi2 K2 bm`2c 2p2M KQ`2, sup σ (K + t) < ∞
UNXkyV
t∈Rm
7Q` HH +QKT +i K ⊂ Rm X HbQ 7Q` f, g : Rm → R- K2 bm` #H2 M/ MQM@M2; iBp2 E [N (f ) N (g)] = (f ∗ gˇ) dσ . UNXkRV Rm
U_2+ HH i?2 MQi iBQM fˇ(t) = f (−t)XV
j99
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
S`QQ7X 6Q` MQM@M2; iBp2 K2 bm` #H2 f : Rm → R g dσ = λE0N [N (g)] , Rm
M/ i?2`27Q`2
g dN E [N (f ) N (g)] =E N (f ) Rm = g (t) N (f ) (ω) N (ω, dt) P (dω) m Ω R =λ g (t) N (f ) (θ−t ω) dt PN0 (dω) Ω Rm =λ g (t) f (t + s) N (ω, ds) dt PN0 (dω) m Rm Ω R =λ f (t + s) g (t) dt N (ω, ds) PN0 (dω) , Ω
Rm
Rm
r?2`2 i?2 i?B`/ 2[m HBiv mb2b J2+F2Ƕb 7Q`KmH X 6BM HHv
0 (f ∗ gˇ) dN = E [N (f ) N (g)] = λEN Rm
Rm
(f ∗ gˇ) dσ.
q2 ? p2 i?2`27Q`2 T`Qp2/ UNXkRVX AM Q`/2` iQ T`Qp2 UNXkyV- b2H2+i ϕ : Rm → R- MQM@M2; iBp2 7mM+iBQM rBi? +QKT +i bmTTQ`i bm+? i? i ϕ ∗ ϕˇ ≥ 1K X h?2M σ (K + t) ≤ (ϕ ∗ ϕ) ˇ (s − t) σ (ds) = ((ϕ ◦ τt ) ∗ ϕ) ˇ (s) σ (ds) Rm
Rm
=E [N (ϕ ◦ τt ) N (ϕ)] = E [(N (ϕ) ◦ θt ) (N (ϕ))] . h?2`27Q`2- #v a+?r `xǶb BM2[m HBiv- σ (K + t) ≤ E N (ϕ)2 - }MBi2 [m MiBiv bBM+2 ϕ Bb #QmM/2/ rBi? +QKT +i bmTTQ`i M/ N Bb b2+QM/@Q`/2` TQBMi T`Q+2bbX
h?2Q`2K NXkXRy G2i N #2 b #Qp2X h?2`2 2tBbib mMB[m2 MQM@M2; iBp2 HQ+ HHv }MBi2 K2 bm`2 σ QM (R, B(R)) bm+? i? i- B7 f Bb +QMiBMmQmb M/ #Qi? f M/ Bib 6Qm`B2` i` Mb7Q`K f `2 O 1/ |t|2 b |t| → ∞- i?2M f (ν) σ (dν) = f (t) σ (dt) UNXkkV R
R
M/- B7 g b iBb}2b i?2 b K2 +QM/BiBQMb b f E [N (f ) N (g)] = λ f (ν) gˇ (ν) σ (dν) . R
S`QQ7XN G2i 7Q` a > 0 N
(L2p2m- RNdd)- S`QTX AAXk9- TX j83X
UNXkjV
NXkX h>1 "_hG1hh aS1*h_G J1al_1 ha (t) :=
j98
1 , 1 + 4π 2 a2 t2
7mM+iBQM rBi? 6Qm`B2` i` Mb7Q`K ha (ν) = P#b2`p2 i? i
1 −a|ν| e . 2a
2 ha/2 ∗ ha/2 = ha a
UNXk9V
bBM+2 BM i?2 6Qm`B2` /QK BM- (ha/2 )2 = a2 ha X HbQ- MQi2 i? i Bi Bb 2[mBp H2Mi iQ b v i? i f Bb O 1/ |t|2 b |t| → ∞ Q` i? i f Bb #QmM/2/ #v KmHiBTH2 Q7 h1 X "v G2KK NXkX3- 7Q` HH a > 0- σa (dt) = ha (t) σ (dt) /2}M2b }MBi2 K2 bm`2 QM (R, B(R))X Aib 6Qm`B2` i` Mb7Q`K e−2iπνt σa (dt) σa (ν) = R
Bb i?2`27Q`2 #QmM/2/ M/ +QMiBMmQmbX lbBM; UNXk9V- r2 ? p2 & &2 & & a −2iπνt E && ha/2 (t) e N (dt)&& = ha (t) e−2iπνt σ (dt) = σa (ν) 2 R R M/ i?2`27Q`2 σa (ν) ≥ 0 7Q` HH a > 0 M/ HH ν ∈ RX h?2`27Q`2- i?2 K2 bm`2 σa (ν) h1 (ν) dν Bb MQM@M2; iBp2 }MBi2 K2 bm`2 QM (R, B(R))X Aib +? ` +i2`BbiB+ 7mM+iBQM UrBi? t b `;mK2MiV Bb e−2iπνt σa (ν) h1 (ν) dν = e−2iπνt ha (s) e−2iπνs σ (ds) h1 (ν) dν R R R = e−2iπν(t+s) h1 (ν) dν ha (s) σ (ds) = h1 (t + s) ha (s) σ (ds) UNXk8V
R
R
R
U6m#BMBǶb i?2Q`2K Bb TTHB+ #H2 ?2`2 #2+ mb2 (ν, s) →1 (ν) ha (s) Bb ×σ@BMi2;` #H2VX h?2`27Q`2 −2iπνt σa (ν) h1 (ν) dν = lim h1 (t + s) ha (s) σ (ds) lim e a↓0 R a↓0 R = h1 (t + s) σ (ds) < ∞ R
U#v /QKBM i2/ +QMp2`;2M+2 bBM+2 h1 (t + s) ha (s) ≤ h1 (t + s)- σ@BMi2;` #H2 7mM+@ iBQM #v G2KK NXkX3VX h?2 HBKBi Bb +QMiBMmQmb BM t U#v /QKBM i2/ +QMp2`;2M+2 ; BMVX h?2`27Q`2- #v i?2 2ti2MbBQM iQ }MBi2 K2 bm`2b Q7 S mH GûpvǶb i?2Q`2KRy i?2 K2 bm`2 σa (ν) h1 (ν) dν +QMp2`;2b r2 FHv b a ↓ 0 iQ }MBi2 K2 bm`2 r?B+? r2 K v Hr vb r`Bi2 b h1 (ν) σ (dν) bBM+2 h1 (ν) > 0- 7Q` HH ν ∈ RX q2 F +QMp2`;2M+2 K2 Mb i? i 7Q` Mv #QmM/2/ M/ +QMiBMmQmb 7mM+iBQM ϕ : R → R Ry
a22 7Q` BMbi M+2 ("`ûK m/- kyR9)- h?2Q`2K kXjX8X
j9e
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
ϕ (ν) h1 (ν) σa (ν) dν =
lim a↓0
R
ϕ (ν) h1 (ν) σ (ν) dν . R
AM T `iB+mH `- B7 f Bb +QMiBMmQmb M/ #QmM/2/ #v KmHiBTH2 Q7 h1 lim f (ν) σa (ν) dν = f (ν) σ (ν) dν . a↓0
R
R
PM i?2 Qi?2` ? M/- #v +QKTmi iBQM bBKBH ` iQ i? i BM UNXk8V f (ν) σa (ν) dν = f (t) ha (t) σ (dt) , R
R
M/ i?2`27Q`2 f (t) σ (dt) , lim f (ν) σa (ν) dν = lim f (t) ha (t) σ (dt) = a↓0
R
a↓0
R
R
r?2`2 f Bb #QmM/2/ #v KmHiBTH2 Q7 h1 Ui?Bb BKTHvBM; i? i R |f |(t)σ (dt) < ∞VX h?Bb T`Qp2b UNXkkVX A7 f M/ g `2 O 1/ |t|2 b r2HH b i?2B` 6Qm`B2` i` Mb7Q`Kb- bQ Bb f ∗ gˇ- M/ i?2`27Q`2 (f ∗ gˇ) dσ = f gˇ dσ . E [N (f ) N (g)] = Rm
Rm
Ai `2K BMb iQ T`Qp2 mMB[m2M2bbX AM pB2r Q7 UNXkkV- i?Bb 7QHHQrb 7`QK i?2 7 +i i? i irQ HQ+ HHv }MBi2 K2 bm`2b ;`22BM; QM HH 7mM+iBQMb i? i `2- iQ;2i?2` rBi? i?2B` 6Qm`B2` i` Mb7Q`Kb- Q7 i?2 Q`/2` Q7 |t|−2 i BM}MBiv- `2 B/2MiB+ HX h?2 7QHHQrBM; Bb `2bi i2K2Mi M/ M Q#pBQmb 2ti2MbBQM iQ i?2 +QKTH2t + b2 Q7 h?2Q`2K NXkXRyX *Q`QHH `v NXkXRR .2MQi2 #v B i?2 +QHH2+iBQM Q7 +QMiBMmQmb 7mM+iBQMb ϕ : Rm → C i? i `2 O 1/ |t|2 b r2HH b i?2B` 6Qm`B2` i` Mb7Q`KX G2i N #2 bBKTH2 rB/2@ b2Mb2 bi iBQM `v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λX h?2M- i?2`2 2tBbib mMB[m2 σ@}MBi2 K2 bm`2 μN U= σ − λ2 ε0 V bm+? i? i 7Q` HH ϕ : Rm → C BM B +Qp (N (ϕ) , N (ψ)) = ϕ(ν)ψ ∗ (ν)μN (dν) . Rm
h?2Q`2K NXkXRy M/ Bib +Q`QHH `v `2 ;2M2` H `2bmHib +QM+2`MBM; i?2 2tBbi2M+2 M/ mMB+Biv Q7 i?2 " `iH2ii bT2+i` H K2 bm`2X h?2 M2ti 2t KTH2b ;Bp2 2tTHB+Bi 7Q`@ KmH b 7Q` i?2 bT2+i` H K2 bm`2 M/ /2b+`BTiBQM Q7 bT +2 Q7 i2bi 7mM+iBQMb BM i?`22 BKTQ`i Mi bT2+B H + b2bX h?Bb +QHH2+iBQM Q7 2tTHB+Bi 2t KTH2b rBHH #2 +QMbB/2` #Hv 2MH `;2/ BM a2+iBQM NX9X 1t KTH2 NXkXRk, h?2 " `iH2ii aT2+i`mK Q7 i?2 ai iBQM `v _2;mH ` :`B/X *QMbB/2` i?2 TQBMi T`Q+2bb QM R2 r?Qb2 TQBMib 7Q`K `2;mH ` (T1 , T2 )@;`B/ QM R2 rBi? ` M/QK Q`B;BM- i? i Bb (T1 > 0 , T2 > 0) , N = (n1 T1 + U1 , n2 T1 + U2 ) , (n1 , n2 ) ∈ Z2
NXkX h>1 "_hG1hh aS1*h_G J1al_1
j9d
r?2`2 U1 - U2 `2 BM/2T2M/2Mi mMB7Q`K ` M/QK p `B #H2b QM [0, T1 ] M/ [0, T2 ] `2@ bT2+iBp2HvX h?2 TQBMi T`Q+2bb Bb b2+QM/@Q`/2` bi iBQM `v rBi? p2` ;2 BMi2MbBiv λ = 1/(T1 T2 )X Aib " `iH2ii bT2+i` H K2 bm`2 Bb μN =
1
ε( T 1 , T 2 ) 1 2 T12 T22 (n1 ,n2 )=(0,0) n
M/ QM2 + M i F2 7Q` i?2 i2bi 7mM+iBQM bT +2 BN := ϕ ∈ L1C (R2 ) ∩ L2C (R2 ) M/ n1 ,n2 ∈Z
n
UNXkeV
,
& & & & n n 1 2 &Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
ϕ (t) N (dt) R2 = E [ϕ (U1 + n1 T1 , U2 + n2 T2 )]
E
n1 ,n2 ∈Z
=
1 T1 T2
T1
T2
ϕ (u1 + n1 T1 , u2 + n2 T2 ) du = 0
0
1 T1 T2
ϕ (t) dt = R2
1 ϕ (0, 0) . T1 T2
h?2`27Q`2
ϕ (t) N (dt)
Var R2
&& n1 n2 &&2 1 & − 1 |ϕ (0, 0)|2 &ϕ = 2 2 , T1 T2 n ,n ∈Z & T1 T2 & T12 T22 1 2 & &2 & n1 n2 && 1 & = 2 2 &ϕ T1 T2 & T1 T2 (n1 ,n2 )=(0,0) = |ϕ (ν1 , ν2 )|2 μN (dν1 × dν2 ) , R
R
r?2`2 μN Bb ;Bp2M #v UNXkeVX
1t KTH2 NXkXRj, h?2 " `iH2ii aT2+i`mK Q7 ai iBQM `v *Qt S`Q+2bbX G2i N #2 *Qt TQBMi T`Q+2bb QM Rm rBi? biQ+? biB+ BMi2MbBiv {λ(t)}t∈Rm X _2K2K#2` i? i i?Bb K2 Mb i?2 7QHHQrBM;X 6B`biHv- {λ(t)}t∈Rm Bb MQM@M2; iBp2 HQ@ + HHv BMi2;` #H2 T`Q+2bb- M/ b2+QM/Hv- +QM/BiBQM HHv QM i?Bb T`Q+2bb- N Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t)X amTTQb2 i? i {λ(t)}t∈Rm Bb rbb T`Q+2bb rBi? K2 M λ M/ TQr2` bT2+i` H K2 bm`2 μλ X h?2M i?2 " `iH2ii bT2+i`mK Q7 N Bb μN (dν) = μλ (dν) + λdν ,
UNXkNV
M/ r2 + M i F2 BN = L1C (Rm ) ∩ L2C (Rm )X 1p2M KQ`2- BM i?Bb + b2 BN = L2N (M2 ) λ S`QQ7X .2MQi2 #v F∞ i?2 σ@}2H/ ;2M2` i2/ #v i?2 biQ+? biB+ T`Q+2bb {λ(t)}t∈R X LQi2 i? i B7 ϕ ∈ L2C (Rm )
E |ϕ(t)|2 λ(t)dt = λ |ϕ(t)|2 dt < ∞ Rm
Rm
M/ i?2`27Q`2- HKQbi bm`2Hv Rm
|ϕ(t)|2 λ(t)dt < ∞ .
aBKBH `Hv- bBM+2 ϕ ∈ L1C (Rm )- HKQbi bm`2Hv |ϕ(t)| λ(t)dt < ∞ . Rm
"v i?2 7Q`KmH b Q#i BM2/ BM h?2Q`2K jXkXR-
NXkX h>1 "_hG1hh aS1*h_G J1al_1
2
E Rm
λ |F∞
|ϕ(t)| N (dt)
j9N
|ϕ(t)| λ(t)dt +
2
2
= Rm
Rm
|ϕ(t)| λ(t)dt
. UNXjyV
q2 i?2`27Q`2 ? p2 E Rm
2
|ϕ(t)| N (dt)
|ϕ(t)| dt + E
2
2
=λ Rm
Rm
|ϕ(t)| λ(t)dt
2 ≥λ |ϕ(t)| dt + E |ϕ(t)| λ(t)dt Rm Rm 2 2 =λ |ϕ(t)| dt + λ |ϕ(t)| dt .
2
Rm
Rm
h?2`27Q`2 B7 ϕ ∈ L2N (M2 )- i?2M ϕ ∈ L1C (Rm ) ∩ L2C (Rm )X HbQ- 7`QK
2
E Rm
|ϕ(t)| N (dt)
2
|ϕ(t)| dt + E 2
=λ Rm
M/ i?2 7 +i i? i B7 ϕ ∈ L1C (Rm )- i?2M E
Rm
|ϕ(t)| λ(t)dt
Rm L1C (Rm )
|ϕ(t)| λ(t)dt
2
,
< ∞ Ui?Bb Bb #2+ mb2
∩ i?2M ϕ ∈ L2N (M2 )X {λ(t)}t∈R Bb rbbV- r2 b22 i? i B7 ϕ ∈ h?2`27Q`2 BM i?2 + b2 Q7 *Qt T`Q+2bb2b rBi? rbb +QM/BiBQM H BMi2MbBiv- ϕ ∈ L1C (Rm ) ∩ L2C (Rm ) Bb 2[mBp H2Mi iQ ϕ ∈ L2N (M2 )X L2C (Rm )-
"v i?2 +QM/BiBQM H p `B M+2 7Q`KmH o `
Rm
ϕ(t)N (dt) & &
& λ & λ = E o ` + o ` E ϕ(t)N (dt)&& F∞ ϕ(t)N (dt)&& F∞ m Rm
R =E ϕ(t)2 λ(t)dt + o ` ϕ(t)λ(t)dt Rm Rm =λ ϕ(t)2 dt + o ` ϕ(t)λ(t)dt . Rm
Rm
"v /2}MBiBQM Q7 i?2 TQr2` bT2+i` H K2 bm`2- 7Q` ϕ ∈ L1C (Rm ) o `
ϕ(t)λ(t)dt
=
Rm
Rm
|ϕ(ν)|2 μλ (dν) ,
M/ #v i?2 SH M+?2`2HĜS `b2p H 7Q`KmH
2
ϕ (t)dt = Rm
Rm
|ϕ(ν)|2 dν .
j8y
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
h?2`27Q`2 o `
ϕ(t)N (dt)
=
Rm
Rm
|ϕ(ν)|2 (μλ (dν) + λdν) ,
r?B+? T`Qp2b UNXkNVX
1t KTH2 NXkXR9, h?2 " `iH2ii aT2+i`mK Q7 i?2 ai iBQM `v _2M2r H SQBMi S`Q+2bbXRj AM Q`/2` iQ +QKTmi2 i?2 " `iH2ii bT2+i`mK Q7 bi iBQM `v `2M2r H TQBMi T`Q+2bb rBi? BMi2MbBiv λ M/ MQM@H iiB+2 `2M2r H /Bbi`B#miBQM F /2}M2 F (2iπν) := e−2iπνt dF (t) . R+
LQi2 i? i- bBM+2 F Bb MQM@H iiB+2- F (ν) = 1- 2t+2Ti 7Q` ν = 0X h?2 +Qp `B M+2 K2 bm`2 Bb ;Bp2M #v i?2 7Q`KmH Γ(dx) = λE0 [N (dx)] − λ2 (dx) , r?2`2 E0 /2MQi2b 2tT2+i iBQM rBi? `2bT2+i iQ i?2 T`Q# #BHBiv K FBM; Q7 N M mM/2H v2/ `2M2r H T`Q+2bbX h?2 K2 bm`2 E0 [N (dx)] Bb i?2 bmK Q7 .B` + K2 bm`2 i 0- ε(dx)- M/ Q7 bvKK2i`B+ K2 bm`2 U (dx)- ;Bp2M- 7Q` dx ⊂ (0, ∞)- #v U (dx) = F ∗n (dx) . n≥1
h?2 K2 bm`2 U rBHH MQr #2 bmTTQb2/ iQ /KBi /2MbBiv u rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2 M/ r2 bbmK2 i? i ∞ |u(t) − λ|dt < ∞ . UNXjRV 0
.2}MBM;
g(ν) :=
∞
e−2iπνt (u(t) − λ) dt
0
M/ i FBM; BMiQ ++QmMi i?2 bvKK2i`v Q7 u e−2iπνt (u(t) − λ)dt = g(ν) + g ∗ (ν) . R
Ai rBHH #2 T`Qp2/ #2HQr i? i g(ν) =
F (2iπν) 1 − F (2iπν)
+
1 . 2iπν
UNXjkV
*QK#BMBM; i?2 #Qp2 `2bmHib b?Qrb i? i i?2 " `iH2ii bT2+i`mK Q7 N /KBib i?2 /2MbBiv F (2iπν) fN (ν) = λ(1 + Re 1 − F (2iπν) Rj
(. H2v M/ o2`2@CQM2b- RN33)- 1t KTH2 RRXk U#V- TX 9RkX
NXjX GPL:@_L:1 .1S1L.1L*1
j8R
6Q` i?2 T`QQ7 Q7 UNXjkV- r`Bi2 7Q` θ > 0 ∞ ∞ e−(θ+2iπν)t (u(t) − λ)dt = e−(θ+2iπν)t F ∗n (dt) − 0
n≥1
=
0
e−(θ+2iπν)t λdt
0
F (θ + 2iπν)n −
n≥1
=
∞
F (θ + 2iπν) 1 − F (θ + 2iπν)
−
λ θ + 2iπν λ . θ + 2iπν
6Q` ν = 0- H2iiBM; θ i2M/ iQ 0 BM i?2 }`bi i2`K Q7 i?2 #Qp2 2[m HBiv- r2 Q#i BM #v ∞ /QKBM i2/ +QMp2`;2M+2 0 e−2iπνt (u(t) − λ)dtX G2iiBM; θ i2M/ iQ 0 BM F (θ + 2iπν)r2 Q#i BM F (2iπν)- ; BM #v /QKBM i2/ +QMp2`;2M+2X >2M+2 i?2 MMQmM+2/ `2bmHiX lM/2` i?2 +QM/BiBQMb Q7 i?2 #Qp2 2t KTH2- QM2 + M i F2 7Q` i?2 i2bi 7mM+iBQMb bT +2 BN = L1C (R) ∩ L2C (R)X AM/22/- 7Q` 7mM+iBQM ϕ ∈ L2N (M2 ) UBM T `iB+mH `ϕ ∈ L1C (R) ∩ L2C (R)V ∗ ϕ (t) N (dt) = ϕ (t) ϕ (t + s) dt ΓN (ds) , o ` Rm
r?2`2
Rm
Rm
ΓN (dt) = λR(dt) − λ2 dt = λ(u(t) − λ) dt + λε(dt) .
UNXjjV
∩ h?2`27Q`2- 7Q` ϕ ∈ ϕ(t) N (dt) o ` R 2 −2iπνs 2 −2iπνs = |ϕ(ν)| e dν (u(s) − λ) ds + λ |ϕ(ν)| e dν ε(ds) R R R R = |ϕ(ν)|2 e−iπνs (u(s) − λ) ds dν + λ |ϕ(ν)|2 dν Rm Rm Rm = |ϕ(ν)|2 f (ν) dν . L1C (R)
L2C (R)-
Rm
_2K `F NXkXR8 LQi2 i? i BM HH i?`22 Q7 i?2 #Qp2 2t KTH2b- i?2 +QHH2+iBQMb BN +QMi BM B M/ i?2`27Q`2 `2 ;2MmBM2 i2bi 7mM+iBQM bT +2b UH `;2 2MQm;? iQ ;m ` Mi22 mMB+Biv Q7 i?2 bT2+i` H K2 bm`2b i?2`2BMc b22 _2K `F NXkXjVX
NXj
GQM;@` M;2 .2T2M/2M+2
h?2 MQiBQM Q7 HQM;@` M;2 /2T2M/2M+2 ? b BKTQ`i Mi BKTHB+ iBQMb BM i?2 i?2Q`v Q7 biQ+? biB+ T`Q+2bb2b- BM+Hm/BM; TQBMi T`Q+2bb2bX Ai Bb +QMp2MB2Mi iQ BMi`Q/m+2 Bi BM i2`Kb Q7 i?2 +Qp `B M+2 7mM+iBQM Q7 rB/2@b2Mb2 biQ+? biB+ T`Q+2bb2bX h?2 dz+H bbB+ HǴ KQ/2Hb Q7 rbb biQ+? biB+ T`Q+2bb2b ? p2 +Qp `B M+2 7mM+iBQM r?B+? Bb ;2QK2i`B@ + HHv #QmM/2/- i? i Bb,
j8k
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 C(|τ |) ≤ Ke−α|τ | ,
r?2`2 K > 0 M/ α > 0X HQM; K2KQ`v rbb biQ+? biB+ T`Q+2bb Bb QM2 7Q` r?B+? 2tTQM2MiB H /2+ v Bb `2TH +2/ #v TQHvMQKB H /2+ vX JQ`2 T`2+Bb2HvC(|τ |) ∼ K|τ |2d−1
b τ → ∞,
UNXj9V
r?2`2 K > 0 M/ d < 12 X PM2 bQK2iBK2b K F2b i?2 /BbiBM+iBQM #2ir22M BMi2`K2@ /B i2 K2KQ`v M/ HQM; K2KQ`v ++Q`/BM; iQ r?2i?2` d < 0- BM r?B+? + b2 C(τ ) dτ < ∞ , R
Q` 0 < d < 12 - BM r?B+? + b2
R
C(τ ) dτ = ∞ .
h?2 i2`KBMQHQ;v Bb MQi v2i bi M/ `/Bx2/R9 M/ bQK2iBK2b QM2 T`272`b HQM;@` M;2 /2T2M/2M+2 iQ HQM;@K2KQ`vX >2`2- HQM;@` M;2 /2T2M/2Mi biQ+? biB+ T`Q+2bb rBHH #2 QM2 r?Qb2 +Qp `B M+2 7mM+iBQM Bb MQi BMi2;` #H2X AM T `iB+mH ` HQM;@K2KQ`v biQ+? biB+ T`Q+2bb Bb HQM;@` M;2 /2T2M/2MiX GQM;@` M;2 /2T2M/2M+2 + M HbQ #2 TT`Q +?2/ 7`QK i?2 bT2+i` H TQBMi Q7 pB2rX AM/22/- bmTTQb2 i? i rbb biQ+? biB+ T`Q+2bb /KBib TQr2` bT2+i` H /2MbBiv f r?Qb2 #2? pBQm` i i?2 Q`B;BM Bb i?2 7QHHQrBM;, lim f (ν)ν β = r > 0 ,
UNXj8V
ν→0
7Q` bQK2 β ∈ (0, 1)X h?2M r`BiBM; +∞ C(τ )τ 1−β = f (ν)τ 1−β e+2iπντ dν = −∞
M/ Q#b2`pBM; i? i
+∞ −∞
1 ν τ 1−β f dν τ τ
1 ν r lim τ 1−β f = β τ ↑∞ τ τ ν
r2 ? p2 i? i UrBi? KBH/ //BiBQM H +QM/BiBQMb QM f ;m ` Mi22BM; i? i G2#2b;m2Ƕb /QKBM i2/ +QMp2`;2M+2 i?2Q`2K Bb TTHB+ #H2V ∞ +2iπν e 1−β lim C(τ )τ =r dν . τ ↑∞ νβ 0 h?2`27Q`2- i BM}MBiv- i?2 +Qp `B M+2 7mM+iBQM #2? p2b b i?2 T`Q+2bb Bb HQM;@` M;2 /2T2M/2MiX
1 τ 1−β
r?2`2 1 − β ∈ (0, 1),
7Q`KmH iBQM BM i2`Kb Q7 i?2 TQr2` bT2+i` H /2MbBiv b BM UNXj8V Bb HbQ / Ti2/ iQ rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb2bX JQ/2Hb 7Q` r?B+? i?Bb T`QT2`iv ?QH/b `2 2 bv iQ Q#i BMX b K ii2` Q7 7 +i- b b?QrM BM 1t KTH2 NXkXRj- *Qt T`Q+2bb r?Qb2 +QM/BiBQM H BMi2MbBiv T`Q+2bb Bb rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb R9
6Q` KQ`2 /2i BH2/ BMi`Q/m+iBQM iQ HQM;@` M;2 /2T2M/2M+2- b22 (a KQ`Q/MBibFv- kyye)X
NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1
j8j
rBi? TQr2` bT2+i` H /2MbBiv fλ /KBib- ++Q`/BM; iQ 7Q`KmH UNXkNV- i?2 " `iH2ii bT2+i` H /2MbBiv fN (ν) = fλ (ν) + λ ,
UNXjeV
M/ Bi Bb i?2M +H2 ` i? i Bib HQM;@` M;2 /2T2M/2M+2 T`QT2`iB2b /B`2+iHv 7QHHQr 7`QK i?Qb2 Q7 i?2 BMi2MbBiv T`Q+2bbX h?2`27Q`2 QM2 + M }M/ b K Mv HQM;@` M;2 /2T2M@ /2Mi bi iBQM `v TQBMi T`Q+2bb2b b i?2`2 `2 MQM@M2; iBp2 HQM;@` M;2 /2T2M/2Mi bi iBQM `v biQ+? biB+ T`Q+2bb2b- M/ i?Bb 7 +i Bb r2HH bmTTQ`i2/ #v BMimBiBQMX h?2 TQBMi T`Q+2bb2b +QMbB/2`2/ BM i?2 T`2pBQmb T ` ;` T? `2 dz/Qm#Hv biQ+? b@ iB+Ǵ- M/ QM2 K v rQM/2` #Qmi i?2 2tBbi2M+2 Q7 dzb2H7@2t+BiBM;Ǵ TQBMi T`Q+2bb 2t?B#BiBM; HQM;@` M;2 /2T2M/2M+2X h?2 Mbr2` Bb 2 bv, i F2 i?2 b K2 *Qt T`Q+2bbX dz"mi Bi Bb MQi b2H7@2t+BiBM;Ǵ K v QM2 `;m2X h?Bb TQBMib QM+2 ; BM iQ i?2 T`Q#H2K `BbBM; r?2M QM2 [m HB}2b TQBMi T`Q+2bb b dzb2H7@2t+BiBM;Ǵ r?2M QM2 b?QmH/ BM@ bi2 / bT2 F Q7 Bib b2H7@2t+BiBM; dz/2b+`BTiBQMǴ Ub22 i?2 /Bb+mbbBQM Q7 a2+iBQM 8XkVX Mv *Qt T`Q+2bb ? b b2H7@2t+BiBM; /2b+`BTiBQM r?B+? Bb i?2 T`QD2+iBQM Q7 i?2 /Qm#Hv biQ+? biB+ BMi2MbBiv QM i?2 M im` H }Hi` iBQMX _Qm;?Hv bT2 FBM;- Bib dzb2H7@2t+BiBM;Ǵ BMi2MbBiv Bb {E λ(t) | FtN }t∈R X * M r2 }M/ HQM;@` M;2 /2T2M/2Mi TQBMi T`Q+2bb Q7 r?B+? i?2`2 Bb MQ Q#pBQmb /Qm#Hv biQ+? biB+ /2b+`BTiBQM #mi QMHv M im` H BMi`BMbB+ /2b+`BTiBQM\ h?Bb Bb p ;m2 [m2biBQM- iQ r?B+? i?2`2 Bb TQbBiBp2 Mbr2`- mbBM; i?2 Hi2`M iBp2 /2}MBiBQM Q7 HQM;@` M;2 /2T2M/2M+2- M K2Hv i? i +QM/BiBQM lim t↑∞
Var N ([0, t]] =∞ t
UNXjdV
#2 b iBb}2/X AM 7 +i-R8 `2M2r H T`Q+2bb b iBb}2b +QM/BiBQM UNXjdV B7 M/ QMHv B7 Bib BMi2`p H /Bbi`B#miBQM ? b BM}MBi2 b2+QM/ KQK2MibX h?2 HBMF rBi? i?2 #2? pBQm` BM i?2 pB+BMBiv Q7 i?2 MmHH 7`2[m2M+v Q7 i?2 /2}MBiBQM UNXjdV Bb K /2 pB . H2vǶb /2}MBiBQM UNXR8V Q7 i?2 " `iH2ii bT2+i` H K2 bm`2X AM a2+iBQM RkX8- r2 b? HH b22 KQ`2 bT2+i +mH ` 2t KTH2- i? i Q7 > rF2b #` M+?BM; TQBMi T`Q+2bb rBi?Qmi M+2biQ`b- 72 im`BM; T? b2 i` MbBiBQM T?2@ MQK2MQMX
NX9
h` Mb7Q`K iBQMb Q7 i?2 aT2+i` H J2 bm`2
h?Bb b2+iBQM 72 im`2b 7Q`KmH i? i HHQrb QM2 iQ T bb 7`QK i?2 bT2+i` H K2 bm`2 Q7 # bB+ rB/2@b2Mb2 bi iBQM `v TQBMi T`Q+2bb iQ i?Qb2 Q7 TQBMi T`Q+2bb2b Q#i BM2/ #v p `BQmb i` Mb7Q`K iBQMb Q7 i?2 BMBiB H TQBMi T`Q+2bbX Ai Bb FBM/ Q7 dzarBbb `Kv FMB72Ǵ- r?2`2 2 +? dz#H /2Ǵ +QM+2`Mb bT2+B}+ i` Mb7Q`K iBQM, i?BMMBM;- i` MbH i@ BM;- DBii2`BM; M/ +Hmbi2`BM;X R8
(h2m;2Hb- RNe3)- (. H2v- RNNN)X
j89
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
lMBp2`b H *Qp `B M+2 6Q`KmH G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2 7 KBHv Q7 BB/ ` M/QK p `B #H2b rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (K, K) rBi? +QKKQM T`Q# #BHBiv /Bbi`B#miBQM Q M/ BM/2T2M/2Mi Q7 N X HbQ bbmK2 i? i N Bb b2+QM/@Q`/2` bi iBQM `v TQBMi T`Q+2bb rBi? " `iH2ii bT2+i` H K2 bm`2 μN M/ i2bi 7mM+iBQM bT +2 BN X G2i Z #2 ` M/QK 2H2K2Mi rBi? /Bbi`B#miBQM QX _2+ HH i?2 MQi iBQM LpC ( × Q) 7Q` i?2 b2i Q7 K2 bm` #H2 7mM+iBQMb ϕ : Rm × K → C bm+? i? i |ϕ(t, z)|p Q(dz) dt = E [|ϕ(t, Z)|p ] dt < ∞ . Rm
Rm
K
AM T `iB+mH `- ϕ(t, Z) ∈ K2 bm`2VX
LpC (P )
7Q` HKQbi HH t ∈ R UrBi? `2bT2+i iQ i?2 G2#2b;m2
G2i ϕ : Rm × K → R #2 K2 bm` #H2 7mM+iBQM bm+? i? i ϕ ∈ L1C ( × Q) .
UNXj3V
AM T `iB+mH `- ϕ(t, Z) ∈ L1C (P ) 7Q` HKQbi HH t ∈ R UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2V M/ i?2 [m MiBiv ϕ(t) := E [ϕ(t, Z)] Bb r2HH /2}M2/ 7Q` HKQbi HH tX Ai HbQ 7QHHQrb 7`QK bbmKTiBQM UNXj3V i? i ϕ ∈ L1C (Rm ) M/ 7Q` Q@ HKQbi HH z ∈ K- ϕ(·, z) ∈ L1C (Rm )X .2MQi2 i?2 6Qm`B2` i` Mb7Q`Kb Q7 i?2b2 irQ 7mM+iBQMb #v ϕ M/ ϕ(·, z) `2bT2+iBp2HvX amTTQb2 i? i ϕ ∈ L2C ( × Q). UNXjNV aBM+2 +QM/BiBQM UNXjNV Bb i?2 b K2 b Rm |E [ϕ(t, Z)]|2 dt < ∞- r2 b22 i? i ϕ ∈ L2C (Rm ) M/ i? i 7Q` Q@ HKQbi HH z ∈ K- ϕ(·, z) ∈ L2C (Rm )X LQi2 i? i ϕ(ν) = E [ϕ(ν, Z)] := ϕ(ν). 6BM HHv- bmTTQb2 i? i
ϕ ∈ BN .
UNX9yV
h?2Q`2K NX9XR URe V G2i N M/ {Zn }n∈N #2 b #Qp2 M/ H2i ϕ, ψ : Rm × K → R b iBb7v +QM/BiBQMb UNXj3V- UNX9yV M/ UNXjNVX h?2M cov
ϕ(Tn , Zn ) ,
n∈N
=
ψ(Tn , Zn )
n∈N ∗
ϕ(ν) ψ (ν)μN (dν) + λ Rm
cov ϕ(ν, Z), ψ ∗ (ν, Z) dν, UNX9RV
Rm
r?2`2 λ Bb i?2 BMi2MbBiv Q7 N M/ Z Bb K@p Hm2/ ` M/QK p `B #H2 rBi? /Bbi`B#miBQM Q- i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX Re
("`ûK m/ M/ J bbQmHBû- kyyk)X
NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1
j88
S`QQ7X AM Q`/2` iQ ;Bp2
KQ`2 `2 / #BHBiv iQ i?2 + H+mH iBQMb #2HQr- i?2 7QHHQrBM; MQi iBQM rBHH #2 mb2/, x∈N ϕ(x, Z(x)) := n∈N ϕ(Xn , Zn )X 6Q`K HHv, E
ϕ(t, Z(t))
t∈N
=E
ψ(t, Z(t))
t∈N
=E
ϕ(t)
=E
∗
ϕ(t)ψ (t ) + E
t,t ∈N,t=t
∗
ψ (t )
−E
t∈N
∗
ϕ(t, Z(t))ψ (t, Z(t))
t∈N
t ∈N
t∈N
ϕ(t, Z(t))ψ(t , Z(t )) + E
t,t ∈N,t=t
∗
ϕ(t, Z)ψ (t, Z)
∗
ϕ(t)ψ (t) +E
t∈N
∗
ϕ(t, Z)ψ (t, Z) .
t∈N
.2MQi2 #v (a)−(b)+(c) i?2 H bi HBM2 /BbTH v2/X h?2 #Qp2 7Q`K H +QKTmi iBQMb `2 DmbiB}2/ #2+ mb2 HH i?`22 i2`Kb `2- r?2M ϕ M/ ψ `2 `2TH +2/ #v i?2B` #bQHmi2 p Hm2b- }MBi2X h?Bb 7QHHQrb 7`QK a+?r `xǶb BM2[m HBiv M/ i?2 7 +ib i? i U7Q` (a) 2 m M/ (b)V ϕ M/ ψ `2 BM L2N (M2 ) M/ BM LC (R )c M/ 7Q` (c) #2+ mb2 Q7 +QM/BiBQM UNXjNVX aBM+2 E t∈N ϕ(t, Z(t)) = E t∈N ϕ(t) cov
ϕ(t, Z(t)),
t∈N
cov
ψ(t, Z(t))
t∈N
t∈N
ϕ(t),
ψ(t)
=
−E
t∈N
∗
ϕ(t)ψ (t) + E
t∈N
∗
ϕ(t, Z)ψ (t, Z) .
t∈N
.2MQi2 #v A- B M/ C i?2 i?`22 i2`Kb QM i?2 `B;?i@? M/ bB/2 Q7 i?2 #Qp2 2[m iBQMr?B+? i?2M `2 /b A−B +CX "v /2}MBiBQM Q7 i?2 " `iH2ii bT2+i`mK M/ ?vTQi?2bBb UNX9yV ∗
A=
ϕ(ν)ψ (ν)μN (dν) . Rm
"v /2}MBiBQM Q7 i?2 BMi2MbBiv λ ∗ B=λ ϕ(t)ψ(t) dt, C = λ Rm
E [ϕ(t, Z)ψ(t, Z)∗ ] dt . Rm
"v i?2 SH M+?2`2HĜS `b2p H B/2MiBiv ϕ(ν)ψ(ν)∗ dν = λ ϕ(ν)ψ(ν)∗ dν B=λ m Rm R =λ E [ϕ(ν, Z)] E ψ(ν, Z)∗ dν , Rm
M/ C = λE
ϕ(ν, Z)ψ(ν, Z)∗ dν ,
Rm
M/ i?2 `2bmHi UNX9RV 7QHHQrbX
j8e
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
1t KTH2 NX9Xk, a?Qi LQBb2X *QMbB/2` i?2 #Qp2 K `F2/ TQBMi T`Q+2bb UrBi? BM/2T2M/2Mi BB/ K `FbV M/ H2i h : Rm × K → R b iBb7v +QM/BiBQMb UNX9yV M/ UNXjNVX G2i X(t) := h(t − Xn , Zn ). n∈N
U_2K2K#2` i? i i?2 biQ+? biB+ T`Q+2bb {X(t)}t∈Rm Bb + HH2/ b?Qi MQBb2XV h?2M h(t)dt,
E [X(t)] = λ Rm
M/
+Qp (X(u), X(v)) =
r?2`2
e2iπν,u−v μX (dν) , Rm
& &2 & & μX (dν) = &E h(ν, Z) & μN (dν) + λo ` h(ν, Z) dν .
UNX9kV
h?Bb 7QHHQrb 7`QK i?2 7mM/ K2Mi H BbQK2i`v 7Q`KmH TTHB2/ iQ ϕ(t, z) = h(u−t, z) M/ ψ(t, z) = h(v − t, z)X 1t KTH2 NX9Xj, h?BMMBM;X G2i N #2 rbb bi iBQM `v TQBMi T`Q+2bb QM Rm rBi? bT2+i` H K2 bm`2 μN M/ i2bi 7mM+iBQM bT +2 BN X G2i {Zn }n∈Z #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b BM/2T2M/2Mi Q7 N - i FBM; i?2B` p Hm2b BM {0, 1} M/ bm+? i? i P (Z1 = 1) = αX .2}M2 i?2 TQBMi T`Q+2bb Nα #v Nα (C) =
(C ∈ B(Rm ) .
Zn 1{Xn ∈C} ,
n≥1
h?mb- Nα Bb Q#i BM2/ #v i?BMMBM; N - TQBMi Q7 N #2BM; `2i BM2/ rBi? T`Q# #BHBiv αX h?2 i?BMM2/ TQBMi T`Q+2bb Nα /KBib i?2 bT2+i` H K2 bm`2 μNα := α2 μN + λα(1 − α) m
UNX9jV
M/ r2 K v i F2 BNα := ϕ ∈ L1C (Rm ) ∩ L2C (Rm ) ∩ BN 7Q` i?2 i2bi 7mM+iBQM bT +2X hQ T`Qp2 i?Bb r2 Kmbi b?Qr i? i 7Q` Mv 7mM+iBQM ϕ˜ ∈ BNα
o ` LQr
ϕ(x) ˜ Nα (dx) = Rm
Rm
ϕ(x) ˜ Nα (dx) = Rm
& & & & ˜ & μNα (dν) . &ϕ(ν)
Zn ϕ(X ˜ n) ,
n≥1
M/ i?2`27Q`2- i?2 `2bmHi 7QHHQrb 7`QK h?2Q`2K NX9XR- #v TTHvBM; 7Q`KmH UNX9RV rBi? ϕ(x, z) = ψ(x, z) = z ϕ(x) rBi? ϕ ∈ BN U1t2`+Bb2 NX8XNVX
NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1
j8d
1t KTH2 NX9X9, CBii2`BM;X *QMbB/2` i?2 K `F2/ TQBMi T`Q+2bb Q7 h?2Q`2K Bb /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib NX9XR- rBi? K = Rm X TQBMi T`Q+2bb N {Xn + Zn }n∈N . h?2M- rBi? λ i?2 BMi2MbBiv Q7 N M/ μN i?2 " `iH2ii bT2+i`mK Q7 N μN (dν) = |ψZ (ν)|2 μN (dν) + λ 1 − |ψZ (ν)|2 dν , r?2`2
ψZ (ν) := E e2iπν,Z
UNX99V UNX98V
Bb i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 ` M/QK /BbTH +2K2Mib /Bbi`B#mi2/ b QX q2 + M i F2 ˜ + Z)] ∈ BN M/ ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm ) . UNX9eV BN˜ = ϕ˜ ; E [ϕ(t 6Q` i?2 T`QQ7- /2}M2 ϕ(t, z) = ϕ(t+z)X *QM/BiBQMb UNXj3V M/ UNXjNV 7Q` i?2 7mM+iBQM ϕ `2 2[mBp H2Mi iQ +QM/BiBQMb ϕ˜ ∈ L1C (Rm ) M/ ϕ˜ ∈ L2C (Rm ) `2bT2+iBp2Hv- bBM+2 7Q` Mv p ≥ 0
E Rm
|ϕ(t, Z)|p dt =
Rm
|ϕ(t)| ˜ p dt .
*QM/BiBQM UNX9yV 7Q` i?2 7mM+iBQM ϕ Bb b iBb}2/ #v i?2 / ?Q+ /2}MBiBQM Q7 BN˜ X q2 K v i?2`27Q`2 TTHv h?2Q`2K NX9XRX q2 ? p2 ϕ(ν, z) = e2iπν,z ϕ(ν), ϕ(ν) = ϕ(ν) = ψZ (ν)ϕ(ν), &2 && & ∗ +Qp (ϕ(ν, Z), ϕ(ν, Z) ) = 1 − |ψZ (ν)|2 &ϕ(ν) & . HbQ-
o `
Rm
(dt) ϕ(t) N
= o `
M/ i?2`27Q`2- TTHvBM; 7Q`KmH UNX9RV (dt) = ϕ(t) N o ` Rm
ϕ(X ˜ n + Zn )
,
n∈N
Rm
& &2 & & & μN (dν) , &ϕ(ν)
r?2`2 μN Bb b BM UNX99VX 1t KTH2 NX9X8, h?2 CBii2`2/ :`B/X q2 +QMbB/2` i?2 + b2 r?2`2 N Bb i?2 ;`B/ Q7 1t KTH2 NXkXRkX q2 + M i F2 && n1 n2 && 1 2 2 2 & & BN˜ = ϕ˜ ; , ) < ∞ M/ ϕ˜ ∈ LC (R ) ∩ LC (R ) . ˜ &ϕ( T1 T2 & n1 ,n2 ∈Z
S`QQ7X AM/22/- Q#b2`pBM; i? i
j83
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 & & & & & & & & ˜ + Z)](ν)& = &ϕ(ν) ˜ &, &E [ϕ(·
r2 b22 i? i +QM/BiBQM E [ϕ(t ˜ + Z)] ∈ BN Bb 2[mBp H2Mi iQ i?2 +QM/BiBQM && n1 n2 && &ϕ( & & ˜ T1 , T2 )& < ∞ . n ,n ∈Z 1
2
1t KTH2 NX9Xe, h?2 CBii2`2/ *Qt S`Q+2bbX G2i N #2 i?2 *Qt T`Q+2bb Q7 1t KTH2 NXkXRjX q2 + M i F2 BN˜ = ϕ˜ ; ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm ) . AM/22/- +QM/BiBQM E [ϕ(t ˜ + Z)] ∈ BN - i? i Bb- BM i?Bb T `iB+mH ` + b2- E [ϕ(t ˜ + Z)] ∈ L1C (Rm ) ∩ L2C (Rm )- Bb 2t +iHv ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm )X 1t KTH2 NX9Xd, h?2 *Hmbi2` SQBMi T`Q+2bbX G2i N #2 rB/2@b2Mb2 bi iBQM@ `v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ > 0 `2T`2b2Mi2/ #v Bib TQBMi b2[m2M+2 {Xn }n≥1 - rBi? TQr2` bT2+i` H K2 bm`2 μN M/ i2bi 7mM+iBQM bT +2 BN X G2i {Zn }n≥1 #2 M BB/ +QHH2+iBQM Q7 TQBMi T`Q+2bb2b QM Rm - BM/2T2M/2Mi Q7 N X G2i Z #2 TQBMi T`Q+2bb QM Rm rBi? i?2 b K2 /Bbi`B#miBQM b i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX .2}M2
ψZ (ν) := E e2iπν,t Z (dt) . Rm
h?2 7mM+iBQM ψZ Bb r2HH /2}M2/ mM/2` i?2 bbmKTiBQM E [Z (Rm )] < ∞. UAM T `iB+mH `- Z Bb HKQbi bm`2Hv }MBi2 TQBMi T`Q+2bbXV q2 MQr /2}M2 irQ TQBMi M/ N -QM Rm #v T`Q+2bb N (C) =N (C) + N Zn (C − Xn ) , N (C) =
n≥1
Zn (C − Xn ) ,
n≥1
rBi? i?2 Q#D2+iBp2 Q7 +QKTmiBM; i?2B` " `iH2ii bT2+i` H K2 bm`2b- bi `iBM; rBi? i?2 }`bi QM2X 6Q`K HHv (dt) =o ` o ` ϕ (Xn ) + ϕ(t)N ϕ (Xn + s) Zn (ds) Rm
=o `
n≥1
n≥1
Rm
ϕ (Xn , Zn ) ,
NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1
j8N
r?2`2 ϕ (x, z) = ϕ (x) +
ϕ (x + s) z (ds) . Rm
q2 ? p2
E [ϕ (x, Z)] = ϕ (x) + E
ϕ (x + s) Z (ds) Rm
ϕ (ν, z) = ϕ (ν) +
Rm
Rm
= ϕ (ν) + R
ϕ (t + s) z (ds) e−2iπν,t dt ϕ (t + s) e−2iπν,t dt z (ds)
Rm
m
ϕ (ν) e2iπν,s z (ds) = ϕ (ν) + m R 2iπν,s = ϕ (ν) 1 + e z (ds) . Rm
LQi2 i? i i?2 2t+? M;2 Q7 Q`/2` Q7 BMi2;` iBQM Bb MQi T`Q#H2K B7 z Bb }MBi2 TQBMi T`Q+2bb- BM T `iB+mH ` B7 z Bb `2TH +2/ #v Bib ` M/QK p2`bBQM ZX HbQ ϕ (ν) = ϕ (ν) (1 + ψZ (ν)) . TTHvBM; 7Q`K HHv h?2Q`2K NX9XR- r2 Q#i BM o `
ϕ (Xn , Zn )
= Rm
n≥1
|ϕ (ν)|2 |1 + ψZ (ν)|2 μN (dν)
+λ Rm
P#b2`p2 i? i o `
|ϕ (ν)| o ` 2
e2iπν,s Z (ds)
1+ Rm
iQ Q#i BM o `
n∈N
ϕ (Xn )
1+
e
2iπν,s
Z (ds) dν .
Rm
e2iπν,s Z (ds)
= o `
Rm
= Rm
r?2`2
|ϕ (ν)|2 μN (dν) ,
μN (dν) = |1 + ψZ (ν)|2 μN (dν) + λo `
e2iπν,s Z (ds) dν
Rm
X aBKBH ` +QKTmi iBQMb H2 / iQ Bb i?2 " `iH2ii bT2+i` H K2 bm`2 Q7 N 2 2iπν,s e Z (ds) dν. μN (dν) = |ψZ (ν)| μN (dν) + λo ` Rm
UNX9dV
UNX93V
jey
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
JmHiBp `B i2 SQBMi S`Q+2bb2b ii2MiBQM rBHH #2 `2bi`B+i2/ iQ #Bp `B i2 TQBMi T`Q+2bb2b- i?2 2ti2MbBQM iQ M `#B@ i` `v MmK#2` Q7 /BK2MbBQMb #2BM; BKK2/B i2X #Bp `B i2 TQBMi T`Q+2bb QM Rm Bb Dmbi T B` (N1 , N2 ) Q7 TQBMi T`Q+2bb2b QM Rm X Ai Bb + HH2/ b2+QM/@Q`/2` #Bp `B i2 TQBMi T`Q+2bb B7 #Qi? N1 M/ N2 `2 b2+QM/@Q`/2` TQBMi T`Q+2bb2bX Ai Bb + HH2/ rbb B7 #Qi? N1 M/ N2 `2 rbb M/ B7 KQ`2Qp2` i?2v `2 DQBMiHv rbb- i? i Bb B7 7Q` HH #QmM/2/ b2ib A, B ∈ B(Rm ) M/ HH t ∈ Rm E [N1 (A + t)N2 (B + t)] = E [N1 (A)N2 (B)] . AM i?Bb bBim iBQM- r2 b? HH b v i? i N1 M/ N2 /KBi i?2 +`Qbb@bT2+i` H K2 bm`2 μN1 ,N2 B7 i?2 H ii2` Bb σ@}MBi2 bB;M2/ K2 bm`2 bm+? i? i 7Q` HH ϕ1 ∈ BN1 - ϕ2 ∈ BN2 cov (N1 (ϕ1 ), N2 (ϕ2 )) = ϕ 21 (ν)2 ϕ2 (ν)∗ μN1 ,N2 (dν) . Rm
1t KTH2 NX9X3, h?2 "Bp `B i2 rbb *Qt S`Q+2bbX G2i N1 M/ N2 #2 rbb *Qt T`Q+2bb2b QM R r?Qb2 `2bT2+iBp2 biQ+? biB+ BMi2MbBiB2b {λ1 (t)}t∈R M/ {λ2 (t)}t∈R `2 DQBMiHv bi iBQM `v rbb biQ+? biB+ T`Q+2bb2b rBi? `2bT2+iBp2 bT2+i` H K2 bm`2 μλ1 M/ μλ2 - M/ +`Qbb@bT2+i` H K2 bm`2 μλ1 ,λ2 X 6Q` ϕ1 , ϕ2 ∈ L1C (R) ∩ L2C (R) bi M/ `/ +QKTmi iBQMb b?Qr i? i E [N1 (ϕ1 )N2 (ϕ2 )∗ ] = E E N1 (ϕ1 )N2 (ϕ2 )∗ |F λ1 ,λ2 = E E N1 (ϕ1 )|F λ1 ,λ2 E N1 (ϕ1 )∗ |F λ1 ,λ2
ϕ1 (t)λ1 (t) dt × ϕ2 (t)∗ λ2 (t) dt =E R
M/
E [N1 (ϕ1 )] E [N2 (ϕ2 )∗ ] = E R
R
ϕ1 (t)λ1 (t) dt × E ϕ2 (t)∗ λ2 (t) dt . R
h?2`27Q`2
cov (N1 (ϕ1 ), N2 (ϕ2 )) = cov ϕ1 (t)λ1 (t) dt, ϕ2 (t)λ2 (t) dt R R ∗ = ϕ 21 (ν)2 ϕ2 (ν) μλ1 ,λ2 (dν) . Rm
h?2`27Q`2 μN1 ,N2 = μλ1 ,λ2 .
h?2 mMBp2`b H +Qp `B M+2 7Q`KmH + M #2 mb2/ iQ +QKTmi2 +`Qbb@bT2+i` H K2 @ bm`2b #2ir22M TQBMi T`Q+2bb M/ Bib i` Mb7Q`K #v i?2 #Qp2 QT2` iBQMb Q7 i?BM@ MBM;- +Hmbi2`BM;- i` MbH iBQM Q` DBii2`BM;- b r2HH b #2ir22M i?2 i` Mb7Q`Kb i?2K@ b2Hp2bX 6Q` BMbi M+2,
NX8X 1s1_*Aa1a
jeR
1t KTH2 NX9XN, *`Qbb@bT2+i` H J2 bm`2 Q7 SQBMi S`Q+2bb M/ Bib CBii2`2/ o2`bBQMX h?2 Q#i2MiBQM Q7 i?2 +`Qbb@bT2+i` H K2 bm`2 Q7 rbb TQBMi T`Q+2bb N := N1 rBi? Bib DBii2`2/ p2`bBQM N2 Q7 *Q`QHH `v NX9X9 `2[mB`2b mb iQ +QKTmi2 cov ϕ(Xn ), ψ(Xn + Zn ) . n∈Z
n∈Z
6Q`KmH UNX9RV `2/m+2b BM i?Bb + b2 iQ ϕ(Xn ) , ψ(Xn + Zn ) = cov n≥1
Rm
n≥1
"mi
ϕ(ν)E ψ(ν + Z)∗ μN (dν).
ψ(t + Z) e−2iπνt dt
ψ(ν + Z) = m R
=
ψ(t) e−2iπν(t−Z) dt = ψ(ν)E e+2iπνZ ,
Rm
r?2`2 i?2 2tT2+i iBQM Bb rBi? `2bT2+i iQ Z ` M/QK p `B #H2 rBi? i?2 +QKKQM T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 K `FbX 6BM HHv ϕ(Xn ), ψ(Xn + Zn ) = ϕ(ν)ψ(ν)∗ E e−2iπνZ μN (dν) , cov n∈Z
n∈Z
M/ i?2`27Q`2
NX8
Rm
μN1 ,N2 (dν) = E e−2iπνZ μN (dν) .
1t2`+Bb2b
1t2`+Bb2 NX8XRX a2+QM/ JQK2Mi J2 bm`2 Q7 SQBbbQM S`Q+2bb G2i N #2 SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2 bm`2 νX UBV a?Qr i? i BM Q`/2` 7Q` N iQ #2 b2+QM/@Q`/2` TQBMi T`Q+2bb- Bi Bb M2+2bb `v M/ bm{+B2Mi i? i i?2 BMi2MbBiv K2 bm`2 #2 HQ+ HHv }MBi2X a?Qr i? i BM i?Bb + b2i?2 b2+QM/ KQK2Mi K2 bm`2 M2 Bb ;Bp2M #v M2 (A × B) = ν(A ∩ B) − ν(A)ν(B) . UBBV a?Qr i? i 7Q` HH ϕ, ψ ∈ L1Rm (ν) ∩ L2Rm (ν) E [N (ϕ)N (ψ)∗ ] = ϕ (t) ψ ∗ (t) ν(dt) + ϕ (t) ν(dt) ψ ∗ (t) ν(dt) . R
R
R
jek
*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1
1t2`+Bb2 NX8XkX h?2 _BTH2v 7mM+iBQM :Bp2 i?2 _BTH2v 7mM+iBQM Q7 i?2 L2vK MĜa+Qii +Hmbi2` TQBMi T`Q+2bbX 1t2`+Bb2 NX8XjX ai iBQM `Biv BM h2`Kb Q7 AMi2;` Hb G2i N #2 b2+QM/@Q`/2` TQBMi T`Q+2bb QM Rm X a?Qr i? i 7Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMb ϕ, ψ : Rm → R- i?2 [m MiBiv
E ϕ(τ + t) N (dt) ψ(τ + t) N (dt) R
R
Bb BM/2T2M/2Mi Q7 τ ∈ RX 1t2`+Bb2 NX8X9X h2bi 6mM+iBQMb 7Q` i?2 :`B/ AM 1t KTH2 NXkXRk- b?Qr i? i i?2 i2bi 7mM+iBQM bT +2 & & n1 n2 && 1 2 2 2 & & BN := ϕ ; ϕ ∈ LC (R ) ∩ LC (R ) M/ &ϕ T1 , T2 & < ∞ n1 ,n2 ∈Z
BM/22/ b iBb}2b i?2 mMB[m2M2bb `2[mB`2K2Mi- i? i Bb, B7 i?2 HQ+ HHv }MBi2 K2 bm`2b μ1 M/ μ2 `2 bm+? i? i 2 |ϕ(ν)| μ1 (dν) = |ϕ(ν)|2 μ2 (dν) Rm
Rm
7Q` HH ϕ ∈ BN - i?2M μ1 ≡ μ2 X 1t2`+Bb2 NX8X8X lMB[m2M2bb BM h?2Q`2K NXkXRy S`QpB/2 i?2 /2i BHb 7Q` i?2 mMB[m2M2bb bi i2K2Mi BM h?2Q`2K NXkXRyX 1t2`+Bb2 NX8XeX " `iH2ii aT2+i`mK Q7 _2M2r H S`Q+2bb- A a?Qr i? i B7 i?2 ? x `/ ` i2 Q7 i?2 `2M2r H /Bbi`B#miBQM U bbmK2/ iQ ? p2 /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2V Bb #QmM/2/ #Qp2 M/ #2HQr #v }MBi2 M/ bi`B+iHv TQbBiBp2 +QMbi Mib- +QM/BiBQM UNXjRV Bb b iBb}2/X 1t2`+Bb2 NX8XdX " `iH2ii aT2+i`mK Q7 _2M2r H S`Q+2bb- AA *QMbB/2` `2M2r H T`Q+2bb r?Qb2 `2M2r H /Bbi`B#miBQM ? b /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2X amTTQb2 i? i i?2 mM/2H v2/ p2`bBQM M/ i?2 bi iBQM `v p2`bBQM `2 +QMbi`m+i2/ pB :`B;2HBQMBbǶ BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXjV 7`QK i?2 b K2 SQBbbQM K2 bm`2 QM R2 M/ i? i i?2v +QmTH2 i i?2 ` M/QK iBK2 T X a?Qr i? i B7 i?2 ? x `/ ` i2 Q7 i?2 `2M2r H /Bbi`B#miBQM Bb #QmM/2/ 7`QK #Qp2+QM/BiBQM UNXjRV Bb b iBb}2/ B7 i?2 2tT2+i2/ +QmTHBM; iBK2 Bb }MBi2X 1t2`+Bb2 NX8X3X _2M2r H S`Q+2bb2b rBi? /7` .Bbi`B#miBQM h?Bb 2t2`+Bb2 +QMiBMm2b 1t KTH2 8XdX9X a?Qr i? i BM i?Bb + b2 ∞ ∞ |u(t) − λ| dt = (u(t) − λ) dt 0
0
NX8X 1s1_*Aa1a
jej
M/ i? i i?Bb [m MiBiv Bb bK HH2` i? M E [N ((0, T1 ))]X *QM+Hm/2X 1t2`+Bb2 NX8XNX h?BMMBM; S`QpB/2 i?2 /2i BHb BM i?2 T`QQ7 Q7 1t KTH2 NX9XjX 1t2`+Bb2 NX8XRyX *Hmbi2`BM; i?2 :`B/ *QMbi`m+i TQBMi T`Q+2bb N QM R b 7QHHQrbX G2i T > 0 M/ H2i U #2 ` M/QK p `B #H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, T ]X G2i L #2 }t2/ TQbBiBp2 BMi2;2`X G2i {Vn, }n∈Z,1≤≤L #2 M BB/ b2[m2M+2 Q7 ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, T ]- BM/2T2M/2Mi Q7 U X h?2 +QHH2+iBQM Q7 TQBMib Q7 N Bb {nT +Vn, }n∈Z,1≤≤L X a?Qr i? i N Bb bi iBQM `v- M/ ;Bp2 Bib " `iH2ii bT2+i` H K2 bm`2 rBi? M bbQ+B i2/ i2bi 7mM+iBQM bT +2X 1t2`+Bb2 NX8XRRX *Hmbi2`BM; i?2 SQBbbQM S`Q+2bb M/ N BM i?2 _272` iQ 1t KTH2 NX9XdX *QKTmi2 i?2 TQr2` bT2+i` H K2 bm`2b Q7 N bT2+B H + b2 r?2`2 N Bb SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv λ M/ i?2 b2[m2M+2 Q7 TQBMib Q7 i?2 ;2M2`B+ +Hmbi2` Z Bb a- 2a- Ę- Θa- r?2`2 a Bb TQbBiBp2 MmK#2` M/ Θ Bb SQBbbQM ` M/QK p `B #H2 rBi? K2 M θ > 0X 1t2`+Bb2 NX8XRkX *`Qbb@bT2+i` *QKTmi2 i?2 +`Qbb@bT2+i` H K2 bm`2b #2ir22M TQBMi T`Q+2bb M/ Bib i` Mb7Q`K #v i?2 QT2` iBQMb Q7 i?BMMBM;- +Hmbi2`BM;- i` MbH iBQM Q` DBii2`BM;- b r2HH b #2ir22M i?2 i` Mb7Q`Kb i?2Kb2Hp2bX
*? Ti2` Ry h?2 AM7Q`K iBQM *QMi2Mi Q7 SQBMi S`Q+2bb2b h?2 }`bi irQ b2+iBQMb Q7 i?Bb +? Ti2` /2KQMbi` i2 i?2 TQr2` Q7 i?2 K `iBM; H2 TT`Q +? iQ TQBMi T`Q+2bb2b #v K2 Mb Q7 bBKTH2 2t KTH2b r?B+? K v b2`p2 b M BMi`Q/m+iBQM iQ i?2 KQ`2 2H #Q` i2 i?2Q`v M/ TTHB+ iBQMbXR h?2 H bi b2+iBQM i`2 ib irQ Bbbm2b i? i +QK2 mT r?2M b KTHBM; rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb i ` M/QK iBK2b 7Q`KBM; TQBMi T`Q+2bb, }`biHv- ?Qr Km+? Q7 i?2 Q`B;BM H TQr2` bT2+i` H K2 bm`2 + M #2 `2+Qp2`2/ 7`QK i?2 ` M/QK b KTH2b\ M/ b2+QM/Hv?Qr 7 ` 7`QK i?2 Q`B;BM H Bb i?2 T`Q+2bb Q#i BM2/ #v HBM2 ` }Hi2`BM; TTHB2/ iQ i?2 b KTH2/ bB;M H\ h?2 +H bbB+ H a? MMQMĜLv[mBbi `2+QMbi`m+iBQM i?2Q`2K Q7 # M/@ HBKBi2/ bB;M Hb Bb T `iB+mH ` + b2 Q7 i?2 ;2M2` H i?2Q`vX
RyXR
6BHi2`BM;
h?2 i2`K }Hi2`BM; `272`b iQ i?2 T`Q#H2K Q7 2biBK iBM; i?2 p Hm2 i Mv iBK2 Q7 bQK2 biQ+? biB+ T`Q+2bb ;Bp2M bQK2 T `iB H BM7Q`K iBQM ; i?2`2/ i i?Bb iBK2X JQ`2 T`2@ +Bb2Hv- H2i {X(t)}t≥0 #2 M Ft @T`Q;`2bbBp2Hv K2 bm` #H2 biQ+? biB+ T`Q+2bb- + HH2/ i?2 bi i2 T`Q+2bb- rBi? p Hm2b BM bQK2 K2 bm` #H2 bT +2 (E, E)X G2i {Ot }t≥0 #2 }Hi` iBQM- + HH2/ i?2 Q#b2`p2/ ?BbiQ`v- bm+? i? i Ot ⊆ Ft Ut ≥ 0VX h?2 σ@}2H/ Ot `2T`2b2Mib i?2 BM7Q`K iBQM p BH #H2 i iBK2 t iQ Q#i BM M 2biBK i2 Q7 i?2 T`Q# @ #BHBiv /Bbi`B#miBQM Q7 i?2 bi i2 T`Q+2bb i iBK2 tX aT2+B}+ HHv QM2 b22Fb iQ + H+mH i2 7Q` Mv K2 bm` #H2 7mM+iBQM ϕ : (E, E) → (R, B(R)) i?2 [m MiBiv E [Z(t) | Ot ]r?2`2 Z(t) := ϕ(X(t)) Bb bbmK2/ BMi2;` #H2X AM i?2 bQ@+ HH2/ BMMQp iBQMb T@ T`Q +? iQ i?2 }Hi2`BM; T`Q#H2K- i?2 biQ+? biB+ T`Q+2bb {Z(t)}t≥0 i F2b i?2 7Q`K Q7 dzb2KB@K `iBM; H2Ǵ M/ i?2 K `iBM; H2 `2T`2b2Mi iBQM i?2Q`2K TH vb TBpQi H `QH2X (N,Z)
AM i?Bb +? Ti2`- i?2 Q#b2`p2/ ?BbiQ`v Bb Q7 i?2 7Q`K Ot = Ft Bb K `F2/ TQBMi T`Q+2bbX R
- r?2`2 (N, Z)
a22 7Q` BMbi M+2 ("Q2H- o ` Bv M/ qQM;- RNd8)- ("`ûK m/- RN3R)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_10
je8
jee
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
a2KB@K `iBM; H2 ai i2 S`Q+2bb h?2 bi i2 T`Q+2bb {Z(t)}t≥0 Bb bbmK2/ iQ #2 `2 H@p Hm2/ biQ+? biB+ T`Q+2bb Q7 i?2 7Q`K t Z(t) = Z(0) + f (s) ds + m(t) , URyXRV 0
r?2`2 UBV {f (t)}t≥0 Bb M Ft @T`Q;`2bbBp2 biQ+? biB+ T`Q+2bb bm+? i? i E ∞ 7Q` HH t ≥ 0- M/
t 0
|f (s)| ds
0- 7`QK i?2 p Hm2 a iQ i?2 p Hm2 b- Q7 TQBMi T`Q+2bb N QM i?2 TQbBiBp2 ? H7@HBM2X JQ`2 T`2+Bb2Hv- N Bb *Qt TQBMi T`Q+2bb rBi? `2bT2+i iQ i?2 σ@}2H/ G := σ(τ ) r?Qb2 Ft @BMi2MbBiv ěr?2`2 Ft := FtN ∨ σ(τ )ě Bb λ(t) := a + (b − a)Z(t) , r?2`2 Z(t) := 1{τ ≤t} . .2MQiBM; #v F i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ - M/ bbmKBM; i? i Bi ? b /2MbBiv f rBi? `2bT2+i iQ i?2 G2#2b;m2 K2 bm`2- Bib 7 BHm`2 ` i2 Bb f (t) 1 − F (t)
h(t) := M/ i?2`27Q`2 i?2 T`Q+2bb
Z(t) −
t∧τ
h(s)ds 0
Bb M Ft @K `iBM; H2X 1[mBp H2MiHv t h(s)(1 − Z(s)) ds + m(t) , Z(t) = 0
r?2`2 m(t) = Ft @K `iBM; H2X
1t KTH2 RyXRXk, J `FQp *? BM a2KB@K `iBM; H2 1[m iBQMX G2i {X(t)}t≥0 #2 bi #H2 M/ +QMb2`p iBp2 ?K+ rBi? /2MmK2` #H2 bi i2 bT +2 E M/ BM}MBi2b@ BK H ;2M2` iQ` AX 6Q` 2 +? T B` (i, j) Q7 /BbiBM+i bi i2b- H2i Ni,j #2 i?2 TQBMi k AM 7 +i- T `iB+mH ` 7Q`K Q7 Bi- #mi r2 b? HH MQi M22/ iQ +QMbB/2` KQ`2 ;2M2` H b2KB@ K `iBM; H2b BM i?Bb +? Ti2`X
RyXRX 6AGh1_AL:
jed
T`Q+2bb +QmMiBM; i?2 i` MbBiBQMb 7`QK i iQ j- r?Qb2 T`2/B+i #H2 biQ+? biB+ FtX @ BMi2MbBiv Bb Ub22 1t KTH2 8XRXRjV λij (t) := qij 1{X(t−)=i} X Ai r b b?QrM BM am#@ b2+iBQM kX8 M/ 1t2`+Bb2 kXdXR8 i? i- /2}MBM; Zi (t) := 1{X(t)=i} M/ Mij (t) := t Nij ((0, t]) − 0 Zi (s)qij ds U M FtX @K `iBM; H2V
t
AZ(s) ds + M (t) ,
Z(t) = Z(0) +
URyXkV
0
r?2`2 Z(t) = {Zi (t)}i∈E - M (t) = {Mi (t)}i∈E - M/ 7Q` 2 +? i ∈ E- {Mi (t)}t≥0 Bb M FtX @K `iBM; H2X 6Q`KmH URyXkV `2 /b BM 2tT M/2/ 7Q`K ⎛
t
Zi (t) = Zi (0) + 0
⎝
⎞ Zj (s)qji − Zi (s)qi ⎠ ds + Mi (t)
(i ∈ E) .
URyXjV
j∈E\i
S`QD2+iBQM Q7 i?2 a2KB@J `iBM; H2 ai i2 QM i?2 P#b2`p iBQM >BbiQ`v PM2 b22Fb iQ +QKTmi2 7Q` 2 +? iBK2 t i?2 2biBK i2 Z(t) := E [Z(t) | Ot ]X h?2 }`bi bi2T Q7 i?2 BMMQp iBQMb K2i?Q/ +QMbBbib BM T`QD2+iBM; 1[MX URyXRV QM i?2 Q#b2`p2/ ?BbiQ`vX G2KK RyXRXj lM/2` i?2 #Qp2 bbmKTiBQMb Z(t) = Z(0) +
t
f (s) ds + m(t) ,
URyX9V
0
r?2`2, U V {f (t)}t≥0 Bb M Ot @T`2/B+i #H2 biQ+? biB+ T`Q+2bb mMB[m2Hv /2}M2/ #v i?2 `2[mB`2K2Mi i? i 7Q` HH t ≥ 0 M/ HH #QmM/2/ Ot @T`2/B+i #H2 biQ+? biB+ T`Q+2bb2b {C(t)}t≥0 t
t
E C(s)f (s) ds = E C(s)f (s) ds . 0
0
U#V {m(t)}t≥0 Bb +2Mi2`2/ Ot @K `iBM; H2X S`QQ7X h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 {f (t)}t≥0 b iBb7vBM; U V 7QHHQrb 7`QK i?2 b K2 _ /QMĜLBFQ/ɷK `;mK2Mib b BM i?2 T`QQ7 Q7 i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 T`2/B+i #H2 BMi2MbBivX h?2 /2i BHb `2 H27i b 1t2`+Bb2 RyXeXRX 6`QK URyXRV
t
E [Z(t) | Ot ] = E [Z(0) | Ot ] + E
f (s) ds | Ot + E [m(t) | Ot ] . 0
h?2 T`Q+2bb
je3
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a m(t) := (E [Z(0) | Ot ] − E [Z(0)]) t
t + E f (s) ds | Ot − f (s) ds 0
0
+ E [m(t) | Ot ] := A + B + C Bb 0@K2 M Ot @K `iBM; H2X h?Bb Bb +H2 `Hv i`m2 7Q` i?2 T `ib A M/ CX hQ T`Qp2 i?Bb 7Q` B r2 ? p2 iQ b?Qr i? i 7Q` HH s U0 ≤ s ≤ tV M/ HH A ∈ Os
s
t t
f (u) du | Ot − E f (u) du | Os = E 1A E 1A E f (u) du . () 0
0
s
h?2 H27i@? M/ bB/2 Q7 UV + M #2 `2r`Bii2M b
t
t s f (u) du − f (u) du = E 1A f (u) du = E E 1A 0
0
s
∞
f (u)C(u) du
0
r?2`2 C(u, ω) := 1A (ω)1(s,t] (u) /2}M2b MQM@M2; iBp2 Ou @T`2/B+i #H2 biQ+? biB+ T`Q+2bbX b 7Q` i?2 `B;?i@? M/ bB/2 Q7 UV- Bi Bb 2[m H iQ t
∞
E 1A f (u) du = E f (u)C(u) du . s
0
h?2 2[m HBiv UV i?2M 7QHHQrb 7`QK i?2 /2}MBiBQM Q7 {f (t)}t≥0 X
Ai Bb ?2M+27Q`i? bbmK2/ i? i i?2 Q#b2`p2/ ?BbiQ`v Bb Q7 i?2 7Q`K (N,Z)
Ot = G ∨ F t
,
r?2`2 (N, Z) Bb bBKTH2 HQ+ HHv }MBi2 K `F2/ TQBMi T`Q+2bb QM R+ rBi? K `Fb BM K M/ bbQ+B i2/ HB7i2/ T`Q+2bb NZ QM R+ × KX h?2`27Q`2 i iBK2 t QM2 Bb iQ # b2 i?2 2biBK i2 Q7 i?2 bi i2 Z(t) QM i?2 Q#b2`p iBQM Q7 (N, Z) #27Q`2 t- THmb Mv //BiBQM H BM7Q`K iBQM +QMi BM2/ BM GX Ai Bb KQ`2Qp2` bbmK2/ i? i NZ /KBib i?2 (P, Ot )@HQ+ H +? ` +i2`BbiB+b (λ(t), Φ(t, dz))X h?2 AMMQp iBQMb : BM h?2 b2+QM/ bi2T BM i?2 BMMQp iBQMb K2i?Q/ /2T2M/b +`m+B HHv QM i?2 K `iBM; H2 `2T`2b2Mi iBQM i?2Q`2K Uh?2Q`2K 8XjX9V- r?B+? ;m ` Mi22b i?2 2tBbi2M+2 Q7 M BM/2t2/ biQ+? biB+ T`Q+2bb K ∈ P(O· ) ⊗ K- + HH2/ i?2 BMMQp iBQMb ; BM- bm+? i? i t |K(s, z)|λ(s) Φ(s, dz) ds < ∞ (t ≥ 0) 0
K
M/
2Z (ds × dz) K(s, z)M
m(t) = (0,t]
(t ≥ 0) ,
K
2Z (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) dsX h?2`27Q`2r?2`2 M
RyXRX 6AGh1_AL:
jeN
t
2Z (ds × dz) . K(s, z)M
f (s) ds +
Z(t) = Z(0) + 0
(0,t]
URyX8V
K
Ai `2K BMb iQ /2i2`KBM2 i?2 BMMQp iBQMb ; BM K BM i2`Kb i? i K v #2 K /2 2tTHB+Bi H i2` QMX h?2 ;2M2` H T`Q+2/m`2 Bb i?2 7QHHQrBM;X aBM+2 Z(t) Bb i?2 +QM/BiBQM H 2tT2+i iBQM Q7 Z(t) ;Bp2M Ot - M2+2bb `BHv E [Z(t)U (t)] = E Z(t)U (t) URyXeV 7Q` HH #QmM/2/ Ot @K `iBM; H2b {U (t)}t≥0 U1t2`+Bb2 RyXeXkVX am+? K `iBM; H2b ? p2 i?2 7Q`K U (t) = H(s, z)M Z (ds × dz) , (0,t]
K
t r?2`2 H Bb P(O. ) × K@K2 bm` #H2 M/ KQ`2Qp2` 0 K |H(s, z)|λ(s) Φ(s, dz) ds < ∞X 1[m HBiv URyXeV rBHH T`QpB/2 M 2tT`2bbBQM Q7 i?2 BMMQp iBQMb ; BMX h?2 /2i BHb rBHH #2 ;Bp2M 7Q` M Q#b2`p iBQM i? i Bb KmHiBp `B i2 TQBMi T`Q+2bb b /2b+`B#2/ BM 1t KTH2 8XjXe- iQ r?B+? r2 `272` 7Q` i?2 MQi iBQMX h?mb Ot = FtN M/ i?2`27Q`2 t 2i (ds) , Ki (s)M f (s) ds + Z(t) = Z(0) + 0
(0,t]
i∈E
K
2i (ds) := Ni (ds) − λi (s) ds M/ r?2`2 {λi (t)}t≥0 Bb i?2 Ot @T`2/B+i #H2 r?2`2 M biQ+? biB+ BMi2MbBiv Q7 Ni X h?2Q`2K RyXRX9 G2i {Ψij (t)}t≥0 #2 Ot @T`2/B+i #H2 T`Q+2bb2b bm+? i? i- 7Q` HH i ∈ E M/ HH MQM@M2; iBp2 #QmM/2/ Ot @T`2/B+i #H2 T`Q+2bb2b {C(t)}t≥0 ,
t
t C(s)Z(s)λi (s) ds = E C(s)Ψi1 (s)λi (s) ds , E 0
t
0
C(s)Ψi2 (s)λi (s) ds
C(s)Z(s)λi (s) ds = E
E 0
t
,
0
M/- rBi? Δg(s) := g(s) − g(s−) t
i E C(s)Δm(s)ΔNi (s) = E C(s)Ψ3 (s)λi (s) ds . 0
0Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
_2K `F RyXRX8 PM2 Hr vb ? b Ψi2 (t) = Z(t−) . HbQ- B7 {m(t)}t≥0 /Q2b MQi ? p2 /Bb+QMiBMmBiB2b i i?2 TQBMib QM N - i?2M Ψi3 ≡ 0X S`QQ7X q2 /Q i?2 mMBp `B i2 + b2 7Q` i?2 b F2 Q7 MQi iBQM H bBKTHB+BivX h?2 T`QQ7 #2HQr Bb BM7Q`K H BM i? i i 2 +? bi2T r2 M2Bi?2` +?2+F B7 i?2 [m MiBiB2b r?Qb2 2tT2+i iBQM Bb i F2M `2 BMi2;` #H2- MQ` B7 i?2 bKQQi?BM; i?2Q`2K Bb TTHB+ #H2X Ai + M #2 K /2 `B;Q`Qmb #v `2TH +BM; t #v t ∧ Sn - r?2`2 {Sn }n≥1 Bb b2[m2M+2 Q7 TT`QT`B i2 biQTTBM; iBK2b i2M/BM; iQ ∞- M/ i?2M H2iiBM; n ↑ ∞X AM ai2T R- QM2 +QmH/ i F2 7Q` BMbi M+2 i?2 Ft @biQTTBM; iBK2b t (|H(s) + 1|λ(s) + f (s))ds ≥ n} , Sn := inf{t ≥ 0 ; N (t) + Z(t−) + 0
M/ BM ai2T k M/ ai2T j- i?2 FtN @biQTTBM; iBK2b t Sn := inf{t ≥ 0 ; N (t) + (1 + |K(s)|)λ(s) ds ≥ n} . 0
h?2 p2`B}+ iBQMb `2 H27i 7Q` i?2 `2 /2`X ai2T RX q2 }`bi +QKTmi2 i?2 H27i@? M/ bB/2 Q7 URyXeV- rBi? 2(ds) , U (t) = H(s)M (0,t]
t r?2`2 i?2 FtN @T`2/B+i #H2 T`Q+2bb {H(t)}t≥0 Bb bm+? i? i 0 |H(s)|λ(s) ds < ∞X 6Q` i?Bb- }`bi /2+QKTQb2 i?2 bi i2 b t Z(t) = Zc (t) + Zd (t) := mc (t) + f (s) ds + md (t) . 0
"v i?2 aiB2HiD2bĜG2#2b;m2 `mH2 Q7 BMi2;` iBQM #v T `ib Zd (s−) dU (s) + U (s) dZd (s) Zd (t)U (t) = (0,t] (0,t] 2(s) + Zd (s−)H(s) dM U (s) (dmd (s) + f (s) ds) =
(0,t]
(0,t] t
Zd (s−)H(s) λ(s) − λ(s) ds
Zd (s−)H(s) dM (s) +
= (0,t]
U (s)f (s) ds +
+ 0
+
0
t
U (s−) dmd (s) (0,t]
H(s)Δmd (s)ΔN (s) .
0Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
G2KK RyXRXRy 6Q` HH t ≥ 0 EQ L(t) | FtN EP Z(t) | FtN = EQ Z(t)L(t) | FtN , Q`- 2[mBp H2MiHvEP
Z(t) | FtN
EQ Z(t)L(t) | FtN = , EQ [L(t) | FtN ]
Q@ XbX
P @ XbX
S`QQ7X h?Bb Bb Dmbi `2T?` bBM; Q7 G2KK jX8XeX
1t KTH2 RyXRXRR, 1biBK iBM; i?2 _ M/QK AMi2MbBiv Q7 >QKQ;2@ M2Qmb *Qt S`Q+2bbX h?2 #Qp2 H2KK HHQrb mb iQ `2TH +2 }Hi2`BM; T`Q#H2K rBi? `2bT2+i iQ P #v QM2 rBi? `2bT2+i iQ Q- r?B+? K v #2 bBKTHB}+ iBQM r?2M Q ? b bBKTH2 bi`m+im`2X 6Q` BMbi M+2- B7 Q Bb T`Q# #BHBiv i? i K F2b i?2 TQBMi T`Q+2bb N SQBbbQM rBi? BMi2MbBiv 1- M/ B7 Λ Bb M BMi2;` #H2 p `B #H2 BM/2T2M/2MimM/2` Q- Q7 N - i?2 K2 bm`2 P /2}M2/ #v dPt = ΛN (t) exp{(1 − Λ)t} dQt K F2b N /Qm#Hv biQ+? biB+ T`Q+2bb rBi? BMi2MbBiv ΛX h?2M ∞ N (t)+1 −λt λ e dF (λ) N E Λ | Ft = 0 ∞ N (t) −λt . λ e dF (λ) 0 U1t2`+Bb2 RyXeX8XV
RyXk a2T ` iBQM Q7 .2i2+iBQM M/ 6BHi2`BM; iBK2 p `vBM; bB;M H {θ(t)}t∈[0,T ] Bb i` MbKBii2/ pB TQBMi T`Q+2bb +? MM2HX JQ`2 T`2+Bb2Hv- dzb2M/2`Ǵ /2bB;Mb +Q/2 7Q` i?Bb bB;M H- i? i Bb- biQ+? biB+ T`Q+2bb {μ(t)}t∈[0,T ] +QMi BMBM; BM7Q`K iBQM #Qmi i?2 bB;M H- M/ dz`2+2Bp2`Ǵ Q#b2`p2b TQBMi T`Q+2bb N rBi? M Ft @biQ+? biB+ BMi2MbBiv Q7 i?2 7Q`K λ(t) = λ + μ(t) , r?2`2 Ft := Ftθ ∨ FtN X h?Bb KQ/2H HHQrb 7Q` dz722/# +FǴ- BM i?2 b2Mb2 i? i μ(t) K v /2T2M/- #2bB/2b i?2 T bi i iBK2 t Q7 i?2 bB;M H- QM i?2 T bi i iBK2 t Q7 i?2 Q#b2`@ p iBQM N X h?2 +Q/BM; T`Q+2bb {μ(t)}t∈[0,T ] Bb KQ`2Qp2` bbmK2/ Ft @T`2/B+i #H2X AM +QKKmMB+ iBQMb +QMi2ti- i?2 +Q/BM; /Q2b MQi 2` b2 BM7Q`K iBQM #Qmi i?2 bB;M H M/ /Q2b MQi + ``v BM7Q`K iBQM MQi +QMi BM2/ BM N Q` i?2 bB;M H- i? i Bb- 7Q` HH t ∈ R+ Ftθ ∨ FtN ≡ Ftμ ∨ FtN . (†) h?2 mM/2`HvBM; T`Q# #BHBiv +Q``2bTQM/BM; iQ +Q/2 μ rBHH #2 /2MQi2/ #v Pμ X Ai Bb +QMbi`m+i2/ b 7QHHQrbX q2 bi `i 7`QK `272`2M+2 T`Q# #BHBiv P0 mM/2` r?B+? N Bb
RyXkX a1S_hAPL P6 .1h1*hAPL L. 6AGh1_AL:
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SQBbbQM T`Q+2bb Q7 ` i2 λ M/ r?2`2 i?2 bB;M H {θ(t)}t∈[0,T ] M/ N `2 BM/2T2M/2MiX h?2 +Q/BM; Bb `2[mB`2/ iQ b iBb7v i?2 dzT2 F +QMbi` BMiǴ 0 ≤ μ(t) ≤ c . h?2 T`Q# #BHBiv Pμ Bb +QMbi`m+i2/ 7`QK P0 b 7QHHQrb BM bm+? r v i? i Pμ P0 X JQ`2 2tTHB+BiHv T dPμ μ(s) N (ds) − |FT := L(T ) := exp log 1 + μ(s) ds . () dP0 λ [0,T ] 0 Ai r b b?QrM BM 1t KTH2 8X8Xj i? i i?Bb 7Q`KmH /2}M2b T`QT2` _ /QMĜLBFQ/ɷK /2`Bp iBp2 M/ i? i mM/2` i?2 T`Q# #BHBiv Pμ bQ /2}M2/- N /KBib QM i?2 BMi2`p H [0, T ] i?2 Ft @BMi2MbBiv {λ(t)}t∈[0,T ] X JQ`2Qp2`- i?2 /Bbi`B#miBQMb Q7 i?2 bB;M H mM/2` Pμ Q` P0 `2 i?2 b K2X AM/22/- 7Q` Mv 2p2Mi A ∈ FTθ Pμ (A) = Eμ [1A ] = E0 [1A L(T )]
L(s−)(λ(s) − λ)(N (ds) − λ(s) ds)) = E0 1A (1 + [0,T ]
1A L(s−)(λ(s) − λ)(N (ds) − λ(s) ds) = P0 (A) , = E0 [1A ] + E0 [0,T ]
r?2`2 i?2 H bi BM2[m HBiv K F2b mb2 Q7 i?2 7 +i i? i {1A L(t−)(λ(t)−λ)}t∈[0,T ] Bb M FTθ ∨ FtN @T`2/B+i #H2 T`Q+2bb M/ i? i mM/2` P0 - N /KBib i?2 FTθ ∨ FtN @BMi2MbBiv λX _2K `F RyXkXR h?2 #QmM/2/M2bb +QM/BiBQMb QM i?2 (P, Ft )@BMi2MbBiv Q7 N ;m `@ Mi22 i? i P (L(T ) > 0) = 1 M/ i?2`27Q`2 P0 M/ Pμ `2 Kmim HHv #bQHmi2Hv +QMiBMmQmbX h?2 Jmim H AM7Q`K iBQM 6Q`KmH "v /2}MBiBQM- i?2 Kmim H BM7Q`K iBQM #2ir22M N QM [0, T ] M/ {θ(t)}t∈[0,T ] Bb i?2 [m MiBiv dPμ |FTN ∨FTθ IT (θ, N ) := Eμ log , dPμ |FTθ dPμ |FTN rBi? i?2 7QHHQrBM; BMi2`T`2i iBQM dPμ |FTN ∨FTθ
dPμ |FTθ dPμ |FTN aBM+2
dPμ | θ dP0 FT
:=
dPμ dPμ dPμ | FT |FTθ | N. dP0 dP0 dP0 FT
= 1 IT (θ, N ) := Eμ log
r?2`2 i?2 H bi 2[m HBiv mb2b U†VX h?2 # bB+ `2bmHi Bb,
dPμ |FTN ∨FTθ dPμ |FTN
= Eμ log
dPμ |FTN ∨FTμ dPμ |FTN
,
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h?2Q`2K RyXkXk lM/2` i?2 #Qp2 +QM/BiBQMb- i?2 Kmim H BM7Q`K iBQM #2ir22M N QM [0, T ] M/ {θ(t)}t∈[0,T ] Bb ;Bp2M #v i?2 7Q`KmH T
(ϕ(μ(s)) − ϕ(μ(s))) ds , IT (θ, N ) = Eμ 0
r?2`2
ϕ(x) := (λ + x) log(λ + x) − λ log λ .
S`QQ7X Ai /2T2M/b QM i?2 7QHHQrBM; H2KK - + HH2/ i?2 i?2Q`2K Q7 b2T ` iBQM Q7 /2i2+iBQM M/ }Hi2`BM;X9 G2KK RyXkXj dPμ | N = L(T ) := exp dP0 FT
T μ(s) N (ds) − log 1 + μ(s) ds , λ 0 [0,T ]
()
r?2`2 λ(t) = λ + μ(t) Bb i?2 T`2/B+i #H2
FtN @BMi2MbBiv
Q7 N X
S`QQ7X h?2 +QM/BiBQMb Q7 #QmM/2/M2bb 7Q` {λ(t)}t∈[0,T ] `2 HbQ b iBb}2/ 7Q` {λ(t)}t∈[0,T ] mM/2` Pμ - M/ i?2`27Q`2 mM/2` P0 UPμ M/ P0 `2 Kmim HHv #bQHmi2Hv +QMiBMmQmbc b22 _2K `F RyXkXRVX h?2`27Q`2 i?2 T`Q# #BHBiv K2 bm`2 QdQ |F := L(T ) dP0 T Bb bm+? i? i N ? b QM [0, T ] i?2 (Q, FTN )@BMi2MbBiv {λ(t)}t∈[0,T ] X h?2`27Q`2 Q M/ Pμ ;`22 QM FTN - r?B+? b?Qrb i? i dPμ | N = L(T ) . dP0 FT h?2 T`QQ7 Q7 h?2Q`2K RyXkXk rBHH MQr #2 +QKTH2i2/X _2+ HH i? i
dPμ dPμ |FT − log |FTN . IT θ, N ) = Eμ log dP0 dP0 LQr
T dPμ μ(s) N (ds) − | FT = E μ log 1 + μ(s) ds Eμ log dP0 λ [0,T ] 0
T μ(s) λ(s) ds − log 1 + μ(s) ds = Eμ λ [0,T ] 0 9
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M/ bBKBH `Hv
T dPμ μ(s) Eμ log λ(s) ds − |FTN = Eμ log 1 + μ(s) ds . dP0 λ 0 [0,T ] h?2 `2bmHi 7QHHQrb 7i2` Q#b2`pBM; i? i T
μ(s) ds = Eμ Eμ 0
T
μ(s) ds .
0
* T +Biv Q7 i?2 SQBbbQM *? MM2H h?2 + T +Biv Q7 i?2 SQBbbQM +? MM2H Bb- #v /2}MBiBQM- i?2 [m MiBiv C :=
1 sup sup IT (θ, N ) , T θ μ
r?2`2 i?2 supθ #2 `b QM HH bB;M H T`Q+2bb2b {θ(t)}t∈[0,T ] - M/ i?2 supμ #2 `b QM HH +Q/BM;b b iBb7vBM; i?2 +QMbi` BMibX Ai `2T`2b2Mib- BM `Qm;? i2`Kb- i?2 K tBKmK ` i2 Q7 i` MbKBbbBQM Q7 BM7Q`K iBQM rBi?Qmi 2``Q` i?`Qm;? i?2 +? MM2H +QMbB/2`2/ Ua? MMQMǶb + T +Biv i?2Q`2KVX AM i?2 + b2 r?2`2 QMHv M KTHBim/2 +QMbi` BMi Bb BKTQb2/- r2 ? p2 i?2 7QHHQrBM; `2bmHiX8 h?2Q`2K RyXkX9 C=λ
1 e 1+λ/c c λ 1+ log 1 + . − 1+ e λ c λ
()
S`QQ7X "v C2Mb2MǶb BM2[m HBivEμ [ϕ(μ(s))] ≤ ϕ(Eμ [μ(s)]) = ϕ(Eμ [μ(s)]) , M/ i?2`27Q`2
T
IT θ, N ) ≤
(Eμ [ϕ(μ(s))] − ϕ(Eμ [μ(s)]) ds .
URyXRRV
0
G2i A #2 i?2 +QHH2+iBQM Q7 T`Q# #BHBiv K2 bm`2b Q QM [0, c]X h?2 [m MiBiv Eμ [ϕ(μ(s))]− ϕ(Eμ [μ(s)] Bb Q7 i?2 7Q`K ϕ(x) Q(dx) − ϕ x Q(dx) [0,c]
M/ i?2`27Q`2
[0,c]
IT θ, N ) ≤ sup Q∈A
8
(E # MQp- RNd3)X
ϕ(x) Q(dx) − ϕ [0,c]
x Q(dx) [0,c]
.
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.2MQiBM; #v U i?2 `B;?i@? M/ bB/2 Q7 i?2 T`2pBQmb BM2[m HBiv- r2 ? p2 i? i U := max U (v) , 0≤v≤c
r?2`2 U (v) :=
Q∈A;
ϕ(x) Q(dx) − ϕ(v)
max [0,c]
x Q(dx)=v
.
[0,c]
x v ϕ(x) − ϕ(c) Q(dx) = ϕ(x) Q(dx) − ϕ(c) , c c [0,c] [0,c] x r2 Kmbi K tBKBx2 h(v) − [0,c] h(x) Q(dx)- r?2`2 h(x) := c ϕ(c) − ϕ(x)X h?2 7mM+@ iBQM h Bb +QMiBMmQmb- +QM+ p2- TQbBiBp2 2t+2Ti i x = 0 M/ x = c r?2`2 Bi Bb MmHHX h?2 T`Q# #BHBiv Q QM [0, c] +?B2pBM; i?2 K tBKmK Q7 U (v) bm#D2+i iQ i?2 +QMbi` BMi [0,c] x Q(dx) = v i F2b QMHv i?2 irQ p Hm2b 0 M/ c- M/
aBM+2
Q({0}) = 1 −
v v , Q({c}) = . c c
h?2`27Q`2 U = max
0≤v≤c
i? i BbU =λ
ϕ(c) − ϕ(v) v c
,
1 e 1+λ/c c λ 1+ log 1 + . − 1+ e λ c λ
h?2 p Hm2 Q7 v +?B2pBM; i?2 K tBKmK Bb
e 1+λ/c 1 vm = λ 1+ −1 . e λ aQ 7 `- r2 ? p2 T`Qp2/ i? i C Bb H2bb i? i i?2 `B;?i@? M/ bB/2 Q7 UVX AM Q`/2` iQ b?Qr 2[m HBiv- Bi bm{+2b iQ }M/ b2[m2M+2 Q7 bB;M Hb {θ(n) (t)}t≥0 Un ≥ 1V M/ +Q/2 μ `2bT2+iBM; i?2 +QMbi` BMib M/ bm+? i? i lim IT (θ(n) , N ) = C .
n↑∞
h?Bb Bb +?B2p2/ #v HBM2 ` +Q/2 μ(t) = cθ(t) M/ b2[m2M+2 Q7 dzi2H2;` T? bB;M HbǴX S`2+Bb2Hv- {θ(n) (t)}t≥0 Bb +QMiBMmQmb iBK2 ?K+ rBi? bi i2 bT +2 E = {0, 1} M/ BM}MBi2bBK H +? ` +i2`BbiB+b q0,1 = n, ,
q1,0 =
r?2`2
n(1 − p) , p
vm . c 1 +? bB;M H Bb bbmK2/ bi iBQM `v- i? i Bb- i?2 BMBiB H /Bbi`B#miBQM Bb i?2 bi iBQM `v /Bbi`B#miBQM, p=
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P (θ(n) (0) = 1) = E θ(n) (0) = p . 6`QK 1t2`+Bb2 RyXeX9- i?2 2biBK i2 θ(t) := E θ(t) | FtN Q7 i?2 i2H2;` T? bB;M H 2pQHp2b ++Q`/BM; iQ UQKBiiBM; i?2 bmT2`b+`BTiV n t θ(t) = p + (p − θ(s)) ds p 0 cθ(s−)(λ − θ(s−)) (N (ds) − (λ + cθ(s)) ds) . URyXRkV + 1 + cθ(s−) (0,t] aBM+2 ϕ(cθ(t)) = θ(t)ϕ(c) + (1 − θ(t)ϕ(0) = θ(t)ϕ(c) M/ Eμ [θ(t)] = p = ? p2 i? i vm Eμ [ϕ(cθ(t))] = ϕ(c) . c Ai i?2`27Q`2 `2K BMb iQ b?Qr i? i lim Eμ ϕ(cθ(n) (t)) = ϕ(vm ) .
vm c
r2
n↑∞
h?2 /2`Bp iBp2 Q7 ϕ #2BM; #QmM/2/ QM [0, c] #v }MBi2 +QMbi Mi- b v M |ϕ(cθ(t) − ϕ(vm )| ≤ M |cθ(t) − vm | = M c|θ(t) − p| . "v a+?r `xǶb BM2[m HBiv 12 Eμ |ϕ(cθ(t)) − ϕ(vm )| ≤ M cEμ |θ(t) − p|2 . Ai i?2`27Q`2 `2K BMb iQ T`Qp2 i? i limn↑∞ Eμ |θ(n) (t) − p|2 = 0X "v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 + H+mHmb TTHB2/ iQ URyXRkV t c2 [θ(s)(1 − θ(s))]2 2 2 2 θ(s)(p − θ(s)) + θ(t) = p + ds + m(t) , np 0 1 + θ(s) N r?2`2 {m(t)} t∈[0,T ] Bb M Ft @K `iBM; H2X h?2`27Q`2 Eμ |θ(n) (t) − p|2 = Eμ θ(n) (t)2 − p2 t
2n Eμ = θ(s)(p − θ(s)) ds p 0 t 2 c [θ(s)(1 − θ(s))]2 + Eμ ds . λ + θ(s) 0 qBi? z (n) (t) := Eμ θ(n) (t)2 − p2 - r2 ? p2 i? i
z (n) (t) = − r?2`2
2n p
t
z (n) (s) ds + 0
t
g (n) (s) ds , 0
j39
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(n)
(s) := Eμ
c2 [θ(n) (s)(1 − θ(n) (s))]2
λ + θ(n) (s)
Bb #QmM/2/X h?2`27Q`2- #v :`QMr HHǶb H2KK - limn↑∞ z (n) (t) = 0X
q2 MQr BMi`Q/m+2 M //BiBQM H +QMbi` BMi- i?2 ǵ2M2`;v +QMbi` BMiǴ QM i?2 +Q/BM;M K2Hv T μ(s) ds ≤ E0 . 0
.2}MBM; h(x) :=
x ϕ(c) − ϕ(x) , c
r2 ? p2 7`QK URyXRRVIT θ, N ) ≤
1 T
T
h(Eμ [μ(s)] ds . 0
T A7 E0 ≥ vm - bBM+2 h ? b K tBKmK i x = vm - i?2 K tBKmK Q7 0 h(Eμ [μ(s)] ds Bb ii BM2/ 7Q` Eμ [μ(s)] = vm M/ i?2 2M2`;v +QMbi` BMi Bb b iBb}2/X AM i?2 Qi?2` + b2 E0 < vm - i?2 K tBKmK Eμ [μ(s)] = vm HHQr2/ #v i?2 2M2`;v +QMbi` BMi Bb E0 M/ h Bb K tBKBx2/ QM [0, E0 ] i x = E0 X h?2`27Q`2 IT (θ, N ) ≤ T h(E0 ). b2[m2M+2 Q7 bB;M Hb M/ +Q/2b +?B2pBM; i?2 #QmM/ Bb i?2 b K2 b i?2 QM2 mb2/ #27Q`2- rBi? p = Ec0 UBMbi2 / Q7 p = vcm VX AM bmKK `v, h?2Q`2K RyXkX8 Ue V A7 E0 ≥ vm - i?2 T2 F +QMbi` BMi /QKBM i2b M/ i?2 + T +Biv Bb i?2 b K2 b 7QmM/ #27Q`2X A7 E0 < vm - i?2 2M2`;v +QMbi` BMi Kmbi #2 i F2M BMiQ ++QmMi M/ i?2 + T +Biv Bb i?2M ;Bp2M #v E0 C = ϕ(c) − ϕ(E0 ) . c
RyXj >B/BM; AM7Q`K iBQM BM SQBMi S`Q+2bb _2+ HH `2bmHi U_kV Q7 a2+iBQM 8X9X A7 N Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t) τ (t) /2}M2/ #v N ((0, t]) = /2}MBM; τ (t) #v 0 λ(s) ds = t- i?2 TQBMi T`Q+2bb N N ((0, τ (t)]) Bb bi M/ `/ ?TTX h?Bb `2bmHi rBHH #2 2ti2M/2/ iQ i?2 i` Mb7Q`K iBQM Q7 TQBMi T`Q+2bb rBi? ;Bp2M biQ+? biB+ BMi2MbBiv BMiQ bi M/ `/ ?TTX G2i N #2 bBKTH2 HQ+ HHv #QmM/2/ TQBMi T`Q+2bb QM R+ - rBi? i?2 Ft @T`2/B+i #H2 BMi2MbBiv {λ(t)}t≥0 - M/ bmTTQb2 i? i N (0, ∞) = ∞- P @ XbX Q`- 2[mBp H2MiHv Uh?2@ ∞ Q`2K 8XRXReV- 0 λ(s) ds = ∞- P @ XbX .2}M2 7Q` 2 +? t ≥ 0- i?2 MQM@M2; iBp2 ` M/QK p `B #H2 τ (t) #v e
(. pBb- RN3y)X
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τ (t)
URyXRjV
λ(s) ds = t . 0
∞ 6Q` 2 +? t ≥ 0- τ (t) Bb r2HH@/2}M2/ bBM+2 0 λ(s) ds = ∞ Uh?2Q`2K 8XRXReVX 6Q` 2 +? t ∈ R+ - τ (t) Bb Ft @biQTTBM; iBK2X AM/22/- 7Q` Mv a ∈ R a λ(s) ds ≤ t ∈ Fa . {τ (t) ≤ a} = 0
QM R+ #v .2}M2 i?2 bBKTH2 HQ+ HHv #QmM/2/ TQBMi T`Q+2bb N (0, t] := N (0, τ (t)]. N
URyXR9V
(0, a] := N (0, τ (a)] Bb F N @ LQi2 i? i FtN ⊆ Fτ (t) - bBM+2 7Q` HH a ∈ R+ - N τ (a) K2 bm` #H2- M/ i?2`27Q`2 Fτ (a) @K2 bm` #H2X
? b i?2 Fτ (t) @BMi2MbBiv R U M/ i?2`27Q`2 i?2 FtN @BMi2MbBiv RVX h?2Q`2K RyXjXR N S`QQ7X G2i [a, b] ∈ RX q2 Kmbi b?Qr i? i (a, b] = E [1A (b − a)] E 1A N "mi i?2 H27i@? M/ bB/2 Bb Dmbi τ (b)
τ (a)
∞
N (dt) = E 1A
E 1A
(A ∈ Fτ (a) ) .
1(τ (a),τ (b)] (t)N (dt) . 0
aBM+2 i?2 T`Q+2bb 1A 1(τ (a),τ (b)] Bb Ft @T`2/B+i #H2 U#2BM; Ft @ / Ti2/ M/ H27i@+QMiBMmQmbVi?2 `B;?i@? M/ bB/2 Q7 i?2 #Qp2 2[m HBiv Bb- #v i?2 bKQQi?BM; 7Q`KmH
∞
E 1A
τ (b)
λ(t) dt = E [1A (b − a)] .
1(τ (a),τ (b)] (t)λ(t) dt = E 1A 0
τ (a)
Bb ?QKQ;2M2Qmb _2K `F RyXjXk "v q i M #2Ƕb i?2Q`2K Uh?2Q`2K kX9XRV- N (a, b] Bb BM/2T2M/2Mi SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X AM //BiBQM- 7Q` HH [a, b] ∈ R- N Q7 Fτ (a) X _2K `F RyXjXj h?2 `2bmHi Q7 h?2Q`2K RyXjXR K v #2 mb2/ iQ i2bi i? i ;Bp2M TQBMi T`Q+2bb /KBib ;Bp2M ?vTQi?2iB+ H BMi2MbBiv, T2`7Q`K i?2 +Q``2bTQM/BM; +? M;2 Q7 iBK2 M/ b22 B7 i?2 `2bmHi Bb bi M/ `/ SQBbbQM T`Q+2bb- #v mbBM; Mv p BH #H2 bi iBbiB+ H i2bi iQ bb2bb i? i ;Bp2M }MBi2 b2[m2M+2 Q7 ` M/QK p `B #H2b Bb BB/ M/ 2tTQM2MiB HHv /Bbi`B#mi2/ rBi? K2 M 1X h?2 +Q``2bTQM/BM; K2i?Q/ Bb FMQrM b TQBMi T`Q+2bb `2bB/m H M HvbBbXd d
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j3e
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
h?2 7QHHQrBM; [m2biBQM `Bb2b M im` HHv,3 ?Qr Km+? BM7Q`K iBQM Bb HQbi BM +? M;2 Q7 iBK2 b+ H2\ *QMbB/2` 7Q` BMbi M+2 i?2 bBim iBQM Q7 h?2Q`2K RyXjXRX aBM+2 i?2 +? M;2 Q7 iBK2 i` Mb7Q`Kb i?2 Q`B;BM H TQBMi T`Q+2bb BMiQ ?QKQ;2M2Qmb SQBb@ bQM T`Q+2bb- ? p2 r2 2` b2/ HH i?2 BM7Q`K iBQM T`2pBQmbHv +QMi BM2/ BM i?2 biQ+? b@ iB+ BMi2MbBiv\ h?2 Mbr2` Bb, Bi /2T2M/bX 6Q` BMbi M+2 bmTTQb2 i? i N Bb *Qt TQBMi T`Q+2bb rBi? BMi2MbBiv Λ- MQM@M2; iBp2 `2 H@p Hm2/ ` M/QK p `B #H2X AM Qi?2` rQ`/b- N /KBib i?2 biQ+? biB+ Ft @BMi2MbBiv λ(t) ≡ Λ- r?2`2 Ft := FtN ∨ σ(Λ)X A7 /2}M2/ #v URyXR9V r2 T2`7Q`K i?2 iBK2 +? M;2 τ (t) = Λt - i?2 `2bmHiBM; T`Q+2bb N (d) − N (c) Bb BM/2T2M@ Bb bi M/ `/ SQBbbQM T`Q+2bbX JQ`2Qp2`- 7Q` HH 0 ≤ c ≤ d- N /2Mi Q7 Fc = FcN ∨ σ(Λ) M/ BM T `iB+mH ` Q7 ΛX AM i?Bb b2Mb2- i?2 iBK2 +? M;2 ? b 2` b2/ HH BM7Q`K iBQM +QM+2`MBM; Λ- r?2`2 b Λ +QmH/ #2 `2+Qp2`2/ 7`QK N bBM+2#v i?2 bi`QM; H r Q7 H `;2 MmK#2`bΛ = lim t↑∞
N (t) . t
()
AM i?2 + b2 Q7 M BMi`BMbB+ +? M;2 Q7 iBK2 `2 /` K iB+ HHv /Bz2`2MiX h?2 i?BM;b biQ+? biB+ FtN @BMi2MbBiv Q7 N - λ(t) = E Λ | FtN - Bb ;Bp2M #v U1t KTH2 RyXRXRRV, ∞ N (t)+1 −λt λ e dF (λ) . λ(t) = 0 ∞ N (t) −λt λ e dF (λ) 0 hQ #2 2p2M KQ`2 bT2+B}+- bmTTQb2 i? i P (Λ = a) = P (Λ = b) = 0 < a < b- BM r?B+? + b2λ(t) =
1 2
7Q` bQK2
1 + (b/a)N (t)+1 e(a−b)t . 1 + (b/a)N (t) e(a−b)t
S2`7Q`KBM; i?2 iBK2 +? M;2
τ(t)
λ(t) dt = t , 0
/2}M2/ #v N (t) := N (τ (t))- r?B+? Bb bi M/ `/ r2 Q#i BM TQBMi T`Q+2bb N SQBbbQM T`Q+2bbX >Qr2p2`- i?Bb iBK2- Λ + M #2 2MiB`2Hv `2+Qp2`2/ 7`QK BiX AM 7 +i M/ i?2M Λ + M #2 Q#i BM2/ #v b r2 MQr b?Qr- N + M #2 `2+QMbi`m+i2/ 7`QK N UVX AM 7 +i- B7 Tn Bb i?2 n@i? TQBMi Q7 N - i?2M
Tn+1 Tn
λ(t) dt = Tn+1 − Tn
Q`- KQ`2 2tTHB+BiHvTn+1 − Tn = f (n, Tn+1 ) − f (n, Tn ) , r?2`2
f (n, t) = at − ln 1 + (b/a)n+1 e(a−b)t .
*H2 `Hv i?2M- i?2 b2[m2M+2 {Tn }n≥1 + M #2 `2+Qp2`2/ 7`QK {Tn }n≥1 X 3
("`ûK m/- RNd8 +)X
RyX9X LPAau SPALha
j3d
_2K `F RyXjX9 M BMi2`T`2i iBQM Q7 i?2 #Qp2 `2bmHib BM i2`Kb Q7 +`vTiQ;` T?v Bb i?2 7QHHQrBM;X A7 i?2 BM7Q`K iBQM Bb +QMi BM2/ BM Λ- i?2 BMi`BMbB+ iBK2 +? M;2 vB2H/b bi M/ `/ SQBbbQM T`Q+2bb 7`QK r?B+? Λ + M #2 2ti` +i2/ QMHv B7 QM2 FMQrb i?2 dzF2vǴ- i? i Bb- i?2 /Bbi`B#miBQM Q7 ΛX ULQi2 ?Qr2p2` i? i 7`QK }MBi2 i` D2+iQ`v Q7 QM2 + M QMHv Q#i BM M TT`QtBK iBQM Q7 ΛX AM i?Bb b2Mb2- b2+m`2 i` MbKBbbBQM N rQmH/ #2 i i?2 T`B+2 Q7 bQK2 mM`2HB #BHBivX h?Bb mM`2HB #BHBiv + M #2 +QMi`QHH2/ i i?2 2tT2Mb2 Q7 i` MbKBbbBQM ` i2- r?B+? Bb T2`? Tb ++2Ti #H2 B7 QM2 Bb BMi2`2bi2/ QMHv BM biQ` ;2 b2+m`BivXV _2K `F RyXjX8 AM i?2 +QMiBMm iBQM Q7 a2+iBQM 8X3- M/ rBi? i?2 MQi iBQM i?2`2Q7 bBKBH ` +`vTiQ;` T?B+ 2z2+i Bb Q#b2`p2/ BM i?2 bB;M H THmb MQBb2 KQ/2HX i` Mb@ t KBii2` rBb?2b iQ b2M/ BM MQBb2H2bb +? MM2H i?2 BMi2;` i2/ bB;M H { 0 ϕ(s) ds}t≥0 X AMbi2 /- ?2 b2M/b pQHmMi `BHv +Q``mTi2/ p2`bBQM
t
X(t) =
ϕ(s) ds + W (t) ,
()
0
r?2`2 i?2 BMi2;` i2/ MQBb2 Bb BM/2T2M/2Mi Q7 i?2 bB;M Hi?2 `2+2Bp2` 7`QK r?B+? + M 2ti` +i M 2biBK i2 {ϕ(t)}t≥0 - r?2`2 ϕ(t) := E ϕ(t) | FtX Ut ≥ 0VX h?Bb Bb TQbbB#H2 bBM+2 i?2 `2+2Bp2` Bb r `2 Q7 i?2 7 +i i? i i?2 BMi2;` i2/ MQBb2 Bb qB2M2` T`Q+2bb Q7 FMQrM p `B M+2 BM/2T2M/2Mi Q7 i?2 bB;M HX LQr- bmTTQb2 i? i i?2 b2M/2` /2+B/2b iQ T2`7Q`K i?2 b K2 +QKTmi iBQMb iQ Q#i BM i?2 #Qp2 2biBK i2 Ur?B+? ?2 + M /Q- bBM+2 ?2 ?BKb2H7 +`2 i2/ i?2 MQBb2VX >2 i?2M b2M/b i?2 BMMQp iBQMb T`Q+2bb 2(t)}t≥0 7`QK r?B+? i?2 b2M/2` + M 2ti` +i i?2 Q`B;BM H +Q``mTi2/ bB;M H UV M/ {W Q#i BM i?2 2biBK i2 {ϕ(t)}t≥0 X
RyX9
LQBbv SQBMib
h?2 ;2M2` H bBim iBQM bim/B2/ BM i?Bb b2+iBQM Bb i?2 7QHHQrBM; QM2X h?2`2 Bb UbBKTH2 M/ HQ+ HHv }MBi2V TQBMi T`Q+2bb N i? i Bb Q#b2`p2/ M/ r2 FMQr i? i Bi Bb i?2 bmK Q7 N = N1 + N2 Q7 irQ TQBMi T`Q+2bb2b- QM2 Q7 i?2K- b v N2 - #2BM; +QMbB/2`2/ b MQBb2X q? i + M r2 /Q iQ /Bb+`BKBM i2 i?2 MQBbv TQBMib 7`QK i?2 BM7Q`K iBp2 TQBMib Q7 N2 \ JQ`2 T`2+Bb2Hv- H2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM i?2 TQbBiBp2 ? H7@HBM2- rBi? i?2 Ft BMi2MbBiv Q7 i?2 7Q`K λ(t) = λ1 (t) + λ2 (t) (t ≥ 0) . N + M #2 i?Qm;?i Q7 b i?2 bmK N1 M/ N2 rBi? i?2 `2bT2+iBp2 Ft BMi2MbBiB2b {λ1 (t)}t≥0 M/ {λ1 (t)}t≥0 X 1t KTH2 RyX9XR, SQBbbQM LQBb2X >2`2 N1 M/ N2 `2 BM/2T2M/2Mi- M/ N2 Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ2 X h?Bb Bb +H bbB+ H bBim iBQM BM QTiB+ H i` MbKBbbBQM #v T?QiQMbX h?2 M2ti 2t KTH2 /Q2b MQi BM 7 +i BMpQHp2 MQBb2X
j33
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
1t KTH2 RyX9Xk, S`BK `v b?Q+Fb M/ 7i2`b?Q+FbX h?2 b2BbKB+ KQ/2H Q7 1t KTH2 8XRXkd + M #2 2ti2M/2/ iQ BM+Hm/2 7i2`b?Q+Fb,N N (0,t)
λ(t) := eX(0)+ct−
i=1
Zi
t
+ −∞
>2`2 λ1 (t) = e
N (0,t)
X(0)+ct−
i=1
h(t − s) N (ds) .
Zi
M/ λ2 (t) =
t
−∞
h(t − s) N (ds)
`2 i?2 `2bT2+iBp2 biQ+? biB+ BMi2MbBiB2b Q7 i?2 T`BK `v b?Q+Fb M/ Q7 i?2 7i2`@ b?Q+FbX :Bp2M b KTH2 Q7 N QM2 rQmH/ HBF2 iQ b2T ` i2 i?2 7i2`b?Q+Fb 7`QK i?2 T`BK `v b?Q+FbX h?2 #2bi i? i + M #2 /QM2 Bb iQ b2H2+i i?2 TQBMib Q7- b v N2 - BM i?2 K MM2` ;Bp2M #v i?2 7QHHQrBM; `2bmHi Ubi i2/ BM i?2 ;2M2` H bBim iBQMV, h?2Q`2K RyX9Xj h?2 TQBMi T`Q+2bb2b N1 M/ N2 /2}M2/ #v N1 ((a, b]) :=
∞
1U
n=1
λ1 (Tn ) n ≤ λ (T ) 2 n
1
(a,b] (Tn )
M/ N2 = N − N1 - r?2`2 i?2 b2[m2M+2 {Un }n≥1 Bb BB/ M/ mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]- ? p2 i?2 `2bT2+iBp2 Ft @BMi2MbBiB2b {λ1 (t)}t≥0 M/ {λ1 (t)}t≥0 X h?Bb b?Qrb i? i i?2 TQBMib + MMQi #2 b2T ` i2/ BM mMB[m2 K MM2`X :Bp2M N - i?2 T`BK2/ TQBMi T`Q+2bb2b /2T2M/ QMHv QM i?2 ` M/QK b2[m2M+2- M/ QM2 + M `2T2 i i?2 T`Q+2/m`2 iQ Q#i BM b2[m2M+2 Q7 BM/2T2M/2Mi U+QM/BiBQM HHv QM N V T`BK2/ TQBMi T`Q+2bb2bX 6Q` ;Bp2M TQBMi Tk Q7 N - QM2 + M i?2M +QmMi i?2 MmK#2` Q7 iBK2b i?Bb TQBMi ? b #22M bbB;M2/ iQ N1 BM n 2tT2`BK2Mib- M/ B7 i?Bb MmK#2` Bb ;`2 i2` i? M 12 n- mHiBK i2Hv bbB;M i?Bb TQBMi iQ N1 X
RyX8 _ M/QK a KTHBM; a KTHBM; i i?2 iBK2b {Tn }n∈Z Q7 TQBMi T`Q+2bb N Ui?2 b KTH2`V Q7 +QMiBMmQmb@ iBK2 biQ+? biB+ T`Q+2bb {X(t)}t∈R Ui?2 b KTH2/ T`Q+2bbV vB2H/b b2[m2M+2 Q7 ` M@ /QK b KTH2b Ui?2 b KTH2 b2[m2M+2V {X(Tn )}n∈Z X h?2 Tb2m/Q@T`Q+2bb Y (t) =
X(Tn )δ(t − Tn )
n∈Z
r?2`2 δ (t) Bb i?2 .B` + Tb2m/Q@7mM+iBQM- Bb + HH2/ i?2 b KTH2 +QK#X N
(a+?Q2M#2`; M/ "QHi- kyyy)X
RyX8X _L.PJ aJSGAL:
j3N X(t) X(Tn )
×
×
×
Tn
×
× ×
t
×
×
×
×
× ×
t
q2 MQr 7Q`KmH i2 i?2 MQiBQM Q7 ` M/QK b KTHBM; BM i?2 bT iB H + b2XRy aT2+i` H J2 bm`2 Q7 i?2 a KTH2 "`mb? >2`2 i?2 b KTH2/ T`Q+2bb Bb rbb biQ+? biB+ T`Q+2bb {X(t)}t∈Rm rBi? K2 M mX +Qp `B M+2 7mM+iBQM CX - TQr2` bT2+i` H K2 bm`2 μX M/ *` Kû`ĜE?BM+?BM bT2+i` H /2+QKTQbBiBQM ZX , e2iπν,t ZX (dν) + mX ,
X(t) = Rm
r?2`2 i?2 BMi2;` H i?2`2Q7 Bb .QQ# BMi2;` HXRR _2+ HH i? i 7Q` HH 7mM+iBQMb g ∈ L2C (μX )- i?2 .QQ# BMi2;` H Rm g(ν)ZX (dν) Bb r2HH@/2}M2/ 2H2K2Mi Q7 L2C (P )X JQ`2Qp2` & &2 & & E && g(ν)ZX (dν)&& = |g (ν)|2 μX (dν) . URyXR8V m m R
R
h?2 b KTH2` Bb bBKTH2 rbb TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ ∈ (0, ∞)rBi? TQBMi b2[m2M+2 {Vn }n≥1 X h?2 b KTH2/ T`Q+2bb M/ i?2 b KTH2` `2 bbmK2/ BM/2T2M/2Mi M/ rB/2@b2Mb2 bi iBQM `vX h?2 b KTH2 #`mb? Y (t) = X(Vn )δ(t − Vn ) URyXReV n≥1
Bb B/2MiB}2/ rBi? i?2 bB;M2/ K2 bm`2 i aX
n≥1
X(Vn )εVn - r?2`2 εa Bb i?2 .B` + K2 bm`2
.2}M2 i?2 bT2+i` H K2 bm`2 Q7 i?2 b KTH2 #`mb? iQ #2 HQ+ HHv }MBi2 K2 bm`2 μY bm+? i? i- 7Q` Mv ϕ ∈ BY ϕ (t) X(t)N (dt) = |ϕ (ν)|2 μY (dν) , URyXRdV Var Rm
Rm
r?2`2 BY Bb H `;2 2MQm;? p2+iQ` bT +2 Q7 7mM+iBQMb- ?2`2 HbQ + HH2/ i?2 i2bi 7mM+iBQMbX "v dzH `;2 2MQm;?Ǵ- r2 K2 M i? i i?2`2 + MMQi #2 irQ /Bz2`2Mi HQ+ HHv }MBi2 K2 bm`2b μY b iBb7vBM; URyXRdV 7Q` HH ϕ ∈ BY X P#b2`p2 i? i Ry RR
1 `Hv rQ`F BM i?2 + b2 Q7 iBK2 bB;M Hb, ("2miH2` M/ G2M2K M- RNee) M/ (J b`v- RNd3)X a22 7Q` BMbi M+2 ("`ûK m/- kyky)X
jNy
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
ϕ (t) Y (t) dt = Rm
ϕ (t) Rm
=
X (Vn ) δ (t − Vn ) dt
n≥1
ϕ (Vn ) X (Vn ) =
ϕ (t) X(t)N (dt) , Rm
n≥1
bQ i? i 2[m HBiv URyXRdV #2+QK2b- 7Q`K HHv
ϕ (t) Y (t) dt
Var Rm
= Rm
|ϕ (ν)|2 μY (dν) .
Uh?Bb 2tT`2bbBQM `2M/2`b i?2 M HQ;v rBi? i?2 +H bbB+ H "Q+?M2` bT2+i` H K2 bm`2 KQ`2 TT `2MiXV G2i N #2 rB/2@b2Mb2 bi iBQM `v bBKTH2 TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ ∈ (0, ∞)- " `iH2ii bT2+i`mK μN M/ i2bi 7mM+iBQM bT +2 BN X h?2Q`2K RyX8XR URk V amTTQb2 i? i i?2 biQ+? biB+ T`Q+2bb {X(t)}t∈Rm M/ i?2 TQBMi T`Q+2bb N `2 BM/2T2M/2MiX h?2M- i?2 b KTH2/ #`mb? URyXReV /KBib i?2 TQr2` bT2+i` H K2 bm`2 μY = μN ∗ μX + λ2 μX + |mX |2 μN .
URyXR3V
A7 BN Bb bi #H2 rBi? `2bT2+i iQ KmHiBTHB+ iBQMb #v +QKTH2t 2tTQM2MiB H 7mM+iBQMb- r2 + M i F2 7Q` i?2 i2bi 7mM+iBQM bT +2 BY = BN X _2K `F RyX8Xk h?Bb `2bmHi Bb iQ #2 +QKT `2/ rBi? i? i ;BpBM; i?2 bT2+i` H K2 @ bm`2 μY Q7 i?2 T`Q/m+i Q7 irQ BM/2T2M/2Mi rbb biQ+? biB+ T`Q+2bb2b- Y (t) = Z(t)X(t), μY = μZ ∗ μX X h?Bb `2bmHi Bb T `i Q7 i?2 7QHFHQ`2 Q7 i?2 2H2+i`B+ H 2M;B@ M22`BM; HBi2` im`2- r?2`2 Bi Bb T`Qp2/ i i?2 2tT2Mb2 Q7 7Q`K H K MBTmH iBQMbX S`QQ7X 7Q`K H 2t+? M;2 Q7 i?2 Q`/2` Q7 BMi2;` iBQM ;Bp2b ϕ (t) X(t)N (dt) = ϕ (t) e2iπν,t ZX (dν) + mX N (dt) m Rm Rm R 2iπν,t = ϕ (t) e N (dt) ZX (dν) Rm Rm ϕ (t) N (dt) . + mX Rm
aBM+2 i?2 BMi2;` Hb rBi? `2bT2+i iQ N M/ ZX `2 Q7 /Bz2`2Mi M im`2 UQM2 Bb mbm H BM}MBi2 bmK- i?2 Qi?2` Bb qB2M2` BMi2;` HV- i?Bb 2t+? M;2 Kmbi #2 7Q`K HHv DmbiB}2/- r?B+? Bb /QM2 i i?2 +HQb2 Q7 i?2 T`QQ7X "v i?2 +QM/BiBQM H p `B M+2 7Q`KmH - r2 ? p2- /2MQiBM; #v F N i?2 σ@}2H/ ;2M2` i2/ #v i?2 TQBMi T`Q+2bb N Rk h?2 `2bmHi Bb bi i2/ BM 1t KTH2 RRX j U/V Q7 (. H2v M/ o2`2@CQM2b- RN33)X a22 HbQ ("`ûK m/J bbQmHBû M/ _B/QH}- kyy8)X
RyX8X _L.PJ aJSGAL:
jNR
Var ϕ (t) X(t)N (dt) Rm &
& = E Var ϕ (t) e2iπν,t N (dt) ZX (dν) + mX ϕ (t) N (dt)&& F N m m m & R R
R & 2iπν,t + Var E ϕ (t) e N (dt) ZX (dν) + mX ϕ (t) N (dt)&& F N . Rm
Rm
Rm
.2MQi2 #v α + β i?2 `B;?i@? M/ bB/2 Q7 i?Bb 2[m HBivX P#b2`p2 i? i- bBM+2 ϕ ∈ L2 (M2 )& &2 & &2 & & & & 2iπν,t & & ≤& & < ∞ , P @ XbX ϕ (t) e N (dt) |ϕ (t)| N (dt) URyXRNV & m & & m & R R lbBM; i?2 7 +i i? i- r?2M N Bb }t2/- mX Rm ϕ (t) N (dt) Bb /2i2`KBMBbiB+&
& N 2iπν,t & ϕ (t) e N (dt) ZX (dν)& F α = E Var m Rm & R &2 & & 2iπν,t & & =E N (dt)& μX (dν) U#v 1[MbX URyXR8V M/ URyXRNVV & m ϕ (t) e Rm R & &2 & & 2iπν,t & = E & ϕ (t) e N (dt)&& μX (dν) m m R R &
&2 & & 2iπν,t 2iπν,t & = ϕ (t) e N (dt) + &E ϕ (t) e N (dt) && μX (dν) Var m m m R
R
R
& & |ϕ (x − ν)| μN (dx) + && m 2
= Rm
R
ϕ (t) e Rm
2iπν,t
&2 & λdt&& μX (dν)
(?vTQi?2bBb QM BN ) |ϕ (x − ν)|2 μN (dx) μX (dν) + λ2 |ϕ (−ν)|2 μX (dν) = Rm Rm Rm 2 2 = |ϕ (x + ν)| μN (dx) μX (dν) + λ |ϕ (+ν)|2 μX (dν) m m m R R R 2 2 2 = |ϕ (ν)| (μN ∗ μX ) (dν) + λ |ϕ (ν)| μX (dν) .
aBM+2 E
Rm
Rm
Rm
β = Var
mX Rm
6BM HHv-
Rm
ϕ (t) Y (t) dt
Var Rm
Rm
& N = 0N (dt) ZX (dν)& F∞ ϕ (t) N (dt) = |mX |2 |ϕ (ν)|2 μN (dν) (#2+ mb2 ϕ ∈ BN ) .
ϕ (t) e
2iπν,t
= Rm
|ϕ (ν)|2 μN ∗ μX + λ2 μX + |mX |2 μN (dν) ,
b?QrBM; i? i {Y (t)}t∈Rm /KBib M 2ti2M/2/ "Q+?M2` bT2+i` H K2 bm`2 ;Bp2M #v URyXR3VX
jNk
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
G2KK RyX8Xj G2i N #2 bBKTH2 HQ+ HHv #QmM/2/ rbb TQBMi T`Q+2bb /2}M2/ QM Rm M/ /KBiiBM; " `iH2ii bT2+i` H K2 bm`2 μN X G2i M2 #2 Bib b2+QM/ KQK2Mi K2 bm`2X G2i {X(t)}t∈Rm #2 rbb ` M/QK }2H/ rBi? *` Kû`ĜE?BM+?BM /2+QKTQ@ bBiBQM ZX M/ TQr2` bT2+i` H K2 bm`2 μX X h?2M- 7Q` HH ϕ ∈ L2 (M2 ) ϕ (t) X(t)N (dt) = ϕ (t) e2iπ N (dt) ZX (dν) . URyXkyV Rm
Rm
Rm
S`QQ7X h?2 T`QQ7 Bb /QM2 BM i?2 mMBp `B i2 + b2X h?2 KmHiBp `B i2 + b2 7QHHQrb 2t +iHv i?2 b K2 HBM2bX h?2 H27i@? M/ bB/2 Q7 URyXkyV Bb A= ϕ (Tn ) X(Tn ) = lim ϕ (Tn ) X(Tn )1[−c,+c] (Tn ) = lim A (c) c↑∞
n∈Z
c↑∞
n∈Z
r?2`2 i?2 HBKBi Bb BM L1C (P )X AM/22/
E [|A − A (c)|] ≤ E |ϕ (t) X(t)| N (dt) [−c,+c] |ϕ (t)| E [|X(t)|] λdt ≤ λK = [−c,+c]
|ϕ (t)| dt
[−c,+c]
r?2`2 K = supt E [|X(t)|] < ∞ U#v a+?r `xǶb BM2[m HBiv- r2 ? p2 E [|X(t)|] ≤ 1 1 E |X(t)|2 2 = E |X (0)|2 2 VX h?2`27Q`2- bBM+2 ϕ ∈ L1C (R)- limc↑∞ E [|A − A (c)|] = 0X h?2 `B;?i@? M/ bB/2 Bb B = lim ϕ (t) e2iπ N (dt) ZX (dν) = lim B (c) , c↑∞
R
c↑∞
[−c,+c]
r?2`2 i?2 HBKBi Bb BM L2C (P )X AM/22/ & & 2 E |B − B (c)| = E && R
& & & =E & R
ϕ (t) e
[−c,+c]
& & = E E && R
&2 & N (dt) ZX (dν)&&
2iπνt
&2 && && N & ϕ (t) e N (dt) ZX (dν)& & F∞ & [−c,+c] &2 & ϕ (t) e2iπνt N (dt)&& μX (dν) .
2iπνt
[−c,+c]
G2i ϕc (t) := ϕ (t) 1[−c,c] (t)X h?2M & & &2 &2 & & & & 2iπνt 2iπνt & & & E N (dt)& μX (dν) = E & ϕc (t) e N (dt)&& μX (dν) . & ϕc (t) e R
"mi
R
R
R
& &2 2 & & 2iπνt E && ϕc (t) e N (dt)&& ≤ E |ϕc (t)| N (dt) R R = |ϕc (t)| |ϕc (s)| M2 (dt × ds) , R×R
RyX8X _L.PJ aJSGAL:
jNj
[m MiBiv i? i i2M/b iQ 0 b c ↑ ∞- #v /QKBM i2/ +QMp2`;2M+2X .QKBM i2/ +QM@ p2`;2M+2 TTHB2/ iQ i?2 }MBi2 K2 bm`2 μX i?2M vB2H/b i?2 /2bB`2/ +QMp2`;2M+2 BM L2C (P )X "mi ϕ (Tn ) X(Tn )1[−c,+c] A (c) = n∈Z
=
ϕ (Tn )
R
e R
n∈Z
=
2iπνTn
ϕ (Tn ) e
ZX (dν) 1[−c,+c] (Tn )
2iπνTn
1[−c,+c] (Tn ) ZX (dν) = B (c) ,
n∈Z
r?2`2 i?2 7 +i i? i i?2 bmKb BMpQHp2/ `2 }MBi2 ? b #22M i F2M BMiQ ++QmMiX h?2`27Q`2 A BM L1C (P ) lim A (c) = c↑∞ B BM L2C (P ) , 7`QK r?B+? Bi 7QHHQrb i? i A = B- XbX U`2+ HH i? i B7 b2[m2M+2 Q7 ` M/QK p `B@ #H2b +QMp2`;2b BM L1C (P ) Q` L2C (P ) iQ bQK2 ` M/QK p `B #H2- QM2 + M 2ti` +i bm#b2[m2M+2 i? i +QMp2`;2b HKQbi bm`2Hv iQ i?2 b K2 ` M/QK p `B #H2VX 1t KTH2 RyX8X9, *Qt a KTHBM;X G2i N #2 *Qt T`Q+2bb rBi? rbb BMi2MbBiv T`Q+2bb {λ (t)}t∈Rm rBi? bT2+i` H K2 bm`2 μλ X h?2M- `2+ HHBM; 7`QK 1t KTH2 NXkXRj i?2 2tT`2bbBQM μN (dν) = μλ (dν) + λdν- r2 Q#i BM 2 μY = μλ ∗ μX + λ2 μX + |mX |2 μλ + λ σX + |mX |2 m r?2`2 m Bb i?2 G2#2b;m2 K2 bm`2X UAM Q`/2` iQ Q#i BM i?Bb 2tT`2bbBQM- Bi r b MQi2/ i? i μX ∗ m = μX X AM 7 +i- 7Q` HH C ∈ B(Rm )
m (C − ν) μX (dν) (μX ∗ m )(C) = m R 2 =
m (C) μX (dν) = m (C)μX (Rm ) = m (C)σX .) Rm
HbQ 7`QK 1t KTH2 NXkXRj M/ h?2Q`2K RyX8XR- BN = L1C (Rm ) ∩ L2C (Rm ) = BY X AM i?2 T `iB+mH ` + b2 r?2`2 i?2 b KTH2` N Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb M/ i?2 b KTH2/ T`Q+2bb Bb +2Mi2`2/2 m μY = λ2 μX + λσX
.
URyXkRV
1t KTH2 RyX8X8, _2;mH ` a KTHBM;- h F2 RX _2;mH ` b KTHBM; `272`b iQ i?2 + b2 r?2`2 i?2 b KTH2` Bb ;`B/ UBM i?Bb 2t KTH2, QM i?2 `2 H HBM2V rBi? TQBMib b2T ` i2/ #v i?2 /Bbi M+2 T X amTTQb2 7Q` bBKTHB+Biv
i? i i?2 b KTH2/ T`Q+2bb Bb +2Mi2`2/X TTHvBM; 6Q`KmH URyXR3V rBi? μN = T1 n∈Z\{0} ε Tn - M/ Q#b2`pBM; i? i
jN9
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
μX ∗ ε Tn (C) = μX (C − Tn )- QM2 }M/b BM i?2 + b2 r?2`2 i?2`2 2tBbib TQr2` bT2+i` H /2MbBiv fX 7Q` i?2 b KTH2/ T`Q+2bb i?2 7QHHQrBM; 2tT`2bbBQM 7Q` i?2 bT2+i` H /2MbBiv Q7 i?2 b KTH2 +QK# 2 1 n . fX ν − fY (ν) = T T n∈Z h?2`27Q`2- i?2 bT2+i` H /2MbBiv Q7 i?2 b KTH2/ T`Q+2bb + M #2 2MiB`2Hv `2+Qp2`2/ 7`QK i? i Q7 i?2 b KTH2 +QK# T`QpB/2/ i?2 7Q`K2` Bb # M/@HBKBi2/- rBi? # M/ rB/i? 2B < T1 X
1t KTH2 RyX8Xe, SQBbbQM b KTHBM;- AX SQBbbQM b KTHBM; `272`b iQ i?2 + b2 r?2`2 i?2 b KTH2` Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ ∈ (0, ∞)X lM/2` i?2 b K2 +QM/BiBQMb 7Q` i?2 b KTH2/ T`Q+2bb b BM i?2 T`2pBQmb 2t KTH2i?2 bT2+i` H K2 bm`2 Q7 i?2 b KTH2 +QK# /KBib i?2 /2MbBiv 2 fY (ν) = λ2 fX (ν) + λσX .
h?2`27Q`2- r? i2p2` i?2 b KTHBM; 7`2[m2M+v νs = λ- i?2`2 Bb MQ HB bBM;- M/ i?2 bT2+i`mK Q7 i?2 b KTH2/ T`Q+2bb + M #2 `2+Qp2`2/ 7`QK i? i Q7 i?2 b KTH2 +QK#X
_2+QMbi`m+iBQM 7`QK a KTH2b AM i?2 H bi irQ 2t KTH2b- r2 r2`2 BMi2`2bi2/ BM i?2 `2+Qp2`v Q7 i?2 bT2+i` H K2 @ bm`2 Q7 i?2 b KTH2/ T`Q+2bb 7`QK i? i Q7 i?2 b KTH2 +QK#X q2 MQr +QMbB/2` i?2 T`Q#H2K Q7 `2+QMbi`m+iBM; i?2 T`Q+2bb Bib2H7 7`QK Bib b KTH2bX JQ`2 KQ/2biHv- r2 TT`QtBK i2 i?2 b KTH2/ T`Q+2bb #v }Hi2`2/ p2`bBQM Q7 i?2 b KTH2 +QK#, ϕ (t − s) Y (s) ds , Rm
r?2`2 ϕ ∈ L1C Rm ) ∩ L2C Rm )- M/ r2 HbQ bbmK2 i? i BN = L1C Rm ) ∩ L2C Rm )X h?2 /Bz2`2M+2 #2ir22M X (t) M/ Bib TT`QtBK iBQM- i? i Bb- i?2 `2+QMbi`m+iBQM 2``Q`Bb K2 bm`2/ #v & &2 & & ε := E && ϕ (t − u) Y (u) du − X (t)&& . m R
h?2 7QHHQrBM; `2bmHi Bb bi i2/ BM i?2 mMB/BK2MbBQM H + b2 7Q` 2 b2 Q7 MQi iBQMX h?2Q`2K RyX8Xd h?2 `2+QMbi`m+iBQM 2``Q` Bb- r?2M i?2 b KTH2/ T`Q+2bb Bb +2M@ i2`2/, 2 |λϕ (ν) − 1| μX (dν) + λ |ϕ (ν)|2 (μX ∗ μλ ) (dν) . URyXkkV ε= Rm
Rm
RyX8X _L.PJ aJSGAL:
jN8
S`QQ7X a22 1t2`+Bb2 RyXeXeX
q2 MQr ;Bp2 i?2 `2+QMbi`m+iBQM 2``Q`b 7Q` /Bz2`2Mi b KTHBM; b+?2K2bX q2 /2@ p2HQT i?2 +QKTmi iBQMb 7Q` ;2M2` HBx2/ dz# M/@HBKBi2/Ǵ b KTH2/ T`Q+2bb- }Hi2`2/ rBi? dz# b2@# M/Ǵ }Hi2` Q7 i` MbKBii M+2 ϕX JQ`2 T`2+Bb2Hv- /2MQiBM; #v S i?2 bmTTQ`i U bbmK2/ Q7 G2#2b;m2 K2 bm`2 2B < ∞V Q7 i?2 bT2+i` H K2 bm`2 μX ϕ (ν) =
1 1S (ν) , λ
r?2`2 λ Bb i?2 BMi2MbBiv Q7 i?2 b KTH2 +QK#X 1t KTH2 RyX8X3, SQBbbQM a KTHBM;- h F2 kX h?2 bBim iBQM Bb i? i Q7 1t KTH2 RyX8XeX h?2 bT2+i` H K2 bm`2 Q7 i?2 b KTH2 +QK# Bb i?2M ;Bp2M #v URyXkRVX h?2 `2+QMbi`m+iBQM 2``Q` Bb 2 ε= |λϕ (ν) − 1|2 μX (dν) + λσX |ϕ (ν)|2 (dν) . URyXkjV Rm
Rm
AM i?2 dz+H bbB+ HǴ # M/@HBKBi2/ + b2 /2b+`B#2/ #Qp2- r2 ? p2 1 2 2 |ϕ (ν)|2 (dν) = λσX 1 (ν) dν, ε = λσX 2 S R R λ i? i Bb
2B · λ h?2`27Q`2- b KTHBM; i i?2 dzLv[mBbi ` i2Ǵ λ = 2B ;Bp2b p2`v TQQ` T2`7Q`K M+2MQi #2ii2` i? M i?2 2biBK i2 # b2/ QM MQ Q#b2`p iBQM i HHX 2 ε = σX
h?Bb /Q2b MQi K2 M- ?Qr2p2`- i? i #2HQr i?2 ` i2 λ = 2B- i?2`2 Bb MQ BM7Q`K iBQM UQ` BM b2Mb2 b i?2 `2bmHi bm;;2bib dzM2; iBp2 BM7Q`K iBQMǴV +QM+2`MBM; i?2 T`Q+2bb Bib2H7 +QMi BM2/ BM Bib b KTH2bX #2ii2` +?QB+2 Q7 }Hi2` rQmH/ BM/22/ ;Bp2 HBM2 ` 2 2biBK i2 rBi? 2``Q` bK HH2` i? M σX X 6Q` BMbi M+2- B7 r2 H2i ϕ #2 `2 H- r2 }M/ 7Q` i?2 `2+QMbi`m+iBQM 2``Q` 2 ε= (λϕ (ν) − 1)2 fX (ν) + λσX ϕ (ν)2 dν, R
r?2`2 Bi Bb bbmK2/ i? i i?2 b KTH2/ T`Q+2bb ? b i?2 TQr2` bT2+i` H /2MbBiv fX X KBMBKmK Q++m`b 7Q` λfX (ν) ϕ (ν) = 2 2 λ fX (ν) + λσX M/ i?2M
ε=
r?2`2
2 σX
1−
R
λfX (ν) fX (ν) dν 1 + λfX (ν)
fX (ν) fX (ν) fX (ν) := = 2 ) dν σX f (ν X R
,
jNe
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
Bb i?2 MQ`K HBx2/ TQr2` bT2+i` H /2MbBivX h?Bb QTiBK H }Hi2` `2[mB`2b i?2 FMQrH2/;2 Q7 i?2 bT2+i` H /2MbBiv Q7 i?2 b KTH2/ T`Q+2bb- r?B+? Bb BM T`BM+BTH2 p BH #H2 7`QK i?2 b KTH2b- b b?QrM BM 1t KTH2 RyX8XeX 1t KTH2 RyX8XN, _2;mH ` a KTHBM;- h F2 kX q?2M i?2 b KTH2/ +QK# Bb Q#i BM2/ #v `2;mH ` b KTHBM; b BM 1t KTH2 RyX8X8- i?2 `2+QMbi`m+iBQM 2``Q` Bb &2 & & &1 & ϕ (ν) − 1& μX (dν) + 1 ε= |ϕ (ν) − 1|2 dν URyXk9V & & T R R T Ur?2`2 r2 iQQF BMiQ ++QmMi i?2 7 +i i? i ∗ μX = μX (R) VX AM i?2 # M/@HBKBi2/ + b2- B7 r2 +QMbB/2` T = 1/2B- i? i Bb- λ = 2B- 2[m iBQM URyXk9V ;Bp2b M 2``Q` 2[m H iQ x2`QX h?2`27Q`2- i?2 T`Q+2bb Bb T2`72+iHv `2+QMbi`m+i2/ #v X (t) = ϕ (t − s) X (s) N (ds) = X (Tn ) sinc (2B(t − Tn )) , R
n∈Z
r?B+? Bb i?2 mbm H `2+QMbi`m+iBQM 7Q`KmH FMQrM b i?2 a? MMQMĜLv[mBbi 7Q`KmH X 1t KTH2 RyX8XRy, CBii2` BM _2;mH ` Lv[mBbi a KTHBM;X h?2 `2+QMbi`m+@ iBQM 2``Q` 7`QK mMB7Q`K b KTH2b BM i?2 T`2b2M+2 Q7 DBii2` Bb Q#i BM2/ #v THm;;BM; μY ;Bp2M #v UNX99V BMiQ i?2 2``Q` 7Q`KmH URyXkkVX h?2 T`2pBQmb 2t KTH2 b?Qr2/ i? i rBi?BM i?2 dz+H bbB+ HǴ b KTHBM; 7` K2rQ`F i?2 T`Q+2bb K v #2 T2`72+iHv `2+QM@ bi`m+i2/X LQr- BM i?2 T`2b2M+2 Q7 DBii2`Rj i?Bb Bb MQi TQbbB#H2 M/ i?2 `2+QMbi`m+iBQM 2``Q` Bb ;Bp2M #v B 1 2 ε= σX 1 − |ψZ |2 ∗ fX (ν) dν , URyXk8V 2B −B r?2`2 fX Bb i?2 MQ`K HBx2/ TQr2` bT2+i` H /2MbBiv Q7 i?2 T`Q+2bb X (t)X
RyXe 1t2`+Bb2b 1t2`+Bb2 RyXeXRX .2i BHb 7Q` G2KK RyXRXj :Bp2 i?2 /2i BHb Q7 i?2 T`QQ7 Q7 2tBbi2M+2 M/ mMB[m2M2bb Q7 {f (t)}t≥0 b iBb7vBM; U V Q7 G2KK RyXRXjX 1t2`+Bb2 RyXeXkX *? ` +i2`Bx iBQM Q7 Z(t) h?2 MQi iBQM M/ bbmKTiBQMb `2 i?Qb2 Q7 G2KK RyXRXjX a?Qr i? i- 7Q` Z(t) iQ #2 i?2 +QM/BiBQM H 2tT2+i iBQM Q7 Z(t) ;Bp2M Ot - Bi Bb M2+2bb `v M/ bm{+B2Mi i? i E [Z(t)U (t)] = E Z(t)U (t) () Rj
1 `Hv rQ`F , (" H F`Bb?M M- RNek) M/ ("`QrM- RNej)X
RyXeX 1s1_*Aa1a
jNd
7Q` HH #QmM/2/ Ot @K `iBM; H2b {U (t)}t≥0 X 1t2`+Bb2 RyXeXjX "B`i?@ M/@/2 i? S`Q+2bb 1biBK i2/ 7`QK i?2 .2 i? S`Q+2bb G2i {X(t)}t≥0 #2 #B`i?@ M/@/2 i? T`Q+2bb- i? i Bb- M ?K+ rBi? bi i2 bT +2 E = N- M/ r?Qb2 MQM@MmHH 2Mi`B2b Q7 i?2 BM}MBi2bBK H ;2M2` iQ` `2 qn,n+1 = λn M/ qn,n−1 = μn 1{n≥1} . h?2 Q#b2`p iBQM Bb i?2 TQBMi T`Q+2bb N i? i +QmMib i?2 /QrMr `/ i` MbBiBQMb Ui?2 iBK2b t bm+? i? i X(t) = X(t−) − 1VX .2`Bp2 BM /2i BH i?2 }Hi2`BM; 2[m iBQMb
t
Zn (t) = P (X(0) = n) +
(Zn−1 (s)1{n≥0} λn−1 0
− Zn (s)(λn + μn 1{n≥1} ) + Zn+1 (s)μn+1 ) ds N (ds) − ds . μj Zj (s−)) × μn+1 Zn+1 (s−) − Zn (s−)( + (0,t] j≥1 μj Zj (s−) j≥1
1t2`+Bb2 RyXeX9X SQBMi S`Q+2bb JQ/mH i2/ #v h2H2;` T? aB;M H G2i {θ(t)}t≥0 #2 +QMiBMmQmb@iBK2 ?K+ rBi? bi i2 bT +2 E = {0, 1} M/ BM}MBi2b@ BK H +? ` +i2`BbiB+b n(1 − p) q0,1 = n , , q1,0 = p r?2`2 p ∈ (0, 1)X h?Bb ?K+ Bb bbmK2/ bi iBQM `vX G2i N #2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ftθ ∨ FtN @BMi2MbBiv λ(t) := cθ(t)X a?Qr i? i i?2 2biBK i2 θ(t) := E θ(t) | FtN 2pQHp2b ++Q`/BM; iQ θ(t) = p +
n p
t
(p − θ(s)) ds 0
+ (0,t]
cθ(s−)(1 − θ(s−)) 1 + cθ(s−)
(N (ds) − (λ + cθ(s) ds) .
1t2`+Bb2 RyXeX8X *Qt S`Q+2bb S`Qp2 i?2 bi i2K2Mi Q7 1t KTH2 RyXRXRRX 1t2`+Bb2 RyXeXeX _2+QMbi`m+iBQM 1``Q` S`Qp2 h?2Q`2K RyX8XdX 1t2`+Bb2 RyXeXdX >B;? _ i2 SQBbbQM a KTHBM; AM 1t KTH2 RyX8X3 b?Qr i? i i?2 2``Q` ε i2M/b iQ 0 b i?2 b KTHBM; ` i2 λ i2M/b iQ BM}MBivX
jN3
*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a
1t2`+Bb2 RyXeX3X S`[email protected]/2Mi a KTHBM; *QMbB/2` i?2 ;2M2` H bBim iBQM Q7 ` M/QK b KTHBM;- b BM am#b2+iBQM RyX8X q2 bmTTQb2 MQr i? i i?2 b KTHBM; ` i2 /2T2M/b QM i?2 T`Q+2bbX JQ`2 T`2+Bb2Hv- i?2 KQ/2H 7Q` i?2 b KTH2` Bb MQr *Qt T`Q+2bb QM Rd rBi? i?2 +QM/BiBQM H BMi2MbBiv Q7 i?2 7Q`K λ (t) = λ (t, X) , r?2`2 X `2T`2b2Mib bi iBQM `v biQ+? biB+ T`Q+2bb {X(t)}t∈Rd X U6Q` BMbi M+2- BM & &2 & & i?2 mMBp `B i2 + b2- λ (t) = |X(t)|2 - λ (t) = &X˙ (t)& - r?2`2 X˙ Bb i?2 /2`Bp iBp2 i t Q7 t → X(t)X JQ`2 +QKTH2t 7mM+iBQM Hb + M #2 +QMbB/2`2/XV bbmK2 i? i E X(t)2 λ (t, X)2 < ∞ 7Q` HH t ∈ Rm - M/ i? i {λ (t)}t∈Rm Bb HQ+ HHv BMi2;` #H2 T`Q+2bbX G2i μZ #2 i?2 TQr2` bT2+i`mK Q7 i?2 bi iBQM `v T`Q+2bb Z (t) = X(t)λ (t) bbmK2/ bi iBQM `v U b Bb i?2 + b2 BM i?2 2t KTH2b #Qp2VX a?Qr i? i μY (dν) = μZ (dν) + X 2 λdν , r?2`2 X 2 λ := E [X(t)2 λ (t)] UBM/2T2M/2Mi Q7 tVX
*? Ti2` RR SQBMi S`Q+2bb2b BM Zm2m2BM; Zm2m2BM; i?2Q`v Bb Q7 BMi2`2bi 7Q` i?2 T2`7Q`K M+2 2p Hm iBQM Q7 b2`pB+2 bvbi2Kb 72 im`BM; i?2 irQ `2H i2/ T?2MQK2M Q7 +QM;2biBQM M/ /2H v- r?B+? +QmH/ #2 MmBb M+2X h?Bb Bb i?2 + b2 7Q` +QKKmMB+ iBQMb M/ +QKTmi2` bvbi2Kb- BM //BiBQM iQ i?2 +H bbB+ H bvbi2Kb Q7 QT2` iBQMb `2b2 `+? 72 im`BM; iB+F2i #QQi?b- 7`22r v iQHHb M/ i?2 HBF2X 7i2` `2+ HHBM; i?2 # bB+ i?2Q`v Q7 [m2m2b- K BMHv J `FQpB M- i?Bb +? Ti2` rBHH +QM+2Mi` i2 QM i?2 +QMi`B#miBQM Q7 TQBMi T`Q+2bb i?2Q`v- K BMHv i?2 biQ+? biB+ BMi2MbBiv i?2Q`v M/ i?2 S HK i?2Q`v QM i?2 HBM2- iQ i?2 7mM/ K2Mi H `2bmHib bm+? b GBiiH2Ƕb 7Q`KmH M/ i?2 T bi T`QT2`iv M/ i?2B` TTHB+ iBQMbX
RRXR
_2pB2r Q7 J `FQpB M Zm2m2BM; h?2Q`v
h?Bb b2+iBQM Bb bmKK `v Q7 i?2 dz2H2K2Mi `v i?2Q`vǴ Q7 [m2m2b- i? i Bb- i?2 iQTB+b i? i + M #2 i`2 i2/ mbBM; i?2 2H2K2Mi `v i?2Q`v Q7 J `FQp +? BMbX AbQH i2/ J `FQpB M Zm2m2b >2`2 Bb b KTH2 Q7 i?2 bBKTH2bi [m2m2BM; KQ/2HbX 1t KTH2 RRXRXR, JfJfRf∞f7B7QX *QMbB/2` iB+F2i #QQi? rBi? bBM;H2 ii2M@ / Mi- Q` b2`p2`X *mbiQK2`b r Bi BM HBM2 BM 7`QMi Q7 i?2 #QQi?- M/ i?2 7 +BHBiv Bb bQ H `;2 i? i MQ #QmM/ Bb BKTQb2/ QM i?2 MmK#2` Q7 +mbiQK2`b r BiBM; 7Q` b2`pB+2X AM Qi?2` rQ`/b- i?2 r BiBM; `QQK ? b BM}MBi2 + T +BivX am+? bvbi2K rBHH #2 + HH2/ 1/∞ b2`pB+2 bvbi2K- r?2`2 1 Bb 7Q` i?2 MmK#2` Q7 b2`p2`b- M/ ∞ Bb 7Q` i?2 + T +Biv Q7 i?2 r BiBM; `QQKX *mbiQK2` ``Bp Hb `2 KQ/2HH2/ #v ?QKQ;2M2Qmb SQBbbQM T`Q+2bb {Tn }n≥1 QM i?2 TQbBiBp2 ? H7@HBM2- Q7 TQbBiBp2 BMi2MbBiv λX *mbiQK2` n ``BpBM; i iBK2 Tn #`BM;b b2`pB+2 `2[m2bi σn - r?B+? K2 Mb i? i i?2 b2`p2` rBHH i F2 σn mMBib Q7 iBK2 7Q` T`Q@ +2bbBM; i?Bb `2[m2biX h?2 b2`pB+2 b2[m2M+2 {σn }n≥1 Bb bbmK2/ BB/- rBi? 2tTQM2MiB H /Bbi`B#miBQM Q7 K2 M μ−1 X HbQ- i?2 ``Bp H b2[m2M+2 M/ i?2 b2`pB+2 b2[m2M+2 `2 bbmK2/ iQ #2 BM/2T2M/2MiX am+? T ii2`M Q7 ``Bp Hb Bb + HH2/ M JfJ BMTmi T`Q+2bbX AM i?Bb MQi iBQM- BMi`Q/m+2/ #v . pB/ E2M/ HH-R J bi M/b 7Q` dzJ `FQ@ R
(E2M/ HH- RN8j)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_11
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*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
pB MǴX UAM/22/- i?2 SQBbbQM T`Q+2bb Bb J `FQpB M- M/ 2tTQM2MiB H /Bbi`B#miBQMb `2 BMiBK i2Hv +QMM2+i2/ rBi? i?2 J `FQp T`QT2`ivXV h?2 b2`p2` ii2M/b QM2 +mbiQK2` i iBK2 M/ /Q2b MQi `2K BM B/H2 b HQM; b i?2`2 Bb i H2 bi QM2 +mbiQK2` BM i?2 bvbi2K UiB+F2i #QQi? THmb r BiBM; `QQKVX PM+2 i?2 b2`pB+2 Q7 +mbiQK2` Bb bi `i2/ Bi + MMQi #2 BMi2``mTi2/ #27Q`2 +QKTH2iBQMX h?2 #Qp2 bvbi2K Bb + HH2/ M JfJfRf∞ [m2m2X Aib /2b+`BTiBQM +QmH/ #2 +QK@ TH2K2Mi2/ #v M BM/B+ iBQM Q7 i?2 b2`pB+2 /Bb+BTHBM2, 7Q` BMbi M+2- 7B7Q U}`bi@BM@ }`bi@QmiV- r?2`2 i?2 b2`p2`- 7i2` +QKTH2iBQM Q7 b2`pB+2- +?QQb2b ?Bb M2ti +mbiQK2` i i?2 ?2 / Q7 i?2 HBM2X h?2 biQ+? biB+ T`Q+2bb {X(t)}t≥0 - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t- Bb + HH2/ i?2 +QM;2biBQM T`Q+2bbX Ai Bb +QMiBMmQmb@ iBK2 ?K+ rBi? bi i2 bT +2 E = N M/ BM}MBi2bBK H ;2M2` iQ` Ub22 i?2 `2K `F #2HQrV qi,i+1 = λ , qi,i−1 = μ1{i≥1} . Ub mbm H- QMHv i?2 Qz@/B ;QM H 2Mi`B2b Q7 i?2 BM}MBi2bBK H ;2M2` iQ` K i`Bt `2 ;Bp2M M/ i?2 MmHH QM2b `2 QKBii2/XV
_2K `F RRXRXk h?2 7Q`K H T`QQ7b Q7 J `FQpB MBiv M/ Q7 i?2 7Q`KmH b ;BpBM; i?2 BM}MBi2bBK H ;2M2` iQ`b Q7 i?2 [m2m2BM; T`Q+2bb2b M/ Q7 i?2B` M2irQ`Fb rBHH #2 ;Bp2M BM a2+iBQM RRXkX
1t KTH2 RRXRXj, JfJfEf∞f7B7QX h?Bb Bb i?2 JfJfRf∞f7B7Q [m2m2- 2t+2Ti i? i i?2`2 `2 MQr K b2`p2`bX h?2 +QM;2biBQM T`Q+2bb Bb +QMiBMmQmb@iBK2 ?K+ rBi? bi i2 bT +2 E = N M/ BM}MBi2bBK H ;2M2` iQ` qi,i+1 = λ , qi,i−1 = inf(i, K)μ .
1t KTH2 RRXRX9, JfJfEfy- Q` 1`H M; [m2m2Xk h?Bb [m2m2 Bb + HH2/ i?2 1`H M; [m2m2 BM ?QMQ` Q7 i?2 . MBb? i2H2T?QM2 2M;BM22`- r?Q mb2/ Bi b KQ/2H 7Q` i2H2T?QM2 brBi+? rBi? K HBM2b- Q` +? MM2Hb, +mbiQK2` }M/BM; 7`22 HBM2 Bb +QMM2+i2/ M/ ?QH/b i?2 HBM2 7Q` i?2 iBK2 Q7 +QMp2`b iBQMX +mbiQK2` 2Mi2`BM; i?2 bvbi2K Ui? i Bb- ``BpBM; M/ b22BM; QM2 Q` KQ`2 B/H2 +? MM2HbV rBHH #2 bbB;M2/ 7`22 +? MM2H i ` M/QKX +mbiQK2` }M/BM; HH +? MM2Hb #mbv Bb `2D2+i2/ M/ HQbi UQ`- T2`? Tb- `Qmi2/ iQ MQi?2` brBi+? +2Mi2`VX h?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i Mv ;Bp2M iBK2 Bb H2bb i? M Q` 2[m H iQ KX Ai Bb [m2m2 rBi? #HQ+FBM;X 1`H M; r b #H2 iQ Q#i BM ?Bb 7 KQmb #HQ+FBM; 7Q`KmH ;BpBM; i?2 T`Q# #BHBiv i? i- BM bi iBQM `v `2;BK2- ;Bp2M +mbiQK2` }M/b HH HBM2b #mbvX h?Bb Bb i?2 }`bi 7Q`KmH Q7 [m2m2BM; i?2Q`v Ub22 1t KTH2 RRXRXN #2HQrVX k
(1`H M;- RNRd)X
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
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>2`2- X(t) Bb i?2 MmK#2` Q7 #mbv HBM2b U M/ i?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2KV i iBK2 tX h?2 +QM;2biBQM T`Q+2bb Bb +QMiBMmQmb@iBK2 ?K+ rBi? bi i2 bT +2 E = {0, 1, . . . , K} M/ BM}MBi2bBK H ;2M2` iQ` A ;Bp2M #v qi,i+1 = λ1{0≤i≤K−1} , qi,i−1 = iμ1{1≤i≤K} .
1t KTH2 RRXRX8, JfJfRf∞fHB7Q T`22KTiBp2 `2bmK2X h?Bb [m2m2BM; bvb@ i2K `2+2Bp2b /2b+`BTiBQM bBKBH ` iQ i? i Q7 JfJfRf∞f7B7Q- 2t+2Ti 7Q` i?2 b2`pB+2 /Bb+BTHBM2 r?B+? MQr #2+QK2b HB7Q T`22KTiBp2 `2bmK2 Ui?2 ##`2pB iBQM HB7Q bi M/b 7Q` dzG bi AM 6B`bi PmiǴVX +mbiQK2` mTQM ``Bp H ;Q2b `B;?i r v iQ i?2 iB+F2i #QQi?- i?2 +mbiQK2` T`2b2MiHv `2+2BpBM; b2`pB+2 #2BM; b2Mi # +F iQ i?2 r BiBM; `QQK- r?2`2 ?2 rBHH bi M/ BM 7`QMi Q7 i?2 HBM2 U i H2 bi mMiBH i?2 iBK2 r?2M MQi?2` `m/2 +mbiQK2` b?Qrb mT- b2M/BM; i?2 }`bi `m/2 +mbiQK2` iQ i?2 7`QMi Q7 i?2 [m2m2- M/ bQ QMVX h?Bb ivT2 Q7 /Bb+BTHBM2 Bb + HH2/ T`22KTiBp2X h?2 T?` b2 dzT`22KTiBp2 `2bmK2Ǵ K2 Mb i? i T`22KTi2/ +mbiQK2` /Q2b MQi ? p2 iQ bi `i 7`QK b+` i+?, r?2M i?2 b2`p2` b22b ?BK M2ti iBK2- ?2 rBHH `2bmK2 rQ`F r?2`2 Bi r b H27iXj h?2 +QM;2biBQM T`Q+2bb Q7 i?Bb [m2m2 Bb 2t +iHv Q7 i?2 b K2 M im`2 b i? i Q7 i?2 JfJfRf∞f7B7Q [m2m2X AM T `iB+mH ` Bi ? b i?2 b K2 BM}MBi2bBK H ;2M2` iQ`X 1t KTH2 RRXRXe, JfJf∞X h?Bb Bb M JfJf∞f∞ [m2m2- #mi i?2 b2+QM/ ∞ Bb QKBii2/ M/ r2 i?2M bT2 F Q7 M JfJf∞ [m2m2X h?Bb Bb MQi `2 HHv [m2m2BM; bvbi2K- #mi Tm`2 /2H v bvbi2KX h?2 ``Bp H T`Q+2bb- i?2 b2`pB+2 iBK2b b2[m2M+2 M/ i?2 r BiBM; `QQK `2 b BM i?2 JfJfRf∞ KQ/2H- #mi MQr i?2`2 Bb M BM}MBiv Q7 b2`p2`b- M/ i?2`27Q`2 MQ [m2m2BM;- bBM+2 MvQM2 2Mi2`BM; i?2 bvbi2K }M/b M B/H2 b2`p2` Dmbi 7Q` ?BKb2H7X h?2 +QM;2biBQM T`Q+2bb Bb +QMiBMmQmb@iBK2 ?K+ rBi? bi i2 bT +2 E = N M/ BM}MBi2bBK H ;2M2` iQ` qi,i+1 = λ , qi,i−1 = iμ .
1t KTH2 RRXRXd, JfJfRf∞f7B7Q [m2m2 rBi? BMbi Mi M2Qmb 722/# +FX h?Bb Bb M JfJfRf∞ [m2m2 r?2`2 +mbiQK2` }MBb?BM; b2`pB+2 2Bi?2` H2 p2b i?2 bvbi2K rBi? T`Q# #BHBiv 1 − p Q` Bb BKK2/B i2Hv `2+v+H2/ rBi? T`Q# #BHBiv p i i?2 2M/ Q7 i?2 r BiBM; HBM2 Q` i i?2 b2`pB+2 #QQi? B7 i?2`2 Bb M 2KTiv r BiBM; HBM2rBi? M2r BM/2T2M/2Mi 2tTQM2MiB H b2`pB+2 `2[m2biX AM i?Bb 2t KTH2- i?2 iBK2b Q7 j h?Bb ivT2 Q7 /Bb+BTHBM2 Bb MQi b mM7 B` b Bi K v TT2 `X 6B`bi Q7 HH- HH +mbiQK2`b #2BM; 2[m HHv `m/2- i?2v 2M/m`2 b Km+? b i?2v ?m`iX +mbiQK2` rBi? H `;2 b2`pB+2 `2[m2bi bT2M/b H `;2` iBK2 i i?2 iB+F2i #QQi? M/ Bb i?2`27Q`2 KQ`2 2tTQb2/ i? M +mbiQK2` rBi? KQ/2bi `2[m2bi- M/ Bi Bb T`2+Bb2Hv BM i?Bb b2Mb2 i? i i?2 /Bb+BTHBM2 Bb 7 B`X Ai K F2b HQM;2` `2[m2bib r?Q `2 `2bTQMbB#H2 7Q` +QM;2biBQM r Bi HQM;2` BM i?2 bvbi2KX h?2 T`2+Bb2 `2bmHi Bb i? i i?2 2tT2+i2/ bQDQm`M iBK2 Q7 +mbiQK2` BM i?2 bvbi2K- ;Bp2M i? i Bib b2`pB+2 `2[m2bi Bb x- Bb 2[m H iQ x/(1 − ρ)r?2`2 ρ = λ/μX a22 (qQHz- RN3N)X
9yk
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
b2`pB+2 +QKTH2iBQM Q7 `2+v+H2/ +mbiQK2`b /Q MQi +Q``2bTQM/ iQ ;2MmBM2 i` MbBiBQM Q7 i?2 +QM;2biBQM T`Q+2bbX h?2 +QM;2biBQM T`Q+2bb Q7 i?2 JfJfRf∞f7B7Q [m2m2 rBi? BMbi Mi M2Qmb 722/# +F Bb +QMiBMmQmb@iBK2 ?K+ rBi? bi i2 bT +2 E = N M/ BM}MBi2bBK H ;2M2` iQ` ;Bp2M #v qi,i+1 = λ , qi,i−1 = (1 − p)μ1{i≥1} .
AM i?2 [m2m2BM; HBi2` im`2- 2bT2+B HHv r?2M TTHB+ iBQMb iQ +QKKmMB+ iBQMb M/ +QKTmi2` M2irQ`Fb `2 +QMbB/2`2/- [m2m2BM; bvbi2K Bb `2T`2b2Mi2/ #v i?2 TB+iQ;` K Q7 i?2 M2ti };m`2- r?2`2 i?2 BMTmi ``Qr `2T`2b2Mib i?2 ``Bp H bi`2 K Q7 DQ#b U+mbiQK2`bV- i?2 QmiTmi ``Qr `2T`2b2Mib i?2 bi`2 K Q7 +QKTH2i2/ DQ#b Ub2`p2/ +mbiQK2`bV- i?2 +B`+H2 Bb T`Q+2bbQ` Ui?2 b2`pB+2 bvbi2KV- M/ i?2 bi +F Bb #mz2` U r BiBM; `QQK- r?2`2 +mbiQK2`b r Bi 7Q` b2`p2` iQ #2 7`22VX 1 μ
c SQBbbQM λ
/2T `im`2b
K T`Q+2bbQ` #mz2` Ur BiBM; `QQKV UiB+F2i #QQi?V h?2 ?QHv TB+iQ;` K Q7 [m2m2BM; i?2Q`v
h?Bb TB+iQ;` K + M #2 `B+?Hv /Q`M2/X 6Q` BMbi M+2- BM i?2 };m`2- r2 ? p2 bvbi2K rBi? K b2`p2`b- r BiBM; `QQK Q7 + T +Biv c- λ Bb i?2 ``Bp H ` i2 Q7 +mbiQK2`b M/ i?2 /2`Bp iBQM b?Qrb i? i i?2 +mbiQK2`b }M/BM; HH K b2`p2`b #mbv M/ 7mHH r BiBM; `QQK `2 `2D2+i2/X HbQ- i?2`2 Bb M BM/B+ iBQM Q7 i?2 p2` ;2 b2`pB+2 iBK2- μ−1 - M/ Q7 i?2 7 +i i? i i?2 BM+QKBM; bi`2 K Bb SQBbbQM T`Q+2bbX AM i?Bb TB+iQ;` K- QM2 rQmH/ i F2 Bi b BKTHB+Bi i? i i?2 b2`pB+2 iBK2b b2[m2M+2 Bb BB/ M/ BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bb- mMH2bb Qi?2`rBb2 2tTHB+BiHv K2MiBQM2/X PM2 bQK2iBK2b HbQ BM/B+ i2b i?2 b2`pB+2 /Bb+BTHBM2 UHB7Q- 7B7Q- 2i+XVX >2m`BbiB+b h?2 `B;Q`Qmb T`QQ7 i? i i?2 +QM;2biBQM T`Q+2bb2b Q7 i?2 [m2m2BM; bvbi2Kb +QM@ bB/2`2/ #Qp2 `2 +QMiBMmQmb@iBK2 ?K+ rBi? i?2 BM/B+ i2/ BM}MBi2bBK H ;2M2` @ iQ`b + M #2 7QmM/ BM am#b2+iBQM RRXkX i i?Bb TQBMi- r2 b? HH #2 +QMi2Mi rBi? ?2m`BbiB+ /2`Bp iBQM Q7 i?2 BM}MBi2bBK H ;2M2` iQ`X *QMbB/2` 7Q` BMbi M+2 i?2 [m2m2 JfJfEf+- rBi? 7B7Q /Bb+BTHBM2X >2`2 c Bb i?2 + T +Biv Q7 i?2 r BiBM; `QQK- M/ i?2`27Q`2 M ``BpBM; +mbiQK2` b22BM; K + c +mbiQK2`b BM b2`pB+2 Bb `2D2+i2/- Q` `Qmi2/ 2Hb2r?2`2X i ;Bp2M iBK2 t- i?2`2 `2 +2`i BM MmK#2` Q7 2tTQM2MiB H ` M/QK p `B #H2b dzBM +iBpBivǴ- QM2 7Q` 2 +? Q7 i?2 +mbiQK2`b T`2b2MiHv `2+2BpBM; b2`pB+2- M/ QM2 +Q``2bTQM/BM; iQ i?2 iBK2 iQ i?2 M2ti ``Bp H Ur?B+? K v #2 `2@ 7mb2/ B7 i?2 bvbi2K Bb 7mHHVX A7 X(t) = i < K- 2 +? Q7 i?2 i +mbiQK2`b Bb ii2M/2/ #v b2`p2`- M/ r2 ? p2 ` M/QK p `B #H2b X1 , . . . , Xi `2T`2b2MiBM; i?2 `2K BMBM;
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9yj
iBK2b mMiBH +QKTH2iBQM Q7 b2`pB+2X "v i?2 H +F Q7 K2KQ`v T`QT2`iv Q7 i?2 2tTQM2M@ iB H ` M/QK p `B #H2b- i?2b2 `2 BB/ 2tTQM2MiB H ` M/QK p `B #H2b rBi? K2 M μ−1 X 6Q` bBKBH ` `2 bQMb- i?2 iBK2 iQ i?2 M2ti ``Bp H Bb M 2tTQM2MiB H ` M/QK p `B #H2 X0 rBi? K2 M λ−1 - BM/2T2M/2Mi Q7 X1 , . . . , Xi X h?2`27Q`2 P (X(t + h) = i + 1 | X(t) = i) = P (X0 ≤ h, X1 > h, . . . , Xi > h) + o(h), i FBM; BMiQ ++QmMi i? i 7Q` BM}MBi2bBK H h i?2`2 Bb i KQbi QM2 Q7 i?2 2tTQM2MiB Hb X0 , X1 , . . . , Xi BM (0, h)X LQr- iQ i?2 }`bi Q`/2` BM hP (X0 ≤ h, X1 > h, . . . , Xi > h) = (1 − eλh ) × eμh × . . . × eμh λh. _2+ HH i?2 K2 MBM; Q7 qi,i+1 P (X(t + h) = i + 1 | X(t) = i) = qi,i+1 h + o(h). h?2`27Q`2- qi,i+1 = λX aBKBH `Hv- B7 1 ≤ i ≤ KP (X(t + h) = i − 1 | X(t) = i) = P (X0 > h, inf {X1 , . . . , Xi } ≤ h) + o(h). "mi inf {X1 , . . . , Xi } Bb M 2tTQM2MiB H ` M/QK p `B #H2 Q7 K2 M (iμ)−1 - M/ i?2`2@ 7Q`2 P (X0 > h, inf {X1 , . . . , Xi } ≤ h) iμh. *QKT `BM; rBi? P (X(t + h) = i − 1 | X(t) = i) = qi,i−1 h + o(h) r2 ? p2 i? i qi,i−1 = iμX A7 i ≥ K i?2`2 `2 2t +iHv K +mbiQK2`b BM b2`pB+2- M/ i?2 b K2 ?2m`BbiB+b b #Qp2 ;Bp2 qi,i−1 = KμX AM i?2 + b2 i = 0- i?2`2 Bb Q7 +Qm`b2 QMHv QM2 +iBp2 2tTQM2MiB H- M K2Hv X0 - i?2 `2K BMBM; iBK2 iQ i?2 M2ti ``Bp HX q2 ? p2 qi,i−1 = 0- M/ qi,i+1 = λX AM i?2 + b2 i = c + K- MQ ``Bp Hb BM i?2 bvbi2K `2 TQbbB#H2- bBM+2 TQi2MiB H +mbiQK2` rBHH MQi #2 ++2Ti2/X h?2`27Q`2 BM i?Bb + b2qi,i+1 = 0X AM bmKK `vqi,i+1 = λ 1{iSh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
h?2 i`2 iK2Mi /Bz2`b BM i?2 +QKTmi iBQM Q7 qi,i−1 7Q` i > 0X q2 ? p2 iQ BMi`Q/m+2 {0, 1}@p Hm2/ ` M/QK p `B #H2 Z- BM/2T2M/2Mi Q7 X0 , X1 - rBi? P (Z = 1) = pBM/B+ iBM; r?2i?2` i?2 M2ti +mbiQK2` iQ +QKTH2i2 b2`pB+2 rBHH #2 `2+v+H2/ UZ = 1V Q` MQi UZ = 0VX >2`2 qi,i−1 h P (X(t + h) = i − 1 | X(t) = i) P (X0 > h, X1 ≤ h, Z = 0) = P (X0 > h, X1 ≤ h)P (Z = 0) μh × (1 − p), r?2`2 i?2 bvK#QH BM/B+ i2b 2[m HBiv mT iQ i?2 }`bi Q`/2` BM hX h?2`27Q`2 qi,i−1 = μ(1 − p) 1{i ≥ 1}X h?2 *QM;2biBQM S`Q+2bb b "B`i?@ M/@/2 i? S`Q+2bb h?2 BM}MBi2bBK H ;2M2` iQ` Q7 i?2 +QM;2biBQM T`Q+2bb Q7 M JfJfRf∞ [m2m2 Bb BM/2T2M/2Mi Q7 i?2 b2`pB+2 /Bb+BTHBM2 B7 r2 +QMbB/2` QMHv b2`pB+2 /Bb+BTHBM2b bm+? i? i i?2 b2`p2` rQ`Fb i 7mHH bT22/- 2[m H iQ R- r?2M2p2` i?2`2 Bb i H2 bi QM2 +mbiQK2` BM i?2 bvbi2KX h?Bb ;2M2` H 7 +i Bb /m2 iQ i?2 H +F Q7 K2KQ`v Q7 2tTQM2MiB H p `B #H2b M/ Bi + M #2 +?2+F2/ mbBM; i?2 b K2 ?2m`BbiB+b b BM i?2 7B7Q + b2 7Q` /Bb+BTHBM2b rBi?Qmi BMi2``mTiBQM Q7 b2`pB+2- bm+? b HB7Q MQM@T`22KTiBp2- ` M/QK Ui?2 M2ti +mbiQK2` Bb +?Qb2M ` M/QKHv BM i?2 r BiBM; `QQKVX AM i?2 + b2 Q7 BMi2``mTiBQM Q7 b2`pB+2- rBi? `2bmK2/ b2`pB+2 Ui? i Bb- i?2 b2`pB+2 H`2 /v T`QpB/2/ Bb MQi #QHBb?2/ #v i?2 T`22KTiBQMV i?2 ?2m`BbiB+b rBHH ;Bp2 ; BM i?2 b K2 `2bmHib- #2+ mb2 i?2 `2K BMBM; iBK2 Q7 M BMi2``mTi2/ 2tTQM2MiB H ` M/QK p `B #H2 Bb M 2tTQM2MiB H ` M/QK p `B #H2 rBi? i?2 b K2 K2 MX A7 7i2` T`22KTiBQM i?2 b2`pB+2 Bb bi `i2/ 7`QK i?2 #2;BMMBM;- ; BM MQi?BM; +? M;2b b 7 ` b i?2 BM}MBi2bBK H ;2M2` iQ` Bb +QM+2`M2/X h?2 +QM;2biBQM T`Q+2bb2b U+QmMiBM; i?2 +mbiQK2`b BM i?2 bvbi2KV Q7 i?2 [m2m2@ BM; bvbi2Kb i? i r2 Dmbi +QMbB/2`2/ `2 bT2+B H + b2b Q7 #B`i?@ M/@/2 i? T`Q+2bb2b rBi? M BM}MBi2bBK H ;2M2` iQ` Q7 i?2 7Q`K ⎛ ⎞ −λ0 λ0 0 0 ··· ⎜ μ1 −(λ1 + μ1 ) λ1 0 · · ·⎟ ⎜ ⎟ A=⎜ 0 μ2 −(λ2 + μ2 ) λ2 · · ·⎟ ⎝ ⎠ XX XX XX XX X X X X r?2`2 i?2 bi i2 bT +2 Bb E = N- Q` E = {0, 1, . . . , N } 7Q` }MBi2 N X 6Q` HH i?2 [m2m2BM; T`Q+2bb2b r2 +QMbB/2` BM i?2 T`2b2Mi b2+iBQM- λi > 0 7Q` HH i ∈ E- 2t+2Ti i = N r?2M E = {0, 1, . . . , N }- M/ μi > 0 7Q` HH i ∈ E- 2t+2Ti i = 0X h?2b2 +QM/BiBQMb Q#pBQmbHv ;m ` Mi22 B``2/m+B#BHBivX q2 `2+ HH i?2 ;2M2` H `2bmHi QM #B`i?@ M/@/2 i? T`Q+2bb2b- MQiBM; }`bi i? i HH i?2 +? BMb `BbBM; 7`QK i?2 2t KTH2b #Qp2 `2 `2;mH ` DmKT T`Q+2bb2b U1t2`+Bb2 RRXeXRVX h?2 2`;Q/B+Biv +QM/BiBQM Bb, i λn−1 i∈E i≥1
BM r?B+? + b2- 7Q` i ≥ 1-
n=1
μn
< ∞,
URRXRV
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u π(i) = π(0)
i λn−1 n=1
μn
,
9y8 URRXkV
r?2`2 π(0) Bb Q#i BM2/ #v MQ`K HBx iBQMX h?2 2`;Q/B+Biv +QM/BiBQM Bb- Q7 +Qm`b2 miQK iB+ HHv b iBb}2/ r?2M i?2 bi i2 bT +2 Bb }MBi2X 1t KTH2 RRXRX3, JfJfRf∞X 6Q` i?2 JfJfRf∞ [m2m2- E = N- λi ≡ λ > 0 i M/ μi = μ > 0 7Q` HH i ≥ 1X h?2 2`;Q/B+Biv +QM/BiBQM `2 /b i≥1 μλ < ∞- i? i Bbλ ρ := < 1 . μ h?Bb +QM/BiBQM Bb M im` H- bBM+2 ρ = λE[σ1 ]- i?2 i` {+ BMi2MbBiv- Bb i?2 p2` ;2 ` i2 Q7 rQ`F 2Mi2`BM; i?2 bvbi2K T2` mMBi iBK2- M/ b?QmH/ MQi 2t+22/ i?2 K tBK H bT22/ Q7 b2`pB+2- 2[m H iQ 1X h?2 bQHmiBQM Q7 i?2 # H M+2 2[m iBQM Bb π(i) = (1 − ρ)ρi .
1t KTH2 RRXRXN, JfJfEfy- Q` 1`H M; [m2m2X 6Q` i?2 JfJfEfy [m2m2 U1`H M; [m2m2V E = {0, . . . , K} M/ i?2 bQHmiBQM Q7 i?2 # H M+2 2[m iBQMb Bb ρi /i! π(i) = K n n=0 ρ /n! 7Q` 0 ≤ i ≤ KX AM T `iB+mH `ρK /K! π(K) = K n n=0 ρ /n! Bb i?2 #HQ+FBM; T`Q# #BHBiv- i?2 T`Q# #BHBiv Q7 }M/BM; HH +? MM2Hb #mbvX h?2 +Q``2@ bTQM/BM; 7Q`KmH Bb + HH2/ i?2 1`H M; #HQ+FBM; 7Q`KmH X h?2 /Bbi`B#miBQM π Bb + HH2/ SQBbbQM /Bbi`B#miBQM i`mM+ i2/ i K- bBM+2 π(i) = P (Z = i | Z ≤ K), r?2`2 Z Bb SQBbbQM ` M/QK p `B #H2 rBi? K2 M ρX 1t KTH2 RRXRXRy, JfJf∞X 6Q` i?2 JfJf∞ [m2m2- i?2 2`;Q/B+Biv +QM/BiBQM
ρi URRXRV Bb ∞ i=1 i! < ∞ M/ Bb Hr vb b iBb}2/ b HQM; b ρ < ∞X h?2 bi iBQM `v /Bbi`B#miBQM Bb i?2 SQBbbQM /Bbi`B#miBQM rBi? K2 M 2[m H iQ i?2 i` {+ BMi2MbBiv ρπ(i) = e−ρ
ρi . i!
9ye
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
1t KTH2 RRXRXRR, JfJfEf∞X 6Q` i?2 JfJfEf∞ [m2m2- i?2 2`;Q/B+Biv +QM@ /BiBQM URRXRV Bb b iBb}2/ QMHv B7 ρ :=
λ < K. μ
Ai b vb i? i i?2 p2` ;2 BM+QKBM; rQ`F T2` mMBi iBK2 + MMQi 2t+22/ i?2 K tBK H b2`pB+2 bT22/- r?B+? Bb K r?2M HH b2`p2`b `2 #mbvX h?2 bi iBQM `v /Bbi`B#miBQM Bb i?2M- 7Q` 0 ≤ i ≤ Kρi π(i) = π(0) , i! M/ r?2M i ≥ KρK ρ i−K , π(i) = π(0) s! K r?2`2 s−1 i ρ ρK 1 + . π(0)−1 = i! K! 1 − ρ/s i=0 AM i?Bb bvbi2K- i?2 T`Q# #BHBiv Q7 r BiBM; Bb i?2 T`Q# #BHBiv Q7 2Mi2`BM; i?2 bvbi2K r?2M i?2 K b2`p2`b `2 #mbv- i? i Bb π(≥ K) := i≥K π(i)X PM2 Q#i BMb i?2 1`H M; r BiBM; 7Q`KmH π(≥ K) =
ρs 1 K! 1−ρ/K
K−1 ρi ρK 1 i=0 i! + K! 1−ρ/K
.
h?2 JfJfRf∞ h M/2K q2 MQr im`M iQ i?2 bim/v Q7 [m2m2BM; M2irQ`Fb- bi `iBM; rBi? i?2 bBKTH2bi QM2- i?2 i M/2K Q7 JfJfRf∞ [m2m2bX h?2 b2KBM H `2bmHi ?2`2 Bb "m`F2Ƕb QmiTmi i?2Q`2KX9 Ai b vb BM T `iB+mH ` i? i i?2 TQBMi T`Q+2bb Q7 i?2 /2T `im`2 iBK2b Q7 M JfJfRf∞ [m2m2 BM 2[mBHB#`BmK Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? i?2 b K2 BMi2MbBiv b i?2 ``Bp H T`Q+2bbX AM 7 +i- i?2 `2bmHi 2ti2M/b iQ KQ`2 ;2M2` H #B`i?@ M/@/2 i? T`Q+2bb2bX8 *QMbB/2` #B`i?@ M/@/2 i? T`Q+2bb QM N rBi? #B`i? T ` K2i2`b Q7 i?2 7Q`K λi ≡ λ, M/ bmTTQb2 i? i μi > 0 7Q` HH i ≥ 1X h?2 +Q``2bTQM/BM; +? BM Bb B``2/m+B#H2- M/ r2 b? HH bbmK2 i? i Bi Bb 2`;Q/B+X i 2[mBHB#`BmK- i?Bb +? BM Bb `2p2`bB#H2 bBM+2 i?2 /2i BH2/ # H M+2 2[m iBQMb π(i + 1)μi+1 = π(i)λ Ui ∈ NV `2 b iBb}2/- b + M #2 `2 /BHv +?2+F2/ mbBM; i?2 2tTHB+Bi 7Q`K Q7 i?2 bi iBQM `v /Bbi`B#miBQMX h?2 mTr `/ i` MbBiBQMb `2 /m2 iQ ``BpBM; +mbiQK2`b M/ i?2 /QrMr `/ i` M@ bBiBQMb `2 /m2 iQ /2T `iBM; +mbiQK2`bX amTTQb2 i? i i?2 [m2m2 Bb BM 2[mBHB#`BmKX 9 8
("m`F2- RN8e)X h?2 T`QQ7 #2HQr- mbBM; i?2 `2p2`bB#BHBiv `;mK2Mi- Bb i? i Q7 (_2B+?- RN8d)X
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9yd
AM T `iB+mH `- i?2 `2p2`b2/ T`Q+2bb ? b i?2 b K2 /Bbi`B#miBQM b i?2 /B`2+i T`Q+2bbXe q?2M iBK2 Bb `2p2`b2/- /2T `im`2b #2+QK2 ``Bp Hb M/ i?2`27Q`2- BM pB2r Q7 i?2 `2p2`bB#BHBiv T`QT2`iv- i?2 `2p2`b2/ T`Q+2bb Q7 /2T `im`2b Bb M ?TTX LQr- i?2 T`Q#@ #BHBbiB+ M im`2 Q7 M ?TT /Q2b MQi +? M;2 r?2M iBK2 Bb `2p2`b2/X h?2`27Q`2- i?2 /2T `im`2 T`Q+2bb Bb SQBbbQM T`Q+2bbX HbQ- bBM+2 BM /B`2+i iBK2- 7Q` Mv iBK2 t ≥ 0- i?2 bi i2 X(t) Bb BM/2T2M/2Mi Q7 i?2 7mim`2 i iBK2 t Q7 i?2 ``Bp H T`Q+2bb UJ `FQp T`QT2`iv Q7 SQBbbQM T`Q+2bb2bVBi 7QHHQrb 7`QK `2p2`bB#BHBiv i? i X(t) Bb BM/2T2M/2Mi Q7 i?2 T bi i iBK2 t Q7 i?2 /2T `im`2 T`Q+2bbX X(t)
t ``Bp Hb Q7 i?2 iBK2@`2p2`b2/ T`Q+2bb S`QQ7 Q7 "m`F2Ƕb i?2Q`2K #v `2p2`bB#BHBiv
1t KTH2 RRXRXRk, h M/2K Q7 JfJfRf∞ [m2m2bX *QMbB/2` irQ JfJfRf∞ BM bm++2bbBQM- BM i?2 b2Mb2 i? i i?2 /2T `im`2 T`Q+2bb 7`QK i?2 }`bi QM2 Bb i?2 ``Bp H T`Q+2bb Q7 i?2 b2+QM/ QM2X h?2 TQBMi T`Q+2bb Q7 ``Bp Hb BMiQ i?2 }`bi [m2m2 Bb M ?TT Q7 BMi2MbBiv λX h?2 b2`pB+2 b2[m2M+2b `2 BM/2T2M/2Mi BB/ 2tTQM2MiB H b2[m2M+2b rBi? K2 M 1/μ1 M/ 1/μ2 `2bT2+iBp2HvX "Qi? b2`pB+2 b2[m2M+2b `2 BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bbX G2i X1 (t) M/ X2 (t) #2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 }`bi M/ i?2 b2+QM/ [m2m2BM; bvbi2K `2bT2+iBp2HvX
∞
X1 (t)
∞ 1
X2 (t) 1
e ai`B+iHv bT2 FBM;- i?2 `2p2`b2/ +QM;2biBQM T`Q+2bb Bb H27i@+QMiBMmQmb r?2`2 b i?2 Q`B;BM H T`Q+2bb Bb `B;?i@+QMiBMmQmbX >Qr2p2`- +? M;BM; i?2 p Hm2b i- b v- i?2 k@i? /Bb+QMiBMmBiv TQBMi /Q2b MQi +? M;2 i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 `2p2`b2/ T`Q+2bb- bBM+2 i?2 HQ+ iBQM Q7 bm+? /Bb+QMiBMmBiv /KBib T`Q# #BHBiv /2MbBivX
9y3
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
i iBK2 t- X2 (t) /2T2M/b QM i?2 /2T `im`2 T`Q+2bb Q7 i?2 }`bi [m2m2 #27Q`2 iBK2 t M/ QM i?2 b2+QM/ b2`pB+2 b2[m2M+2X aBM+2 #v "m`F2Ƕb i?2Q`2K M/ i?2 BM/2T2M@ /2M+2 T`QT2`iv Q7 i?2 b2`pB+2 b2[m2M+2 i?2 H ii2` `2 BM/2T2M/2Mi Q7 X1 (t)- Bi 7QH@ HQrb i? i X1 (t) M/ X2 (t) `2 BM/2T2M/2MiX "Qi? [m2m2b `2 BM BbQH iBQM JfJfRf∞ rBi? i` {+ BMi2MbBiB2b ρ1 = λ/μ1 M/ ρ2 = λ/μ2 - M/ i?2`27Q`2 M2+2bb `v M/ bm{+B2Mi +QM/BiBQM Q7 2`;Q/B+Biv Q7 i?2 +QMiBMmQmb@iBK2 ?K+ {(X1 (t), X2 (t))}t≥0 Bb λ < inf(μ1 , μ2 ) M/ Bib bi iBQM `v /Bbi`B#miBQM i?2M i F2b i?2 T`Q/m+i 7Q`K π(n1 , n2 ) = π1 (n1 )π2 (n2 ) = (1 − ρ1 )ρn1 1 (1 − ρ2 )ρn1 2
(n1 , n2 ≥ 0) .
h?2 C +FbQM L2irQ`F h?Bb Bb M QT2M M2irQ`F Q7 BMi2`+QMM2+i2/ [m2m2bXd h?2`2 `2 K bi iBQMb- M/ 2 +? bi iBQM ? b Rf∞ b2`pB+2 bvbi2K- i? i Bb- mMB[m2 b2`p2` rQ`FBM; i mMBi bT22/ M/ M BM}MBi2 r BiBM; `QQKX h?2`2 `2 irQ ivT2b Q7 +mbiQK2`b [m2m2BM; i ;Bp2M bi iBQM, URV i?Qb2 r?B+? `2 72/@# +F- i? i Bb- r?Q ? p2 `2+2Bp2/ b2`pB+2 BM MQi?2` Q` i?2 b K2 bi iBQM M/ `2 `2`Qmi2/ iQ i?2 ;Bp2M bi iBQM 7Q` KQ`2 b2`pB+2- M/ UkV i?Qb2 r?Q `2 2Mi2`BM; i?2 M2irQ`F 7Q` i?2 }`bi iBK2 Ub22 i?2 };m`2 #2HQrVX ¯i λ
ri
rij rji ¯j λ
rj
C +FbQM M2irQ`F ¯i - rBi? h?2 2tQ;2MQmb ``Bp Hb BMiQ bi iBQM i U1 ≤ i ≤ KV 7Q`K M ?TT- /2MQi2/ #v N ¯ BMi2MbBiv λi ∈ [0, ∞)X h?2 b2[m2M+2 Q7 b2`pB+2 iBK2b i bi iBQM i `2 2tTQM2MiB H ` M/QK p `B #H2b Q7 K2 M 1/μi ∈ (0, ∞)X h?2 b2`pB+2 iBK2b BM i?2 b K2 M/ BM /Bz2`2Mi bi iBQMb `2 BM/2T2M/2Mi M/ ¯i - M/ i?2 H ii2` ?TTb `2 BM/2T2M/2Mi BM/2T2M/2Mi Q7 i?2 2tQ;2MQmb BMTmi ?TTb N Q7 QM2 MQi?2`X h?2 `QmiBM; Bb Q7 i?2 "2`MQmHHB ivT2X 1 +? +mbiQK2` Dmbi +QKTH2iBM; b2`pB+2 BM bi iBQM i iQbb2b (K +1)@7 +2/ /B2 rBi? T`Q# #BHBiB2b ri,1 , . . . , ri,K , ri rBi? i?2 2z2+i i? i i?2 +mbiQK2` Bb b2Mi iQ bi iBQM j rBi? T`Q# #BHBiv rij Q` H2 p2b i?2 bvbi2K rBi?
T`Q# #BHBiv ri = 1 − K j=1 rij X h?2 K i`Bt d
(C +FbQM- RNej)X
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9yN
R = {rij }1≤i,j≤K Bb i?2 `QmiBM; K i`BtX h?2 bm++2bbBp2 iQbb2b Q7 i?2 `QmiBM; /B+2 Q7 HH bi iBQMb `2 BM/2T2M/2Mi- M/ BM/2T2M/2Mi Q7 i?2 2tQ;2M2Qmb ``Bp H T`Q+2bb2b M/ Q7 HH i?2 b2`pB+2 iBK2bX h?Bb Bb i?2 Q`B;BM H C +FbQM KQ/2H- r?B+? + M #2 2M`B+?2/ #v i?2 BMi`Q/m+iBQM Q7 b2`pB+2 bT22/bX A7 i?2`2 `2 ni +mbiQK2`b BM bi iBQM i U1 ≤ i ≤ KV- i?2 b2`p2` rQ`Fb i bT22/ φi (ni )- r?2`2 φi (0) = 0 M/ φi (ni ) > 0 7Q` HH ni ≥ 1X G2i Xi (t) #2 i?2 MmK#2` Q7 +mbiQK2`b BM bi iBQM i i iBK2 t- M/ H2i X(t) := (X1 (t), . . . , Xk (t)). Ai + M #2 T`Qp2/ UmbBM; i?2 K2i?Q/b Q7 a2+iBQM RRXkV i? i i?2 T`Q+2bb {X(t)}t≥0 Bb `2;mH ` DmKT ?K+ rBi? bi i2 bT +2 E = NK M/ rBi? BM}MBi2bBK H ;2M2` iQ` A = {qn,n }- r?2`2 HH i?2 MQM@MmHH Qz@/B ;QM H i2`Kb `2 ¯i, qn,n+ei = λ qn,n−ei = μi φi (ni )ri 1{ni >0} , qn,n−ei +ej = μi φi (ni )rij 1{ni >0} , r?2`2 n = (n1 , . . . , nK ) ∈ E = NK M/ r?2`2 ei Bb i?2 ii? p2+iQ` Q7 i?2 + MQMB+ H # bBb Q7 RK X h?Bb ;2M2` iQ` Bb BM/2T2M/2Mi Q7 i?2 b2`pB+2 bi` i2;věHB7Q- 7B7Q- Q` T`Q+2bbQ`@b? `BM;X h?2 7Q`K Q7 i?2 BM}MBi2bBK H ;2M2` iQ` + M #2 Q#i BM2/ 7`QK 7mHH /2b+`BTiBQM Q7 i?2 M2irQ`F b SQBbbQM bvbi2K Ub22 a2+iBQM RRXkVX >Qr2p2`- r2 b? HH #2 +QMi2Mi rBi? ?2m`BbiB+ `;mK2MibX 6Q` BMbi M+2- i?2 2tT`2bbBQM 7Q` qn,n−ei +ej 7QHHQrb 7`QK i?2 BMimBiBp2 +QMbB/2` iBQMb #2HQrX A7 i?2 bi i2 i iBK2 t Bb n- i` Mb72` 7`QK bi iBQM i iQ bi iBQM j `2[mB`2b i? i i?2 2tTQM2MiB H ` M/QK p `B #H2 rBi? K2 M 1/μi `2T`2b2MiBM; i?2 `2[mB`2/ b2`pB+2 Q7 i?2 +mbiQK2` #2BM; b2`p2/ i bi iBQM i i iBK2 t #2 i2`KBM i2/ #2ir22M iBK2b t M/ t + h UT`Q# #BHBiv μi φi (ni )h mT iQ i?2 }`bi Q`/2` BM hV M/ i? i i?2 +Q``2bTQM/BM; +mbiQK2` #2 `Qmi2/ iQ bi iBQM j UT`Q# #BHBiv rij )X bbmK2 i?2 +? BM {X(t)}t≥0 B``2/m+B#H2X h?Bb Bb i?2 + b2 r?2M (α) 7Q` HH j ∈ {1, 2, . . . , K}- i?2`2 2tBbi i, i1 , . . . , im ∈ {1, 2, . . . , K} bm+? i? i ¯ i rii ri i · · · rim j > 0 , λ 1 1 2 M/ (β) 7Q` HH j ∈ {1, 2, . . . , K}- i?2`2 2tBbi j1 , j2 , . . . , j , k ∈ {1, 2, . . . , K} bm+? i? i rjj1 rj1 j2 · · · rj k rk > 0 . _2+ HHBM; i? i μi > 0 7Q` HH bi iBQMb- +QM/BiBQM (α) b vb i? i Mv bi iBQM Bb 2tQ;2MQmbHv bmTTHB2/ r?2`2 b +QM/BiBQM (β) b vb i? i Mv bi iBQM ? b M QmiH2iX *QMbB/2` MQr i?2 (K + 1) × (K + 1) K i`Bt
9Ry
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: ⎛
··· ···
r11 r12 ⎜ r21 r22 ⎜ XX X ˜ =⎜ R ⎜ XX X ⎜ ⎝rK1 rK2 0 0
r1K r2K XX X
··· ···
rKK 0
⎞ r1 r2 ⎟ ⎟ XX ⎟ . X ⎟ ⎟ rK ⎠ 1
Ai + M #2 BMi2`T`2i2/ b i?2 i` MbBiBQM K i`Bt Q7 M ?K+ QM i?2 }MBi2 bi i2 bT +2 {1, . . . , K, K+1}X *QM/BiBQM (β) BKTHB2b i? i bi i2 K+1 Bb #bQ`#BM; M/ i?2 bi i2b 1, . . . , K `2 i` MbB2MiX AM T `iB+mH `- 7Q` HH i, j BM i?2 i` MbB2Mi b2i- limn↑∞ pij (n) = 0- i? i Bb- limn↑∞ Rn = 0X aBM+2 (I + R + · · · + Rn )(I − R) = I − Rn+1 , i?2 BMp2`b2 (1 − R)−1 2tBbib M/ Bb 2[m H iQ ( i?2 bvbi2K Q7 2[m iBQMb ¯i + λi = λ
K
λj rji
∞ n=0
Rn )X h?2`27Q`2 i?2 bQHmiBQM Q7
(1 ≤ i ≤ K) ,
URRXjV
j=1
¯ T + λT R- ? b mMB[m2 bQHmiBQM i? i Bb- rBi? Q#pBQmb MQi iBQM- λT = λ ¯ T (1 − R)−1 , λT = λ r?2`2 i?2 H ii2` 2tT`2bbBQM b?Qrb i? i i?Bb bQHmiBQM Bb MQM@M2; iBp2X 1[m iBQMb URRXjV `2 + HH2/ i?2 i` {+ 2[m iBQMb- #2+ mb2 i?2v ;Bp2 M2+2bb `v `2H iBQM #2@ ir22M i?2 p2` ;2 MmK#2`b Q7 +mbiQK2`b λi 2Mi2`BM; bi iBQM i BM bi2 /v bi i2 B7 i?2 ¯ i THmb i?2 M2irQ`F Bb 2`;Q/B+X AM/22/- λi Bb 2[m H iQ i?2 2tQ;2M2Qmb ` i2 Q7 ``Bp Hb λ bmK Q7 HH p2` ;2 ` i2b Q7 i` Mb72` 7`QK Qi?2` bi iBQMbX 6`QK bi iBQM j- i?2 +Q``2@ bTQM/BM; ` i2 Bb αj rji - r?2`2 αj Bb i?2 p2` ;2 ` i2 Q7 +mbiQK2`b }MBb?BM; b2`pB+2 BM bi iBQM jX "mi i 2[mBHB#`BmK αj = λj - bBM+2 i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b BM bi iBQM j `2K BMb +QMbi Mi- r?2M+2 i?2 i` {+ 2[m iBQMb URRXjVX h?Bb ?2m`BbiB+ `;mK2Mi Bb MQi M22/2/- M/ BM T `iB+mH `- QM2 M22/ MQi ii2KTi iQ B/2MiB7v λi BM URRXjV b i?2 p2` ;2 BM+QKBM; ``Bp H ` i2 BM bi iBQM iX h?Bb B/2MiB}+ iBQM Bb TQbbB#H2 B7 2[mBHB#`BmK Bb ;m ` Mi22/ U1t2`+Bb2 RRXeXRRVX >Qr2p2`2p2M BM i?2 MQM@2`;Q/B+ + b2b- i?2 i` {+ 2[m iBQMb ? p2 mMB[m2 bQHmiBQMX q2 MQr +QMbB/2` i?2 + b2 r?2`2 i?2 b2`pB+2 bT22/b Q7 HH i?2 b2`p2`b `2 2[m H iQ R M/ `272` i?2 `2 /2` iQ 1t2`+Bb2 RRXeX9 7Q` i?2 ;2M2` H + b2X h?2 ;HQ# H # H M+2 2[m iBQMb `2 K K K ¯ i 1{n >0} + π(n) (λi + μi ri 1{n >0} ) = π(n − ei )λ π(n + ei )μi ri i
i=1
i
i=1
+
K K
i=1
π(n + ei − ej )μi rij 1{nj >0} .
i=1 j=1
Ai im`Mb Qmi i? i B7 i?2 bQHmiBQM Q7 i?2 i` {+ 2[m iBQM b iBb}2b
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u ρi =
9RR
λi < 1 (1 ≤ i ≤ K) , μi
URRX9V
i?2M i?2 M2irQ`F Bb 2`;Q/B+ rBi? bi iBQM `v /Bbi`B#miBQM π(n) =
K
URRX8V
πi (ni ) ,
i=1
r?2`2 πi Bb i?2 bi iBQM `v /Bbi`B#miBQM Q7 M JfJfRf∞ [m2m2 rBi? i` {+ BMi2MbBiv ρi πi (ni ) = ρni i (1 − ρi ) . URRXeV ˜ QM hQ T`Qp2 i?Bb- TTHv i?2 `2p2`b H i2bi U1t2`+Bb2 RRXeXkVX .2}M2 i?2 ;2M2` iQ` A K E = N #v π(n)˜ qn,n = π(n )qn ,n M/ +?2+F i? i
>2`2 qn =
q˜n,n = qn .
K
λi + μi 1{ni >0} .
i=1
˜ Bb ;Bp2M #v h?2 ;2M2` iQ` A π(n)˜ qn,n+ei = π(n + ei )qn+ei ,n = π(n + ei )μi ri , ¯ i 1{n >0} , π(n)˜ qn,n−ei = π(n − ei )qn−ei ,n = π(n − ei )λ i π(n)˜ qn,n+ei −ej = π(n + ei − ej )qn+ei −ej ,n = π(n + ei − ej )μi rij 1{nj >0} , M/ i?2`27Q`2- i FBM; BMiQ ++QmMi i?2 bT2+B}+ 7Q`K Q7 π(n) ;Bp2M #v URRX8V M/ URRXeVq˜n,n+ei = ρi μi ri = λi ri , ¯i λ q˜n,n−ei = 1{n >0} , ρi i ρi 1 μi rij 1{nj >0} = λi rij 1{nj >0} . q˜n,n+ei −ej = ρj ρj Ai Kmbi #2 p2`B}2/ i? i K
¯ i + μi 1{r >0} ) = (λ i
i=1
K i=1
"v i?2 i` {+ 2[m iBQMi?2 T`2pBQmb 2[m HBiv Bb
K i=1
K i=1
¯i λ 1 λi ri + 1{ni >0} + ρi ρi
K
λj rji
1{ni >0}
.
i=1
¯ i - M/ i?2`27Q`2 i?2 `B;?i@? M/ bB/2 Q7 λj rji = λi − λ
1 λi ri + λi 1{ni >0} ρi
M/ Bi `2K BMb iQ +?2+F i? i
=
K i=1
(λi ri + μi 1{ni >0} ),
9Rk
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: K
¯i = λ
i=1
K
λi ri .
i=1
6Q` i?Bb- Bi bm{+2b iQ bmK i?2 i` {+ 2[m iBQMb URRXjVX 1t KTH2 RRXRXRj, h?2 bTHBiiBM; T ` /QtX3 h?2`2 Bb M TT `2Mi T ` /Qt `2H iBp2 iQ i?2 [m2biBQM Q7 ~Qrb BM [m2m2BM; M2irQ`F- BM 7 +i i?2 bBKTH2bi MQM@ + b+ /2 C +FbQM M2irQ`F, i?2 M/M/1/∞ [m2m2 rBi? BMbi Mi M2Qmb 722/# +F rBi? `2+v+HBM; T`Q# #BHBiv 12 X AM i?Bb + b2- i?2 QmiTmi ~Qr 7`QK i?2 bvbi2K U+mbiQK2`b H2 pBM; 7Q` ;QQ/V Bb SQBbbQM- #mi i?2 `2+v+H2/ ~Qr Bb MQi SQBbbQMX h?2 TT `2Mi T ` /Qt Ui?2 SQBbbQM bTHBiiBM; T ` /QtV Bb i? i r2 ? p2 TQBMi T`Q+2bb U+mbiQK2`b Dmbi H2 pBM; i?2 b2`pB+2 #QQi?V r?B+? Bb bTHBi BM irQ mbBM; M BB/ b2[m2M+2 Q7 7 B` +QBM iQbb2b- M/ ? H7 Q7 Bi `2K BMb SQBbbQM r?BH2 i?2 Qi?2` ? H7 Bb MQi SQBbbQMX h?2 T`QQ7 i? i i?2 722/# +F TQBMi T`Q+2bb Bb MQi M ?TT Bb b 7QHHQrbX h?2 }Hi2`BM; 2[m iBQM 7Q` Z0 (t) := 1{X(t)=0} rBi? `2bT2+i iQ i?2 T bi Q7 i?2 722/# +F TQBMi T`Q+2bb Bb U2t2`+Bb2- mb2 i?2 `2bmHib Q7 a2+iBQM RyXRV,
t
Z0F (s)μ(1 − p) ds −
Z0F (t) = Z0F (0) + 0
Z0F (s−)(N (ds) − μp(1 − Z0F (s)) ds) . (0,t]
amTTQb2 i?2 722/# +F ~Qr r2`2 SQBbbQM- i?2M Bib FtF @T`2/B+i #H2 BMi2MbBiv μp(1 − Z0F (t−)) rQmH/ #2 +QMbi Mi ξX h?2`27Q`2- #v mMB[m2M2bb Q7 i?2 FtF @ T`2/B+i #H2 BMi2MbBiv- P (dω)ξ dt@ X2X M/ P (dω) N (dt)@ X2X Uh?2Q`2K 8XRXj3V μp(1 − Z0F (t−, ω)) = ξ . .2MQiBM; #v τ1 i?2 }`bi TQbBiBp2 2p2Mi iBK2 Q7 F - Z0F (τ1 ) = 0 Ui?Bb Bb BMimBiBp2Hv Q#pBQmb- Q` 2Hb2- mb2 i?2 2tT`2bbBQM 7Q` i?2 ; BM iQ Q#i BM Z0F (τ1 ) = Z0F (τ1 −) + K0F (τ1 ) = Z0F (τ1 −) − Z0F (τ1 −) = 0 ). P#b2`pBM; i? i Z0F (t−) = Z0F (t) B7 t Bb MQi M 2p2Mi iBK2 Q7 N - r2 ? p2 i? i Z0F (t) =
μp − ξ μp
M/ i?2`27Q`2- #v `B;?i@+QMiBMmBivZ0F (τ1 ) =
μp − ξ . μp
X h?Bb BM im`M BKTHB2b i? i P @ XbX- Z0F (t) = 0 7Q` HH t ≥ 0X h?2`27Q`2 0 = μp−ξ μp h FBM; 2tT2+i iBQMb- 7Q` HH t ≥ 0- P (X(t) = 0) = 0X h?Bb Bb BM +QMi` /B+iBQM rBi? λ P (X(t) = 0) = μ(1−p) X 3
("`ûK m/- RNd3)c b22 HbQ (q H` M/ M/ o ` Bv - RNd3)X
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9Rj
h?2 :Q`/QMĜL2r2HH L2irQ`F h?Bb Bb +HQb2/ C +FbQM M2irQ`F- rBi? i?2 7QHHQrBM; bT2+B}+Biv,N ¯ i = 0, ri = 0 λ
(1 ≤ i ≤ K) .
URRXdV
AM Qi?2` rQ`/b- i?2`2 Bb MQ BMH2i M/ MQ QmiH2i- M/ i?2`27Q`2 i?2 MmK#2` N Q7 +mbiQK2`b BM i?2 M2irQ`F `2K BMb +QMbi MiX h?2 bi i2 bT +2 Bb E = {(n1 , . . . , nK ) ∈ NK ;
K
ni = N } .
i=1
h?2 i` {+ 2[m iBQMb `2 MQr
λT = λT R ,
URRX3V
M/ bBM+2 R Bb biQ+? biB+ K i`Bt bbmK2/ B``2/m+B#H2- Bi ? b M BM}MBiv Q7 bQHm@ iBQMb- HH KmHiBTH2b Q7 i?2 b K2 p2+iQ` Ui?2 bi iBQM `v /Bbi`B#miBQM Q7 RVX Ai Bb i`m2 i? i i?2 p2+iQ` Q7 p2` ;2 i` {+b i?`Qm;? i?2 bi iBQMb Bb bQHmiBQM Q7 URRX3VX q2 /Q MQi FMQr r?B+? QM2- BM +QMi` bi rBi? i?2 QT2M M2irQ`F- r?2`2 i?2 bQHmiBQM Q7 i?2 i` {+ 2[m iBQM Bb mMB[m2X >Qr2p2`- 7Q` Mv TQbBiBp2 bQHmiBQMr2 b22 #v BMbT2+iBQM i? i- H2iiBM; ρi := λi /μi i?2M
(1 ≤ i ≤ K) ,
1 ρn i G(N, K) i=1 i K
π(n) :=
(n ∈ E)
URRXNV
Bb bi iBQM `v bQHmiBQMX >2`2 G(N, K) Bb i?2 MQ`K HBxBM; 7 +iQ` G(N, K) =
K
n∈NK n1 +···+nK =N
i=1
ρni i .
URRXRyV
LQi2 i? i mM/2` i?2 B``2/m+B#BHBiv bbmKTiBQM 7Q` i?2 `QmiBM; K i`Bt R- i?2 +? BM {X(t)}t≥0 Bb Bib2H7 B``2/m+B#H2- M/ bBM+2 i?2 bi i2 bT +2 Bb }MBi2- Bi Bb TQbBiBp2 `2+m``2Mi rBi? mMB[m2 bi iBQM `v /Bbi`B#miBQMX AM T `iB+mH `- 7Q` +HQb2/ M2irQ`Fbi?2`2 Bb MQ 2`;Q/B+Biv +QM/BiBQMX T` +iB+ H T`Q#H2K rBi? +HQb2/ C +FbQM M2irQ`Fb `2bB/2b BM i?2 +QKTmi iBQM Q7 i?2 MQ`K HBxBM; 7 +iQ` G(N, K)X "`mi2 7Q`+2 bmKK iBQM pB URRXRyV Bb T` +iB+ HHv BM72 bB#H2 7Q` H `;2 TQTmH iBQMb M/fQ` H `;2 M2irQ`FbX AMbi2 /- QM2 K v T`Q+22/ b 7QHHQrbX .2}M2 G(j, ) iQ #2 i?2 +Q2{+B2Mi Q7 z j BM i?2 TQr2` b2`B2b /2p2HQTK2Mi Q7 ∞ 1 ni ni . = ρi z g (z) = 1 − ρi z n =0 i=1 i=1 i
N
(:Q`/QM M/ L2r2HH- RNed)X
9R9
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
h?2 MQ`K HBxBM; 7 +iQ` Bb BM/22/ 2[m H iQ G(N, K)X aBM+2 g (z) = g−1 (z) + ρ zg (z), r2 }M/ i?2 `2+m``2M+2 2[m iBQM G(j, ) = G(j, − 1) + ρ G(j − 1, )
URRXRRV
rBi? i?2 BMBiB H +QM/BiBQMb G(j, 1) = ρj1 (j ≥ 0),
URRXRkV
G(0, ) = 1 ( ≥ 1). h?Bb T`QpB/2b M H;Q`Bi?K 7Q` +QKTmiBM; i?2 MQ`K HBxBM; 7 +iQ`X Ai Bb HbQ Q7 BMi2`2bi iQ #2 #H2 iQ +QKTmi2 i?2 miBHBx iBQM Q7 b2`p2` i- /2}M2/ #v URRXRjV
Ui (N, K) := P (Xi (t) > 0) .
h?Bb + M #2 mb27mH- 7Q` BMbi M+2 BM +QKTmiBM; i?2 p2` ;2 i?`Qm;?Tmi 7`QK bi iBQM i iQ bi iBQM j- i? i Bb- i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b i` Mb72``2/ 7`QK bi iBQM i iQ bi iBQM j BM QM2 mMBi Q7 iBK2, URRXR9V
dij (N, K) = μi Ui (N, K)rij . aBM+2 Ui (N, K) =
π(n) =
n1 +···+nK =N ni >0
1 G(N, K)
K
n1 +···+nK =N ni >0
j=1
n
ρj j ,
r2 b22 i? i G(N, K)Ui (N, K) Bb i?2 +Q2{+B2Mi Q7 z N BM ⎛ ⎛ ⎞⎞ K ∞ ∞ ⎜ ⎝ nj nj ⎟ n n i i . ρj z ⎠⎠ ρi z g˜N,i (z) = ⎝ j=1 j=i
nj =0
ni =1
LQrg˜N,i (z) = gN (z)ρi z, M/ i?2`27Q`2 G(N, K)Ui (N, K) = ρi G(N − 1, K)- i? i BbUi (N, K) = ρi
G(N, K) . G(N − 1, K)
URRXR8V
J `FQpB M M HvbBb Q7 LQM@J `FQpB M Zm2m2b J `FQpB M KQ/2Hb `2 2 bv iQ ? M/H2- #mi i?2v `2 MQi Hr vb BM ;`22K2Mi rBi? `2 HBivX h?2 SQBbbQMB M bbmKTiBQM 7Q` i?2 BMTmi Bb p2`v Q7i2M DmbiB}2/X >Qr2p2`i?2 2tTQM2MiB H bbmKTiBQM 7Q` i?2 b2`pB+2 iBK2 /Bbi`B#miBQM Bb mbm HHv mM`2 HBbiB+X >Qr2p2`- i?2 bim/v Q7 bm+? KQ/2Hb + M #2 `2+ bi BM i?2 7` K2rQ`F Q7 /Bb+`2i2@iBK2 ?K+bXRy Ry
(E2M/ HH- RN8j)X
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9R8
1t KTH2 RRXRXR9, Jf:AfRf∞f7B7QX h?Bb [m2m2 Bb 2t +iHv HBF2 M JfJfRf∞f7B7Q [m2m2- QMHv rBi? ;2M2` H /Bbi`B#miBQM 7Q` i?2 BB/ b2`pB+2 iBK2b b2[m2M+2, G(x) := P (σ1 ≤ x)X h?2 +Q``2bTQM/BM; +QM;2biBQM T`Q+2bb {X(t)}t≥0 Bb MQ HQM;2` J `FQpB MX AMbi2 /- r2 +QMbB/2` i?2 T`Q+2bb {Xn }n≥0 - r?2`2 τ0 = 0 M/ 7Q` n ≥ 1 Xn := X(τn ) , r?2`2 τn Bb i?2 ni? /2T `im`2 iBK2 Ub22 i?2 };m`2 #2HQrVX aBM+2 i?2 +QM;2biBQM T`Q@ +2bb Bb i F2M iQ #2 `B;?i@+QMiBMmQmb- Xn Bb i?2 MmK#2` Q7 +mbiQK2`b i? i +mbiQK2` n H2 p2b #2?BM/ ?BK mTQM +QKTH2iBM; b2`pB+2X
Xn+1
Xn+1
Xn Xn = 0 τn+1
τn
τn
Tn
τn+1
# h?2 2K#2//2/ T`Q+2bb i /2T `im`2 iBK2b
;H M+2 i i?2 };m`2 #Qp2 `2p2 Hb i? i Xn+1 = (Xn − 1)+ + Zn+1 ,
URRXReV
r?2`2 Zn+1 Bb i?2 MmK#2` Q7 +mbiQK2`b ``BpBM; BM i?2 BMi2`p H (αn , αn + σn+1 ]r?2`2 αn = τn B7 Xn > 0 M/ αn = Tn B7 Xn = 0X h?2 iBK2 αn Bb M FtA ∨ Fnσ @biQTTBM; iBK2 Ur?2`2 A Bb i?2 ``Bp H SQBbbQM T`Q+2bb M/ Fnσ Bb i?2 σ@}2H/ ;2M2` i2/ #v i?2 }`bi n b2`pB+2 `2[m2bib σ1 , . . . , σn VX HbQ- i?2 (n + 1)bi b2`pB+2 `2[m2bi σn+1 Bb BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bb #27Q`2 αn M/ Q7 i?2 T`2pBQmb `2[m2bib (σ1 , . . . , σn )X AM T `iB+mH `- #v i?2 bi`QM; J `FQp T`QT2`iv Q7 ?TTbZn+1 = A(αn , αn + σn+1 ] Bb BM/2T2M/2Mi Q7 X0 , . . . , Xn - M/ {Zn }n≥1 Bb M BB/ b2[m2M+2 /Bbi`B#mi2/ b Z = N (0, σ]- r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ BM/2T2M/2Mi Q7 i?2 ` M/QK p `B #H2 σ rBi? b2`pB+2 iBK2 /Bbi`B#miBQM GX h?2`27Q`2- {Xn }n≥0 Bb /Bb+`2i2@iBK2 ?K+- + HH2/ i?2 2K#2//2/ +? BM i /2T `im`2 iBK2bX Aib i` MbBiBQM K i`Bt Bb ⎞ ⎛ a0 a1 a2 a3 . . . ⎜ a0 a1 a2 a3 . . . ⎟ ⎟ ⎜ ⎟ ⎜ P = ⎜ 0 a0 a1 a2 . . . ⎟ , ⎜ 0 0 a0 a1 . . . ⎟ ⎠ ⎝ XX XX XX XX X X X X r?2`2
9Re
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: σ5
σ2
σ1
σ3 T3
T2
T1
σ6
σ4 T4
T5
T6
σ5
σ6
W (t) σ2
σ1
σ3 τ1
τ2
σ4
τ3
τ4
τ5
X(t)
``Bp H T`Q+2bb- 7B7Q rQ`FHQ /- M/ +QM;2biBQM T`Q+2bb2b
∞
ai = P (Zn+1 = i) = P (N (0, σ] = i) =
P (N (0, t] = i) dG(t), 0
i? i Bb
∞
ai =
e−λt
0
(λt)i dG(t). i!
URRXRdV
h?Bb +? BM Bb B``2/m+B#H2 M/ bm{+B2Mi +QM/BiBQM Q7 TQbBiBp2 `2+m``2M+2 Bb E[Z] < 1- r?2`2 b B7 E[Z] > 1- Bi Bb i` MbB2MiXRR HbQ- BM i?2 TQbBiBp2 `2+m``2Mi + b2- i?2 ;2M2` iBM; 7mM+iBQM Q7 i?2 bi iBQM `v /Bbi`B#miBQM Bb ∞
π(i)z i = (1 − E[Z])
i=0
AM pB2r Q7 URRXRdV
∞
E [Z] = 0
∞
∞
gZ (z) =
e 0
i!
i=0
M/
ie
−λt (λt)
−λt
∞ i=0
∞ 0
i
(z − 1)gZ (z) . z − gZ (z)
∞
λ t dG(t) = λE [σ]
dG(t) =
(λt)i i z i!
URRXR3V
0
dG(t) =
∞
e−λt(1−z) dG(t).
0
AM i?2 Jf:AfRf∞f7B7Q [m2m2BM; bvbi2K- i?2 MmK#2` Q7 +mbiQK2`b Xn = X(τn ) H27i #2?BM/ #v i?2 n@i? +mbiQK2` r?2M ?2 H2 p2b i?2 bvbi2K Bb 2t +iHv i?2 RR
a22 Mv i2ti QM J `FQp +? BMb- Q` r Bi 7Q` i?2 KQ`2 ;2M2` H `2bmHib Q7 a2+iBQM RRXjX
RRXRX _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u
9Rd
MmK#2` Q7 +mbiQK2`b ``BpBM; /m`BM; ?2` bQDQm`M- i? i Bb- BM i?2 BMi2`p H #2ir22M ?2` ``Bp H iBK2 Tn M/ ?2` /2T `im`2 iBK2 τn = Tn + Vn - r?2`2 Vn Bb ?2` bQDQm`M iBK2, Xn = A(Tn , Tn + Vn ]. AMpQFBM; i?2 bi`QM; J `FQp T`QT2`iv Q7 ?TTb M/ MQiBM; i? i Vn /2T2M/b QMHv QM (σ1 , . . . , σn ) M/ i?2 T bi Q7 i?2 ``Bp H T`Q+2bb A i iBK2 Tn - Bi 7QHHQrb i? i i 2[mBHB#`BmK∞ π(i)z i = E z N ((0,Vn ]) , i=0
r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ BM/2T2M/2Mi Q7 Vn X LQr ∞ ∞ N ((0,Vn ]) N ((0,v]) E z = /FVn (v) = E z eλv(z−1) /FVn (v), 0
0
r?2`2 FVn (v) Bb i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 Vn X h?Bb b?Qrb BM T `iB+mH ` i? i i?2 /Bbi`B#miBQM Q7 Vn Bb BM/2T2M/2Mi Q7 n M/ i? i- /2MQiBM; #v V Mv ` M/QK p `B #H2 rBi? i?2 b K2 /Bbi`B#miBQM b Vn ∞
π(i)z i =
eλs(z−1) /FV (s).
URRXRNV
i=0
AM bmKK `v, i 2[mBHB#`BmK- Xn Bb /Bbi`B#mi2/ b X = N (V )- r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ- M/ V Bb ` M/QK p `B #H2 BM/2T2M/2Mi Q7 N rBi? i?2 /Bbi`B#miBQM Q7 i?2 bi iBQM `v bQDQm`M iBK2X 1t KTH2 RRXRXR8, :AfJfRf∞f7B7QX h?2 :AfJfRf∞f7B7Q [m2m2 Bb Q7 i?2 b K2 M im`2 b i?2 JfJfRf∞f7B7Q [m2m2- 2t+2Ti 7Q` i?2 ``Bp H T`Q+2bb- r?B+? Bb MQi SQBbbQMB M #mi `2M2r H, i?2 BMi2` ``Bp H iBK2b 7Q`K M BB/ b2[m2M+2 rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM F (x) M/ K2 M λ−1 X h?2 b2`pB+2 iBK2b `2K BM 2tTQM2MiB H rBi? K2 M μ−1 X b BM i?2 Jf:AfRf∞f7B7Q [m2m2- i?2 +QM;2biBQM T`Q+2bb {X(t)}t≥0 Bb MQi J `FQpB M #mi M2p2`i?2H2bb K2M #H2 iQ J `FQpB M M H@ vbBbX AM/22/- B7 r2 H2i Xn := X(Tn −) #2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K b22M mTQM ``Bp H #v i?2 ni? +mbiQK2`r2 ? p2 i?2 `2H iBQM Xn+1 = (Xn + 1 − Zn+1 )+ , r?2`2 ivTB+ H Zn+1 Bb i?2 MmK#2` Q7 /2T `im`2b BM i?2 BMi2`p H (Tn , Tn+1 ]X h?2 b2[m2M+2 {Zn }n≥1 Bb BB/- BM/2T2M/2Mi Q7 X0 M/ /Bbi`B#mi2/ b Z = N (0, τ ]- r?2`2 τ M/ N `2 BM/2T2M/2Mi M/ N Bb M ?TT rBi? BMi2MbBiv μ M/ i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ Bb F X h?2`27Q`2 {Xn }n≥0 Bb M ?K+ M/
∞
bk := P (Zn+1 = k) = 0
e−μt
(μt)k dF (t) . k!
h?2 ii? `Qr Q7 i?2 i` MbBiBQM K i`Bt P Bb i?2`27Q`2
9R3
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: (1 −
i
bk , bi , bi−1 , . . . , b0 , 0, 0, . . .).
k=0
aBM+2 bk > 0 7Q` HH k ≥ 0- P Bb B``2/m+B#H2 M/ T2`BQ/B+X P#b2`p2 i? i ∞
kbk = ρ−1 ,
k=0
r?2`2 ρ =
λ μ
Bb i?2 i` {+ BMi2MbBivX AM/22/∞
kbk =
k=0
∞ k
0 k=0 ∞ ∞
=
∞
0
e−μt
ke−μt
k=0
(μt)k dF (t) k!
(μt)k dF (t) = k!
∞
μt dF (t) = 0
μ . λ
A7 ρ < 1- i?2 +? BM Bb TQbBiBp2 `2+m``2MiX hQ T`Qp2 i?Bb- Bi bm{+2b iQ T`Qp2 i?2 2tBbi2M+2 Q7 bi iBQM `v /Bbi`B#miBQMX q2 K F2 i?2 2/m+ i2/ ;m2bb i? i π ? b i?2 7Q`K π(i) = ξ i (1 − ξ) URRXkyV 7Q` bQK2 ξ ∈ (0, 1)X hQ p2`B7v i?2 +Q``2+iM2bb Q7 i?Bb ;m2bb- r2 Kmbi }M/ ξ ∈ (0, 1) bm+? i? i ∞ ξ j−i+1 bj−i+1 = ξ (i ≥ 1) () j=i−1
M/
∞ ∞ ( bk )ξ j = 1 ,
()
j=0 k=j+1
bBM+2 i?2b2 2[m iBQMb `2 i?2 # H M+2 2[m iBQMb r?2M π Bb ;Bp2M #v URRXkyVX 1[m @ iBQMb UV HH `2/m+2 iQ ξ = gZ (ξ), URRXkRV
∞ k r?2`2 gZ (ξ) = k=0 bk ξ Bb i?2 ;2M2` iBM; 7mM+iBQM Q7 Z1 X aBM+2 HH i?2 bk Ƕb `2 TQbBiBp2- i?2`2 Bb mMB[m2 bQHmiBQM ξ0 Q7 URRXkRV BM (0, 1) B7 M/ QMHv B7 gZ (1) =
∞
kbk > 1 ,
k=1
i? i Bb- B7 M/ QMHv B7 ρ < 1X q2 Kmbi HbQ b iBb7v UVX h?2 H27i@? M/ bB/2 2[m Hb k−1 ∞ k=1
∞
1 − ξk 1 bk ξ = bk ( )= 1 − ξ 1 − ξ j=0 k=1 j
M/ BM pB2r Q7 URRXkRV- i?Bb 2[m Hb
1 (1 1−ξ
1 − b0 −
∞ k=1
− b0 − (ξ − b0 )) = 1X
bk ξ
k
,
RRXkX SPAaaPL auah1Ja
9RN
1t KTH2 RRXRXRe, h?2 r BiBM; iBK2 /Bbi`B#miBQM Q7 :AfJfRf∞f7B7QX M ``BpBM; +mbiQK2` }M/b Xn = X(Tn− ) +mbiQK2`b BM 7`QMi Q7 ?BK- M/ i?2`27Q`2BM i?2 7B7Q /Bb+BTHBM2- ?Bb r BiBM; iBK2 Wn Ui?2 iBK2 #2ir22M ?Bb ``Bp H iBK2 M/ i?2 iBK2 r?2M ?2 bi `ib `2+2BpBM; b2`pB+2V Bb i?2 iBK2 M22/2/ 7Q` i?2 b2`p2` iQ i F2 + `2 Q7 i?2 Xn +mbiQK2`b T`2b2Mi i iBK2 Tn −X AM T `iB+mH `Wn =
Xn
Yj ,
j=1
r?2`2 i?2 Yj `2 BB/ 2tTQM2MiB Hb Q7 K2 M μ−1 M/ +QKKQM +? ` +i2`BbiB+ 7mM+iBQM E[eiuY ] =
μ , μ − iu
M/ `2 BM/2T2M/2Mi Q7 Xn X HbQ- P (Xn = k) = (1 − ξ0 )ξ0k X h?2`27Q`2- BM bi2 /v bi i2E[eiuWn ] = E[ =
∞
eiu
k
j=1
Yj
1{Xn =k} ]
k=0 ∞
∞
k=0
k=0
E[eiuY ]k P (Xn = k) =
(
μ )k (1 − ξ0 )ξ0k . μ − iu
q2 }M/ 7Q` i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 bi iBQM `v r BiBM; iBK2 μ(1 − ξ0 ) (1 − ξ0 ) + ξ0 . μ(1 − ξ0 ) + iu h?Bb Bb i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 ` M/QK p `B #H2 i? i Bb MmHH rBi? T`Q# #BHBiv 1 − ξ0 M/ 2tTQM2MiB H Q7 K2 M [μ(1 − ξ0 )]−1 rBi? T`Q# #BHBiv ξ0 X
_2K `F RRXRXRd h?2 #Qp2 M HvbBb Q7 MQM@J `FQpB M [m2m2b ;Bp2b i?2 bi iBQM@ `v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb 2Bi?2` Dmbi #27Q`2 i?2 ``Bp H iBK2b UM/GI/1/∞V- Q` Dmbi 7i2` i?2 /2T `im`2 iBK2b UGI/M/1/∞VX q? i #Qmi i?2 /Bbi`B#miBQMb Q7 i?2 bi iBQM `v T`Q+2bb Dmbi 7i2` /2T `im`2 iBK2 7Q` i?2 M/GI/1/∞ [m2m2 Q` Dmbi #27Q`2 i?2 ``Bp H iBK2b 7Q` i?2 GI/M/1/∞ [m2m2\ M/ 7Q` #Qi?- r? i #Qmi i?2 bi iBQM `v /Bbi`B#miBQM i M `#Bi` `v }t2/ iBK2\ h?Bb rBHH #2 Mbr2`2/ BM am#b2+iBQM RRX9 BM p2`v ;2M2` H i2`KbX
RRXk
SQBbbQM avbi2Kb
h?2 T`QQ7 Q7 J `FQpB MBiv M/ i?2 +QKTmi iBQM Q7 i?2 BM}MBi2bBK H ;2M2` iQ`b Q7 J `FQpB M [m2m2BM; bvbi2Kb- 2p2M i?2 bBKTH2bi QM2b- /Q2b MQi BKK2/B i2Hv 7QHHQr 7`QK dzM im` HǴ /2b+`BTiBQM BM i2`Kb Q7 MmK#2` Q7 b2`p2`b- b2`pB+2 /Bb+B@ THBM2b- T`BQ`Biv bbB;MK2Mib- 2i+X 6Q` i?2 [m2m2BM; bvbi2Kb +QMbB/2`2/ bQ 7 ` Bi r b /KBii2/ i? i i?2B` +QM;2biBQM T`Q+2bb2b r2`2 ?K+ M/ ?2m`BbiB+ /2`Bp iBQM Q7
9ky
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
i?2B` BM}MBi2bBK H ;2M2` iQ`b r b ;Bp2MX q2 MQr ;Bp2 i?2 7Q`K H T`QQ7b BM [mBi2 ;2M2` H 7` K2rQ`F i? i HbQ ++QKKQ/ i2b [m2m2BM; M2irQ`Fb M/ KQ`2 +QKTH2t bvbi2KbXRk q2 bi `i rBi? +QMbi`m+iBQM bHB;?iHv KQ`2 ;2M2` H i? M M22/2/ 7Q` J `FQpB M [m2m2BM; bvbi2Kb UBM T `iB+mH ` i?2 b2`pB+2 iBK2b Q` BMi2` ``Bp H iBK2b M22/ MQi #2 2tTQM2MiB H ` M/QK p `B #H2bVX h?Bb Bb i?2 ;bKT U:2M2` HBx2/ a2KB@J `FQp S`Q+2bb2bVRj +QMbi`m+iBQMX h?2 ;bKT *QMbi`m+iBQM h?2 ` iBQM H2 #2?BM/ i?Bb +QMbi`m+iBQM Bb i? i i?2`2 `2 bQm`+2b Q7 2p2Mib i? i + M #2 2Bi?2` +iBp2 Q` BM +iBp2 /2T2M/BM; QM i?2 bi i2 Q7 i?2 T`Q+2bbX +iBp2 bQm`+2b i?2M +QKT2i2 iQ T`Q/m+2 M 2p2Mi i? i rBHH #2 i?2 + mb2 7Q` bi i2 i` MbBiBQMX h?2`2 `2 irQ BM;`2/B2Mib, U V M BMTmi- r?B+? +QMbBbib Q7 ` M/QK p `B #H2b Q` TQBMi T`Q+2bb2b- M/ U#V i` MbBiBQM K2+? MBbK- i? i i` Mb7Q`Kb i?2 BMTmi BMiQ i?2 bi i2 T`Q+2bbX AM i?2 JfJfRf∞ [m2m2 7Q` BMbi M+2- i?2 BMTmi +QMbBbib Q7 i?2 BMi2` ``Bp H iBK2b M/ i?2 b2`pB+2 iBK2b- i?2 i` MbBiBQM K2+? MBbK /2b+`B#2b i?2 7mM+iBQMBM; Q7 i?2 bvbi2K UBM+Hm/BM; T`BQ`BiB2b- b2`pB+2 /Bb+BTHBM2- M/ bQ QMV M/ i?2 bi i2 T`Q+2bb Bb i?2 +QM;2biBQM T`Q+2bbX AM i?Bb bBKTH2 2t KTH2 i?2`2 `2 irQ bQm`+2b, QM2 ;2M2` i2b i?2 BMi2` ``Bp H iBK2b- i?2 Qi?2` T`Q/m+2b i?2 b2`pB+2 iBK2bX q2 b? HH MQr T`Q+22/ rBi? i?2 ;2M2` H + b2- M/ }`bi /2b+`B#2 i?2 i` MbBiBQM K2+? MBbKX G2i S M/ E #2 irQ +QmMi #H2 b2ib- i?2 b2i Q7 2p2Mi bQm`+2b UbQm`+2b 7Q` b?Q`iV M/ i?2 b2i Q7 bi i2b- `2bT2+iBp2HvX qBi? 2 +? bQm`+2@bi i2 T B` (s, i) Bb bbQ+B i2/ MmK#2` c(s, i) ≥ 0 K2 bm`BM; i?2 BMi2MbBiv Q7 +iBpBiv Q7 bQm`+2 s r?2M i?2 bi i2 Bb i UBM [m2m2BM; bvbi2Kb i?Bb HHQrb mb iQ +QMbB/2`- 7Q` BMbi M+2- i?2 b2`pB+2 bT22/ Q7 b2`p2`VX M 2p2Mi bQm`+2 s Bb b B/ iQ #2 +iBp2 BM bi i2 i B7 c(s, i) > 0X h?2 +QHH2+iBQM Q7 bQm`+2b s +iBp2 BM bi i2 i rBHH #2 /2MQi2/ #v A(i)X qBi? 2 +? bQm`+2@bi i2 T B` (s, i) Bb bbQ+B i2/ T`Q# #BHBiv /Bbi`B#miBQM p(s, i, ·) QM EX h?Bb /Bbi`B#miBQM i2HHb mb ?Qr iQ b2H2+i i?2 M2ti bi i2 r?2M M 2p2Mi Bb i`B;;2`2/ #v i?2 bQm`+2 s +iBp2 BM bi i2 iX h?2 [m /`mTH2 (E, S, p(·, ·, ·), c(·, ·)) URRXkkV Bb + HH2/ i?2 i` MbBiBQM K2+? MBbKX h?2 +QHH2+iBQM T Q7 i`BTH2b (s, i, j) bm+? i? i c(s, i)p(s, i, j) > 0 Bb + HH2/ i?2 b2i Q7 HHQr2/ i` MbBiBQMbX A7 (s, i, j) ∈ T - QM2 b vb i? i i` MbBiBQM i → j + M #2 i`B;;2`2/ #v M 2p2Mi T`Q/m+2/ #v bQm`+2 s i? i Bb +iBp2 BM bi i2 iX q2 MQr /2b+`B#2 i?2 BMTmiX 1 +? bQm`+2 s ∈ S + M ;2M2` i2 b K Mv BM/2T2M@ /2Mi ` M/QK p `B #H2b b M22/2/ rBi? i?2 b K2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Rk h?Bb r?QH2 b2+iBQM + M ?Qr2p2` #2 bFBTT2/ B7 QM2 Bb b iBb}2/ #v i?2 ?2m`BbiB+ `;mK2Mib Q7 i?2 }`bi b2+iBQM Q7 i?2 +m``2Mi +? Ti2`X Rj (J ii?2b- RNek)X
RRXkX SPAaaPL auah1Ja
9kR
Gs (x)X AM //BiBQM- HH i?2 ` M/QK p `B #H2b ;2M2` i2/ #v i?2 p `BQmb bQm`+2b `2 BM/2T2M/2MiX h?2 bi i2 T`Q+2bb Bb Q#i BM2/ 7`QK i?2 BMTmi M/ i?2 i` MbBiBQM K2+? MBbK b 7QHHQrbX h?2 bi i2 T`Q+2bb Bb BM 7 +i DmKT T`Q+2bb- i FBM; M2r p Hm2b i i?2 bm++2bbBp2 i` MbBiBQM iBK2b τ˜n Un ≥ 0V r?2`2 τ˜0 = 0X i 2 +? i` MbBiBQM iBK2 τ˜n i?2 bBim iBQM Q7 i?2 bvbi2K Bb /2b+`B#2/ #v i?2 bi i2 ˜n ∈ E X (˜ τn ) = X M/ +QHH2+iBQM Q7 +QMbmK #H2 ` M/QK p `B #H2b Ys (n)
(s ∈ S) .
h?2 M2ti i` MbBiBQM Q++m`b i iBK2 τ˜n+1 = τ˜n +
Y (n) . s inf ˜ ˜n s∈A(Xn ) c s, X
URRXkjV
˜ n i?2 bQm`+2 `2 HBxBM; i?2 BM}KmK BM URRXkjV-R9 V i?2 M2ti .2MQiBM; #v s˜n ∈ A X ˜ n+1 Bb +?Qb2M ++Q`/BM; iQ bi i2 X ˜ n = i, s˜n = s = p (s, i, j) , ˜ n+1 = j | X URRXk9V P X ˜ n−1 , s˜n−1 ;Bp2M X ˜ n M/ s˜n X ˜ 0 , s˜0 , . . . , X M/ Bb BM/2T2M/2Mi Q7 X i i?2 Q`B;BM Q7 iBK2 τ˜0 = 0- i?2 b2i Ys (0) Us ∈ SV Bb M BM/2T2M/2Mi +QHH2+iBQM Q7 ` M/QK p `B #H2b rBi? i?2 `2bT2+iBp2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Gs (x) Us ∈ SVX h?2 b2i Q7 +QMbmK #H2 ` M/QK p `B #H2b Ys (n + 1) Us ∈ SV p BH #H2 i i` MbBiBQM iBK2 τ˜n+1 Bb `2 +im HBx2/ b 7QHHQrb, ˜n Ys (n + 1) = Ys (n) B7 s ∈ A X Ui?2 +QMbmK #H2 p `B #H2b Q7 M BM +iBp2 bQm`+2 `2 MQi +im HHv +QMbmK2/V M/ ˜ n (˜ ˜ n s = s˜n Ys (n + 1) = Ys (n) − c s, X τn+1 − τ˜n ) B7 s ∈ A X Ui?2 +QMbmK #H2 p `B #H2b Q7 M +iBp2 bQm`+2 s /2TH2i2 i i?2 ` i2 c(s, i) r?2M i?2 bvbi2K Bb BM bi i2 iV- M/ 7Q` s = s˜n - i?2 +QMbmK2/ ` M/QK p `B #H2b Ys (n) Bb `2TH +2/ #v M2r ` M/QK p `B #H2 Ys (n + 1) rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Gs M/ BM/2T2M/2Mi Q7 Ys (k) Uk ≤ n, s ∈ SV Ui?2 +QMbmK #H2 p `B #H2 Q7 i?Bb T `iB+mH ` bQm`+2 r b 2MiB`2Hv +QMbmK2/ M/ i?2`27Q`2 ? b iQ #2 `2TH +2/VX q2 b? HH MQr brBi+? iQ SQBbbQM bvbi2Kb- r?B+? `2 +HQb2Hv `2H i2/ iQ ;bKTb r?2M i?2 BMTmi ` M/QK p `B #H2b ? p2 M 2tTQM2MiB H /Bbi`B#miBQMX R9
bbmK2/ mMB[m2X h?Bb ? b iQ #2 p2`B}2/ BM 2 +? + b2X Ai Bb i`m2 Q7 i?2 SQBbbQM bvbi2Kb #2HQrX
9kk
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
SQBbbQM avbi2Kb h?2 i` MbBiBQM K2+? MBbK Bb i?2 b K2 b BM i?2 ;bKT +QMbi`m+iBQMX h?2 BMTmi Bb 7Q`K HHv UMQi 2bb2MiB HHvV /Bz2`2Mi- BM i? i Bi +QMbBbib Q7 ?TTb- HbQ + HH2/ i?2 /`BpBM; ?TTbX qBi? 2 +? bQm`+2 s ∈ S Bb bbQ+B i2/ M BMi2MbBiv- Q` ` i2- λs > 0- M/ i?2`2 Bb ;Bp2M 7 KBHv Q7 BM/2T2M/2Mi ?TTb Ns,i,j U(s, i, j) ∈ T V rBi? i?2 `2bT2+iBp2 BMi2MbBiB2b as,i,j := λs c(s, i)p(s, i, j) (s, i, j) ∈ T ) . G2i aij :=
λs c(s, i)p(s, i, j) (i, j ∈ E)
URRXk8V
s∈S
M/ ai :=
aij
(i ∈ E)
URRXkeV
j∈E
M/ BMi`Q/m+2 i?2 bi #BHBiv bbmKTiBQM, ai < ∞ (i ∈ E) .
URRXkdV
h?2 +QMbi`m+iBQM Q7 i?2 bi i2 T`Q+2bb 7`QK i?2 # bB+ /`BpBM; ?TTb Bb bBKBH ` iQ i?2 QM2 mb2/ BM am#b2+iBQM kX8 7Q` DmKT ?K+b- QMHv rBi? KQ`2 /2i BHb +Q``2@ bTQM/BM; iQ i?2 dzM im` HǴ /2b+`BTiBQM Q7 i?2 bvbi2K +QMbB/2`2/X h?2 bi i2 T`Q+2bb {X(t)}t≥0 i F2b Bib p Hm2b BM E ∪{Δ}- r?2`2 Δ Bb M 2H2K2Mi QmibB/2 E- M/ Bb +QM@ bi`m+i2/ b 7QHHQrbX Uh?2 `2 /2` b?QmH/ ? p2 BM KBM/ h?2Q`2K kXRX8 +QM+2`MBM; i?2 bmT2`TQbBiBQM Q7 BM/2T2M/2Mi ?TTb- h?2Q`2K kXRXe- i?2 bQ@+ HH2/ +QKT2iBiBQM i?2@ Q`2K- M/ h?2Q`2K kXRXd- i?2 bi`QM; J `FQp T`QT2`iv Q7 ?TTbXV h?2 BMBiB H bi i2 X(0) Bb ` M/QK p `B #H2 BM/2T2M/2Mi Q7 i?2 ?TTb Ns,i,j Ƕb bm+? i? i (s, i, j) ∈ T X amTTQb2 i? i i?2 bi i2 i iBK2 t Bb X(t) = i ∈ EX h?2 +iBp2 bQm`+2b `2 i?2M HH i?2 bQm`+2b s ∈ A(i)X h?2 ?TTb +QKT2iBM; 7Q` i?2 /2i2`KBM iBQM Q7 i?2 M2ti i` MbBiBQM 7i2` t `2 i?2 Ns,i,j Ƕb bm+? i? i s ∈ A(i) M/ p(s, i, j) > 0X Ns0 ,i,j0 Bb i?2 ?TT T`Q/m+BM; i?2 }`bi 2p2Mi bi`B+iHv 7i2` t- b v i iBK2 t + τ˜X h?2 bi i2 `2K BMb mM+? M;2/ BM [t, t + τ˜]- M/ i` MbBiBQM 7`QK bi i2 i iQ bi i2 j0 Q++m`b i iBK2 t + τ˜X h?2 bi #BHBiv +QM/BiBQM ai < ∞ 2Mbm`2b i? i τ˜ > 0- bBM+2 ai Bb i?2 bmK Q7 i?2 BMi2MbBiB2b Q7 i?2 Ns,i,j BM +QKT2iBiBQM r?2M i?2 bi i2 Bb iX h?Bb `2+m`bBp2 T`Q+2/m`2 bi `i2/ 7`QK iBK2 0 ;Bp2b b2[m2M+2 Q7 i` MbBiBQM ˜1, X ˜ 2 , . . .- r?2`2 X ˜ n = X(˜ iBK2b τ˜1 , τ˜2 , . . .X h?2 2K#2//2/ T`Q+2bb Bb X τn )X SQbbB#Hvi?2`2 Bb ` M/QK BM/2t K bm+? i? i τ˜K = ∞- BM r?B+? + b2 r2 H2i τ˜k = ∞ ˜k = X ˜ K 7Q` k ≥ KX h?Bb T`Q+2/m`2 /2}M2b X(t) 7Q` HH t ∈ [0, τ˜∞ )- r?2`2 M/ X τ˜∞ = limn↑∞ τ˜n Bb i?2 2tTHQbBQM iBK2X 6Q` t ≥ τ˜∞ - b2i X(t) = ΔX 6`QK MQr QM- r2 b? HH bbmK2 i? i τ˜∞ = ∞- M/ i?2`27Q`2 i? i {X(t)}t≥0 Bb `2;mH ` DmKT ?K+ QM EX UAM HH i?2 [m2m2BM; M2irQ`Fb i? i r2 b? HH +QMbB/2`i?Bb bbmKTiBQM ?QH/b i`m2- rBi? M 2 bv T`QQ7XV Ai Bb i?2M +H2 ` i? i {X(t)}t≥0 Bb `2;mH ` J `FQp DmKT ?K+ rBi? bi i2 bT +2 EX q2 MQr HQQF BM /2i BH i i?2 }`bi i` MbBiBQM- Q++m``BM; i iBK2 τ˜1 - r?B+? Bb ivT@ B+ H Q7 HH Qi?2` i` MbBiBQMbX amTTQb2 i? i X(0) = iX ai `iBM; 7`QK i- i?2 +QKT2iBM; T`Q+2bb2b `2 i?2 Ns,i,j r?2`2 s ∈ A(i) M/ r?2`2 j b iBb}2b p(s, i, j) > 0X A7 QM2 Bb
RRXkX SPAaaPL auah1Ja
9kj
MQi BMi2`2bi2/ BM i?2 T `iB+mH ` bQm`+2 i`B;;2`BM; i?2 i` MbBiBQM- QM2 + M HmKT HH i?2 Ns,i,j Us ∈ A(i)V iQ 7Q`K bBM;H2 ?TT Nij Q7 BMi2MbBiv aij X q2 `2 i?2M BM i?2 b K2 bBim iBQM b BM am#b2+iBQM kX8- rBi? QM2 2t+2TiBQM, aii K v #2 bi`B+iHv TQbB@ ˜ n BMbi2 / Q7 Xn , aij BMbi2 / Q7 iBp2X h?Bb KQiBp i2b i?2 MQi iBQMb τ˜n BMbi2 / Q7 τn , X qij - 2i+X- r?B+? `2 i?2`2 iQ r `M mb i? i Tb2m/Q@i` MbBiBQM + M Q++m`- i? i Bb- dzi` MbBiBQMǴ 7`QK i iQ i?2 b K2 bi i2 iX Pi?2`rBb2- i?2 bBim iBQM Bb bBKBH ` iQ i? i ˜ 0 := X(0), Q7 am#b2+iBQM kX8- M/ r2 + M ;Bp2 i?2 7QHHQrBM; `2bmHi- H2iiBM; X ˜ n }n≥0 Bb /Bb+`2i2@iBK2 ?K+ rBi? p Hm2b BM E M/ i` MbBiBQM K i`Bt (α) {X {˜ pij }i,j∈E ;Bp2M #v a p˜ij = aiji B7 ai > 0 URRXk3V p˜ii = 1 B7 ai = 0. (β) 6Q` HH n ≥ 1- x ∈ R+ ˜n, X ˜ n−1 , · · · , X ˜ 0 , τ˜n , τ˜n−1 , · · · , τ˜1 ) = e−ai X˜n . P (˜ τn+1 − τ˜n ≥ x | X
URRXkNV
Sb2m/Q@i` MbBiBQMb aQ 7 ` i?2`2 Bb MQi?BM; 2bb2MiB HHv M2r rBi? `2bT2+i iQ i?2 +QMbi`m+iBQM Q7 bm#@ b2+iBQM kX8 2t+2Ti i? i r2 HHQr Tb2m/Q@i` MbBiBQMb r?B+? + M TH v `QH2 BM bQK2 bBim iBQMb- BM 722/# +F [m2m2b- 7Q` BMbi M+2 Ub22 1t KTH2 RRXkX9VX G2i {τn }n≥0 #2 i?2 b2[m2M+2 Q7 dzi`m2Ǵ i` MbBiBQM Ui? i Bb- 7`QK bQK2 bi i2 iQ /Bz2`2Mi bi i2V iBK2b Q7 i?2 bi i2 T`Q+2bb- M/ H2i {Xn }n≥0 #2 i?2 2K#2//2/ T`Q+2bb i i?2b2 i`m2 i` MbBiBQM iBK2bX A7 QM2 Bb BMi2`2bi2/ QMHv BM i`m2 i` MbBiBQMb- Bi bm{+2b iQ ;2i `B/ Q7 HH i?2 ?TTb Ns,i,i #2+ mb2 i?2v /Q MQi z2+i i?2 bi i2 T`Q+2bbX q2 i?2`27Q`2 `2@ i`B2p2 i?2 bBim iBQM Q7 bm#b2+iBQM kX8X h?2 bi i2 T`Q+2bb Bb M ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` qij = λs c(s, i)p(s, i, j), URRXjyV s∈S
r?2`2 j = i- M/ qi =
λs c(s, i)p(s, i, j).
URRXjRV
j∈E,j=i s∈S
q2 MQr T`Q+22/ iQ K F2 i?2 +QMM2+iBQM rBi? i?2 dzM im` HǴ ;bKTX AM pB2r Q7 i?Bb- r2 }`bi ;Bp2 M Hi2`M iBp2 r v Q7 HQQFBM; i i?2 ;bKT +QMbi`m+iBQM- mbBM; K `F2`bX J `F2`b qBi? 2 +? bQm`+2 s ∈ S Bb bbQ+B i2/ M mM/2H v2/ `2M2r H TQBMi T`Q+2bb Ns ≡ s s {Tns }n≥0 - /2}M2/ 7Q` n = 0 #v T0s = 0 M/ 7Q` n ≥ 0 #v Tn+1 − Tns := Sn+1 s r?2`2 {Sn }n≥1 Bb M BB/ b2[m2M+2 rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Gs (x)X h?2 7 KBHv Q7 TQBMi T`Q+2bb2b {Ns }s∈S Bb BM/2T2M/2MiX 1 +? TQBMi T`Q+2bb ? b K `F2` i? i 2Bi?2` `2K BMb biBHH Q` KQp2b +QMiBMmQmbHv iQ i?2 `B;?i UrBi? p `B #H2 bT22/VX i iBK2 t- Bi ? b `2 +?2/ TQbBiBQM t ˜ (u) du. Ms (t) = c s, X 0
9k9
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
AM Qi?2` rQ`/b- i?2 K `F2` QM Ns KQp2b i bT22/ c (s, i) r?2M i?2 ;bKT Bb BM bi i2 iX i iBK2 τ˜n - i?2 b2i Q7 +QMbmK #H2 (Ys (n + 1) , s ∈ S) Bb ;Bp2M #v i?2 7Q`r `/ `2+m``2M+2 iBK2b Ub22 i?2 };m`2 #2HQrV Ys (n + 1) = inf {Tks ; Tks > Ms (τn )} − Ms (τn ) . k
URRXjkV
Ms1 (˜ τn ) Ys1 (n + 1) s1 τn ) Ms2 (˜ Ys2 (n + 1) = 0 s˜n = s2 Ms3 (˜ τn ) Ys3 (n + 1) s3 Ms4 (˜ τn ) Ys4 (n + 1) s4 t=0 J `F2`b M/ +QMbmK #H2 p `B #H2b i ;Bp2M BMbi Mi t- QMHv i?2 K `F2`b QM i?2 TQBMi Ns bm+? i? i T`Q+2bb2b ˜ ˜ s ∈ A X(t) `2 KQpBM; i i?2 `2bT2+iBp2 bT22/b c s, X(t) - M/ i?2 }`bi QM2 i? i K22ib i?2 M2ti 2p2Mi Q7 Bib TQBMi T`Q+2bb T`QpQF2b i` MbBiBQMX h?2 iBK2 Q7 ˜ n+1 `2 ;Bp2M #v URRXkjV M/ URRXk9V- `2bT2+@ i` MbBiBQM τ˜n+1 M/ i?2 M2r bi i2 X iBp2Hv- r?2`2 s˜n Bb i?2 bQm`+2 +Q``2bTQM/BM; iQ i?2 K `F2` i? i }`bi `2 +?2/ M 2p2Mi Q7 Bib TQBMi T`Q+2bbX *QMM2+iBM; ;bKT M/ SQBbbQM avbi2Kb q?2M Gs (x) = 1 − e−λs x 7Q` HH s ∈ S- i?2 ;bKT +QMbi`m+iBQM ;Bp2b i?2 b K2 bi i2 T`Q+2bb b i?2 Tm`2Hv SQBbbQMB M /2b+`BTiBQMX AM/22/- BM i?Bb +QMbi`m+iBQM- i iBK2 τ˜n - i?2`2 Bb +QKT2iBiBQM KQM; i?2 SQBbbQM T`Q+2bb2b {Sτ˜n N(s,i,j) }(s,j)∈T (i) , r?2`2 T (i) = {(s, j) ; (s, i, j) ∈ T }X "v i?2 bi`QM; J `FQp T`QT2`iv 7Q` ?TTbi?2b2 ?TTb `2 BM/2T2M/2Mi M/ /Bbi`B#mi2/ b {N(s,i,j) }(s,j)∈T (i) ,
URRXjjV
M/ KQ`2Qp2`- i?2v `2 BM/2T2M/2Mi Q7 r? i ? TT2M2/ #27Q`2 iBK2 τ˜n X q2 + M i?2`27Q`2 `2bi`B+i Qm` ii2MiBQM iQ i?2 + b2 τ˜n = τ˜0 = 0 rBi?Qmi HQbb Q7 ;2M2` HBivX
RRXkX SPAaaPL auah1Ja
9k8
h?2 }`bi i` MbBiBQM iBK2 τ˜1 Bb Q#i BM2/ #v +QKT2iBiBQM KQM; i?2 ?TTb BM ˜ 0 = iX h?Bb +QKT2iBiBQM + M #2 Q`; MBx2/ BM irQ URRXjjV- r?2`2 r2 bmTTQb2 i? i X bi ;2bX 6B`bi i?2`2 Bb +QKT2iBiBQM KQM; i?2 {N(s,i) }s∈A(i) r?2`2 N(s,i) :=
N(s,i,j) ,
j ; (s,i,j)∈ T
r?B+? T`Q/m+2b i?2 i` MbBiBQM iBK2 τ˜1 = inf Z(s,i) (0), s∈A(i)
r?2`2 Z(s,i) (0) Bb i?2 }`bi TQBMi Q7 N(s,i) - M 2tTQM2MiB H ` M/QK p `B #H2 rBi? K2 M (λs c (s, i))−1 X PM2 + M r`Bi2 Z(s,i) (0) =
Ys (0) , c (s, i)
r?2`2 Ys (0) Bb M 2tTQM2MiB H ` M/QK p `B #H2 rBi? K2 M λ−1 s - M/ i?2`27Q`2 τ˜1 − τ˜0 =
inf ˜0 ) s∈A(X
Y (0) s , ˜0 c s, X
M/ r2 `2i`B2p2 URRXkjV Q7 i?2 ;bKT +QMbi`m+iBQMX G2i s˜0 #2 i?2 bQm`+2 ;BpBM; i?2 ˜ 1 Bb +?Qb2M #v +QKT2iBiBQM KQM; i?2 SQBbbQM T`Q+2bb2b BM}KmKX h?2 M2r bi i2 X Ns˜0 ,i,j - j ∈ E, p(˜ s0 , i, j) > 0X h?2`27Q`2 ˜ 0 = i, s˜0 = s = p (s, i, j) , ˜1 = j | X P X M/ r2 `2i`B2p2 URRXk9V Q7 i?2 ;bKT +QMbi`m+iBQMX J `FQpB M Zm2m2b b SQBbbQM avbi2Kb q2 b? HH MQr `2pBbBi bQK2 2t KTH2b BM Q`/2` iQ b?Qr ?Qr i?2v + M #2 KQ/2HH2/ b SQBbbQM bvbi2KbX h?Bb rBHH BM T `iB+mH ` T`Qp2 i? i i?2 [m2m2b i?2`2Q7 `2 ?K+ M/ T`QpB/2 K i?2K iB+ HHv `B;Q`Qmb /2`Bp iBQM Q7 i?2B` BM}MBi2bBK H ;2M2` iQ`X 1t KTH2 RRXkXR, JfJfRf∞f7B7QX q2 i F2 S = {α, δ}- r?2`2 α bi M/b 7Q` ``Bp H M/ δ 7Q` /2T `im`2X h?2 bi i2 bT +2 Bb E = N = {0, 1, 2, . . .}- i ∈ E `2T`2b2MiBM; i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K UBM i?2 r BiBM; HBM2 Q` #2BM; ii2M/2/ #v i?2 b2`p2`VX A7 i > 0- A(i) = {α, δ}- K2 MBM; i? i r?2M i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K Bb bi`B+iHv TQbBiBp2- M ``Bp H Q` /2T `im`2 +QmH/ Q++m`X A7 i = 0- A(0) = {α}- K2 MBM; i? i B7 i?2 bvbi2K Bb 2KTiv- QM2 + MMQi 2tT2+i /2T `im`2- QMHv M ``Bp HX HH i?2 +iBpBiv BMi2MbBiB2b i? i `2 MQi MmHH `2 i F2M 2[m H iQ 1X HbQ- λα = λ M/ λδ = μ- #2+ mb2 r2 r Mi i?2 BMi2` ``Bp H iBK2b iQ #2 2tTQM2MiB H ` M/QK p `B #H2b rBi? K2 M λ−1 M/ i?2 b2`pB+2 `2[m2bib iQ #2 2tTQM2MiB H rBi? K2 M μ−1 X
9ke
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
A7 BM bi i2 i ≥ 0 i` MbBiBQM Bb i`B;;2`2/ #v M 2p2Mi Q7 bQm`+2 s0 = α ∈ A(i)i?2 M2ti bi i2 Bb j = i + 1 U M ``Bp H BM+`2 b2b i?2 MmK#2` Q7 +mbiQK2`b #v 1Vc r?2`2 b BM bi i2 i > 0- B7 s0 = δ ∈ A(i)- i?2 M2ti bi i2 Bb j = i − 1X h?mbp(α, i, i + 1) = 1 7Q` HH i ≥ 0- M/ p(δ, i, i − 1) = 1 7Q` HH i > 0X h?2 };m`2 #2HQr /2TB+ib ivTB+ H 2pQHmiBQM Q7 i?2 bi i2 T`Q+2bb- bi `iBM; 7`QK bi i2 i0 = 0X α U ``Bp HbV
δ X(t) 2 1 0 t=0
t ;bKT +QMbi`m+iBQM Q7 JfJfRf∞f7B7Q
6Q`KmH URRXjyV K F2b i?2 BM}MBi2bBK H ;2M2` iQ` `2 / #H2 7`QK i?2 M im` H /2b+`BTiBQM BM i2`Kb Q7 i?2 ` i2b λs - i?2 bT22/b c(s, i)- M/ i?2 i` MbBiBQM T`Q# @ #BHBiB2b p(s, i, j)X 6Q` i?2 #Qp2 JfJfRf∞f7B7Q [m2m2- Bi + M #2 `2 /BHv +?2+F2/ i? i qi,i+1 = λ, qi,i−1 = μ1{i≥0} .
PM2 b?QmH/ TQBMi Qmi i? i 2p2M 7Q` i?2 bBKTH2bi 2t KTH2ěi?2 JfJfRf∞ [m2m2ěi?2 SQBbbQM bvbi2K /2b+`BTiBQM /Bz2`b 7`QK i?2 mbm H `2;2M2` iBp2 /2b+`BT@ iBQM Q7 i?2 +Q``2bTQM/BM; J `FQp +? BMX h?2 SQBbbQM bvbi2K /2b+`BTiBQM ;Bp2b HH i?2 /2i BHb #Qmi i?2 K2+? MBbK Q7 ;2M2` iBQM Q7 i` MbBiBQMb- M/ Bib +QHH Tb2 BMiQ i?2 mbm H `2;2M2` iBp2 /2b+`BTiBQM Bb ++QKT MB2/ #v HQbb Q7 BM7Q`K iBQM +QM+2`MBM; i?2 }M2 /2i BHbX AM/22/- 7`QK i?2 TQBMi Q7 pB2r Q7 i?2B` /Bbi`B#miBQMb- QM2 + MMQi /BbiBM;mBb? #2ir22M M JfJfRf∞f7B7Q [m2m2 i? i ? b M BM}MBi2bBK H ;2M2` iQ` Q7 i?2 7Q`K A = (λ + μ)(K − I), r?2`2
⎛
0
⎜ μ ⎜ λ+μ K=⎜ 0 ⎝ XX X
1 0 μ λ+μ
XX X
0 λ λ+μ
0 XX X
0 0 λ λ+μ
XX X,
⎞ ··· · · ·⎟ ⎟ , · · ·⎟ ⎠
RRXkX SPAaaPL auah1Ja
9kd
M/ i?2 mMB7Q`K J `FQp +? BM +QMbi`m+i2/ 7`QK SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ + μ M/ ` M/QK r HF rBi? `2~2+iBM; # ``B2` 0 rBi? i?2 i` MbBiBQM K i`Bt KX Ai Bb +im HHv MQi 2 bv iQ b22 7`QK i?2 mMB7Q`K /2b+`BTiBQM i? i QM2 Bb /2 HBM; rBi? M JfJfRf∞ [m2m2X h?Bb /B{+mHiv BM+`2 b2b rBi? i?2 bBx2 M/ i?2 +QKTH2tBiv Q7 i?2 bvbi2K U[m2mBM; M2irQ`Fb- 7Q` BMbi M+2VX 1t KTH2 RRXkXk, JfJfEfy- Q` 1`H M; [m2m2X PM2 TQbbB#H2 i` MbBiBQM K2+? MBbK Bb i?2 7QHHQrBM;, S E A(i) c(s, i)
= = = =
{0, 1, . . . , K}, { HH bm#b2ib Q7 S +QMi BMBM; y}, { HH 2p2Mi bQm`+2b BM i}, 1 7Q` HH s ∈ A(i).
h?2 i` MbBiBQM T`Q# #BHBiB2b p(·, ·, ·) rBHH #2 /2b+`B#2/ BM 72r HBM2bX h?2 SQBbbQM T`Q+2bb2b +Q``2bTQM/BM; iQ 2p2Mi bQm`+2b 1, 2, . . . , K HH ? p2 i?2 b K2 BMi2MbBiv μ > 0- r?2`2 b i?2 SQBbbQM T`Q+2bb Q7 2p2Mi bQm`+2 0 ? b i?2 BMi2MbBiv λ > 0X aQm`+2 0 +Q``2bTQM/b- b BM i?2 T`2pBQmb 2t KTH2- iQ i?2 ``Bp HbX 6Q` Mv bi i2 i ∈ E, i − {0} Bb bm#b2i Q7 {1, . . . , K}- `2T`2b2MiBM; i?2 b2`p2`b i? i `2 #mbv r?2M i?2 bvbi2K Bb BM i? i bi i2X A7 X(t) = i- i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K Bb |i|−1- r?2`2 |i| /2MQi2b i?2 + `/BM HBiv Q7 i?2 b2i iX M 2p2Mi bQm`+2 s s U1 ≤ s ≤ KV +Q``2bTQM/b iQ b2`p2` s M/ i?2 b2[m2M+2 Sns = Tns − Tn−1 (n ≥ 1) Bb i?2 b2[m2M+2 Q7 bm++2bbBp2 b2`pB+2 iBK2b T`QpB/2/ #v i?Bb b2`p2`X q2 b? HH MQr ;Bp2 i?2 i` MbBiBQM T`Q# #BHBiB2b p(·, ·, ·)X 6Q` i?Bb r2 M22/ iQ /BbiBM;mBb? irQ bi i2b, i0 = {0} M/ iK = {0, 1, . . . , K} +Q``2bTQM/BM; `2bT2+iBp2Hv iQ M 2KTiv M/ 7mHH bvbi2KX A7 i = i0 M/ i` MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 s ∈ i r?2`2 s > 0- i?Bb K2 Mb i? i b2`p2` s `2H2 b2b +mbiQK2`- M/ i?2 M2ti bi i2 Bb i?2M i − {s}X A7 i = iK M/ i` MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 0 Ur?B+? K2 Mb i? i M2r +mbiQK2` ``Bp2bV- i?2 M2ti bi i2 Bb i + {r}- r?2`2 r Bb +?Qb2M i ` M/QK BM S − i- i?2 b2i Q7 B/H2 b2`p2`b BM bi i2 iX A7 i = iK M/ i` MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 0 U M2r +mbiQK2` ``Bp2bV- i?2 bi i2 /Q2b MQi +? M;2 Ui?2 M2r +mbiQK2` Bb `2D2+i2/ bBM+2 HH i?2 b2`p2`b `2 #mbvVX h?2 BM}MBi2bBK H ;2M2` iQ` + M #2 +QKTmi2/ 7`QK 7Q`KmH URRXjyV M/ i?Bb ;Bp2b 1 B7 i ∈ E, r ∈ {1, . . . , K}, r ∈ i, λ K+1−|i| qi,i+{r} = qi,i−{r} = μ B7 i ∈ E, r ∈ {1, . . . , K}, r ∈ i. .2}MBM; Q(t) = |X(t)| − 1 Ui?2 MmK#2` Q7 #mbv b2`p2`bV- r2 + M TTHv h?2Q`2K kX8Xe iQ T`Qp2 i? i i?Bb Bb `2;mH ` DmKT ?K+ rBi? bi i2 bT +2 E˜ = {0, 1, . . . , K} ˜ ;Bp2M #v M/ BM}MBi2bBK H ;2M2` iQ` A q˜n,n+1 = λ B7 0 ≤ n ≤ K − 1 q˜n,n−1 = nμ B7 1 ≤ n ≤ K.
9k3
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
;Bp2M bi i2 n ∈ E˜ Bb Q#i BM2/ #v ;`QmTBM; i?2 bi i2b i ∈ E bm+? i? i |i| − 1 = nX hQ b?Qr i? i q˜n,n−1 = μn 7Q` 1 ≤ n ≤ K- r2 Kmbi +QMbB/2` i?2 i` MbBiBQMb Q7 {X(t)} 7`QK i bm+? i? i |i| − 1 = n iQ j bm+? i? i |j| − 1 = n − 1X 6BtBM; i bm+? i? i |i| − 1 = n- i?2`2 `2 2t +iHv n bi i2b j bm+? i? i qij > 0 M/ |j| − 1 = n − 1M K2Hv HH bi i2b Q7 i?2 7Q`K j = i − {r}- r?2`2 r ∈ {1, . . . , K} M/ r ∈ iX h?2 +Q``2bTQM/BM; bmK j qij = nμ M/ Bb BM/2T2M/2Mi Q7 i bm+? i? i |i| − 1 = nX h?mb +QM/BiBQM UkXR8V Q7 h?2Q`2K kX8Xe Bb b iBb}2/ 7Q` α = n, β = n − 1X h?2 T`QQ7 Q7 q˜n,n+1 = λ Bb bBKBH `X 1t KTH2 RRXkXj, JfJfRf∞fHB7Q T`22KTiBp2 `2bmK2X h?2 +QMbi`m+iBQM b SQBbbQM bvbi2K Bb b 7QHHQrb S = N, E = { HH }MBi2 bm#b2ib Q7 S +QMi BMBM; 0}, M/- /2MQiBM; bi i2 i ∈ E b i = {0, i1 , i2 , . . . , in(i) }A(i) = {0, in(i) }, c(s, i) = 1 B7 s ∈ A(i), p(0, i, i + {in(i) + 1}) = 1, p(in(i) , i, i − {in(i) }) = 1. aQm`+2 0 Bb i?2 ``Bp H bQm`+2- b mbm HX h?2 H bi T`2b+`BTiBQM Bb i? i r?2M M2r +mbiQK2` b?Qrb mT Ui? i Bb- i` MbBiBQM Bb i`B;;2`2/ QM bQm`+2 0)- i?2 bi i2 #2BM; i M2r bQm`+2 in(i) + 1 Bb //2/ iQ i iQ 7Q`K i?2 M2ti bi i2- M/ i?Bb bQm`+2 #2+QK2b BKK2/B i2Hv +iBp2X q?2M bQm`+2 in(i) i`B;;2`b M 2p2Mi- i?Bb K2 Mb /2T `im`2 7`QK i?2 [m2m2- M/ i?2 bQm`+2 in(i) /Bb TT2 `b 7`QK bi i2 i- M/ i?2 bQm`+2 in(i)−1 Bb `2 +iBp i2/X h?2 ?TT {Tn0 }n≥1 ? b BMi2MbBiv λ > 0- M/ 7Q` HH k ≥ 1, {Tnk }n≥1 ? b BMi2MbBiv μ > 0X q?2M k ≥ 1- i?2 BMi2`2p2Mi iBK2b BM {Tnk }n≥1 +Q``2bTQM/ iQ b2`pB+2 iBK2bX q2 b22 i? i BM bi i2 i- i?2`2 `2 n(i) +mbiQK2`b BM i?2 bvbi2KX h?mb i?2 +QM@ ;2biBQM T`Q+2bb i iBK2 t Bb |X(t)| − 1- r?2`2 |X(t)| Bb i?2 + `/BM HBiv Q7 i?2 b2i X(t)X 1t KTH2 RRXkXR r b +?Qb2M 7Q` Bib bBKTHB+BivX 1t KTH2 RRXkXk b?Qrb i? i i?2 bi i2 bT +2 E b?QmH/ bQK2iBK2b #2 +?Qb2M H `;2` i? M i?2 bT +2 BM r?B+? T`Q+2bb Q7 BMi2`2bi- ?2`2 i?2 +QM;2biBQM T`Q+2bb- i F2b Bib p Hm2bc ?Qr2p2`- i?2 H ii2` b?QmH/ #2 7mM+iBQM Q7 i?2 bi i2 T`Q+2bb- M/ i?Bb Bb r?v i ∈ E Bb bQK2iBK2b + HH2/ K +`Qbi i2X 1t KTH2 RRXkXj +QMi BMb i?2 b K2 i2 +?BM;b b 1t KTH2 9Xk- THmb i?2 72 im`2 i? i HH bQm`+2b BM i?2 UK +`QV bi i2 i M22/ MQi #2 +iBp2X 1t KTH2 RRXkX9, JfJfRf∞f7B7Q [m2m2 rBi? BMbi Mi M2Qmb 722/# +FX Hi?Qm;? QM2 + M // M/ bmTT`2bb Tb2m/Q@i` MbBiBQMb i rBHH rBi?Qmi Hi2`BM; i?2 bi i2 T`Q+2bb- Tb2m/Q@i` MbBiBQMb `2 MQi Hr vb K2 MBM;H2bbX 6Q` BMbi M+2- +QMbB/2` M JfJfRf∞ [m2m2 rBi? BMbi Mi M2Qmb 722/# +F- r?2`2 +mbiQK2` }MBb?BM; b2`@ pB+2 2Bi?2` H2 p2b i?2 bvbi2K rBi? T`Q# #BHBiv 1 − p Q` Bb BKK2/B i2Hv `2+v+H2/
RRXjX h>1 :f:fRf∞ Zl1l1
9kN
rBi? T`Q# #BHBiv p i i?2 2M/ Q7 i?2 r BiBM; HBM2 Q` i i?2 b2`pB+2 #QQi? B7 i?2`2 Bb M 2KTiv r BiBM; HBM2- rBi? M2r BM/2T2M/2Mi 2tTQM2MiB H b2`pB+2 `2[m2bi Ub22 i?2 };m`2 #2HQrVX h?2M i?2 iBK2b Q7 b2`pB+2 +QKTH2iBQM Q7 `2+v+H2/ +mbiQK2`b /Q MQi +Q``2bTQM/ iQ ;2MmBM2 i` MbBiBQM Q7 {X(t)}t≥0 - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 tX
RRXj
h?2 :f:fRf∞ Zm2m2
h?Bb Bb p2`v ;2M2` H KQ/2H Q7 [m2m2 rBi? QM2 b2`p2` M/ BM}MBi2 + T +Biv i? i b2`p2b b # bBb 7Q` 2t KTH2b Q7 TTHB+ iBQMX G2i σ M/ τ #2 BMi2;` #H2 MQM@M2; iBp2 ` M/QK p `B #H2b /2}M2/ QM i?2 T`Q#@ #BHBiv bT +2 (Ω, F, P 0 ) M/ H2i θ : (Ω, F) → (Ω, F) #2 K2 bm` #H2 K T rBi? K2 bm` #H2 BMp2`b2X bbmK2 i? i (P 0 , θ) Bb 2`;Q/B+X GBM/H2v T`Q+2bbR8 bbQ+B i2/ rBi? i?2b2 ` M/QK p `B #H2b Bb biQ+? biB+ T`Q+2bb {Wn }n∈T - r?2`2 T = N Q` Zb iBb7vBM; i?2 `2+m`bBQM 2[m iBQM Wn+1 := (Wn + σn − τn )+ r?2`2
σn := σ ◦ θn ,
(n ∈ T) ,
URRXj9V
τn := τ ◦ θn .
h?Bb 2[m iBQM ? b i?2 7QHHQrBM; BMi2`T`2i iBQM BM i2`Kb Q7 [m2m2BM;X .2}M2 TQBMi T`Q+2bb A QM R #v Bib 2p2Mi iBK2b b2[m2M+2 {Tn }n∈Z , T0 ≡ 0 M/ Tn+1 − Tn := τn
(n ∈ Z) .
AMi2`T`2i Tn b i?2 ``Bp H iBK2 BM [m2m2BM; bvbi2K Q7 +mbiQK2` n M/ σn b i?2 KQmMi Q7 b2`pB+2 UBM iBK2 mMBibV `2[mB`2/ #v i?Bb +mbiQK2`X .2}M2 i?2 i` {+ BMi2MbBiv E0 [σ] ρ := 0 . E [τ ] a2`pB+2 Bb T`QpB/2/ i mMBi ` i2 r?2M2p2` i?2`2 `2K BMb i H2 bi QM2 +mbiQK2` BM i?2 bvbi2KX Pi?2`rBb2 i?2`2 Bb MQ 7m`i?2` T`2b+`BTiBQM b iQ b2`pB+2 /Bb+BTHBM2- T`BQ`BiB2b M/ bQ QMX A7 Wn Bb i?2 iQi H b2`pB+2 `2K BMBM; iQ #2 /QM2 Dmbi #27Q`2 +mbiQK2` n ``Bp2b Ui? i Bb- i iBK2 Tn −V- i?2M GBM/H2vǶb `2+m``2M+2 URRXj9V Bb b iBb}2/X AM i?2 [m2m2BM; BMi2`T`2i iBQM- i?2 GBM/H2v T`Q+2bb Bb mbm HHv + HH2/ i?2 rQ`FHQ / T`Q+2bbX q?2M T = N- i?2 GBM/H2v T`Q+2bb Bb BKK2/B i2Hv + H+mH i2/ 7`QK i?2 BMBiB H rQ`FHQ / W0 X AM i?2 + b2 T = Z- r2 ? p2 MQr?2`2 iQ bi `i i?2 `2+m`bBQMX h?Bb + b2 +Q``2bTQM/b iQ i?2 bBim iBQM Q7 [m2m2BM; bvbi2K i? i ? b #22M QT2` iBM; 7`QK i?2 BM}MBi2 T biX q2 K v 2tT2+i i? i mM/2` +2`i BM +B`+mKbi M+2b UQ7 +Qm`b2 ;QQ/ ;m2bb Bb i? i ρ < 1 rBHH /QV i?2 rQ`FHQ / T`Q+2bb ? b bi iBQM `v p2`bBQMX PM2 Bb i?2`27Q`2 H2/ iQ TQb2 i?2 T`Q#H2K BM i?2 7QHHQrBM; i2`Kb, 2t?B#Bi }MBi2 MQM@M2; iBp2 ` M/QK p `B #H2 W bm+? i? i- H2iiBM; Wn := W ◦ θ−n , R8
(GBM/H2v- RN8k)X
9jy
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
i?2 GBM/H2v `2+m`bBQM URRXj9V Bb b iBb}2/ 7Q` HH n ∈ ZX 1[mBp H2MiHv, r2 i`v iQ }M/ }MBi2 MQM@M2; iBp2 ` M/QK p `B #H2 W bm+? i? i W ◦ θ = (W + σ − τ )+ .
URRXj8V
h?Bb 2[m iBQM Bb + HH2/ GQvM2bǶ 2[m iBQMX h?2Q`2K RRXjXR URe V A7 ρ < 1- i?2`2 2tBbib mMB[m2 }MBi2 MQM@M2; iBp2 bQHmiBQM W Q7 URRXj8VX S`QQ7X 6Q` n ≥ 0- /2}M2 Mn iQ #2 i?2 rQ`FHQ / 7QmM/ #v +mbiQK2` 0 bmTTQbBM; i? i +mbiQK2` − n 7QmM/ M 2KTiv [m2m2 mTQM ``Bp HX PM2 +?2+Fb #v BM/m+iBQM i? i + m Mn = max (σ−i − τ−i ) . URRXjeV 1≤m≤n
i=1
AM T `iB+mH `- Mn Bb BMi2;` #H2 7Q` HH n ∈ NX h?2 GQvM2bǶ b2[m2M+2 {Mn }n≥0 b iBb}2b i?2 `2+m``2M+2 `2H iBQM Mn+1 ◦ θ = (Mn + σ − τ )+
URRXjdV
M/ URRXjeV b?Qrb i? i Bi Bb MQM@/2+`2 bBM;X .2}M2 i?2 GQvM2bǶ p `B #H2 UTQbbB#Hv i FBM; BM}MBi2 p Hm2bV iQ #2 M∞ := lim ↑ Mn = n→∞
sup n≥1
n
+ (σ−i − τ−i )
.
URRXj3V
i=1
G2iiBM; n ;Q iQ ∞ BM URRXjdV- r2 b22 i? i M∞ Bb MQM@M2; iBp2 ` M/QK p `B #H2 b iBb7vBM; GQvM2bǶ 2[m iBQM, M∞ ◦ θ = (M∞ + σ − τ )+ .
URRXjNV
lbBM; i?2 B/2MiBiv (a − b)+ = a − a ∧ b- 1[m HBiv URRXjdV #2+QK2b Mn+1 ◦ θ = Mn − Mn ∧ (τ − σ)
URRX9yV
M/ i?2`27Q`2- bBM+2 P 0 Bb θ@BMp `B Mi M/ bBM+2 {Mn }n≥1 Bb MQM@/2+`2 bBM; M/ BMi2;` #H2E0 [Mn ∧ (τ − σ)] = E0 [Mn − Mn+1 ◦ θ] = E0 [Mn − Mn+1 ] ≤ 0 UmbBM; G2KK dXjXe 7Q` i?2 b2+QM/ 2[m HBivVX Ai 7QHHQrb #v KQMQiQM2 +QMp2`;2M+2 i? i E0 [M∞ ∧ (τ − σ)] ≤ 0 . URRX9RV 1[m HBiv URRXjNV b?Qrb i? i i?2 2p2Mi {M∞ = ∞} Bb θ@BMp `B Mi U`2+ HH i? i σ M/ τ `2 }MBi2VX "v 2`;Q/B+Biv- P 0 (M∞ = ∞) Bb 2Bi?2` 0 Q` 1X AM pB2r Q7 URRX9RVRe
(GQvM2b- RNek)X
RRXjX h>1 :f:fRf∞ Zl1l1
9jR
P 0 (M∞ = ∞) = 1 BKTHB2b E0 [τ − σ] ≤ 0X h?2`27Q`2- i?2 +QM/BiBQM E0 [σ] < E0 [τ ] BKTHB2b i? i M∞ < ∞- P 0 @ XbX GQvM2bǶ p `B #H2 M∞ Bb i?2 KBMBK H MQM@M2; iBp2 bQHmiBQM Q7 GQvM2bǶb 2[m @ iBQMX h?Bb Bb T`Qp2/ #v BM/m+iBQMX AM/22/- bi `iBM; rBi? MQM@M2; iBp2 bQHmiBQM W Q7 URRXj8V- W ≥ Mn BKTHB2b W ≥ Mn+1 Ui?Bb Bb i`m2 #2+ mb2 Mn+1 ◦ θ = (Mn + σ − τ )+ ≤ (W + σ − τ )+ = W ◦ θVX h?2 }`bi i2`K Q7 i?2 BM/m+iBQM Bb p2`B}2/ bBM+2 W ≥ 0 = M0 X Ai `2K BMb iQ T`Qp2 mMB[m2M2bb Q7 }MBi2 bQHmiBQM Q7 GQvM2bǶ 2[m iBQMX G2i W #2 MQi?2` }MBi2 bQHmiBQM- i?2M σ − τ ≤ W ◦ θ − W ≤ σ, M/ BM T `iB+mH ` W ◦ θ − W Bb BMi2;` #H2X h?2`27Q`2- #v G2KK dXjXeE0 [W ◦ θ − W ] = 0X q2 b22F iQ T`Qp2 i? i B7 ρ < 1- i?2M W = M∞ Bb i?2 mMB[m2 MQM@M2; iBp2 }MBi2 bQHmiBQM Q7 URRXj8VX q2 ? p2 b22M i? i M∞ Bb i?2 KBMBK H bQ@ HmiBQMX h?2`27Q`2 7Q` Mv MQM@M2; iBp2 bQHmiBQM W - {W = 0} ⊆ {W = M∞ }X h?2 H ii2` 2p2Mi Bb θ@+QMi` +iBM; bBM+2 #Qi? W M/ M∞ b iBb7v URRXj8VX aBM+2 (P 0 , θ) Bb 2`;Q/B+- r2 Kmbi i?2M ? p2 P 0 (W = M∞ ) = 0 Q` 1X Ai Bb i?2`27Q`2 2MQm;? iQ b?Qr i? i P 0 (W = 0) > 0- r?B+? 7QHHQrb 7`QK i?2 M2ti H2KK X G2KK RRXjXk A7 P 0 (W = 0) = 0- 7Q` bQK2 }MBi2 bQHmiBQM W Q7 URRXj8V- i?2M ρ = 1X S`QQ7X AM/22/- B7 W ◦ θ > 0 P 0 @ XbX- W ◦ θ = W + σ − τ - M/ E0 [W ◦ θ − W ] = 0 BM pB2r Q7 G2KK dXjXe- M/ i?Bb BKTHB2b E0 [σ] = E0 [τ ]X _2+m``2M+2 iQ 0 h?2Q`2K RRXjXj h?2 bi #BHBiv +QM/BiBQM ρ < 1 Bb bbmK2/ iQ ?QH/X G2i W = M∞ #2 i?2 mMB[m2 MQM@M2; iBp2 bQHmiBQM Q7 URRXj8VX h?2`2 2tBbib M BM}MBiv Q7 M2; iBp2 U`2bTX TQbBiBp2V BM/B+2b n bm+? i? i W ◦ θn = 0X S`QQ7X AM pB2r Q7 i?2 2`;Q/B+Biv Q7 (P 0 , θ) Bi bm{+2b iQ b?Qr i? i P 0 [M∞ = 0] > 0.
URRX9kV
h?Bb 7QHHQrb 7`QK G2KK RRXjXk- BKTHvBM; i? i r2 + MMQi ? p2 P 0 [M∞ = 0] = 0 B7 ρ < 1X h?2 7QHHQrBM; Bb T `iB H +QMp2`b2 Q7 h?2Q`2K RRXjXRX h?2Q`2K RRXjX9 A7 E0 [σ] > E0 [τ ]- i?2M MQ }MBi2 bQHmiBQM Q7 URRXj8V 2tBbibX
9jk
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
S`QQ7X hQ T`Qp2 i?Bb- Bi Bb 2MQm;? iQ b?Qr i? i P 0 (M∞ = ∞) = 1 bBM+2 M∞ Bb i?2 KBMBK H MQM@M2; iBp2 bQHmiBQM Q7 URRXj8VX h?Bb 7QHHQrb 7`QK 1 (σ−i − τ−i ) = E0 [σ − τ ] > 0, n→∞ n i=1 n
lim
bBM+2 i?Bb BM im`M BKTHB2b M∞ =
sup n
n
+ (σ−i − τ−i )
= ∞.
i=1
h?2 :f:fRf∞ Zm2m2 BM *QMiBMmQmb hBK2 aQ 7 `- r2 ? p2 bim/B2/ i?2 :f:fRf∞ [m2m2 i i?2 ``Bp H iBK2b Q`- BM Qi?2` rQ`/bmM/2` i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? i?2 ``Bp H T`Q+2bbX q2 MQr bi ;2 HH i?Bb BM i?2 UiBK2@V bi iBQM `v 7` K2rQ`FX G2i UΩ, F, P ) #2 T`Q# #BHBiv bT +2 rBi? K2 bm` #H2 ~Qr {θt }t∈R - bm+? i? i (P, θt ) Bb 2`;Q/B+X G2i A #2 bBKTH2 TQBMi T`Q+2bb /2}M2/ QM (Ω, F)- +QKT iB#H2 rBi? i?2 ~Qr M/ rBi? }MBi2 p2` ;2 ` i2, λ := E[A((0, 1])] < ∞ . Ai Bb + HH2/ i?2 ``Bp H UTQBMiV T`Q+2bb M/ Bib n@i? TQBMi Tn Bb BMi2`T`2i2/ b #27Q`2 b i?2 ``Bp H iBK2 Q7 +mbiQK2` n BM i?2 b2`pB+2 bvbi2KX _2+ HH i?2 +QMp2MiBQM Tn < Tn+1 Un ∈ ZV M/ T0 ≤ 0 < T1 X h?2 BMi2`@ ``Bp H iBK2 #2ir22M +mbiQK2`b n M/ (n + 1) Bb τn := Tn+1 − Tn . *mbiQK2` n + ``B2b M KQmMi Q7 UMQM@M2; iBp2V `2[mB`2/ b2`pB+2 UQ` b2`pB+2 iBK2V /2MQi2/ #v σn - r?2`2 i?2 b2[m2M+2 {σn }n∈Z Bb bbmK2/ iQ #2 b2[m2M+2 Q7 K `Fb Q7 i?2 ``Bp H T`Q+2bbX G2iiBM; PA0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? P M/ A- r2 /2}M2 i?2 i` {+ BMi2MbBiv ρ := λE0A [σ0 ] . h?2 b2[m2M+2 {(Tn , σn )}n∈Z /2b+`B#2b i?2 BMTmi BMiQ bQK2 [m2m2BM; bvbi2K U G/G BMTmi- BM E2M/ HHǶb i2`KBMQHQ;vVX Ai Bb bi iBQM `v BM irQ /BbiBM+i U#mi `2H i2/V b2Mb2b, mM/2` i?2 S HK T`Q# #BHBiv PA0 - i?2 b2[m2M+2 {(τn , σn )}n∈Z Bb bi iBQM `v- M/ mM/2` i?2 bi iBQM `v T`Q# #BHBiv P - i?2 K `F2/ TQBMi T`Q+2bb (A, σ) Bb bi iBQM `vX Ai r b T`Qp2/ BM i?2 T`2pBQmb bm#b2+iBQM i? i B7 i?2 i` {+ BMi2MbBiv ρ Bb bi`B+iHv bK HH2` i? M 1- i?2M i?2`2 2tBbib mMB[m2 }MBi2 b2[m2M+2 {Wn }n∈Z bm+? i? i Wn = W0 ◦ θn M/ i? i Wn+1 = (Wn + σn − τn )+
(n ∈ Z),
PA0 @ XbX
()
RRXjX h>1 :f:fRf∞ Zl1l1
9jj
"v h?2Q`2K dXeXRy- UV HbQ ?QH/b P @ XbX G2i {W (t)}t∈R #2 i?2 U+QMiBMmQmb@iBK2V biQ+? biB+ T`Q+2bb /2}M2/ #v W (t) = (Wn + σn − (t − Tn ))+
(t ∈ [Tn , Tn+1 ))
URRX9jV
Ui?2 KQmMi Q7 rQ`F `2K BMBM; i iBK2 tVX LQi2 i? i Wn = W (Tn −)X h?Bb /2}M2b θt @+QKT iB#H2 rQ`FHQ / T`Q+2bb {W (t)}t∈R X LQi2 HbQ i? i bBM+2 Wn = 0 7Q` M BM}MBiv Q7 BM/B+2b n ∈ Z,
PA0 @ XbX ,
#v h?2Q`2K dXeXRy ; BM- i?Bb `2K BMb i`m2 P @ XbX _2+ TBimH iBM;, lM/2` i?2 bi #BHBiv +QM/BiBQM ρ < 1- i?2`2 2tBbib mMB[m2 }MBi2 MQM@M2; iBp2 θt @+QKT iB#H2 T`Q+2bb {W (t)}t∈R b iBb7vBM; 2[m iBQM URRX9jVX JQ`2Qp2`- i?2`2 `2 M BM}MBi2 MmK#2` Q7 M2; iBp2 BM/B+2b n M/ M BM}MBi2 MmK#2` Q7 TQbBiBp2 BM/B+2b n bm+? i? i W (Tn −) = 0X h?2 a2+QM/ `v S`Q+2bb2b G2i mb bbmK2 i? i ρ < 1X h?2 TQBMi T`Q+2bb R /2}M2/ #v 1C (Tn )1{0} (W (Tn −)), R(C) =
URRX99V
n∈Z
+QmMiBM; i?2 Tn Ƕb i r?B+? M ``BpBM; +mbiQK2` }M/b M 2KTiv [m2m2- Bb θt @ +QKT iB#H2X G2i {Un }n∈Z #2 i?2 b2[m2M+2 Q7 TQBMib Q7 R- rBi? i?2 mbm H +QMp2MiBQM U0 ≤ 0 < U 1 . 6Q` 2 +? n ∈ Z- H2i Vn+1 #2 i?2 }`bi iBK2 t 7i2` Un i r?B+? W (t) = 0X h?2 BMi2`p H [Un , Un+1 ) Bb + HH2/ i?2 n@i? +v+H2- [Un , Vn+1 ) Bb i?2 n@i? #mbv T2`BQ/ M/ [Vn+1 , Un+1 ) Bb i?2 n@i? B/H2 T2`BQ/X h?2 Q#D2+iBp2 Bb iQ +QMbi`m+i `B;?i@+QMiBMmQmb +QM;2biBQM T`Q+2bb {X(t)}t∈R - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b BM i?2 [m2m2@ BM; bvbi2K i iBK2 t- #2BM; 2Bi?2` BM i?2 r BiBM; `QQK Q` i i?2 b2`pB+2 #QQi?X * HH {X(t)}t∈R i?2 bvbi2K +QM;2biBQM T`Q+2bb- iQ /BbiBM;mBb? Bi 7`QK i?2 r BiBM; `QQK +QM;2biBQM T`Q+2bb {Q(t)}t∈R +QmMiBM; i?2 +mbiQK2`b BM i?2 r BiBM; `QQKX AM G/G/1/∞ [m2m2- i?2 bvbi2K +QM;2biBQM T`Q+2bb M/ i?2 r BiBM; `QQK +QM;2biBQM T`Q+2bb `2 `2H i2/ #v Q(t) = (X(t) − 1)+ . URRX98V h?2 #Qp2 +QM;2biBQM T`Q+2bb /2T2M/b QM i?2 T`BQ`Biv `mH2b M/ i?2 b2`pB+2 /Bb+B@ THBM2X >2`2 r2 b? HH bmTTQb2 i? i i?2`2 2tBbib QMHv QM2 +H bb Q7 +mbiQK2`b bQ i? i MQ [m2biBQM Q7 T`BQ`Biv `Bb2bX h?2 b2`pB+2 /Bb+BTHBM2 Bb 2K#Q/B2/ BM `mH2 /2+B/BM; r?B+? Q7 i?2 +mbiQK2`b T`2b2Mi i iBK2 t b?QmH/ #2 b2`p2/X AKTQ`i Mi /Bb+BTHBM2b #2bB/2b 7B7Q- HB7Q M/ ` M/QK `2 Ç a?Q`i2bi S`Q+2bbBM; hBK2 UbTiVX h?2 b2`p2` i F2b i?2 +mbiQK2` BM HBM2 rBi? i?2 b?Q`i2bi `2[mB`2/ b2`pB+2 iBK2X Ç a?Q`i2bi _2K BMBM; S`Q+2bbBM; hBK2 Ub`TiVX h?Bb Bb T`22KTiBp2 `2bmK2 /Bb+BTHBM2- r?2`2 i 2 +? BMbi Mi t i?2 b2`p2` ii2M/b i?2 +mbiQK2` rBi? i?2 b?Q`i2bi `2K BMBM; T`Q+2bbBM; iBK2X T`22KTiBQM i?2`27Q`2 Q++m`b B7 M/ QMHv B7 i?2 M2rHv ``Bp2/ +mbiQK2` ? b `2[mB`2/ b2`pB+2 bK HH2` i? M i?2 `2K BMBM; b2`pB+2 iBK2 Q7 i?2 +mbiQK2` #2BM; b2`p2/X
9j9
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
HH i?2 #Qp2 b2`pB+2 /Bb+BTHBM2b #2HQM; iQ i?2 +H bb Q7 HQ+ H /Bb+BTHBM2bX "v i?Bb r2 K2 M i? i B7 t ∈ [Un , Un+1 )- i?2 +?QB+2 Q7 i?2 +mbiQK2` #2BM; b2`p2/ i iBK2 t /2T2M/b mTQM i?2 ?BbiQ`v Q7 i?2 bvbi2K QMHv i?`Qm;? r? i ? TT2M2/ BM i?2 iBK2 BMi2`p H [Un , t)X AM //BiBQM- QM2 + M /BbiBM;mBb? #2ir22M i?Qb2 /Bb+BTHBM2b r?B+? /2T2M/ mTQM i?2 b2`pB+2 iBK2b UHBF2 bTi Q` b`TiV- M/ i?Qb2 r?B+? /Q MQi UHBF2 7B7Q- 7B7Q Q` ` M/QKVX *H2 `Hv- 7Q` Mv HQ+ H b2`pB+2 /Bb+BTHBM2- i?2 +QM;2biBQM T`Q+2bb {X(t)} Ut ∈ [Un , Vn+1 )V + M #2 +QMbi`m+i2/ 7`QK {W (t)} Ut ∈ [Un , Vn+1 )V M/ 7`QK i?2 FMQrH@ 2/;2 Q7 i?2 b2`pB+2 /Bb+BTHBM2 UK v#2 iQ;2i?2` rBi? i?2 ?2HT Q7 ` M/QK 2tT2`B@ K2Mi- b BM i?2 ` M/QK /Bb+BTHBM2VX JQ`2Qp2`- i?Bb +QMbi`m+iBQM `2bmHib BM +QM@ ;2biBQM T`Q+2bb {X(t)}t∈R i? i Bb θt @+QKT iB#H2X 1t KTH2 RRXjX8, *QM;2biBQM T`Q+2bb 7Q` i?2 7B7Q /Bb+BTHBM2X "v +QM@ bi`m+iBQM X(t) = 1{Tn ≤t} 1{Tn +W (Tn −)+σn >t} , n∈Z
M/ i?2`27Q`2- bBM+2 i?2 b2[m2M+2b {W (Tn −)}n∈Z M/ {σn }n∈Z `2 b2[m2M+2b Q7 K `Fb Q7 A- {X(t)}t∈R Bb θt @+QKT iB#H2 T`Q+2bbX 1t KTH2 RRXjXe, *QM;2biBQM T`Q+2bb 7Q` i?2 ` M/QK /Bb+BTHBM2X >2`2 i?2 # bB+ BMTmi T`Q+2bb Kmbi #2 m;K2Mi2/ rBi? M BB/ b2[m2M+2 {Zn }n∈Z Q7 [0, 1]N @ p Hm2/ ` M/QK p `B #H2b BM/2T2M/2Mi Q7 {(Tn , σn )}n∈Z X h?2 +QQ`/BM i2b Zn1 , Zn2 , . . . Q7 Zn `2 BB/ M/ mMB7Q`KHv /Bbi`B#mi2/ BM i?2 BMi2`p H [0, 1]X q?2M i?2 n@i? b2`pB+2 Bb Dmbi +QKTH2i2/- B7 i?2`2 `2 k +mbiQK2`b UrBi? BM/B+2b (n1 , . . . , nk )V r BiBM; BM i?2 [m2m2- i?2 b2`p2` +?QQb2b +mbiQK2` ni B7 Zni < Znj - 7Q` HH 1 ≤ j ≤ k- j = iX EMQrH2/;2 Q7 {(Tn , σn , Zn )}n∈Z M/ Q7 i?2 b2[m2M+2 Q7 +QMbi`m+iBQM TQBMib {Un }n∈Z bm{+2b iQ +QMbi`m+i i?2 +QM;2biBQM T`Q+2bb2bX 6Q` i?2 b K2 BMTmi {(Tn , σn )}n∈Z - i?2 rQ`FHQ / T`Q+2bb `2K BMb i?2 b K2 r?2`2 b /Bz2`2Mi b2`pB+2 /Bb+BTHBM2b mbm HHv ;Bp2 /Bz2`2Mi +QM;2biBQM T`Q+2bb2b b i?2 `2 /2` K v p2`B7v rBi? bBKTH2 2t KTH2bX 1t KTH2 RRXjXd, G/G BMTmi bi`2 K rBi? T`BQ`Biv +H bb2bX *QMbB/2` G/G BMTmi bi`2 K {(Tn , σn , Un }n∈Z - r?2`2 σn Bb i?2 KQmMi Q7 b2`pB+2 `2[mB`2/ #v +mbiQK2` n M/ Un Bb Bib T`BQ`Biv +H bb, Un ∈ {1, 2, . . . , M }X G2i σ ˜ (t) := σn B7 t ∈ [Tn , Tn+1 ) . G2i {Ti,n }n∈Z #2 i?2 b2[m2M+2 bbQ+B i2/ rBi? i?2 ``Bp H T`Q+2bb Ai Q7 +mbiQK2`b Q7 +H bb i- rBi? i?2 mbm H +QMp2MiBQM Ti,0 ≤ 0 < Ti,1 X h?2 [m MiBiv σi,n := σ(Ti,n ) Bb i?2 KQmMi Q7 b2`pB+2 `2[mB`2/ #v +mbiQK2` n Q7 ivT2 i ``BpBM; i iBK2 Ti,n X bbmK2 i? i λ = E[A((0, 1])] < ∞ M/ BM T `iB+mH ` λi = E[Ai ((0, 1])] < ∞ U1 ≤ i ≤ M VX h?2`27Q`2 QM2 K v /2}M2 i?2 S HK T`Q# #BHBiB2b PA0 M/ PA0 i X 6`QK 1t KTH2 dXkXRy-
RRX9X Sah- GAhhG1- 1h*X
9j8 λi = PA0 (U0 = i). λ
*QMbB/2` i?2 i` {+ BMi2MbBiB2b ρi = λi E0Ai [σ0 ] = λi E0Ai [σi,0 ]X _2+ HH i? i PA0 i (Ti,0 = 0) = 1 M/ i?2`27Q`2 σi,0 = σ0 , PA0 i @ XbX h?2 BM/BpB/m H i` {+ BMi2M@
bBiB2b M/ i?2 ;HQ# H i` {+ BMi2MbBiv Bb ρ = ki=1 ρi Ub22 1t KTH2 dXkXRyVX
RRX9 T bi - GBiiH2- 2i+X h?2 T bi S`QT2`iv h?Bb +H bbB+ H `2bmHi Q7 [m2m2BM; i?2Q`v bi i2b i? i B7 i?2 ``Bp H TQBMi T`Q+2bb Bb SQBbbQM T`Q+2bb- QT2` iBQM H +? ` +i2`BbiB+b Q7 i?2 bvbi2K +QKTmi2/ Dmbi #27Q`2 ``Bp H iBK2b M/ i `#Bi` `v iBK2b `2 i?2 b K2 UT bi Bb i?2 ##`2pB iBQM Q7 dzSQBbbQM ``Bp Hb a22 hBK2 p2` ;2bǴVX >2`2 Bb T`2+Bb2 bi i2K2MiX G2i #2 ;Bp2M K2 bm` #H2 ~Qr {θt }t∈R M/ θt @BMp `B Mi T`Q# #BHBiv P QM (Ω, F)X G2i A #2 θt @+QKT iB#H2 TQBMi T`Q+2bb rBi? }MBi2 BMi2MbBiv λ Ur2 mb2 i?2 MQi iBQM A #2+ mb2 i?2 T bi i?2Q`2K Bb KQbiHv TTHB2/ iQ ``Bp H TQBMi T`Q+2bb2bVX amTTQb2 i? i 7Q` bQK2 ?BbiQ`v {Ft }t∈R - A Bb M Ft @SQBbbQM T`Q+2bb Q7 BMi2MbBiv λX G2i {Z(t)}t∈R #2 +Q`HQH θt @+QKT iB#H2 Ft @ / Ti2/ T`Q+2bb rBi? p Hm2b BM bQK2 iQTQHQ;B+ H K2 bm` #H2 bT +2 (K, K)X G2i f : K → R #2 MQM@M2; iBp2 K2 bm` #H2 7mM+iBQMX h?2ME0A [f (Z(0−))] = E[f (Z(0))] .
URRX9eV
LQi2 i? i- #v i?2 +`Qbb@2`;Q/B+ i?2Q`2K Uh?2Q`2K dXeXRkV- B7 (P, θt ) Bb 2`;Q/B+URRX9eV BKTHB2b i? i P @ HKQbi bm`2Hv M/ P @ HKQbi bm`2Hv N 1 1 T f (Z(Tn −)) = lim f (Z(s))ds = E[f (Z(0))] . N →∞ N T →∞ T 0 n=1
E0A [f (Z(0−))] = lim
h?2 7QHHQrBM; mb27mH 2ti2MbBQM Q7 i?2 #Qp2 +H bbB+ H T bi T`QT2`iv Bb /B`2+i +QMb2[m2M+2 Q7 i?2 /2}MBiBQMb Q7 biQ+? biB+ BMi2MbBiv M/ Q7 S HK T`Q# #BHBivX h?2Q`2K RRX9XR G2i A- P - {θt }t∈R M/ {Ft }t∈R #2 b #Qp2- rBi? i?2 /Bz2`2M+2 i? i A Bb MQi M2+2bb `BHv SQBbbQM T`Q+2bb #mi ? b θt @+QKT iB#H2 biQ+? biB+ Ft @ BMi2MbBiv {λ(t)}t∈R X aiBHH bbmK2 i? i Bib p2` ;2 BMi2MbBiv λ Bb }MBi2X G2i PA0 #2 i?2 S HK T`Q# #BHBiv bbQ+B i2/ rBi? (A, θt , P )X h?2M- 7Q` Mv θt @+QKT iB#H2 +Q`HQH biQ+? biB+ T`Q+2bb {Z(t)}t∈R i FBM; Bib p Hm2b BM bQK2 K2 bm` #H2 bT +2 (K, K) M/ Mv MQM@M2; iBp2 7mM+iBQM f : K → RλE0A [f (Z(0−))] = E [λ(0)f (Z(0))] .
URRX9dV
9je
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
S`QQ7X URd V
f (Z(s−)) N (ds) = E f (Z(s−))λ(s) ds (0,1] (0,1]
f (Z(s))λ(s) ds = E [f (Z(s))λ(s)] ds =E (0,1] (0,1] E [f (Z(0))λ(0)] ds = E [f (Z(0))λ(0)] . =
λE0A [f (Z(0−))] = E
(0,1]
AM i?2 SQBbbQM + b2 Uλ(t) = λV r2 `2+Qp2` i?2 +H bbB+ H T bi T`QT2`ivX >Qr@ 2p2` i?2 #Qp2 ;2M2` HBx iBQM ? b mb27mH TTHB+ iBQMb b i?2 7QHHQrBM; 2t KTH2 rBHH /2KQMbi` i2X 1t KTH2 RRX9Xk, h?2 ǯDQ#@Q#b2`p2`ǰ T`QT2`ivXR3 *QMbB/2` :Q`/QMĜL2r2HH M2irQ`F BM bi iBQM `v `2;BK2X h?2 TQBMi T`Q+2bb Aij +QmMiBM; i?2 i` Mb72`b 7`QK bi iBQM i iQ bi iBQM j /KBib i?2 FtX @BMi2MbBiv λij (t) = μi rij 1{Xi (t)>0} . Aib U p2` ;2V BMi2MbBiv Bb i?2`27Q`2 λij = μi rij P (Xi (0) > 0)X "v h?2Q`2K RRX9XRλij PA0 ij [X(0−) = n] = E[μi rij 1{Xi (0)>0} 1{X(0)=n} ] . h?2`27Q`2 PA0 ij (X(0−)
P (X(0) = n) = n)P (Xi (0) > 0) = 0
B7 ni > 0c Qi?2`rBb2X
AM pB2r Q7 i?2 2tT`2bbBQM URRXRyV 7Q` i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 :Q`/QMĜ L2r2HH M2irQ`F !K nl 1 αini −1 B7 ni > 0c α l=1, l=i l PA0 ij (X(0−) = n) = C 0 Qi?2`rBb2r?2`2 C Bb +QMbi Mi Q#i BM2/ #v MQ`K HBx iBQM,
K
αlnl
αini −1 = C.
n1 +···+nK =M,ni >0 l=1, l=i
7i2` i?2 +? M;2 Q7 bmKK iBQM p `B #H2 ni → ni − 1- i?2 H27i@? M/ bB/2 Q7 i?2 #Qp2 2[m HBiv #2+QK2b n1 +···+nK =M −1 Rd R3
K
αlnl = G(K, M − 1).
l=1
("`ûK m/- RN3N)X (_2Bb2` M/ G p2M#2`;- RN3y)- (a2p+BF M/ JBi` MB- RN3R)X
RRX9X Sah- GAhhG1- 1h*X
9jd
h?2`27Q`2- 7Q` n bm+? i? i ni > 0PA0 ij (X(0−)
1 = n) = G(K, M − 1)
K
αlnl
αini −1 .
l=1, l=i
q?2M +mbiQK2` Bb i` Mb72``2/ 7`QK i iQ j i iBK2 t- i?2 bBim iBQM ?2 b22b /m`BM; ?Bb i` Mb72` Ur?2M ?2 ? b H27i i #mi MQi v2i `2 +?2/ jV 7Q` i?2 `2bi Q7 i?2 M2irQ`F Bb MQi X(t−) #mi X(t−) − ei X h?2`27Q`2- i?2 bi i2 Q7 i?2 M2irQ`F #v i?Bb !K nQ#b2`p2/ 1 l +mbiQK2` U2t+Hm/BM; ?BKV Bb n rBi? T`Q# #BHBiv G(K,M α X Ai Bb i?2 b K2 l=1 l −1) b i?2 bi i2 Q7 i?2 b K2 M2irQ`F rBi? M − 1 +mbiQK2`b Q#b2`p2/ i M `#Bi` `v iBK2 #v M 2ti2`M H Q#b2`p2`X 1t KTH2 RRX9Xj, AMb2MbBiBpBiv Q7 Jf:AfRf∞ HB7Q T`22KTiBp2X dzAMb2M@ bBiBpBivǴ `272`b iQ bBim iBQMb r?2`2 ;Bp2M p2` ;2 /2T2M/BM; QM i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 QM2 Q` b2p2` H ` M/QK p `B #H2b +im HHv /2T2M/b QM i?2b2 /Bbi`B@ #miBQMb QMHv i?`Qm;? i?2B` K2 MbX h?Bb Bb i?2 + b2 7Q` BMbi M+2 7Q` i?2 Jf:Af∞ /2H v bvbi2K Q7 1t KTH2 jXjXk BM bi iBQM `v `2;BK2- 7Q` r?B+? i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K /2T2M/b QM i?2 /2H v QMHv i?`Qm;? Bib K2 MX h?2`2 `2 K Mv Qi?2` BMbi M+2b Q7 i?Bb T?2MQK2MQM BM [m2m2BM; i?2Q`v Ui?2 /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb Q7 i?2 bi iBQM `v 1`H M; [m2m2 ? b i?Bb T`QT2`ivVXRN * HH X(t) i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K i iBK2 t BM M Jf:AfRf∞ HB7Q T`22KTiBp2 [m2m2X 6Q` }t2/ k ≥ 1- /2MQi2 #v Nk i?2 TQBMi T`Q+2bb +QmMiBM; i?2 ``Bp Hb i? i K F2 i?2 +QM;2biBQM T`Q+2bb `2 +? H2p2H k, Nk (C) = 1C (Tn )1{X(Tn −)=k−1} . n∈Z (k)
(k)
(k)
G2i {Tn } #2 i?2 b2[m2M+2 Q7 TQBMib Nk - rBi? i?2 mbm H +QMp2MiBQM T0 ≤ 0 < T1 X (k) (k) +mbiQK2` ``BpBM; i iBK2 Tn `2[mB`2b i?2 b2`pB+2 σn X h?2 7QHHQrBM; 7 +i Bb (k) i`m2, {σn }n∈Z Bb BB/ rBi? i?2 b K2 /Bbi`B#miBQM b {σn }n∈Z Ui?2 T`QQ7 Bb H27i 7Q` (k) i?2 `2 /2`VX "2+ mb2 Q7 i?2 HB7Q T`22KTiBp2 `mH2- +mbiQK2` Tn `2+2Bp2b HH ?Bb b2`pB+2 r?2M i?2 [m2m2 Bb i H2p2H k Ub22 i?2 };m`2 #2HQrVX (k)
σn
k
(k)
Tn RN
(" `#Qm`- RNde)- (EƺMB; M/ C Mb2M- RNde)X
9j3
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
aBM+2 k Bb `2+m``2Mi UmM/2` i?2 bi #BHBiv ?vTQi?2bBbV- i?2 H r Q7 H `;2 MmK#2`b ;Bp2b t 1 ds 0 {X(s)=k} lim = μ−1 XbX t→∞ 1 N (ds) (0,t] {X(s−)=k−1} "v 2`;Q/B+Biv1 t→∞ t
t
1{X(s)=k} ds = π(k) ,
lim
0
r?2`2 π Bb i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bbX JQ`2Qp2` 1 N (t) 1 lim 1{X(s−)=k−1} N (ds) = lim 1{X(s−)=k−1} N (ds). t→∞ t (0,t] t→∞ t N (t) (0,t] "mi limt→∞
N (t) t
1 lim t→∞ N (t)
= λ- M/ BM pB2r Q7 i?2 T bi T`QT2`iv 1 1{X(s−)=k−1} N (ds) = lim t→∞ t (0,t]
bQ i? i
h?Bb H2 /b iQ
π(k) =ρ π(k − 1) π(k) = (1 − ρ)ρk
t
1{X(s)=k−1} ds = π(k − 1) , 0
(k ≥ 1) . (k ≥ 0) ,
r?B+? b?Qrb BM T `iB+mH ` i?2 BMb2MbBiBpBiv Q7 i?2 M/GI/1/∞ HB7Q T`22KTiBp2 [m2m2- i? i Bb- π /2T2M/b QM G QMHv i?`Qm;? Bib K2 MX h?2 T`QQ7 Q7 BMb2MbBiBpBiv + M #2 2ti2M/2/ iQ i?2 bBim iBQM r?2`2 i?2 BMTmi T`Q+2bb N /KBib i?2 biQ+? biB+ (P, Ft )@BMi2MbBiv {λX(t) }t∈R - r?2`2 {Ft }t∈R Bb i?2 ?BbiQ`v `2+Q`/BM; i?2 T bi Q7 N i iBK2 t M/ i?2 r?QH2 b2[m2M+2 {σn }n∈N X LQrr2 ? p2 iQ mb2 i?2 2ti2M/2/ p2`bBQM Q7 T bi t 1 t 1 t 1{X(s−)=k−1} N (ds) = lim 1{X(s)=k−1} λk−1 ds lim t→∞ N (t) 0 t→∞ N (t) t 0 λk−1 = π(k − 1) , λ r?B+? T`Qp2b BMb2MbBiBpBivX
Zm2m2 G2M;i? i .2T `im`2b Q` ``Bp Hb AM i?2 Jf:fRf∞ Q` :fJfRf∞ bvbi2Kb- r2 r2`2 #H2 iQ +QKTmi2 i?2 bi iBQM `v [m2m2 /Bbi`B#miBQMb i /2T `im`2 iBK2b M/ Dmbi #27Q`2 M ``Bp H iBK2 `2bT2+iBp2HvX 6Q` i?2 ;2M2` H :f:fRf∞ [m2m2- i?2`2 Bb p2`v ;2M2` H `2H iBQM #2ir22M i?2 bi iBQM `v /Bbi`B#miBQMb i /2T `im`2 iBK2b M/ Dmbi #27Q`2 M ``Bp H iBK2X h?2 bBim iBQM Bb i?2 7QHHQrBM;, q2 ? p2 K2 bm` #H2 ~Qr {θt }t∈R - irQ bBKTH2 TQBMi T`Q+2bb2b A M/ D rBi?Qmi +QKKQM TQBMib M/ biQ+? biB+ T`Q+2bb {X(t)}t∈R - HH θt @+QKT iB#H2 M/ bm+? i? i 7Q` HH t ≥ 0
RRX9X Sah- GAhhG1- 1h*X
9jN
X(t) = X(0) + A((0, t]) − D((0, t]) ≥ 0 . h?mb {X(t)}t∈R Bb i?2 +QM;2biBQM T`Q+2bb U[m2m2 H2M;i?V Q7 [m2m2- A M/ D #2BM; `2bT2+iBp2Hv i?2 ``Bp H M/ /2T `im`2 T`Q+2bb2bX HbQ bbmK2 i? i A ? b }MBi2 BMi2MbBiv λA = λ M/ i? i X(0) U M/ i?2`27Q`2 HbQ X(t)V ? b }MBi2 2tT2+i @ iBQMX h?2M- i FBM; 2tT2+i iBQM BM i?2 #Qp2 2pQHmiBQM 2[m iBQM 7Q` i?2 +QM;2biBQM T`Q+2bb- r2 }M/ i? i E[A((0, t]] = E[D((0, t])] M/ i?2`27Q`2 i?2 BMi2MbBiv Q7 i?2 /2T `im`2 T`Q+2bb Bb λD = λX G2i MQr f : N → R #2 #QmM/2/ 7mM+iBQMX 6Q` HH t ≥ 0 f (X(t)) = f (X(0)) + (0,t]
{f (X(s)) − f (X(s−))}A(ds) {f (X(s)) − f (X(s−))}D(ds) . + (0,t]
i M 2p2Mi iBK2 Q7 A- f (X(t)) = f (X(t−) + 1) M/ i M 2p2Mi iBK2 Q7 Df (X(t−)) = f (X(t) + 1)X h?2`27Q`2 f (X(t)) = f (X(0)) + (0,t]
{f (X(s−) + 1) − f (X(s−))}A(ds) {f (X(s−) − 1) − f (X(s−))}D(ds) . + (0,t]
.BpB/BM; #v λt- i FBM; 2tT2+i iBQMb rBi? `2bT2+i iQ P - Q#b2`pBM; i? i E[f (X(t))] = E[f (X(0))]- r2 }M/ i? i E0A [f (X(0−) + 1) − f (X(0−))] + E0D [f (X(0)) − f (X(0) + 1)] = 0 . qBi? f (x) = 1{n} (x)- r?2`2 n ≥ 1- M/ H2iiBM; 0 0 (n) := PA0 (X(0−) = n) , πD (n) := PD0 (X(0) = n) , πA− 0 0 0 0 (n − 1) − πA− (n) = πD (n) − πD (n + 1)X 6Q` n = 0- Q#b2`pBM; i? i r2 Q#i BM πA− 0 0 X(0−) ≥ 0 M/ X(0−) > 0 B7 y Bb /2T `im`2 iBK2- πA− (0) = πD (1)X "v bmKKBM; i?2 H bi irQ 2[m HBiB2b 7Q` HH i ≥ 0, 0 0 πA− (i) = πD (i + 1) .
AM rQ`/b, i?2 bi iBQM `v +QM;2biBQM T`Q+2bb2b Q#b2`p2/ Dmbi #27Q`2 M ``Bp H M/ Dmbi 7i2` /2T `im`2 ? p2 i?2 b K2 /Bbi`B#miBQMX 1t KTH2 RRX9X9, * b2 Q7 M Jf:AfRf∞ [m2m2X h?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 MmK#2` Q7 +mbiQK2`b Dmbi 7i2` i?2 /2T `im`2 iBK2b Q7 bi #H2 Jf:fRf∞ [m2m2 Bb 2[m H iQ i?2 bi iBQM `v /Bbi`B#miBQMX h?Bb Bb M BKK2/B i2 +QMb2[m2M+2 Q7 i?2 #Qp2 `2bmHi M/ Q7 i?2 T bi T`QT2`ivX
99y
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
GBiiH2Ƕb 6Q`KmH G2i (Ω, F, P ) #2 T`Q# #BHBiv bT +2 2M/Qr2/ rBi? K2 bm` #H2 ~Qr {θt }t∈R X bbmK2 i? i (P, θt ) Bb 2`;Q/B+X *QMbB/2` TQBMi T`Q+2bb N +QKT iB#H2 rBi? i?2 ~Qr M/ i?2`27Q`2 bi iBQM `vX bbmK2 Bib BMi2MbBiv λ Bb }MBi2X SQBMi Tn Bb BMi2`T`2i2/ b i?2 ``Bp H iBK2 BMiQ bQK2 dzbvbi2KǴ Q` dz#H +F #QtǴ Ub v- b2`pB+2 bvbi2K Q` T`Q+2bbQ`V Q7 dz+mbiQK2`Ǵ Q` dzDQ#Ǵ- i? i `2K BMb BM i?2 bvbi2K 7Q` i?2 ` M/QK iBK2 Wn X h?2 b2[m2M+2 {Wn }n∈Z Bb bbmK2/ θt @+QKT iB#H2 U M/ i?2`27Q`2 bi iBQM `v mM/2` i?2 S HK T`Q# #BHBiv PN0 VX
λ
X(t) Wn
λ
h?2 MmK#2` X(t) Q7 +mbiQK2`b- Q` DQ#b- BM i?2 bvbi2K i iBK2 t Bb X(t) =
1(−∞,t] (Tn )1(t,−∞) (Tn + Wn ).
n∈Z
h?2 T`Q+2bb {X(t)}t∈R Bb #v +QMbi`m+iBQM θt @+QKT iB#H2 M/ i?2`27Q`2 bi iBQM `vX G2iiBM; t = 0 M/ i FBM; 2tT2+i iBQM ;Bp2b U#v J2+F2Ƕb 7Q`KmH M/ MQiBM; i? i W n = W 0 ◦ θTn V E [X(0)] = E 1(−∞,0] (Tn )1(0,−∞) (Tn + W0 ◦ θTn ) n∈Z
=
λE0N
0 −∞
1(0,−∞) (s + W0 )ds
0
=λ −∞ ∞ =λ
PN0 (s + W0 > 0)ds PN0 (W0 > u)du = λE0N [W0 ] .
0
q2 ? p2 Dmbi Q#i BM2/ i?2 +2H2#` i2/ GBiiH2Ƕb 7Q`KmH ,ky E [X(0)] = λE0N [W0 ] . A7 (P, θt ) Bb 2`;Q/B+- i?2 +`Qbb@2`;Q/B+ i?2Q`2K Uh?2Q`2K dXeXRkV HHQrb mb iQ r`Bi2 i?2 #Qp2 2[m HBiv b 1 Wk n↑∞ n k=1 n
E [X(0)] = λ lim
P @ XbX
h?Bb 7Q`KmH TT2 `b BM i?2 TTHB2/ HBi2` im`2 mM/2` i?2 7Q`K Q = λW X Ai Bb ` i?2` p2`b iBH2 7Q`KmH - i?2 +?QB+2 Q7 i?2 dz#H +F #QtǴ /2HBKBiBM; i?2 bvbi2K #2BM; H27i i QM2Ƕb BK ;BM iBQMX ky
(GBiiH2- RNeR)X
RRX9X Sah- GAhhG1- 1h*X
99R
1t KTH2 RRX9X8, 1KTiBM2bb T`Q# #BHBiv Q7 i?2 G/G/1/∞ [m2m2X *QM@ bB/2` i?2 bi iBQM `v G/G/1/∞ [m2m2 Q7 a2+iBQM RRXj rBi? b2`pB+2 /Bb+BTHBM2 rBi?Qmi T`22KTiBQM U b2`pB+2 QM+2 bi `i2/ + MMQi #2 BMi2``mTi2/VX .2MQi2 #v X(t) i?2 MmK#2` Q7 +mbiQK2`b BM i?2 r?QH2 bvbi2K Ur BiBM; `QQK THmb b2`pB+2 #QQi?VX h F2 7Q` i?2 #H +F #Qt i?2 b2`pB+2 #QQi?- M/ H2i Wn #2 i?2 bQDQm`M iBK2 Q7 +mbiQK2` n UiBK2 bT2Mi BM r BiBM; THmb iBK2 bT2Mi BM b2`pB+2VX 1
h?2 MmK#2` Q7 T2QTH2 BM i?2 #QQi? Bb i KQbi R- +im HHv- 2[m H iQ 1{X(t)=1} i?2 2tT2+i iBQM Q7 r?B+? Bb 1 − π(0)- r?2`2 π(0) Bb i?2 bi iBQM `v T`Q# #BHBiv Q7 }M/BM; i?2 bvbi2K 2KTiv i Mv ;Bp2M iBK2 tX h?2 bQDQm`M iBK2 Q7 +mbiQK2` n BM i?2 #QQi? Bb σn X h?2 θt @+QKT iB#H2 TQBMi T`Q+2bb Q7 ``Bp Hb BM i?2 #QQi? ? b i?2 p2` ;2 BMi2MbBiv λX "v GBiiH2Ƕb 7Q`KmH π(0) = 1 − ρ , r?2`2 i?2 i` {+ BMi2MbBiv ρ Bb ;Bp2M #v i?2 7Q`KmH 1 σk . n↑∞ n k=1 n
ρ = λ lim
h?Bb Bb mMBp2`b H 7Q`KmH i? i r2 ? p2 H`2 /v 2M+QmMi2`2/ BM p `BQmb T `iB+mH ` + b2bX 1t KTH2 RRX9Xe, J2 M@p Hm2 M HvbBb Q7 +HQb2/ C +FbQM M2irQ`FbX *HQb2/ C +FbQM M2irQ`F KQ/2Hb `Bb2 7Q` BMbi M+2 BM i?2 bBim iBQM r?2`2 M QT2M M2irQ`F Bb QT2` i2/ rBi? #HQ+FBM; /KBbbBQM TQHB+v, B7 i?2`2 `2 H`2 /v N +mb@ iQK2`b BM i?2 M2irQ`F- i?2 M2r+QK2`b r Bi i ; i2 UBM [m2m2V mMiBH bQK2 +mbiQK2` Bb `2H2 b2/ 7`QK i?2 M2irQ`F- i r?B+? iBK2 QM2 KQM; i?2 #HQ+F2/ +mb@ iQK2`b- B7 Mv- Bb /KBii2/X AM i?2 M2irQ`F- i?2`2 `2 i KQbi N +mbiQK2`bX i dzb im` iBQMǴ- i?2`2 Bb Hr vb QM2 +mbiQK2` `2 /v iQ `2TH +2 /2T `iBM; +mbiQK2`#v /2}MBiBQM Q7 b im` iBQMX h?2`27Q`2- i b im` iBQM- Q` 7Q` HH T` +iB+ H Tm`TQb2b M2 ` b im` iBQM- 2p2`vi?BM; HQQFb b B7 /2T `iBM; +mbiQK2` r b #2BM; BKK2/B i2Hv `2+v+H2/X a
b
≤N ∞ ≥N
a
b
99k
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: N a
b
Ai Bb BKTQ`i Mi BM T` +iB+2 iQ #2 #H2 iQ +QKTmi2 i?2 p2` ;2 MmK#2` Q7 +mb@ iQK2`b d(N ) T bbBM; i?`Qm;? i?2 2Mi` M+2 TQBMi a T2` mMBi Q7 iBK2X h?Bb [m MiBiv Bb i?2 K tBKmK i?`Qm;?Tmi M/ Bb i?2`27Q`2 `2H i2/ iQ i?2 2{+B2M+v Q7 i?2 bvbi2K 7`QK i?2 TQBMi Q7 pB2r Q7 i?2 QT2` iQ` Ur?Q K F2b KQM2v rBi? +mbiQK2`bVX 6`QK i?2 TQBMi Q7 pB2r Q7 i?2 [m HBiv Q7 b2`pB+2- M BKTQ`i Mi T ` K2i2` Bb W (N )- i?2 p2` ;2 iBK2 bT2Mi #v +mbiQK2` #2ir22M a M/ bX b K ii2` Q7 7 +i- N - W (N ) M/ d(N ) `2 `2H i2/ #v d(N )W (N ) = N , () T `iB+mH ` + b2 Q7 GBiiH2Ƕb 7Q`KmH X h?2 7QHHQrBM; H;Q`Bi?K- + HH2/ K2 M p Hm2 M HvbBb-kR K F2b mb2 Q7 GBiiH2Ƕb 7Q`KmH M/ Q7 i?2 DQ#@Q#b2`p2` T`QT2`iv 7Q` +QKTmiBM; d(N )X G2i vi /2MQi2 i?2 p2` ;2 MmK#2` Q7 pBbBib i? i +mbiQK2` 2Mi2`BM; BM a K F2 BM bi iBQM i #27Q`2 H2 pBM; i?2 M2irQ`F pB bX G2i Li (N ) #2 i?2 bi iBQM `v p2` ;2 MmK#2` Q7 +mbiQK2`b BM bi iBQM i- H2i Wi (N ) #2 i?2 +mbiQK2`@ p2` ;2 bQDQm`M iBK2 BM bi iBQM i M/ H2i λi (N ) #2 i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b b2`p2/ #v bi iBQM i T2` mMBi Q7 iBK2X q2 ? p2 λi (N ) = d(N )vi M/ 7`QK GBiiH2Ƕb 7Q`KmH Li (N ) = λi (N )Wi (N ) = d(N )vi Wi (N ). aBM+2 i?2 iQi H MmK#2` Q7 +mbiQK2`b Bb N - r2 ? p2 N=
K
Li (N ) = d(N )
i=1
N
vi Wi (N ).
i=1
G2i Yi (N ) /2MQi2 i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b BM bi iBQM i b22M #v +mbiQK2` ``BpBM; i bi iBQM iX "v i?2 DQ#@Q#b2`p2` T`QT2`iv- Yi (N ) = Li (N − 1)X HbQ- #v q H/Ƕb B/2MiBivYi (N ) + 1 Wi (N ) = , μi r?2`2 μi Bb i?2 b2`pB+2 ` i2 i bi iBQM iX *QK#BMBM; i?2 #Qp2 `2H iBQMb- r2 Q#i BM i?2 7QHHQrBM; `2+m``2M+2 `2H iBQMb, 1 (1 + Li (N − 1)) , μi N , T (N ) = N v i=1 i Wi (N ) Li (N ) = vi T (N )Wi (N ) .
Wi (N ) =
kR
(_2Bb2` M/ G p2M#2`;- RN3y)- (a2p+BF M/ JBi` MB- RN3R)X
URRX93V
RRX9X Sah- GAhhG1- 1h*X
99j
PM2 + M i?2`27Q`2 +QKTmi2 d(N ) M/ i?2M W (N ) pB UV- M/ i?2M +?QQb2 M QT2` iBM; TQBMi N i? i T`QpB/2b i?2 `2[mB`2/ # H M+2 #2ir22M i?2 QT2` iQ`Ƕb T`Q}i M/ i?2 +mbiQK2`Ƕb +QK7Q`iX
6mM+iBQM H 6Q`K Q7 J2+F2Ƕb 6Q`KmH AM J2+F2Ƕb 7Q`KmH QM2 + M i F2 /p Mi ;2 Q7 i?2 TQbbB#BHBiv 7Q` Zn iQ i F2 p H@ m2b BM M `#Bi` `v K2 bm` #H2 bT +2 (K, K)- 7Q` BMbi M+2 K = D(R; R)- i?2 b2i Q7 +Q`HQH 7mM+iBQMb 7`QK R BMiQ R- rBi? Mv bmBi #H2 iQTQHQ;v M/ K i?2 "Q`2H }2H/ bbQ+B i2/ rBi? i?Bb iQTQHQ;vX M 2H2K2Mi Q7 i?Bb bT +2 Bb 7mM+iBQM z := (z(t), t ∈ R)X G2i Zn := (Zn (t, ω), t ∈ R)X aBM+2 Bi Bb θt @+QKT iB#H2 K `FZn = Z0 ◦ θTn := {Z0 (t − Tn }t∈R X J2+F2Ƕb 7Q`KmH i?2M i F2b i?2 7Q`K
E Zn (Tn ) = λ E0N [Z0 (t)] dt = λE0N Z0 (t)dt . URRX9NV R
n∈Z
R
h?Bb 2ti2MbBQM Q7 GBiiH2Ƕb 7Q`KmH Bb + HH2/ i?2 H = λG 7Q`KmH Xkk 1t KTH2 RRX9Xd, h?2 SQHH +x2FėE?BMi+?BM2 7Q`KmH X *QMbB/2` G/G/s/c [m2m2 BM bi iBQM `v `2;BK2- rBi? MQM@T`22KTiBp2 b2`pB+2 /Bb+BTHBM2, QM+2 bi `i2/i?2 b2`pB+2 Q7 +mbiQK2` + MMQi #2 BMi2``mTi2/X h?2 +QMi`B#miBQM Q7 +mbiQK2` n Un ≤ 0V iQ i?2 rQ`FHQ / W (0) i iBK2 0 Bb Zn (0 − Tn )- r?2`2 + Zn (t) = σn 1{Tn ≤t≤Tn +V˜n } + σn − (t − Tn − V˜n ) 1{t≥Tn +V˜n } M/ r?2`2 V˜n Bb i?2 r BiBM; iBK2 Q7 +mbiQK2` nX Zn (t)
Zn (t-Tn )
σn
Zn (-Tn ) 0
V˜n
V˜n +σn
t
Tn
Tn +V˜n
Tn +V˜n +σn
h?2 rQ`FHQ / i iBK2 y + M #2 2tT`2bb2/ b Zn (0 − Tn )1(−∞,0] (Tn ) W (0) = n≤0
M/ i?2`27Q`2- #v i?2 7mM+iBQM bT +2 J2+F2 7Q`KmH
0 Z0 (t)dt . E[W (0)] = λEA R
AM pB2r Q7 i?2 b? T2 Q7 Z0 (t) M/ mbBM; i?2 7 +i i? i PA0 (T0 = 0) = 1kk
("`mK2HH2- RNdR)X
t
999
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: E0A
1 Z0 (t)dt = E0A [σ0 V˜0 + σ02 ]. 2 R
h?Bb `2bmHib BM i?2 7Q`KmH , 1 E[W (0)] = λE0A [σ0 V˜0 ] + λE0A [σ02 ]. 2 q?2M σ0 M/ V˜0 `2 BM/2T2M/2Mi- 7Q` BMbi M+2 BM GI/GI/1/∞ 7B7Q Q` HB7Q MQM@T`22KTiBp2 [m2m2 U#mi MQi 7Q` bTi 7Q` BMbi M+2V 1 E[W (0)] = λE0A [σ0 ]E0A [V˜0 ] + λE0A [σ02 ] . 2
()
q2 MQr bT2+B HBx2 i?2 #Qp2 iQ M/GI/1/∞ 7B7Q [m2m2X "2+ mb2 Q7 i?2 7B7Q /Bb+BTHBM2 M/ bBM+2 i?2`2 Bb QMHv QM2 b2`p2`- i?2 r BiBM; iBK2 V˜n Q7 +mbiQK2` n Bb 2[m H iQ i?2 rQ`FHQ / b?2 b22b BM 7`QMi Q7 ?2` mTQM ``Bp H- i? i Bb- W (Tn− )X "v i?2 T bi T`QT2`ivE0A [W (Tn −)] = E[W (0)] M/ i?2`27Q`2- mbBM; UV M/ i?2 T`2pBQmb `2K `Fb E0A [V˜0 ] =
1 λE0A [σ02 ] 2
URRX8yV
1−ρ
Q` 2[mBp H2MiHv E[W (0)] =
1 λE[σ02 ] 2
1−ρ
.
"v GBiiH2Ƕb 7Q`KmH - λE0A [V˜0 ] = E[Q(0)]X HbQ- E[Q(0)] = E[(X(0) − 1)+ ] = E[X(0)] − 1 + P (X(0) = 0). h?2`27Q`2 BM pB2r Q7 URRX98V- E[Q(0)] = E[X(0)] − ρX *QK#BMBM; HH i?2 #Qp2 `2K `Fb- r2 }M/ i?2 2tT2+i2/ MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K M/GI/1/∞ 7B7Q, E[X(0)] = ρ +
1 2 λ E[σ02 ] 2
1−ρ
.
q`BiBM; λ2 E[σ02 ] = λ2 (Var(σ0 ) + (E[σ0 ])2 ) = λ2 Var(σ0 ) + ρ2 BM i?2 T`2pBQmb 2[m HBiv- r2 Q#i BM i?2 7Q`KmH kj E[X(0)] = ρ +
kj
(SQHH +x2F- RNjy)- (E?BMi+?BM- RNjk)X
1 λ2 1 ρ2 + Var(σ0 ). 21−ρ 21−ρ
RRX9X Sah- GAhhG1- 1h*X
998
1t KTH2 RRX9X3, EH2BM`Q+Fǰb +QMb2`p iBQM H rXk9 h?Bb 2t KTH2 72 im`2b i?2 G/G/s/∞ [m2m2BM; bvbi2K rBi? M T`BQ`Biv +H bb2b Q7 1t KTH2 RRXjXdX A7 r2 TTHv i?2 SQHH +x2FĜE?BM+?BM K2 M p Hm2 7Q`KmH iQ 2 +? Q7 i?2 T`BQ`Biv +H bb2br2 Q#i BM- rBi? i?2 MQi iBQM BMi`Q/m+2/ BM 1t KTH2 RRXjXd1 2 E[Wi (0)] = λi E0Ai [σi,0 V˜i,0 ] + λi E0Ai [σi,0 ]. 2
URRX8RV
h?Bb 7Q`KmH TTHB2b r?2M i?2 b2`pB+2 Q7 +mbiQK2` Q7 Mv ivT2 + MMQi #2 BMi2`@ `mTi2/ M/ i?2 b2`pB+2 ` i2 Bb 1X h?2 QMHv `2[mB`2K2Mi 7Q` i?2 b2`pB+2 /Bb+BTHBM2 KQM; +mbiQK2`b Q7 i?2 b K2 +H bb M/ 7Q` i?2 T`BQ`Biv bbB;MK2Mi #2ir22M i?2 M +H bb2b Bb i? i HH i?2 Q#D2+ib +QMbB/2`2/ }i i?2 θt @7` K2rQ`FX h?2 bBKTH2bi + b2 rQmH/ #2 7B7Q /Bb+BTHBM2 KQM; i?2 K2K#2`b Q7 i?2 b K2 +H bb- M/ MQM@ T`22KTiBp2 T`BQ`Biv `mH2 rBi? ;Bp2M Q`/2` Q7 T`BQ`Biv- b v 1 ( 2 ( · · · ( M - r?2`2 i ( j K2 Mb i? i
i ? b T`BQ`Biv Qp2` jX amKKBM; mT i?2 M 2[m iBQMb URRX8RV M/ Q#b2`pBM; i? i M i=1 Wi (t) Bb i?2 iQi H rQ`FHQ /- M/ i? i 1 1 2 λi E0Ai [σi,0 ] = λE0A [σ02 ], 2 i=1 2 M
2 ] = λ λλi E0Ai [σ02 ] = λPA0 (U0 = i)E0A [σ02 | U0 = i]V- r2 Q#i BM UbBM+2 λi E0Ai [σi,0 M i=1
1 λi E0Ai [σi,0 V˜i,0 ] = E[W (0)] − λE0A [σ02 ] . 2
()
h?2 `B;?i@? M/ bB/2 /Q2b MQi /2T2M/ mTQM i?2 b2`pB+2 /Bb+BTHBM2b KQM; K2K#2`b Q7 i?2 b K2 +H bb Q` i?2 T`BQ`Biv bbB;MK2Mi b HQM; b i?2 +QM/BiBQMb bi i2/ Dmbi 7i2` 7Q`KmH URRX8RV `2 BM 7Q`+2X h?Bb ;Bp2b i?2 +QMb2`p iBQM H r, M
λi E0Ai [σi,0 V˜i,0 ] = +QMbi Mi.
i=1
P7 +Qm`b2- B7 7Q` 2 +? +H bb i- σi,0 M/ V˜i,0 `2 BM/2T2M/2Mi UmM/2` P Q` PA0 i - i?Bb Bb 2[mBp H2MiV- UV i F2b i?2 7Q`K M i=1
1 ρi E0Ai [V˜i,0 ] = E[W (0)] − λE0A [σ02 ]. 2
h?Bb 7Q`KmH ?QH/b 7Q` BMbi M+2 BM M M/GI/1/∞ bvbi2K rBi? T`BQ`Biv +H bb2b- MQ T`22KTiBQM- M/ 7B7Q /Bb+BTHBM2 KQM; i?2 K2K#2`b Q7 ;Bp2M +H bbX k9
(EH2BM`Q+F- RNe8)X
99e
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
RRX8 a2H2+i2/ TTHB+ iBQMb *? ` +i2`BbiB+ 6mM+iBQM Q7 i?2 qQ`FHQ / G2i {W (t)}t∈R #2 i?2 bi iBQM `v rQ`FHQ / T`Q+2bb Q7 bi iBQM `v G/G/1/∞ [m2m2 UrBi? i` {+ BMi2MbBiv ρ < 1VX "v HQQFBM; b2T ` i2Hv i r? i ? TT2Mb i i?2 DmKT iBK2b M/ r? i ? TT2Mb #2ir22M i?2 DmKTb- QM2 + M p2`B7v i?2 7QHHQrBM; 2pQHmiBQM 2[m iBQM, t eiuW (t) = eiuW (0) − iu eiuW (s) 1{W (s)>0} ds 0 + {eiuW (Tn ) − eiuW (Tn −) }1{0≤Tn ≤t} . n∈Z
h FBM; 2tT2+i iBQMb M/ Q#b2`pBM; i? i E[eiuW (t) ] = E[eiuW (0) ] M/ eiuW (s) 1{W (s)>0} = eiuW (s) − 1{X(s)=0} −iuE[eiuW (0) ] + iuP (X(0) = 0) + λE0A [eiuW (0) − eiuW (0−) ] = 0 . amTTQb2 i? i σn Bb BM/2T2M/2Mi Q7 W (Tn −) Ur?B+? Bb i?2 + b2 BM GI/GI/1/∞ [m2m2 7Q` BMbi M+2VX h?2M E0A [eiuW (0) − eiuW (0−) ] = E0A [eiuW (0−) (eiuσ0 − 1)] = E0A [eiuW (0−) ](E0A [eiuσ0 ] − 1) . AM GI/GI/1/∞ [m2m2- E0A [eiuσ0 ] = E[eiuσ0 ]- M/ i?2`27Q`2- i FBM; BMiQ ++QmMi P (X(0) = 0) = 1 − ρiuE[eiuW (0) ] = λE0A [eiuW (0−) ](E[eiuσ0 ] − 1) + iu(1 − ρ) . AM i?2 bT2+B H + b2 Q7 SQBbbQM ``Bp H T`Q+2bb (M/GI/1/∞)- i?2 T bi T`QT2`iv ;Bp2b E0A [eiuW (0−) ] = E[eiuW (0) ]- M/ i?2`27Q`2 E[eiuW (0) ] =
iu(1 − ρ) , iu − λ(Ψσ (u) − 1)
r?2`2 Ψσ (u) Bb i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 σ0 X JmHiBTH2 o + iBQMb G2i MQr {W (t)}t∈R #2 i?2 rQ`FHQ / T`Q+2bb Q7 M/GI/1/∞ bi iBQM `v [m2m2 rBi? KmHiBTH2 b2`p2` p + iBQMb, b bQQM b i?2 [m2m2 #2+QK2b 2KTiv- i?2 b2`p2` i F2b p + iBQMX h?2 b2[m2M+2 Q7 p + iBQM iBK2b {Vk }k∈Z Bb bbmK2/ BB/ M/ BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bbX Ai Bb bbmK2/ i? i i i?2 2M/ Q7 p + iBQM- i?2 b2`p2` }M/BM; i?2 [m2m2 biBHH 2KTiv i F2b MQi?2` p + iBQMc M/ bQ QM- b HQM; b i?2 [m2m2 `2K BMb 2KTivX * HHBM; tk i?2 bi `iBM; iBK2b Q7 i?2 p + iBQMb σn 1{Tn ≤t} + Vk 1{tk ≤t} − t . () W (t) = W (0) + n≥1
k≥1
RRX8X a1G1*h1. SSGA*hAPLa
99d
A7 2[mBHB#`BmK Bb bbmK2/- E[W (t)] = E[W (0)]- M/ i?2`27Q`2- /2MQiBM; #v λV i?2 BMi2MbBiv Q7 i?2 p + iBQM T`Q+2bb NV - r2 ? p2 ρt + λV tE[V0 ] − t = 0- r?B+? ;Bp2b λV =
1−ρ . E[V0 ]
URRX8kV
6`QK UV 7QHHQrb i?2 2pQHmiBQM 2[m iBQM, exp(iu W (t)) = exp(iu W (0)) + exp(iu W (Tn −))(eiu σn − 1)1{Tn ≤t} n≥1
+
exp(iu W (tk −))(e
iu Vk
− 1)1{tk ≤t} − iu
k≥1
t
eiu W (s) 1{W (s)>0} ds . 0
P#b2`pBM; i? i W (tk −) = 0 M/ i? i i?2 G2#2b;m2 K2 bm`2 Q7 {t; W (t) = 0} Bb XbX MmHH- i?2 7QHHQrBM; 7Q`KmH Bb Q#i BM2/ 7i2` +QKTmi iBQMb bBKBH ` iQ i?Qb2 T2`7Q`K2/ BM i?2 #Qp2 /2`Bp iBQM Q7 i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 rQ`FHQ / bbmKBM; i?2 2tBbi2M+2 Q7 bi iBQM `v bi i2, E[exp(iu W (0))] =
λV (ΨV (u) − 1) , iu − λ(Ψσ (u) − 1)
r?2`2 ΨV (u) M/ Ψσ (u) `2 i?2 +? ` +i2`BbiB+ 7mM+iBQMb Q7 V0 M/ σ0 `2bT2+iBp2HvX h?2`27Q`2- mbBM; 2tT`2bbBQM URRX8kV 7Q` i?2 BMi2MbBiv Q7 i?2 p + iBQM T`Q+2bbE[exp(iu W (0))] =
ΨV (u) − 1 1−ρ × . iu − λ(Ψσ (u) − 1) E[V0 ]
URRX8jV
(u)−1 P#b2`p2 i? i ΨVE[V Bb i?2 +? ` +i2`BbiB+ 7mM+iBQM Q7 i?2 7Q`r `/ UQ` 2[mBp @ 0] H2MiHv- Q7 i?2 # +Fr `/V `2+m``2M+2 iBK2 Q7 bi iBQM `v U/2H v2/V `2M2r H T`Q+2bb rBi? BMi2`@`2M2r H /Bbi`B#miBQM FV - i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 V0 X 6`QK 2tT`2bbBQM URRX8jV M/ i?Bb `2K `F- M/ HbQ Q#b2`pBM; i? i BM i?2 p + iBQM n Bb i?2 r BiBM; iBK2 #27Q`2 bvbi2K b r2HH b BM i?2 Q`B;BM H bvbi2K- W (Tn −) = V b2`pB+2 r?2M 7B7Q /Bb+BTHBM2 Bb bbmK2/- QM2 ? b- rBi? Q#pBQmb MQi iBQM-
ΨV˜0 (u)|F IF O,
V AC
= ΨV 0 (u)|F IF O,
N OV AC
× ΨY (u),
r?2`2 Y Bb ` M/QK p `B #H2 ? pBM; i?2 b K2 /Bbi`B#miBQM b i?2 7Q`r `/ `2@ +m``2M+2 iBK2 Q7 bi iBQM `v `2M2r H T`Q+2bb +Q``2bTQM/BM; iQ i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM FV X AM rQ`/b-k8 h?2Q`2K RRX8XR h?2 r BiBM; iBK2 Q7 M/GI/1/∞ 7B7Q bvbi2K rBi? KmHiBTH2 BM/2T2M/2Mi p + iBQMb Q7 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM FV Bb /Bbi`B#mi2/ b i?2 bmK Q7 irQ BM/2T2M/2Mi ` M/QK p `B #H2b X M/ Y - r?2`2, Ç X Bb /Bbi`B#mi2/ b i?2 r BiBM; iBK2 BM i?2 b K2 M/GI/1/∞ 7B7Q bvbi2K rBi?Qmi p + iBQMbc Ç Y Bb /Bbi`B#mi2/ b i?2 7Q`r `/ `2+m``2M+2 iBK2 Q7 bi iBQM `v `2M2r H T`Q+2bb +Q``2bTQM/BM; iQ i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM FV X k8
(6m?`K MM- RN39)- (6m?`K MM M/ *QQT2`- RN38)X
993
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
S`BQ`BiB2b BM Jf:AfRf∞ h?2 p2` ;2 r BiBM; iBK2 Q7 +mbiQK2` Q7 ;Bp2M ivT2 BM Jf:AfRf∞ bvbi2K rBi? M T`BQ`Biv +H bb2b rBHH MQr #2 +QKTmi2/X JQ`2 T`2+Bb2Hv- i?2 bBim iBQM Bb i? i Q7 1t KTH2 RRXjXd- rBi? i?2 //BiBQM H 72 im`2 i? i i?2 M BMTmi ~Qrb +Q``2bTQM/BM; iQ i?2 M +H bb2b Q7 T`BQ`Biv `2 BM/2T2M/2Mi M/ Q7 i?2 M/GI ivT2X h?2`2 Bb QM2 b2`p2` QT2` iBM; i mMBi bT22/- i?2 b2`pB+2 /Bb+BTHBM2 Bb MQM@T`22KTiBp2- 7B7Q 7Q` i?2 +mbiQK2`b Q7 i?2 b K2 +H bb- M/ T`BQ`Biv Bb ;Bp2M iQ +mbiQK2`b Q7 +H bb i Qp2` +mbiQK2`b Q7 +H bb j B7 i < jX 1[mBHB#`BmK Bb bbmK2/- M/ BM T `iB+mH ` ρ=
M
ρi < 1 .
i=1
_2+ HH i?2 /2}MBiBQM σi (t) = σi,n - B7 t ∈ [Ti,n , Ti,n+1 )X .2}M2 i?2 ?BbiQ`v {Fi,t } #v M 3 σi ,Ai σk ,Ak Fi,t = Ft . URRX89V ∨ F∞ k=1, k=i
h?2 K `F2/ TQBMi T`Q+2bb- (Ai , {σi,n }) /KBib i?2 Fi,t @BMi2MbBiv F2`M2H λi Gi (dσ)i? i Bb iQ b v E H(t, σ)Ni (dt × dσ) = E H(t, σ)λi Gi (dσ)dt , R
R+
7Q` HH MQM@M2; iBp2 P(Fi,t )
Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: E0Ai [V˜i,0 ] = E0A [V˜0 | +mbiQK2` ``BpBM; i T0 = 0 Bb Q7 ivT2 i],
URRX8NV
r?2`2 V˜n Bb i?2 r BiBM; iBK2 Q7 +mbiQK2` ``BpBM; i iBK2 Tn X _2K `F, AM Q`/2` iQ b?Qr i? i E[V˜l (0)] = E0Al [V˜l,0 ], mbBM; i?2 T bi T`QT2`ivbQK2 + `2 Kmbi #2 2t2`+Bb2/X AM/22/ Al Bb M Fl,t @SQBbbQM T`Q+2bb rBi? BMi2MbBiv λl M/ V˜l,0 = V˜l (Tl,0 ) = V˜l (0) PA0 l @a.s. >Qr2p2`- {V˜l (t)} Bb MQi M Fl,t @T`2/B+i #H2 T`Q+2bb- r?2M Fl,t Bb /2}M2/ #v URRX89VX h?Bb T`Q#H2K Bb 2 bBHv +B`+mKp2Mi2/ #v BMi`Q/m+BM; i?2 σ@}2H/ σl Gl,t = Fl,t ∨ F∞ Ur2 // iQ Fl,t i?2 7mim`2 Q7 i?2 b2`pB+2 T`Q+2bbVX h?2 TQBMi T`Q+2bb Bb Gl,t @SQBbbQM T`Q+2bb Q7 BMi2MbBiv λl - M/ i?Bb iBK2 {V˜l (t)} Bb H27i@+QMiBMmQmb T`Q+2bb / Ti2/ iQ Gl,t - M/ i?2`27Q`2 Gl,t @T`2/B+i #H2X h?Bb }MBb?2b i?2 T`QQ7 Q7 *Q#? KǶb 7Q`KmH X 1t KTH2 RRX8Xk, Jf:AfRf∞ bTi MQM@T`22KTiBp2X AM i?Bb /Bb+BTHBM2 i?2 +mb@ iQK2` rBi? bK HH2bi `2[mB`2/ b2`pB+2 UbK HH2bi T`Q+2bbBM; iBK2V ? b MQM@T`22KTiBp2 T`BQ`Biv Qp2` i?2 Qi?2`bX G2iiBM; G #2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 `2[mB`2/ b2`pB+2 iBK2 E0A V˜0 | σ0 ∈ (x − h, x] =
1−λ
1 λE[σ 2 ] 2 0
(0,x−h]
ydG(y)
1−λ
.
(x−h,x]
URRXeyV
ydG(y)
S`QQ7X *QMbB/2` i?`22 T`BQ`Biv +H bb2b, BM +H bb R- Tmi HH +mbiQK2`b rBi? `2[mB`2/ b2`pB+2 bK HH2` i? M Q` 2[m H iQ x − h- BM +H bb k i?Qb2 rBi? `2[mB`2/ b2`pB+2 BM (x − h, x]- M/ BM +H bb j i?2 `2bi Q7 i?2 +mbiQK2`bX h?mb UB;MQ`BM; +H bb jV ρ1 = λ ydG(y), ρ2 = λ ydG(y) . (0,x−h]
(x−h,x]
TTHvBM; URRX83VĜURRX8NV- r2 Q#i BM URRXeyVX AM i?2 + b2 r?2M x Bb MQi /Bb+QMiB@ MmBiv TQBMi Q7 G(x)- H2iiBM; h ;Q iQ 0 BM URRXeyV- r2 Q#i BM E0A [V˜0 | σ0 = x] =
1 λE[σ02 ] 2 . (1 − λ (0,x] ydG(y))2
AM i?2 T`22KTiBp2 + b2- i?2 pB`im H bQDQm`M iBK2 Q7 +mbiQK2` Q7 +H bb i ``BpBM; i iBK2 t Bb Vi (t) = Fi (t) + Gi (t) , URRXeRV r?2`2 Fi (t) Bb i?2 iBK2 #2ir22M t M/ i?2 }`bi iBK2 i?2 +mbiQK2` `2+2Bp2b ii2MiBQM 7`QK i?2 b2`p2`- M/ Gi (t) Bb i?2 bmK Q7 i?2 b2`pB+2 σ ˜i (t) M/ Q7 i?2 b2`pB+2 Q7 HH +mbiQK2`b Q7 +H bb 1, . . . , i − 1 ``BpBM; BM i?2 bvbi2K BM i?2 BMi2`p H (t + Fi (t), t + Fi (t) + Gi (t)]X *H2 `Hv BM i?2 +QKTmi iBQM Q7 E[Fi (t)]- i?2 +mbiQK2`b Q7 +H bb i +
RRX8X a1G1*h1. SSGA*hAPLa
98R
1, . . . , K /Q MQi TH v `QH2- M/ i?2`27Q`2 E[Fi (t)] Bb i?2 p2` ;2 r BiBM; iBK2 Q7 +H bb i pB`im H +mbiQK2` ``BpBM; i iBK2 t BM bvbi2K rBi? QMHv +mbiQK2`b Q7 ivT2 1, . . . , i- BM i?2 MQM@T`22KTiBp2 + b2ke
i 1 2 k=1 λk E[σk ] 2 . E[Fi (t)] =
i−1
i 1 − k=1 ρk 1 − k=1 ρk h?2 i2`K Gi (t) + M #2 2tT`2bb2/ b ˜i (t) + Gi (t) = σ
i−1 k=1
R×R+
σ1(t+Fi (t),t+Fi (t)+Gi (t)] (u)Nk (du × dσ)
M/ `;mBM; b BM i?2 MQM@T`22KTiBp2 + b2- r2 Q#i BM σi (t)] + E[Gi (t)] = E[˜
i−1
λk E[σk ]E[Gi (t)] .
k=1
h?2`27Q`2- Q#b2`pBM; i? i E[˜ σi (t)] = E[σi ] UT bi V- r2 Q#i BM E[Gi (t)] =
1−
E[σi ]
i−1 k=1
ρk
.
URRXekV
*QK#BMBM; URRXeRVĜURRXekV- r2 Q#i BM
i 1 2 E[σi ] k=1 λk E[σk ] 2 + . E[Vi (t)] =
i−1
i−1
i 1 − 1 − k=1 ρk 1 − k=1 ρk k=1 ρk >2`2 HbQ- BM pB2r Q7 T bi E[Vi (t)] = E0Ai [Vi,0 ] = E0A [V0 | i?2 +mbiQK2` ``BpBM; i T0 = 0 Bb Q7 ivT2 i] , r?2`2 Vi,n Bb i?2 bQDQm`M iBK2 Q7 i?2 +mbiQK2` Q7 ivT2 i ``BpBM; i iBK2 Ti,n M/ Vn Bb i?2 bQDQm`M iBK2 Q7 i?2 +mbiQK2` UQ7 Mv ivT2V ``BpBM; i iBK2 Tn X 1t KTH2 RRX8Xj, h?2 c/ρ `mH2X *QMbB/2` i?2 M/GI/1/∞ bvbi2K rBi? M T`B@ Q`Biv +H bb2b rBi? MQM@T`22KTiBp2 T`BQ`Biv b /2b+`B#2/ i i?2 #2;BMMBM; Q7 i?2 T`2b2Mi bm#b2+iBQMX 6Q` bBKTHB+Biv- /2MQi2 E[V˜i (0)] #v V˜i X h?2 Q#D2+iBp2 BM i?Bb 2t KTH2 Bb iQ KBMBKBx2 i?2 7mM+iBQM H C=
M
ci V˜i ,
i=1
r?2`2 ci > 0- KQM; HH TQbbB#H2 ` MFBM;b UBM i2`Kb Q7 T`BQ`BivV Q7 i?2 M +H bb2bX Ai im`Mb Qmikd i? i i?2 +H bb2b b?QmH/ #2 ` MF2/ ++Q`/BM; iQ i?2 p Hm2 Q7 c/ρ- i? i ke kd
(*Q#? K- RN89)X (a2p+BF- RNd9)X
98k
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
Bb- MQM@T`22KTiBp2 T`BQ`Biv Bb ;Bp2M iQ +H bb i Qp2` +H bb j B7 M/ QMHv B7 ρcii > c i?2 + b2 Q7 2[m HBiv ρcii = ρjj - ;Bp2 ?B;?2bi T`BQ`Biv iQ i Q` j BM/Bz2`2MiHvVX
cj ρj
UBM
hQ T`Qp2 i?Bb `2bmHi- +QKT `2 irQ ` MFBM;b A M/ B /Bz2`BM; QMHv BM irQ /D @ +2Mi +H bb2bX JQ`2 T`2+Bb2Hv- bmTTQb2 U7Q` +QMp2MB2M+2V i? i ` MFBM; A +Q``2bTQM/b iQ 1 ( 2 ( · · · ( M, r?2`2 i ( j K2 Mb i? i +H bb i ? b UMQM@T`22KTiBp2V T`BQ`Biv Qp2` +H bb j- M/ bmTTQb2 i? i ` MFBM; B 2t+? M;2b +H bb2b i M/ i + 1 1 ( 2 ( · · · ( i − 1 ( i + 1 ( i ( i + 2 ( · · · ( M. * HH V˜jA M/ V˜jB i?2 p2` ;2 r BiBM; iBK2 7Q` +mbiQK2` Q7 +H bb j mM/2` i?2 ` MFBM;b A M/ B `2bT2+iBp2HvX HbQ H2i CA =
M
cj V˜jA ,
CB =
j=1
M
cj V˜jB .
j=1
6`QK i?2 2tT`2bbBQM URRX83V- r2 b22 i? i V˜kA = V˜kB - 7Q` HH k, 1 ≤ k ≤ M, k = i, k = i + 1- bQ i? i A B C A − C B = ci (V˜iA − V˜iB ) + ci+1 (V˜i+1 − V˜i+1 ).
URRXejV
"v EH2BM`Q+FǶb +QMb2`p iBQM H r Ua2+iBQM RRX9X3VM
ρj V˜jA =
M
j=1
ρj V˜jB ,
j=1
i? i Bb- BM pB2r Q7 i?2 T`2pBQmb `2K `F A B ρi (V˜iA − V˜iB ) = −ρi+1 (V˜i+1 − V˜i+1 ).
URRXe9V
6`QK URRXejV M/ URRXe9V- r2 Q#i BM C − C = ρi (V˜iA − V˜iB ) A
B
ci ci+1 − ρi ρi+1
.
LQr V˜iA − V˜iB < 0 Ui?BMF- Q` mb2 6Q`KmH URRX83VV M/ i?2`27Q`2 C A − C B < 0 B7 M/ QMHv BM ρcii > ρci+1 X i+1 h?Bb T`Qp2b i?2 `2bmHi bBM+2- 7Q` Mv ` MFBM; MQi b iBb7vBM; i?2 ρc +QM/BiBQM- r2 + M }M/ irQ /D +2Mi +H bb2b i M/ i + 1 pBQH iBM; i?2 +QM/BiBQM- M/ 2t+? M;BM; i?2b2 irQ +H bb2b rQmH/ ;Bp2 #2ii2` ` MFBM;X 1t KTH2 RRX8X9, h?2 μc `mH2X amTTQb2 r2 rBb? iQ KBMBKBx2 C=
M i=1
ci Qi ,
RRX8X a1G1*h1. SSGA*hAPLa
98j
r?2`2 Qi = E[Qi (t)]- i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b Q7 +H bb i BM i?2 r BiBM; `QQK i iBK2 tX 6`QK GBiiH2Ƕb 7Q`KmH C=
M
ci λi V˜i
i=1
M/ i?2`27Q`2- 7`QK i?2 T`2pBQmb `2bmHib- r2 b22 i? i T`BQ`Biv b?QmH/ #2 ;Bp2M iQ +H bb i Qp2` +H bb j B7 μi ci > μj cj X 1t KTH2 RRX8X8, PTiBK HBiv Q7 bTi MQM@T`22KTiBp2X 6Q` [m2m2BM; bvb@ i2K Q7 i?2 M/GI/1/∞ ivT2- KQM; HH i?2 MQM@T`22KTiBp2 b2`pB+2 /Bb+BTHBM2b # b2/ QM i?2 `2[mB`2/ b2`pB+2 iBK2- bTi MQM@T`22KTiBp2 KBMBKBx2b i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2KX h?Bb `2bmHi 7QHHQrb 7`QK i?2 μc `mH2 Q7 1t@ KTH2 RRX8X9 rBi? ci ≡ 1 UKBMBKBxBM; i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b E[Q(t)] BM i?2 r BiBM; `QQK Bb i?2 b K2 b KBMBKBxBM; i?2 p2` ;2 MmK#2` E[X(t)] Q7 +mbiQK2`b BM i?2 bvbi2K- r?2M i?2`2 Bb QM2 b2`p2`VX h?mb BM 1t KTH2 RRX8X9- BM Q`/2` iQ KBMBKBx2 E[Q(t)]- r2 Kmbi ;Bp2 ?B;?2bi T`BQ`Biv iQ i?2 +H bb rBi? bK HH2bi p2` ;2 b2`pB+2 iBK2X HBKBiBM; `;mK2Mi UrBi? +H bb i 7Q`K2/ Q7 i?Qb2 +mbiQK2`b rBi? `2[mB`2/ b2`pB+2 #2ir22M ih M/ (i + 1)hV ;Bp2b i?2 QTiBK HBiv Q7 bTi MQM@ T`22KTiBp2X 1t KTH2 RRX8Xe, ii BM2/ b2`pB+2 BM i?2 G/GI/1/∞ T`Q+2bbQ` b? `BM; [m2m2X AM i?Bb 2t KTH2- i?2 ``Bp H T`Q+2bb Bb Q7 ;2M2` H M im`2 M/ i?2 b2`pB+2 iBK2 b2[m2M+2 Bb BB/- BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bbX h?2 [m2m2BM; /Bb+BTHBM2 Bb T`Q+2bbQ` b? `BM; UTbV, 2 +? +mbiQK2` T`2b2Mi BM i?2 bvbi2K `2+2Bp2b b2`pB+2 i ` i2 BMp2`b2Hv T`QTQ`iBQM H iQ i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2KX h?mb +mbiQK2` ``BpBM; i iBK2 Tn < 0 M/ `2[mB`BM; b2`pB+2 σn rBHH #2 T`2b2Mi i iBK2 y M/ rBHH ? p2 ii BM2/ b2`pB+2 MQi ;`2 i2` i? M a B7 M/ QMHv B7 0 1 ds ≤ σn ∧ a , Tn X(s) r?2`2 X(t) /2MQi2b i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi i iBK2 tX .2MQiBM; #v Xa (t) i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi i iBK2 t M/ rBi? ii BM2/ b2`pB+2 MQi ;`2 i2` i? M a, Xa (0) = 1Tn ≤0 1{ 0 1 ds≤σn ∧a} . Tn X(s) n∈Z
J2+F2Ƕb 7Q`KmH ;Bp2b E[Xa (0)] =
∞
λE0A
t
0
1{
1 0 X(s) ds≤σ0 ∧a
} dt .
G2i X ∗ (t) = X(t) B7 t ≤ V0 UV0 Bb i?2 bQDQm`M iBK2 Q7 +mbiQK2` yV M/ X ∗ (t) = X(t) + 1 B7 t > V0 - M/ 7Q` HH u > 0- H2i r(u) #2 /2}M2/ #v r(u) 1 ds = u . ∗ (s) X 0
989
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
h?2M r(σ0 ) = V0
PA0 @ XbX
M/ r(u) Bb i?2 bQDQm`M iBK2 Q7 +mbiQK2` y ;Bp2M Bib `2[mB`2/ b2`pB+2 Bb uX h?2 T`Q+2bb {r(u)}u≥0 Bb + HH2/ i?2 `2bTQMb2 iBK2 T`Q+2bb Q7 +mbiQK2` yX HbQ PA0 @ XbX ∞ ∞ t 1{ 1 ds≤σ0 ∧a} dt = 1{t≤r(σ0 ∧a)} dt = r(σ0 ∧ a). 0
0 X(s)
0
LQr- i?2 b2`pB+2 b2[m2M+2 Bb BB/ M/ BM/2T2M/2Mi Q7 i?2 ``Bp H T`Q+2bb- M/ r(t) /2T2M/b QM i?2 BMTmi T`Q+2bb 2t+2Ti σ0 X h?2`27Q`2 B7 G /2MQi2b i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 σ0 r(x ∧ a)G(dx) = r¯(x)G(dx) + r¯(a)(1 − G(a)), E0A [r(σ0 ∧ a)] = E0A [0,∞)
[0,a]
r?2`2 r¯(x) = E0A [r(x)] Bb i?2 p2` ;2 `2bTQMb2 iBK2 iQ `2[mB`2/ b2`pB+2 tX "mi r¯(x)G(dx) + r¯(a)(1 − G(a)) = (1 − G(x))¯ r(dx), [0,a]
[0,a]
bQ i? i }M HHv
(1 − G(x))¯ r(dx).
E[Xa (0)] = λ
URRXe8V
[0,a]
1t KTH2 RRX8Xd, _2bTQMb2 iBK2 Q7 i?2 M/GI/1/∞ T`Q+2bbQ` b? `BM; [m2m2X G2i i?2 bi i2 i iBK2 t Q7 M/GI/1/∞ T`Q+2bbQ` b? `BM; [m2m2 +QMbBbi Q7 URV i?2 MmK#2` X(t) Q7 +mbiQK2`b BM i?2 bvbi2K- M/ UkV i?2 2H Tb2/ M/ `2bB/m H b2`pB+2 iBK2b Q7 i?2b2 +mbiQK2`bX h?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 bi i2 mM/2` P UQ7 +Qm`b2 r2 bbmK2 ρ < 1V + M #2 +QKTmi2/ #v ;2M2` H K2i?Q/b Q7 S HK i?2Q`vX Ai Bb Q#i BM2/ b 7QHHQrb, Ç 6B`bi b2H2+i i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K iQ #2 n rBi? T`Q# #BHBiv ρn (1 − ρ)X Ç h?2M b2H2+i n BM/2T2M/2Mi T B`b Q7 2H Tb2/@`2bB/m H b2`pB+2 iBK2b /Bbi`B#mi2/ b T B` Q7 # +Fr `/@7Q`r `/ `2+m``2M+2 iBK2b Q7 `2M2r H T`Q+2bb rBi? +X/X7X G- i?2 +X/X7X Q7 i?2 b2`pB+2 iBK2bX AM T `iB+mH ` i?2 `2bB/m H b2`pB+2 iBK2b M/ i?2 ∞ 2H Tb2/ b2`pB+2 iBK2b ? p2 i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Ge (x) = μ x (1 − G(x))dx- r?2`2 μ−1 Bb i?2 K2 M b2`pB+2 iBK2X q2 + M mb2 i?Bb `2bmHi iQ Q#i BM i?2 p2` ;2 `2bTQMb2 iBK2X AM/22/ i?2 T`Q# #BHBiv 7Q` ;Bp2M +mbiQK2` iQ ? p2 ii BM2/ b2`pB+2 MQ ;`2 i2` i? M a Bb Ge (a)- M/ i?2`27Q`2 ρ Ge (a) . E[Xa (0)] = E[X(0)]Ge (a) = 1−ρ
RRXeX 1s1_*Aa1a
988
*QKT `BM; rBi? URRXe8V ;Bp2b a a λ (1 − G(t))dt = λ (1 − G(t))d¯ r(t) 1−ρ 0 0 M/ i?2`27Q`2-k3 bBM+2 r¯(0) = 0r¯(t) =
RRXe
t . 1−ρ
1t2`+Bb2b
1t2`+Bb2 RRXeXRX _2;mH `Biv Q7 [m2m2b M/ i?2B` M2irQ`Fb a?Qr i? i HH i?2 ?K+b `BbBM; 7`QK i?2 [m2m2BM; KQ/2Hb Q7 a2+iBQM RRXR `2 `2;mH ` DmKT T`Q+2bb2bX 1t2`+Bb2 RRXeXkX h?2 `2p2`b H i2bi G2i A = {qij }i,j∈E #2 bi #H2 M/ +QMb2`p iBp2 ;2M2` iQ` QM i?2 +QmMi #H2 bi i2 bT +2 E- M/ H2i π #2 bi`B+iHv TQbBiBp2 T`Q# #BHBiv /Bbi`B#miBQM QM EX .2}M2 ˜ = {˜ A qij }i,j∈E #v π(i)˜ qij = π(j)qji . S`Qp2 i? i B7 7Q` HH i ∈ E-
q˜ij = qi ,
j∈E,j=i
i?2M π T A = πX 1t2`+Bb2 RRXeXjX JfJfRf∞fSa AM i?Bb JfJfRf∞ [m2m2- 2 +? Q7 i?2 +mbiQK2`b T`2b2Mi i iBK2 t `2+2Bp2b b2`pB+2 1 i bT22/ X(t) X :Bp2 ;bKT /2b+`BTiBQM Q7 i?Bb T`Q+2bbX q? i `2 i?2 BM}MBi2b@ BK H ;2M2` iQ`- i?2 2`;Q/B+Biv +QM/BiBQM- M/ i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb\ 1t2`+Bb2 RRXeX9X C +FbQMǰb M2irQ`F rBi? bi i2@/2T2M/2Mi b2`pB+2 bT22/b h?2 7QHHQrBM; KQ/B}+ iBQM Q7 i?2 # bB+ C +FbQM M2irQ`F Bb +QMbB/2`2/X 6Q` HH i1 ≤ i ≤ K- i?2 b2`p2` i bi iBQM i ? b bT22/ Q7 b2`pB+2 φi (ni ) r?2M i?2`2 `2 ni +mbiQK2`b T`2b2Mi BM bi iBQM i- r?2`2 φi (k) > 0 7Q` HH k ≥ 1 M/ φi (0) = 0X h?2 M2r BM}MBi2bBK H ;2M2` iQ` Bb Q#i BM2/ 7`QK i?2 bi M/ `/ QM2 #v `2TH +BM; μi #v μi φ(ni )X *?2+F i?Bb M/ b?Qr i? i B7 7Q` HH i- 1 ≤ i ≤ KAi := 1 +
∞ Ai =1
k3
(a F i - LQ;m+?B M/ PBxmKB- RNeN)X
ρni !ni i k=1 φ(k)
< ∞,
98e
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
r?2`2 ρi = λi /μi M/ λi Bb i?2 bQHmiBQM Q7 i?2 i` {+ 2[m iBQM- i?2M i?2 M2irQ`F Bb 2`;Q/B+ rBi? bi iBQM `v /Bbi`B#miBQM π(n) =
K
πi (ni ),
i=1
r?2`2 πi (ni ) =
ρn i 1 !ni i . Ai k=1 φi (k)
1t2`+Bb2 RRXeX8X *E *QMbB/2` i?2 +HQb2/ C +FbQM M2irQ`F #2HQr r?2`2 HH b2`pB+2 iBK2b i /Bz2`2Mi [m2m2b ? p2 i?2 b K2 U2tTQM2MiB HV /Bbi`B#miBQM Q7 K2 M 15 X *QKTmi2 7Q` N = 5 i?2 p2` ;2 iBK2 bT2Mi #v +mbiQK2` iQ ;Q 7`QK i?2 H27iKQbi TQBMi A iQ i?2 `B;?iKQbi TQBMi B- M/ i?2 p2` ;2 MmK#2` Q7 +mbiQK2`b T bbBM; #v A T2` mMBi iBK2X
1t2`+Bb2 RRXeXeX *HQb2/ C +FbQM rBi? bBM;H2 +mbiQK2` *QMbB/2` i?2 +HQb2/ C +FbQM M2irQ`F Q7 i?2 i?2Q`v- rBi? N = 1 +mbiQK2`X G2i {Y (t)}t≥0 #2 i?2 T`Q+2bb ;BpBM; i?2 TQbBiBQM Q7 i?Bb +mbiQK2`- i? i Bb- Y (t) = i B7 b?2 Bb BM bi iBQM i i iBK2 tX a?Qr i? i {Y (t)}t≥0 Bb `2;mH ` DmKT ?K+- B``2/m+B#H2 M/ ;Bp2 Bib bi iBQM `v /Bbi`B#miBQMX 1t2`+Bb2 RRXeXdX AK#2//2/ ?K+ Q7 M Jf:AfRf∞f7B7Q [m2m2 a?Qr i? i i?2 BK#2//2/ ?K+ Q7 M Jf:AfRf∞f7B7Q Bb B``2/m+B#H2 U b HQM; b i?2 b2`pB+2 iBK2b `2 MQi B/2MiB+ HHv MmHHVX 1t2`+Bb2 RRXeX3X Z = N (0, τ ] S`Qp2 i?2 bi i2K2Mi BM 1t KTH2 RRXRXR8 +QM+2`MBM; i?2 b2[m2M+2 {Zn }n≥1 - M K2Hvi? i Bi Bb BB/ M/ BM/2T2M/2Mi Q7 X0 - M/ Bb /Bbi`B#mi2/ b Z = N (0, τ ]- r?2`2 τ M/ N `2 BM/2T2M/2Mi- N Bb M ?TT rBi? BMi2MbBiv μ M/ i?2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ Bb F X 1t2`+Bb2 RRXeXNX h` MbB2Mi :AfJfRf∞ [m2m2 a?Qr i? i i?2 :AfJfRf∞ [m2m2 Q7 1t KTH2 RRXRXR8 Bb i` MbB2Mi B7 ρ > 1X 1t2`+Bb2 RRXeXRyX *QM7BM2/ ?K+ G2i {X(t)}t≥0 #2 M B``2/m+B#H2 TQbBiBp2 `2+m``2Mi ?K+ rBi? BM}MBi2bBK H ;2M2` iQ` A M/ bi iBQM `v /Bbi`B#miBQM πX amTTQb2 i? i (A, π) Bb `2p2`bB#H2X G2i S #2 bm#b2i ˜ #v Q7 i?2 bi i2 bT +2 EX .2}M2 i?2 BM}MBi2bBK H ;2M2` iQ` A
RRXeX 1s1_*Aa1a
98d q˜ij =
αqij B7 i ∈ S, j ∈ E − S qij Qi?2`rBb2
r?2M i = jX h?2 +Q``2bTQM/BM; ?K+ Bb B``2/m+B#H2 B7 α > 0X A7 α = 0- i?2 bi i2 bT +2 rBHH #2 `2/m+2/ iQ S- iQ K BMi BM B``2/m+B#BHBivX a?Qr i? i i?2 +QMiBMmQmb ˜ /KBib i?2 bi iBQM `v /Bbi`B#miBQM π iBK2 ?K+ bbQ+B i2/ iQ A ˜ ;Bp2M #v αC × π(i) B7 i ∈ S, π ˜i = C × απ(i) B7 i ∈ E − S, rBi? i?2 Q#pBQmb KQ/B}+ iBQM r?2M α = 0X 1t2`+Bb2 RRXeXRRX p2` ;2 i` 77B+ BM C +FbQMǰb M2irQ`F a?Qr i? i BM 2[mBHB#`BmK- i?2 i` {+ 2[m iBQMb 7Q` i?2 C +FbQM M2irQ`F `2+2Bp2 i?2 7QHHQrBM; BMi2`T`2i iBQM, λi = E[Ai (0, 1]],
λi ri j = E[Ai,j (0, 1]]
r?2`2 Ai Bb i?2 TQBMi T`Q+2bb +QmMiBM; i?2 ``Bp Hb U2ti2`M H M/ BMi2`M HV BMiQ bi iBQM i- M/ Ai,j Bb i?2 TQBMi T`Q+2bb +QmMiBM; i?2 i` Mb72`b 7`QK bi iBQM i iQ bi iBQM jX 1t2`+Bb2 RRXeXRkX E2HHvǰb M2irQ`F *QMbB/2` i2H2+QKKmMB+ iBQMb M2irQ`F rBi? K `2H vbX M BM+QKBM; + HH +?QQb2b dz`Qmi2Ǵ r KQM; b2i RX h?2 M2irQ`F i?2M `2b2`p2b i?2 b2i Q7 `2H vb r1 , . . . , rk(r) +Q``2bTQM/BM; iQ i?Bb `Qmi2- M/ T`Q+2bb2b i?2 + HH iQ /2biBM iBQM BM M 2tTQM2MiB H iBK2 Q7 K2 M μ−1 r X h?2 BM+QKBM; + HHb rBi? `Qmi2 r 7Q`K ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λr X Ai Bb bbmK2/ i? i HH i?2 `2H vb `2 mb27mH- BM i? i i?2v `2 T `i Q7 i H2 bi QM2 `Qmi2 BM RX kN
URV h?2 + T +Biv Q7 i?2 bvbi2K Bb 7Q` i?2 iBK2 #2BM; bbmK2/ iQ #2 BM}MBi2- i? i Bbi?2 MmK#2` Xr (t) Q7 + HHb QM `Qmi2 r i iBK2 t + M i F2 Mv BMi2;2` p Hm2X HH i?2 mbm H BM/2T2M/2M+2 ?vTQi?2b2b `2 K /2, i?2 T`Q+2bbBM; iBK2b M/ i?2 SQBbbQM T`Q+2bb2b `2 BM/2T2M/2MiX :Bp2 i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +QMiBMmQmb iBK2 ?K+ {X(t)}t≥0 - r?2`2 X(t) = (Xr (t), r ∈ R)X UkV h?2 + T +Biv Q7 i?2 bvbi2K Bb MQr `2bi`B+i2/ b 7QHHQrbX *QMbB/2` ;Bp2M T B` (a, b) Q7 `2H vbX Ai `2T`2b2Mib dzHBMFǴ BM i?2 M2irQ`FX h?Bb HBMF ? b }MBi2 + T +Biv Cab X h?Bb K2 Mb i? i i?2 iQi H MmK#2` Q7 + HHb mbBM; i?Bb HBMF + MMQi 2t+22/ i?Bb + T +Biv- rBi? i?2 +QMb2[m2M+2 i? i M BM+QKBM; + HH `2[mB`BM; `Qmi2 T bbBM; i?`Qm;? i?Bb HBMF rBHH #2 HQbi B7 i?2 HBMF Bb b im` i2/ r?2M Bi ``Bp2bX h?2 T`Q+2bb {X(t)}t≥0 i?2`27Q`2 ? b 7Q` bi i2 bT +2 ˜ = {n = (nr ; r ∈ R); E nr ≤ Cab 7Q` HH HBMFb (a, b)}. r∈R,(a,b)∈r
q? i Bb i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +? BM {X(t)}t≥0 \ kN
(E2HHv- RNdN)X
983
*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:
1t2`+Bb2 RRXeXRjX o + iBQM [m2m2b a?Qr i? i BM i?2 M/GI/1/∞ 6A6P bvbi2K rBi? KmHiBTH2 BM/2T2M/2Mi p + iBQMbi?2 p2` ;2 r BiBM; iBK2 Bb ;Bp2M #v i?2 7Q`KmH λ E[σ 2 ] 1 E[V 2 ] + . 2 1 − ρ 2 E[V ] 1t2`+Bb2 RRXeXR9X qQ`FHQ / Q7 JfJfRf∞ a?Qr i? i i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 rQ`FHQ / T`Q+2bb Q7 M JfJfRf∞ [m2m2 rBi? ``Bp H ` i2 λ M/ K2 M b2`pB+2 iBK2 μ−1 bm+? i? i ρ = μλ < 1 Bb FW (x) = 1 − ρ (1 − exp{−(μ − λ)x}) .
1t2`+Bb2 RRXeXR8X hrQ 2tT`2bbBQMb 7Q` i?2 i` 77B+ BMi2MbBiv o2`B7v i?2 7QHHQrBM; irQ 2tT`2bbBQMb 7Q` i?2 i` {+ BMi2MbBivX 6B`bi BM i2`Kb Q7 P , N 1 σk , N →∞ N k=1
ρ = λ lim M/ i?2M BM i2`Kb Q7 PA0 ,
ρ=
P @ XbX ,
E0A [σ0 ] . E0A [τ0 ]
1t2`+Bb2 RRXeXReX h?2 JfJfRf∞ [m2m2 G2i A M/ D #2 irQ BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λ M/ μ- M/ H2i X(0) #2 M BMi2;2`@p Hm2/ ` M/QK p `B #H2 BM/2T2M/2Mi Q7 i?2 #Qp2 TQBMi T`Q+2bb2bX .2}M2 7Q` HH t ≥ 0 X(t) = X(0) + A(0, t] − 1{X(s−)>0} dD(s) . (0,t]
a?Qr i? i {X(t)}t≥0 Bb i?2 +QM;2biBQM T`Q+2bb Q7 M JfJfRf∞ [m2m2 rBi? ``Bp H ` i2 λ M/ b2`pB+2 iBK2b rBi? 2tTQM2MiB H /Bbi`B#miBQM Q7 K2 M μX 1t2`+Bb2 RRXeXRdX _2bB/m H b2`pB+2 iBK2 P#i BM i?2 `2bB/m H b2`pB+2 iBK2 7Q`KmH 1 E[R(0)] = λE0A [σ02 ], 2
URRXeeV
p HB/ 7Q` G/G/1/∞ [m2m2 rBi? MQM@T`22KTiBp2 /Bb+BTHBM2X 1t2`+Bb2 RRXeXR3X *QMbi Mi b2`pB+2 iBK2 KBMBKBx2b +QM;2biBQM a?Qr i? i 7Q` }t2/ i` {+ BMi2MbBiv ρ- +QMbi Mi b2`pB+2 iBK2b KBMBKBx2 p2` ;2 +QM;2biBQM BM i?2 M/GI/1/∞ 7B7Q [m2m2X
RRXeX 1s1_*Aa1a
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1t2`+Bb2 RRXeXRNX q BiBM; iBK2 Q7 i?2 G/G/1/∞ [m2m2 AM bi iBQM `v G/G/1/∞ [m2m2 UλE0A [σ0 ] < 1V- H2i Vn #2 i?2 r BiBM; iBK2 UbT2Mi BM i?2 r BiBM; `QQKV Q7 i?2 n@i? +mbiQK2`X :Bp2 i?2 2tT`2bbBQM Q7 1 Vk n↑∞ n k=1 n
lim
BM i2`Kb Q7 λ- π(0) M/ E [X(0)]X 1t2`+Bb2 RRXeXkyX h?2 bi iBQM `v Jf:fRf∞ [m2m2 h?2 M/GI/1/∞ [m2m2 Bb bT2+B H + b2 Q7 i?2 G/G/1/∞ [m2m2X h?2 b2`pB+2 iBK2 b2[m2M+2 {σn }n∈Z Bb BB/ rBi? +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM G(x) M/ K2 M μ−1 M/ i?2 BMTmi T`Q+2bb N Bb SQBbbQM rBi? BMi2MbBiv λX G2i π /2MQi2 i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 MmK#2` Q7 +mbiQK2`b Dmbi 7i2` i?2 /2T `im`2 iBK2b Q7 bi #H2 Jf:fRf∞ [m2m2X a?Qr i? i i?2 bi iBQM `v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb Bb 2[m H iQ π, P (X(0) = i) = π(i)- 7Q` HH iX
*? Ti2` Rk > rF2b SQBMi S`Q+2bb2b > rF2b T`Q+2bb2b K v #2 `2; `/2/ b i?2 biQ+? biB+ 7Q`K Q7 i?2 GQiF ĜoQHi2`` KQ/2H Q7 1t KTH2 8XRXke Q` b T `iB+mH ` 7Q`K Q7 #` M+?BM; TQBMi T`Q+2bb- BM im`M T `iB+mH ` + b2 Q7 +Hmbi2` TQBMi T`Q+2bbXR h?2v + M KQ/2HBx2 i?2 ;`Qri? Q7 M 2TB/2Kv Q` b2BbKB+ +iBpBiv- BM r?B+? + b2 Bi Bb BKTQ`i Mi iQ ? p2 2biBK i2b Q7 i?2 T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 iBK2 iQ bi iBQM `Biv U7Q` BMbi M+2- i?2 iBK2 iQ 2t@ iBM+iBQMVX M BMi2`2biBM; bT2+i Q7 i?2b2 TQBMi T`Q+2bb2b Bb i?2 2tBbi2M+2 Q7 T? b2 i` MbBiBQM T?2MQK2MQM U#` M+?BM; T`Q+2bb2b dzrBi?Qmi M+2biQ`bǴ- a2+iBQM RkX8VX bQK2r? i `2H i2/ +H bb Q7 TQBMi T`Q+2bb2b Bb i?2 bQ@+ HH2/ MQM@HBM2 ` > rF2b T`Q+2bb2b i? i rBHH #2 bim/B2/ BM i?Bb +? Ti2` BM i2`Kb Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQM /`Bp2M #v irQ@/BK2MbBQM H SQBbbQM T`Q+2bbX
RkXR
b "` M+?BM; SQBMi S`Q+2bb
.2}MBiBQM RkXRXR TQBMi T`Q+2bb Z Bb + HH2/ #` M+?BM; TQBMi T`Q+2bb rBi? bBM;H2 M+2biQ` TQBMi i 0 B7 Z = z 0 + z1 + z2 + · · · r?2`2 z0 := ε0 Ui?2 TQBMi T`Q+2bb rBi? bBM;H2 TQBMi i 0V M/ 7Q` HH n ≥ 0- zn+1 Bb i?2 +Hmbi2` TQBMi T`Q+2bb rBi? ;2`K TQBMi T`Q+2bb zn M/ +Hmbi2` αn - bBKTH2 }MBi2 TQBMi T`Q+2bb bm+? i? i αn ({0}) = 0- M/ r?2`2 i?2 b2[m2M+2 {αn }n≥1 Bb BB/X AM i?2 MQi iBQM BMi`Q/m+2/ BM am#b2+iBQM RX8- 1t KTH2 RX8Xk i?2`2BM, zn+1 = zn ∗ αn+1 X
h?2 b2[m2M+2 { nk=0 zk (E)}n≥0 Bb i?2`27Q`2 : HiQMĜq ibQM T`Q+2bb rBi? bBM;H2 M+2biQ` M/ ivTB+ H T`Q;2Mv bBx2 α(E) U ` M/QK p `B #H2 rBi? i?2 +QKKQM T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 αn (E)VX AM T `iB+mH `- B7 E [α(E)] < 1- Z Bb }MBi2 TQBMi T`Q+2bb Q7 p2` ;2 bBx2 E [Z(E)] = R
1 < ∞. 1 − E [α(E)]
(> rF2b- RNdR)- (> rF2b M/ P F2b- RNd9)X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_12
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.2}MBiBQM RkXRXk ;2M2` H #` M+?BM; T`Q+2bb Bb +Hmbi2` TQBMi T`Q+2bb rBi? i?2 7QHHQrBM; bT2+B}+BiB2b, U V E = Rm U#V i?2 BMi2MbBiv K2 bm`2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb ν0 - M/ U+V Z := Z1 Bb i?2 #` M+?BM; TQBMi T`Q+2bb rBi? M+2biQ` TQBMi i 0 Dmbi /2b+`B#2/r?2`2 Bi Bb bbmK2/ i? i E [α(Rm )] < 1X h?2 ;2M2`B+ +Hmbi2` α Bb + HH2/ i?2 ;2M2`B+ T`Q;2Mv Q7 i?2 #` M+?BM; +Hmbi2` TQBMi T`Q+2bbX h?Bb TQBMi T`Q+2bb Bb HbQ + HH2/ M Bi2` i2/ +Hmbi2` T`Q+2bb bBM+2 Bi +QMbBbib Q7 bm++2bbBQM Q7 ;2M2` iBQMb- N0 , N1 , N2 , . . .- r?2`2 7Q` k ≥ 1- Nk Bb Q#i BM2/ #v α@+Hmbi2`BM; Q7 Nk−1 - i? i Bb- Nk = Nk−1 ∗ αk X h?2 }M H TQBMi T`Q+2bb Bb N= Nk . k≥0
h?2 BMi2MbBiv K2 bm`2b Q7 i?2 bm++2bbBp2 ;2M2` iBQMb `2 ν0 , ν1 , ν2 , . . .- r?2`2 νk = νk−1 ∗ να X AM T `iB+mH `- 7Q` HH k ≥ 1 νk (C) = νk−1 (C − x) να (dx) . Rm
AM i?2 bT2+B H + b2 r?2`2 ν0 = λ0 m - r2 ? p2 i? i νk = λk m r?2`2 λk = λ0 να (E)k X h?2M- bBM+2 να (E) < 1- i?2 BMi2MbBiv K2 bm`2 Q7 N Bb ν(dx) =
λ0
m (dx) . 1 − να (E)
1t KTH2 RkXRXj, h?2 P`B;BM H > rF2b S`Q+2bbX AM i?Bb + b2- i?2 ;2M2`B+ α Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM h(t) bm+? i? i R+ h(t) dt < ∞X 1t KTH2 RkXRX9, TT`QtBK i2 a KTH2b Q7 "` M+?BM; SQBMi S`Q+2bbX AM i?Bb 2t KTH2- r2 +QMbB/2` i?2 T`Q#H2K Q7 b KTHBM; i?2 /Bbi`B#miBQM Q7 i?2 ;2M2` H #` M+?BM; T`Q+2bb Q7 .2}MBiBQM RkXRXk BM +QKT +i dzrBM/QrǴ W ∈ B(Rm )X h?Bb `2[mB`2b mb iQ +QMbi`m+i i?2 #` M+?BM; T`Q+2bb2b ii +?2/ iQ HH i?2 TQBMib Q7 i?2 ;2`K T`Q+2bb- r?Qb2 MmK#2` Bb TQbbB#Hv BM}MBi2X h?Bb Bb BM ;2M2` H Qmi Q7 `2 +? U+ b2b r?2`2 i?Bb Bb 72 bB#H2 rBHH #2 +QMbB/2`2/ H i2`VX 6Q` i?2 iBK2 #2BM;- bmTTQb2 i? i BMbi2 / Q7 N - QM2 bm++22/b BM b KTHBM; Bib TT`QtBK iBQM N (n) =
n
Nk ,
k=0
r?Qb2 BMi2MbBiv K2 bm`2 Bb- bbmKBM; i? i i?2 BMi2MbBiv K2 bm`2 Q7 i?2 ;2`K TQBMi m n α (R ) m T`Q+2bb Bb λ0 m (dx)- λ0 1−ν1−|ν
(dx)X AM T `iB+mH `| α E (N − N (n) )(W ) = |να |n
λ0
m (W ) , 1 − |να |
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M/ i?2`27Q`2- bBM+2 7Q` Mv BMi2;2`@p Hm2/ ` M/QK p `B #H2 Y - P (Y > 0) ≤ E [Y ]P ((N − N (n) )(W ) > 0) ≤ γ|να |n , r?2`2 γ=
λ0
m (W ) . 1 − |να |
h?Bb b vb i? i i?2 T`Q# #BHBiv i? i N (n) ≡ N QM W Bb bK HH2` i? M γνα (Rm )n X AM biBHH Qi?2` rQ`/b- /2MQiBM; #v NW i?2 `2bi`B+iBQM Q7 N iQ W (n)
dV (NW , NW ) ≤ γ|να |n , r?2`2 dV Bb i?2 p `B iBQM /Bbi M+2X AM/22/, (n)
|P (NW ∈ Γ) − P (NW ∈ Γ)| (n)
(n)
(n)
(n)
=|P (NW ∈ Γ, NW ≡ NW ) + P (NW ∈ Γ, NW ≡ NW ) (n)
(n)
− P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| (n)
(n)
(n)
= |P (NW ∈ Γ, NW ≡ NW ) + P (NW ∈ Γ, NW ≡ NW ) (n)
(n)
− P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| (n)
(n)
(n)
(n)
= |P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| ≤ P (NW ≡ NW ) . >Qr2p2`- i?2 /B{+mHiv K2MiBQM2/ i i?2 #2;BMMBM; Q7 i?2 +m``2Mi 2t KTH2 `2K BMbX h?2 b KTHBM; Q7 N (n) `2[mB`2b i FBM; BMiQ ++QmMi TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb r?Qb2 MmK#2` + M BM T`BM+BTH2 #2 BM}MBi2X >Qr2p2`- B7 i?2 bmTTQ`i Q7 α Bb }MBi2i? i Bb- B7 7Q` bQK2 R < ∞- P (α({x ∈ Rm ; ||x|| ≥ R}) = 0) = 1- HBiiH2 i?Qm;?i b?QmH/ +QMpBM+2 QM2 i? i BM Q`/2` iQ Q#i BM N (n) QM W - Bi bm{+2b iQ +QMbB/2` QMHv i?2 #` M+?BM; T`Q+2bb2b ii +?2/ iQ i?2 ;2`K TQBMib i /Bbi M+2 H2bb i? i nR 7`QK W X aBM+2 i?2B` MmK#2` Bb }MBi2- r2 i?2`27Q`2 Q#i BM M TT`QtBK iBQM Q7 i?2 b KTH2 r2 HQQF2/ 7Q`- #mi i?2 [m HBiv Q7 i?Bb TT`QtBK iBQM BM i2`Kb Q7 i?2 p `B iBQM /Bbi M+2 + M #2 +QMi`QHH2/ M/ K /2 b ;QQ/ b /2bB`2/ #v T`QT2` +?QB+2 Q7 nX
h?2 > rF2b SQBMi S`Q+2bb > rF2b TQBMi T`Q+2bb2b ? p2 #22M T`QTQb2/ b KQ/2Hb Q7 b2BbKB+ +iBpBiv Q` Q7 T`QT ; iBQM Q7 2TB/2KB2b- 7Q` BMbi M+2X 1t KTH2 RkXRX8, 7i2`b?Q+FbX amTTQb2 i? i i?2`2 Bb Dmbi QM2 M+2biQ` i iBK2 0- i? i Bb- N0 ≡ ε0 X h?2 7QHHQrBM; BMi2`T`2i iBQM Bb QM2 KQM; Qi?2`b,k i?2`2 Bb M 2 `i?[m F2 i iBK2 0 Ui?2 dzTQBMi T`Q+2bbǴ ε0 V r?B+? ;2M2` i2b 7i2`b?Q+Fb 7Q`KBM; i?2 TQBMi T`Q+2bb Q7 7i2`b?Q+Fb N − ε0 X k
(> rF2b M/ / KQTQmHQb- RNd9)X
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h?Bb ivT2 Q7 KQ/2H 7Q` b2BbKB+ +iBpBiv ? b `2+2Bp2/ +QMbB/2` #H2 ii2MiBQM- M/ Bb FMQrM b M 1ha U1TB/2KB+ hvT2 7i2`@a?Q+FV KQ/2HXj 1t KTH2 RkXRXe, 1TB/2KB2bX MQi?2` BMi2`T`2i iBQM Bb i?2 T`QT ; iBQM Q7 bK HH 2TB/2Kv bi `iBM; 7`QK bBM;H2 BM72+i2/ BM/BpB/m H i iBK2 0X U"v dzbK HHǴ- Bi Bb mM/2`biQQ/ i? i i?2 BM72+i2/ TQTmH iBQM `2K BMb M2;HB;B#H2 rBi? `2bT2+i iQ i?2 TQTmH iBQM Q7 bmb+2TiB#H2b- bQ i? i Mv BM/BpB/m H + M QMHv #2 BM72+i2/ QM+2XV >2`2 ; BM- QM2 b22Fb 2biBK i2b Q7 i?2 iBK2 Q7 2tiBM+iBQM Q7 i?2 2TB/2KvX M Bbbm2 Q7 BMi2`2bi BM #Qi? BMi2`T`2i iBQMb ;Bp2M BM i?2 #Qp2 2t KTH2b Bb iQ 2biBK i2 i?2 2tiBM+iBQM iBK2- i? i Bb- i?2 /m` iBQM Q7- `2bT2+iBp2Hv- i?2 7i2`b?Q+F b2[m2M+2 M/ i?2 2TB/2KvX h?2 2tiBM+iBQM Bbbm2 rBHH #2 /2 Hi rBi? BM a2+iBQM RkXkX AM M 2TB/2Kv Ub22 1t KTH2 RkXRXeV- Bi Bb BKTQ`i Mi iQ i F2 BMiQ ++QmMi i?2 /Bz2`2M+2 #2ir22M i?2 TQTmH iBQMb Q7 bmb+2TiB#H2b- #v BMi`Q/m+BM; ivT2b- 2 +? ivT2 +Q``2bTQM/BM; iQ b2i Q7 72 im`2b- bm+? b p ++BM iBQM bi imb- bQ+B H T` +iB+2 M/ ;2M2iB+ # +F;`QmM/X h?Bb KQiBp i2b i?2 BMi`Q/m+iBQM Q7 i?2 KmHiBp `B i2 > rF2b T`Q+2bb- KmHiBivT2 #` M+?BM; TQBMi T`Q+2bb r?Qb2 +QMbi`m+iBQM Bb bBKBH ` iQ i?2 mMBp `B i2 > rF2b T`Q+2bbX Ai +QMbBbib Q7 7 KBHv {N (i) }1≤i≤M r?2`2 2 +? N (i) Bb TQBMi T`Q+2bb QM Rm +QMbi`m+i2/ b bmT2`TQbBiBQM Q7 TQBMi T`Q+2bb2b, (i) Nk , N (i) := k≥0 (i)
r?2`2 N0 U1 ≤ i ≤ M V `2 bBKTH2 HQ+ HHv }MBi2 TQBMi T`Q+2bb2b rBi? `2bT2+iBp2 (i) BMi2MbBiv K2 bm`2 ν0 U1 ≤ i ≤ M V + HH2/ i?2 M+2biQ`b TQBMi T`Q+2bb Q7 ivT2 i M/ (i) Nk Uk ≥ 1V,Bb i?2-k@i? ;2M2` iBQM TQBMi T`Q+2bb Q7 ivT2 i /2}M2/ `2+m`bBp2Hv b 7QHHQrbX G2i
(i,j)
Zn,k
n∈N,k∈N
U1 ≤ i, j ≤ M V #2 7 KBHv Q7 BB/ ` M/QK p `B #H2b-
BM/2T2M/2Mi Q7 N0 - rBi? p Hm2b BM K2 bm` #H2 bT +2 (K, K) M/ rBi? +QKKQM (i) (i) /Bbi`B#miBQM Qij X h?2 TQBMi T`Q+2bb2b Nk ≡ {Xn,k }n∈N `2 i?2 # b2 TQBMi T`Q+2bb , (i,j) (i,j) Q7 i?2 K `F2/ TQBMi T`Q+2bb2b N k QM Rm × K rBi? i?2 BB/ K `Fb Zn,k n∈N
i? i Bb(i,j)
N k (C × L) =
(i)
(i,j)
1C (Xn,k )1L (Zn,k )
(C ⊆ B(Rm ), L ∈ K) .
n∈N
G2i #2 ;Bp2M MQM@M2; iBp2 72`iBHBiv ` i2 7mM+iBQMb hij : Rm × K → R U1 ≤ i, j ≤ M V M/ /2}M2
t
hij (Z (i,j) , t) dt
aij := E
(1 ≤ i, j ≤ M ) ,
0
r?2`2 Z (i,j) Bb K@p Hm2/ ` M/QK p `B #H2 rBi? /Bbi`B#miBQM Qij X Ai Bb bbmK2/ i? i i?2 K i`Bt A := {aij }M i,j=1 j (P; i - RN33 M/ RNN3)- b22 (. H2vĜo2`2@CQM2b- kyyj)- 1t KTH2 eX9 U/V 7Q` /2i BH2/ /2b+`BT@ iBQMX
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Bb B``2/m+B#H2 M/ ? b bT2+i` H ` /Bmb ρ(A) < 1X .2MQiBM; #v Fk i?2 σ@}2H/ `2+Q`/BM; HH i?2 2p2Mib `2H iBp2 iQ N 0 , . . . , N k - i?2 k@i? ;2M2` iBQM TQBMi T`Q+2bb (i) Q7 ivT2 i- Nk - Bb +QM/BiBQM HHv QM Fk−1 *Qt TQBMi T`Q+2bb QM Rm rBi? i?2 ` M/QK BMi2MbBiv 7mM+iBQM (i)
λk (t) :=
M j=1
Rm
(j)
hji (t − s, z)N k−1 (ds × dz) = K
(j)
(j,i)
hji (t − Xn,k−1 , Zn,k−1 ) .
n∈N (i)
(i)
h?2 BMi2`T`2i iBQM Bb i?2 7QHHQrBM;, 2 +? TQBMi Xn,k−1 ∈ Nk−1 Q7 ;2M2` iBQM k − 1 ;Bp2b #B`i? iQ /2b+2M/ Mib Q7 ivT2 j 7Q` i?2 M2ti ;2M2` iBQM ++Q`/BM; iQ *Qt (i) (i,j) TQBMi T`Q+2bb Q7 BMi2MbBiv hij (t − Xn,k−1 , Zn,k−1 )X h?2`27Q`2 2 +? M+2biQ` U TQBMi (i)
(i)
Q7 N0 7Q` bQK2 1 ≤ i ≤ M V M/ 2 +? /2b+2M/2Mi U TQBMi Q7 Nk 7Q` bQK2 k ≥ 1 M/ bQK2 1 ≤ i ≤ M V ? p2 aij /B`2+i /2b+2M/ Mib Q7 ivT2 j QM i?2 p2` ;2X G2i N := Nk , N := Nk . k≥0
k≥0
A7 i?2 M+2biQ`b TQBMi T`Q+2bb2b `2 DQBMiHv bi iBQM `v- rBi? `2bT2+iBp2 p2` ;2 (i) BMi2MbBiv λ0 - i?2 bm#b2[m2Mi ;2M2` iBQMb `2 bi iBQM `v- M/ 7`QK i?2 #Qp2 2t@ (i) T`2bbBQM Q7 λk (t) M/ * KT#2HHǶb 7Q`KmH (i)
λk =
M j=1
(j)
λk−1
Rm
M (j) E h(t, Z (j,i) ) dt = λk−1 aij ,
URkXRV
j=1
(i) (i) rBi? λk := E λk (t) - M/ i?2`27Q`2 i?2 p2` ;2 BMi2MbBiv p2+iQ` λ Q7 N b iBb}2b λ = λ0 + Aλ , (i)
r?2`2 λ0 := {λ0 }1≤i≤M X h?2`27Q`2- B7 λ0 Bb MQi MmHH M/ bBM+2 A Bb B``2/m+B#H2 rBi? bT2+i` H ` /Bmb bi`B+iHv H2bb i? M 1- N ? b }MBi2 BMi2MbBiv p2+iQ` M/ 2 +? M+2biQ` Bb i?2 `QQi Q7 M 2p2Mim HHv 2tiBM;mBb?BM; KmHiBivT2 #` M+?BM; T`Q+2bbX 1t KTH2 RkXRXd, ai`2bb _2H2 b2 M/ 7i2`b?Q+FbX h?2 bi`2bb `2H2 b2 KQ/2H Q7 1t KTH2 8XRXkd ? b #22M 2ti2M/2/ BM 1t KTH2 RyX9Xk BM Q`/2` iQ BM+Hm/2 7i2`b?Q+FbX AM i?Bb + b2 t N (0,t) λ(t) := eX(0)+ct− i=1 Zi + h(t − s) N (ds) , −∞
r?2`2 h : R+ → R+ Bb MQM@M2; iBp2 7mM+iBQM bm+? i? i q2 b? HH +QMbB/2` i?2 KQ/2H λ(t) = ϕ(X(t)) + Y (t), r?2`2 X(t) = X0 + ct −
N (t) n=1
Zn
∞ 0
h(t) dt < 1X
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M/ Y (t) = Y0 e−αt + k
e−α(t−s) dN (s) ,
(0,t]
r?2`2 ϕ Bb TQbBiBp2 7mM+iBQMX h?Bb Bb KQ/B}+ iBQM Q7 i?2 T`2pBQmb KQ/2H- r?2`2 ϕ `2TH +2b i?2 2tTQM2MiB H- M/ h Bb MQr `2bi`B+i2/ iQ #2 M 2tTQM2MiB HX h?2 T`Q#H2K Bb iQ }M/ M2+2bb `v +QM/BiBQM 7Q` i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 +Q``2bTQM/BM; bi iBQM `v T`Q+2bb M/- 7Q` Mv BMBiB H /Bbi`B#miBQM Q7 X0 M/ Y0 - Q7 i?2 +QMp2`;2M+2 iQ i? i /Bbi`B#miBQM- M/ iQ T`Qp2 7Q`K HHv i? i Bi Bb HbQ bm{+B2Mi UmM/2` 7m`i?2` bKQQi?M2bb +QM/BiBQM QM i?2 /Bbi`B#miBQM Q7 Zi VX h?2 2tBbi2M+2 Q7 M 2`;Q/B+ bQHmiBQM Bb T`Qp2/9 BM i?2 + b2 k < 1. α
URkXkV
h?Bb Bb BM/22/ M im` H UBMimBiBp2V +QM/BiBQM M/ KQ`2Qp2`- Bi Bb M2+2bb `v B7 r2 b22F QMHv i?Qb2 bQHmiBQMb 7Q` r?B+? i?2 bi2 /v@bi i2 p2` ;2 BMi2MbBiv λ := E[λ(t)] b iBb}2b 0 < λ < ∞X h?2 T`Q+2bb (X(t), Y (t)), t ≥ 0 Bb iBK2@?QKQ;2M2Qmb J `FQp T`Q+2bb rBi? BMBiB H p Hm2 (X0 , Y0 )- M/ λ(t) = ϕ(X(t)) + Y (t). 6m`i?2`- 2`;Q/B+Biv K2 Mb- BM T `iB+mH `- i? i i?2`2 2tBbib bi iBQM `v p2`bBQM Q7 i?2 T`Q+2bb (X(t), Y (t))X 6Q` bm+? bi iBQM `v p2`bBQM- H2i λ1 = E[ϕ(X(t))] M/ λ2 = EY (t)X h?2M λ = λ1 + λ2 - bQ i?2 }MBi2M2bb Q7 λ BKTHB2b i? i Q7 λ1 M/ Q7 λ2 X "v * KT#2HHǶb 7Q`KmH E[Y (t)] = E[Y (0)]e−αt + kE[ e−α(t−s) N (ds)] = E[Y (0)]e−αt + k
(0,t] t
e−α(t−s) λds]
0
k = E[Y (0)]e−αt + λ (1 − e−αt ) . α h?2`27Q`2- #v bi iBQM `Biv- E[Y (0)] = λ αk = E[Y (t)]X h?2M λ = λ1 + λ
k ≡ λ1 + λ2 . α
h?2 bmT2`+`BiB+ H + b2X amTTQb2- BM pB2r Q7 +QMi` /B+iBQM- i? i αk > 1X h?2 H bi 2[m HBiv i?2M BKTHB2b i? i λ = ∞- r?B+? r2 2t+Hm/2/- Q` i? i λ = 0- M/ i?2M λ1 = E[ϕ(X(t))] = 0X aBM+2 ϕ(X(t)) ≥ 0- i?Bb BKTHB2b P (ϕ(X(t)) = 0) = 1- i? i Bb P (X(t) = −∞) = 1X aBKBH `Hv P (Y (t) = 0) = 1X h?2 +`BiB+ H + b2X amTTQb2 MQr- ; BM BM pB2r Q7 +QMi` /B+iBQM- i? i αk = 1X h?2 H bi /BbTH v2/ 2[m HBiv BKTHB2b i?2M i? i λ = ∞ U2t+Hm/2/V Q` λ1 = 0 M/ i?2`27Q`2 P (ϕ(X(t)) = 0) = 1X h?2M 9
("`ûK m/ M/ 6Qbb- kyRy)X
RkXRX a "_L*>AL: SPALh S_P*1aa λ(t) = Y (0)e
−αt
9ed
e−α(t−s) N (ds).
+k (0,t]
q2 b?Qr i? i Mv TQBMi T`Q+2bb N rBi? i?Bb biQ+? biB+ BMi2MbBiv M/ rBi? }MBi2 p2` ;2 BMi2MbBiv Bb M2+2bb `BHv MmHH UrBi? BMi2MbBiv 2[m H iQ 0VX amTTQb2 i? i λ > 0X *H2 `HvP (N (R+ ) = 0) = E[P (N (R+ ) = 0|Y (0))] = E[e−
∞ 0
Y (0)e−αt dt
1 −E[Y (0)] α
≥e
=e
]
1 −λ α
>0
M/ i?2`27Q`2- bBM+2 r2 bbmK2/ λ < ∞- r2 ? p2 i? i P (N (R+ ) = 0) > 0X LQr{N (R+ ) = 0} ⊆ θt {N (R+ ) = 0} = {N ([t, ∞)) = 0}X h? i Bb- {N (R+ ) = 0} Bb 2tT M/2/ #v i?2 U2`;Q/B+V b?B7i- M/ i?2`27Q`2 Bi ? b T`Q# #BHBiv 0 Q` 1X "v i?2 #Qp2- i?Bb T`Q# #BHBiv Kmbi #2 1X q2 +QM+Hm/2 i? i λ = 0- +QMi` /B+iBQMX h?2`27Q`2 BM i?2 +`BiB+ H + b2 i?2`2 Bb MQ bQHmiBQM 2t+2Ti i?2 i`BpB H QM2 UMQ 2 `i?[m F2bVX h?2`2 Bb M BMi2`2biBM; 72 im`2 Q7 i?2 KQ/2HX q2 bbmK2 ?2`2 ; BM 2`;Q/B+Biv M/ i?2 +QM/BiBQM 0 < λ < ∞X q2 +QMiBMm2 iQ +QMbB/2` i?2 KQ/2H BM i?2 bi iBQM `v `2;BK2X q`BiBM; ⎛ ⎛ ⎞⎞ N (t) N (t) 1 ϕ(X(t)) = ϕ ⎝X(0) + t ⎝c − Zn ⎠⎠ t N (t) n=1 = ϕ(X(0) + t(c − λE[Z1 ] + ε(t))), r?2`2 limt↑∞ ε(t) = 0 XbX G2i Δ := c − λE[Z1 ]X G2i τ #2 i?2 U XbX }MBi2V ` M/QK iBK2 bm+? i? i t ≥ τ BKTHB2b |ε(t)| ≤ 12 |Δ|X amTTQb2 i? i c − λE[Z1 ] > 0X q2 ? p2 1 E[ϕ(X(t))] ≥ E[ϕ(X(t))1{t≥τ } ] ≥ E[ϕ(X(0) + t Δ)1{t≥τ } ] . 2 "mi ϕ(X(0) + 12 Δ)t1{t≥τ } ↑ ∞ b t → ∞ M/ i?2`27Q`2 λ1 = E[ϕ(X(t))] → ∞ , BKTHvBM; i? i λ = ∞- r?B+? Bb 2t+Hm/2/X amTTQb2 i? i c − λE[Z1 ] < 0X q2 b?Qr i? i i?Bb Bb BKTQbbB#H2X >2`2 r2 ? p2 limt↑∞ ϕ(X(t)) = 0 #v bBKBH ` `;mK2MiX q2 T`Qp2 i? i limt↑∞ E[ϕ(X(t))] = 0λ1 = 0 M/ i?2`27Q`2 λ = 0- r?B+? Bb BKTQbbB#H2X 6Q` i?2 T`QQ7 i? i limt↑∞ E[ϕ(X(t))] = 0 r2 K F2 mb2 Q7 i?2 7QHHQrBM; H2KK Ui FBM; + `2 Q7 #Qi? bBim iBQMb r?2M c − λE[Z1 ] = 0VX G2KK RkXRX3 A7 i?2 bi iBQM `v biQ+? biB+ T`Q+2bb {Z(t)}t≥0 Bb bm+? i? i Bi i2M/b HKQbi bm`2Hv iQ /2i2`KBMBbiB+ +QMbi Mi c b t ↑ ∞- i?2M Bi Bb HKQbi bm`2Hv 2[m H iQ i?Bb +QMbi MiX
9e3
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
S`QQ7X 6Bt ε > 0- M/ +QMbB/2` i?2 b2i C = {ω; Z(t, ω) ∈ [c − ε, c + ε] 7Q` HH t ≥ 0}. h?2M 7Q` HH a > 0θa C = {ω; Z(t, ω) ∈ [c − ε, c + ε] 7Q` HH t ≥ a}. "mi θa C ↑ Ω- M/ i?2`27Q`2 P (C) = P (θa C) ↑ 1X aQ i? i P (C) = 1X aBM+2 i?Bb Bb i`m2 7Q` HH ε > 0P {Z(t) = c} = 1, 7Q` HH t ≥ 0. h?2`27Q`2- M2+2bb `BHv λ=
c . E[Z1 ]
h?2`27Q`2- BM i?Bb KQ/2H- i?2 ` i2 Q7 Q++m``2M+2b Q7 2 `i?[m F2b Bb ;Bp2M #v i?2 T?vbB+b Q7 bi`2bb #mBH/ mT Ui?2 +QMbi Mi cV M/ bi`2bb `2H2 b2 UE[Z1 ]VX Ai Bb BM/2T2M@ /2Mi Q7 i?2 2tBbi2M+2 Q7 7i2`b?Q+Fb5 h?2 ;HQ# H ` i2 Bb b? `2/ KQM; T`BK `v M/ b2+QM/ `v 2 `i?[m F2b ++Q`/BM; iQ i?2 T?vbB+b Q7 i?2 7i2`b?Q+Fb Uα M/ kVX h?2 7i2`b?Q+Fb /2+B/2 ?Qr iQ b? `2 i?2 ` i25
RkXk _ i2b Q7 1tiBM+iBQM M/ Q7 AMbi HH iBQM AM i?Bb b2+iBQM- r2 +QMbB/2` 2biBK i2b `2H i2/ iQ i?2 ` i2 Q7 +QMp2`;2M+2 iQ 2[mB@ HB#`BmK Q7 K `F2/ > rF2b T`Q+2bb2bX8 6B`biHv- i?2 bi iBQM `v T`Q+2bb Bb i?2 2KTiv T`Q+2bb- BM r?B+? + b2 r2 bT2 F Q7 i?2 ` i2 Q7 2tiBM+iBQMX a2+QM/Hv- i?2 bi iBQM `v T`Q+2bb Bb i?2 mMB[m2 bi iBQM `v M/ MQM@i`BpB H K `F2/ > rF2b T`Q+2bb rBi? ;Bp2M 72`iBHBiv ` i2- BM r?B+? + b2 r2 bT2 F Q7 i?2 ` i2 Q7 BMbi HH iBQMX _2+ HH i?2 7QHHQrBM; bBKTH2 dz2TB/2KB+Ǵ KQ/2HX h?2 iBK2b Q7 TT2 ` M+2 Q7 M BM72+i2/ BM/BpB/m H 7Q`K TQBMi T`Q+2bb N QM R rBi? bbQ+B i2/ b2[m2M+2 Q7 2p2Mib {Tn }n∈Z X qBi? 2 +? Tn Bb bbQ+B i2/ K `F- i? i Bb- ` M/QK p `B #H2 Zn rBi? p H@ m2b BM bQK2 K2 bm` #H2 bT +2 (K, K) `2T`2b2MiBM; bQK2 ii`B#mi2 Q7 i?2 BM/BpB/m H r?B+? #2;BMb iQ #2 BM72+i2/ i iBK2 Tn U;2M/2`- r2B;?i- p ++BM iBQM bi imbVX h?2 b2[m2M+2 {Zn }n∈Z Bb bbmK2/ BB/- rBi? /Bbi`B#miBQM QX qBi? i?2 /Qm#H2 b2[m2M+2 {(Tn , Zn )}n∈Z Bb bbQ+B i2/ i?2 TQBMi T`Q+2bb NZ QM (R × K, B(R) ⊗ K) /2}M2/ #v NZ (A) = 1A (Tn , Zn ) (A ∈ B(R) ⊗ K) . n∈Z
G2i {Ft }t∈R #2 ?BbiQ`v Q7 NZ X h?2 /vM KB+b Q7 i?2 2TB/2Kv +Q``2bTQM/b iQ i?2 Ft @BMi2MbBiv F2`M2H λ(t) dt × Q(dz) r?2`2 λ(t) = ν(t) + h(t − s, z)NZ (ds × dz) , URkXjV (−∞,t)×K 8
(hQ``BbB- kyyy M/ kyyk)- ("`ûK m/- L TTQ M/ hQ``BbB- kyyk)X
RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL
9eN
r?2`2 {ν(t)}t∈R Bb HQ+ HHv BMi2;` #H2 MQM@M2; iBp2 biQ+? biB+ T`Q+2bb M/ h : R × K → R+ Bb MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM bm+? i? i (t, z) ∈ (−∞, 0] × L =⇒ h(t, z) = 0 .
URkX9V
h?2 T `iB+mH ` /vM KB+b /2b+`B#2/ #v URkXjV `2T`2b2Mi i?2 bBim iBQM r?2`2 i?2`2 Bb M dzBKTQ`i2/Ǵ ` i2 Q7 BM72+iBQM ν(t) MQi /m2 iQ +QMi ;BQMX h?2 T `i h(t − s, z)NZ (ds × dz) (−∞,t)×K
+Q``2bTQM/b iQ BMi2`M H +QMi ;BQMX q2 +QMbB/2` TQBMi T`Q+2bb2b N i? i `2 2KTiv QM (−∞, 0)X lM/2` +QM/BiBQMb ∞ E[h(t, Z1 )]dt < 1 URkX8V 0
M/
∞
ν(t)dt < ∞,
URkXeV
0
i?2 b?B7i2/ T`Q+2bb St N +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 2KTiv T`Q+2bb- M/ B7 KQ`2Qp2` ∞ tE[h(t, Z1 )]dt < ∞ , URkXdV 0
i?2 +QMp2`;2M+2 Bb BM p `B iBQM Ui?Bb rBHH #2 +QMb2[m2M+2- BM 7 +i T `iB+mH ` + b2- Q7 i?2 KQ`2 ;2M2` H h?2Q`2K RkXeXN QM r?B+? r2 MiB+BT i2VX _ i2 Q7 1tiBM+iBQM, i?2 GB;?i@i BH * b2 q2 +QMbB/2` i?2 + b2 r?2M i?2`2 r b MQ + b2 Q7 BM72+iBQM `2+Q`/2/ BMbB/2 i?2 dz+QHQMvǴ Ui?2 TQTmH iBQM Q7 BMi2`2biV #27Q`2 U 0 rBi? ∞ eβt E[h(t, Z1 )]dt = 1. URkX3V 0
Ai Bb HbQ bbmK2/ i? i HKQbi bm`2Hv t → E[h(t, Z1 )] Bb HQ+ HHv #QmM/2/ QM R+ .
URkXNV
9dy
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
h?2Q`2K RkXkXR lM/2` bbmKTiBQMb URkX3V- URkXNV- M/ B7 KQ`2Qp2` i?2 7mM+iBQM eβt ν(t) Bb /B`2+iHv _B2K MM BMi2;` #H2 QM R+ - i?2M 7Q` HH K bm+? i? i ∞ βt e ν(t)dt URkXRyV K > ∞ 0 βt β 0 te E[h(t, Z1 )]dt i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt .
URkXRRV
AM i?2 #Qp2 BM2[m HBiv (12.10) i?2 `B;?i@? M/ bB/2 Bb /2}M2/ iQ #2 0 B7 i?2 /2MQK@ BM iQ` Bb BM}MBi2X S`QQ7X 6Q` HH t > 0- r2 ? p2
P (T > t) = P (N (t, ∞) ≥ 1) ≤ E[N (t, ∞)] = E
∞
λ(u)du =
t
∞
λ(u)du . t
HbQ- #v i?2 bKQQi?BM; 7Q`KmH U`2+ HHBM; i? i N Bb 2KTiv QM (−∞, 0]V
λ(t) = ν(t) + E h(t − s, z)λ(s)Q(dz)ds , (0,t)×L
M/ i?2`27Q`2- λ(t) b iBb}2b i?2 /272+iBp2 `2M2r H 2[m iBQM t λ(t) = ν(t) + E[h(t − s, Z1 )]λ(s)ds,
URkXRkV
0
rBi? i?2 HB;?i@i BH +QM/BiBQM URkX3VX AM T `iB+mH `- Q#b2`pBM; i? i λ(t) Bb HQ+ HHv #QmM/2/ QM R+ - ν(t) Bb #QmM/2/ M/ λ(t) Bb BMi2;` #H2 QM R+ ∞ βt e ν(t)dt βt , lim e λ(t) = ∞ 0βt t→∞ te E[h(t, Z1 )]dt 0
7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX
lM/2` bbmKTiBQMb URkX8V M/ URkXeV- λ(t) Bb BMi2;` #H2 QM R+ M/ i?2`27Q`2i?2 ∞ 2tT2+i2/ MmK#2` Q7 BM72+i2/ T2QTH2 BM i?2 2TB/2Kv Bb }MBi2 bBM+2 E[N (0, ∞)] = λ(s)dsX JQ`2Qp2`0 ∞ ν(t)dt ∞0 . E[N (0, ∞)] = 1 − 0 E[h(t, Z1 )]dt S`QQ7X AM 7 +i- #v 1[X URkXRkV r2 ? p2 t t t λ(s)ds = ν(s)ds + 0
t
0
0
t−u
λ(s)ds E[h(u, Z1 )]du .
0
h?2`27Q`2- 0 λ(s)ds b iBb}2b `2M2r H 2[m iBQM Q7 i?2 /272+iBp2 ivT2- M/ i?2`27Q`2 Uam#b2+iBQM 9Xk- h?2Q`2K 9XRXR8V
RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL t→∞
∞
t
λ(s)ds =
lim
9dR
0
1−
∞0 0
ν(t)dt . E[h(t, Z1 )]dt
1t KTH2 RkXkXk, h?2 2772+i Q7 p ++BM iBQM M/ +QM7BM2K2MiX *QMbB/2` i?2 2TB/2KB+b KQ/2H Q7 1t KTH2 RkXRXe rBi? MQM@` M/QK HQ+ HHv #QmM/2/ HB;?i@ i BH 72`iBHBiv ` i2 h- bQ i? i i?2 2tBbi2M+2 Q7 β > 0 b iBb7vBM; ∞ eβt h(t)dt = 1 0
Bb ;m ` Mi22/ M/ i?2 +QM+HmbBQMb Q7 h?2Q`2K RkXkXR TTHvX q2 MQr BMi`Q/m+2 p ++BM iBQM BM i?2 KQ/2H #v `2TH +BM; h(t) #v Z1 h(t)- r?2`2 Z1 Bb i?2 p ++BM iBQM bi imb Q7 i?2 ivTB+ H BM/BpB/m H- rBi? E[Z1 ] = p ∈ (0, 1)X h?2 M2r 2tiBM+iBQM ` i2 Bb MQr β- ;Bp2M #v ∞
p
eβt h(t)dt = 1 .
0
*H2 `Hv- b 2tT2+i2/- β > βX h?2 2z2+i Q7 BbQH iBQM Q7 M BM72+i2/ BM/BpB/m H + M #2 bBKBH `Hv 2p Hm i2/X AbQH iBQM K2 Mb i? i i?2 72`iBHBiv ` i2 Bb 0 7i2` bQK2 iBK2X AM i?2 bBKTH2 + b2 Q7 /2i2`KBMBbiB+ 72`iBHBiv ` i2 h(t)- BbQH iBQM Bb KQ/2HH2/ #v M2r 72`iBHBiv ` i2 h(t)1t≤Z1 X >2`2 Z1 `2T`2b2Mib i?2 ivTB+ H iBK2 i r?B+? QM2 Bb BbQH i2/ 7i2` BM72+iBQM Ui? i Bb- +QKTH2i2Hv BbQH i2/- #mi vQm K v r2HH BK ;BM2 KQ/2Hb rBi? H2bb bi`BM;2Mi +QM}M2K2MiVX
_ i2 Q7 1tiBM+iBQM, i?2 am#2tTQM2MiB H * b2 q2 MQr +QMbB/2` i?2 + b2 Q7 K2 M BM72+iBpBiv ` i2 rBi? bHQr /2+ vX JQ`2 T`2+Bb2Hvi?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv E[h(t, Z1 )] g(t) = ∞ E[h(t, Z1 )]dt 0 Bb bbmK2/ iQ #2 bm#2tTQM2MiB H-e i? i Bb- 7Q` HH n ∈ N- limt→∞ G(t) := 1 − G(t)- r2 ? p2 i?2 7QHHQrBM; `2bmHi,
URkXRjV 1−G∗n (t) 1−G(t)
= nX qBi?
h?2Q`2K RkXkXj lM/2` bbmKTiBQMb (12.5)- (12.7) M/ (12.9)- B7 KQ`2Qp2` G Bb bm#2tTQM2MiB H M/ i?2 7mM+iBQM ν Bb #QmM/2/ M/ bm+? i? i B = lim sup t→∞
ν(t) < ∞, G(t)
URkXR9V
i?2M 7Q` HH K bm+? i? i e a22 .2}MBiBQM 9XkXky M/ i?2 HBM2b 7QHHQrBM; Bi 7Q` i?2 K BM T`QT2`iB2b Q7 i?Bb ivT2 Q7 /Bbi`B@ #miBQMX
9dk
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a K>
1−
∞ 0
B E[h(t, Z1 )]dt
i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 ∞ P (T > t) ≤ K G(s)ds. t
UP#b2`p2 i? i +QM/BiBQM (12.14) ;m ` Mi22b i? i ν(t) p MBb?2b i H2 bi b 7 bi b G(t), i?2 2z2+i Q7 i?2 BMBiB H BM72+iBQM Bb H2bb T2`bBbi2Mi i? M i?2 BM72+iBQM ` i2 Q7 M BM/BpB/m HXV S`QQ7X q2 }`bi Q#b2`p2 i? i i?2 #QmM/2/M2bb Q7 ν(t)- (12.7) M/ (12.14) BKTHv +QM/BiBQM URkXeVX AM/22/- #v URkXR9V Bi 7QHHQrb i? i 7Q` HH > 0 i?2`2 2tBbib t = t() bm+? i? i ν(t) ≤ (B + )G(t) 7Q` HH t ≥ t , URkXR8V M/ i?2`27Q`2
∞
ν(t)dt ≤ (B + )
t
∞
G(t)dt.
URkXReV
0
∞ aBM+2 #v +QM/BiBQM URkXdV r2 ? p2 0 G(t)dt < ∞- i?2 BMi2;` #BHBiv Q7 ν(t) 7QHHQrb #v (12.16) M/ i?2 #QmM/2/M2bb Q7 ν(t)X LQr- `;mBM; b BM i?2 T`QQ7 Q7 h?2Q`2K ∞ RkXkXR- 7Q` HH t > 0- P (T > t) ≤ t λ(s)ds- M/
t
λ(t − s)dG(s) ,
λ(t) = ν(t) + r
URkXRdV
0
r?2`2
∞
r :=
E[h(t, Z1 )]dt .
URkXR3V
0
h?2 mMB[m2 HQ+ HHv #QmM/2/ bQHmiBQM Q7 URkXRdV Bb Ub22 am#b2+iBQM 9XkV i ∗i (t). λ(t) = ν ∗ rG
URkXRNV
i≥0
lbBM; BM2[m HBiv URkXR8V r2 Q#i BM 0
t
t ν(t − s) ∗i ν(t − s) ∗i dG (s) + dG (s) G(t) G(t) 0 t−t t t−t G(t − s) ∗i ν(t − s) ∗i ≤ (B + ) dG (s) + dG (s) G(t) G(t) 0 t−t t−t G(t − s) ∗i G∗i (t − t) − G∗i (t) ≤ (B + ) dG (s) + V , G(t) G(t) 0
ν(t − s) ∗i dG (s) = G(t)
t−t
r?2`2 V Bb bm+? i? i ν(t) ≤ V QM R+ X "mi G∗i (t − t) − G∗i (t) = 0. t→∞ G(t) lim
RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL AM/22/
9dj
G∗i (t − t) − G∗i (t) G∗i (t − t) G(t − t) G∗i (t) = − , G(t) G(t − t) G(t) G(t) lim
t→∞
G∗i (t − t) G∗i (t) = lim =i t→∞ G(t − t) G(t)
UG2KK 9XkXkjV M/ UG2KK 9XkXkkV lim
t→∞
G(t − t) = 1. G(t)
6m`i?2`KQ`2
t 0
G(t − s) ∗i dG (s) = G(t)
t 0
1 − G(t − s) ∗i dG (s) G(t)
G∗i+1 (t) − G∗i (t) G∗i (t) − G∗i+1 (t) = G(t) G(t)
= M/ i?2`27Q`2
t
lim
t→∞
0
G(t − s) ∗i dG (s) = (i + 1) − i = 1 . G(t)
h?2`27Q`2-
t lim sup t→∞
0
ν(t − s)dG∗i (s) G(t)
≤ B.
URkXkyV
6BM HHv- #v URkXRNV M/ URkXkyV r2 ? p2 i? i- 7Q` HH K > B i?2`2 2tBbib t0 = t0 (K) bm+? i? i t ≥ t0 BKTHB2b t K λ(t) = ri ν(t − s)dG∗i (s) ≤ G(t) 1−r 0 i≥0 7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX
_ i2 Q7 AMbi HH iBQM, i?2 GB;?i@i BH * b2 q2 MQr im`M iQ bim/v i?2 ` i2 Q7 +QMp2`;2M+2 iQ 2[mBHB#`BmK Q7 i` MbB2Mi Ui? i Bb MQM@bi iBQM `vV TQBMi T`Q+2bb2b N - r?B+? `2 2KTiv 7Q` MQM@TQbBiBp2 iBK2b- rBi? HBM2 ` /vM KB+b QM R+ Q7 ivT2 URkXjV rBi? ν(t) = ν > 0X h?2 HB;?i@i BH +QM/BiBQM (12.8) 7Q` i?2 /Bbi`B#miBQM rBi? /2MbBiv URkXRjV Bb bbmK2/X h?2Q`2K RkXkX9 lM/2` bbmKTiBQM (12.8)- M/ B7 KQ`2Qp2` i?2 7mM+iBQM eβt H(t) Bb /B`2+iHv _B2K MM@BMi2;` #H2 QM R+ - r?2`2 ∞ ν ∞ H(t) = E[h(u, Z1 )]du, URkXkRV 1 − 0 E[h(s, Z1 )]ds t i?2M 7Q` HH K bm+? i? i
9d9
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a ∞
K>
eβt H(t)dt ∞0 βt β 0 te E[h(t, Z1 )]dt
URkXkkV
i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt . AM i?2 #Qp2 BM2[m HBiv URkXkkV i?2 `B;?i@? M/ bB/2 Bb /2}M2/ b x2`Q B7 i?2 /2MQK@ BM iQ` Bb BM}MBivX bm{+B2Mi +QM/BiBQM 7Q` eβt H(t) #2BM; /B`2+iHv _B2K MM@BMi2;` #H2 QM R+ Bb ∞ teβt E[h(t, Z1 )]dt < ∞. URkXkjV 0
h?Bb 7QHHQrb #v h?2Q`2K 9XkXRy #2+ mb2 eβt H(t) Bb +QMiBMmQmb 7mM+iBQM QM R+ r?B+? Bb H2bb i? M Q` 2[m H iQ ∞ ν ∞ z(t) := eβu E[h(u, Z1 )]du . 1 − 0 E[h(s, Z1 )]ds t h?2`27Q`2- Bi Bb #QmM/2/ #v bbmKTiBQM (12.8)X JQ`2Qp2`- z(t) Bb /B`2+iHv _B2K MM@ BMi2;` #H2 bBM+2 Bi Bb MQM@BM+`2 bBM; M/ G2#2b;m2 BMi2;` #H2 #v bbmKTiBQM URkXkjVX S`QQ7X "v BK#2//BM; M/ +QmTHBM;X G2i N #2 SQBbbQM T`Q+2bb QM R × L × R+ rBi? K2 M K2 bm`2 dt × Q(dz) × dsX *QMbi`m+i MQM@bi iBQM `v TQBMi T`Q+2bb N r?B+? Bb 2KTiv QM (−∞, 0]- M/ bi iBQM `v QM2 N - #Qi? ? pBM; /vM KB+b (12.3) rBi? ν(t) = ν- #v
NZ (dt × dz) = N (dt × dz × [0, λ (t)]) ((t, z) ∈ R+ × L) , λ (t) = ν + h(t − s, z)NZ (ds × dz) (t ∈ R+ ) ,
URkXk9V
(0,t)×L
NZ (dt × dz) = N (dt × dz × [0, λ(t)]), (t, z) ∈ R × L , h(t − s, z)NZ (ds × dz) (t ∈ R) . λ(t) = ν +
URkXk8V
(−∞,t)×L
G2i
NZ
Ft := FtN ∨ Ft
(t ∈ R) .
"v i?2 BK#2//BM; +QMbi`m+iBQM
N ⊂ N,
λ (t) ≤ λ(t) (t > 0) ,
URkXkeV
{λ (t)}t>0 Bb M Ft @T`2/B+i #H2 biQ+? biB+ BMi2MbBiv Q7 N QM R+ M/ {λ(t)}t∈R Bb M Ft @T`2/B+i #H2 biQ+? biB+ BMi2MbBiv Q7 N QM RX h?2 TQBMi T`Q+2bb N − N /KBib QM R+ i?2 Ft @biQ+? biB+ BMi2MbBiv {λ(t) − λ (t)}t>0 X "v i?2 bKQQi?BM; 7Q`KmH 7Q` TQBMi T`Q+2bb2b rBi? biQ+? biB+ BMi2MbBiv ∞ E[(N − N )(t, ∞)] = δ(u)du, 7Q` HH t > 0 , t
RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL
9d8
r?2`2 δ(t) = E[λ(t) − λ (t)]X h?2`27Q`2-
P (T > t) ≤
∞
URkXkdV
δ(u)du. t
AM/22/ M/
P (T > t) = P ((N − N )(t, ∞) ≥ 1),
7Q` HH t > 0
P ((N − N )(t, ∞) ≥ 1) ≤ E[(N − N )(t, ∞)].
ai` B;?i7Q`r `/ +QKTmi iBQMb mbBM; URkXk9V M/ URkXk8V vB2H/ λ(t)−λ (t) = h(t−s, z)NZ (ds×dz)+ h(t−s, z)(NZ −NZ )(ds×dz). (−∞,0]×L
(0,t)×L
"v i?2 bKQQi?BM; 7Q`KmH ∞ t E[h(u, Z1 )]du + E[h(t − s, Z1 )]δ(s)ds δ(t) = λ t
0
M/ i?2`27Q`2- #v Q#b2`pBM; i? i λ=
1−
∞ 0
ν , E[h(t, Z1 )]dt
r2 Q#i BM i?2 /272+iBp2 `2M2r H 2[m iBQM t δ(t) = H(t) + δ(t − s)E[h(s, Z1 )]ds.
URkXk3V
URkXkNV
0
h?2M- #v i?2 HB;?i@i BH +QM/BiBQM URkX3V M/ i?2 `2bmHib Q7 am#b2+iBQM 9Xk UQ#b2`p2 i? i #Qi? i?2 7mM+iBQMb δ(t) M/ H(t) `2 #QmM/2/V ∞ βt e H(t)dt lim eβt δ(t) = ∞ 0 βt , t→∞ te E[h(t, Z1 )]dt 0
7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX _ i2 Q7 AMbi HH iBQM, i?2 am#2tTQM2MiB H * b2
q2 MQr +QMbB/2` i?2 + b2 r?2M i?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv URkXRjV Bb bm#2tTQM2MiB HX h?2Q`2K RkXkX8 lM/2` bbmKTiBQMb (12.5) M/ (12.7)- B7 KQ`2Qp2` G Bb bm#2t@ TQM2MiB H- i?2M 7Q` HH K bm+? i? i ∞ ν 0 E[h(t, Z1 )]dt K> 2 ∞ 1 − 0 E[h(t, Z1 )]dt i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 ∞ P (T > t) ≤ K G(u)du. t
URkXjyV
9de
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
S`QQ7X `;mBM; 2t +iHv b BM i?2 T`QQ7 Q7 h?2Q`2K RkXkX9 r2 ? p2 P (T > t) ≤
∞
δ(u)du . t
G2i 7Q` HH t > 0- λ (t) := E[λ (t)] M/ λ := E[λ(t)]X "v i?2 bKQQi?BM; 7Q`KmH i FBM; i?2 K2 M Q7 λ (t) M/ λ(t) Bi 7QHHQrb i? i
t
λ (t) = ν + r
λ (t − s)dG(s) 0
M/ λ = ν + λr- r?B+? ;Bp2b ν = λ(1 − r) r?2`2 r Bb ;Bp2M #v URkXR3VX .BpB/BM; λ (t) #v λ- r2 ? p2 λ (t) = (1 − r) + r λ
t 0
λ (t − s) dG(s) . λ
URkXjRV
aBM+2 λ (t) Bb #QmM/2/ #v λ Ub22 URkXkeVV M/ r < 1 #v bbmKTiBQM- r2 Q#i BM #v Bi2` iBM; 1[X URkXjRV λ (t) rn G∗n (t) . = (1 − r) λ n≥0 ai` B;?i7Q`r `/ +QKTmi iBQMb H2 / iQ 1−
λ (t) rn G∗n (t) . = (1 − r) λ n≥0
URkXjkV
*QM/BiBQM r < 1 2Mbm`2b HbQ i?2 2tBbi2M+2 Q7 TQbBiBp2 bm+? i? i r(1 + ) < 1X JQ`2Qp2` UG2KK 9XkXk9V i?2`2 2tBbib D > 0 bm+? i? i- 7Q` HH BMi2;2`b n ≥ 0rn
G∗n (t) ≤ D[r(1 + )]n . G(t)
aBM+2 n≥0 [r(1 + )]n < ∞- Bi 7QHHQrb #v i?2 /2}MBiBQM Q7 bm#2tTQM2MiB H /Bbi`B@ #miBQM M/ #v /QKBM i2/ +QMp2`;2M+2 i? i lim
t→∞
JmHiBTHvBM; URkXjkV #v
λ G(t)
n≥0
rn
G∗n (t) r . = (1 − r)2 G(t)
M/ i FBM; i?2 HBKBi b t → ∞- r2 }M HHv ;2i lim
t→∞
δ(t) νr , = (1 − r)2 G(t)
7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX
RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa
RkXj
9dd
h?2 " `iH2ii aT2+i`mK Q7 i?2 > rF2b S`Q+2bb
h?2 TQr2` bT2+i` H /2MbBiv Q7 i?2 > rF2b #` M+?BM; TQBMi T`Q+2bb BM i?2 mMBp `B i2 M/ KmHiBp `B i2 + b2b rBi? /2i2`KBMBbiB+ 72`iBHBiv ` i2b M/ M M+2biQ` T`Q+2bb i? i Bb M ?TT r b +QKTmi2/ BM > rF2bǶ b2KBM H rQ`FXd h?Bb `2bmHi Bb MQr 2t@ i2M/2/ BM irQ /B`2+iBQMbX 6B`biHv- i?2 72`iBHBiv ` i2 Bb ` M/QK M/ b2+QM/Hv- i?2 M+2biQ` T`Q+2bb Bb ;2M2` H b2+QM/@Q`/2` bi iBQM `v TQBMi T`Q+2bb rBi? FMQrM TQr2` bT2+i` H K2 bm`2X G2i N #2 > rF2b T`Q+2bb rBi? ` M/QK 72`iBHBiv ` i2 h b iBb7vBM;- BM //BiBQM
E h (t, Z) dt < 1, URkXjjV
iQ
Rm
i?2 +QM/BiBQM
2 h (t, Z) dt
E Rm
< ∞.
URkXj9V
h?2 TQBMi T`Q+2bb Q7 M+2biQ`b N0 Bb bbmK2/ rB/2@b2Mb2 bi iBQM `v rBi? TQr2` bT2+i` H K2 bm`2 μ0 M/ bbQ+B i2/ bT +2 Q7 i2bi 7mM+iBQMb BN0 X h?2 M2ti i?2Q`2K ;Bp2b i?2 TQr2` bT2+i` H K2 bm`2 Q7 i?Bb TQBMi T`Q+2bb iQ;2i?2` rBi? /2b+`BTiBQM Q7 i?2 bbQ+B i2/ i2bi 7mM+iBQMbX h?2Q`2K RkXjXR U3 V lM/2` +QM/BiBQM URkXj9V- i?2 " `iH2ii bT2+i` H K2 bm`2 Q7 i?2 > rF2b T`Q+2bb N Bb
λ0 1 ρλ0 μN (dν) = & dν + Var h(ν, Z)dν , &2 μ0 (dν) + & & 1−ρ 1−ρ &1 − E h(ν, Z) & URkXj8V r?2`2 λ = λ0 /(1 − ρ)X JQ`2Qp2`- QM2 + M i F2 7Q` i?2 i2bi 7mM+iBQM bT +2 BN i?2 b2i Q7 7mM+iBQMb f ∈ L1C (Rm ) ∩ L2C (Rm ) bm+? i? i i?2 bQHmiBQM Q7 h(s − t, z)E [ϕ(s, Z)] ds = f (t) URkXjeV ϕ(t, z) − Rm
b iBb}2b E [ϕ(·, Z)] ∈ BN0 ,
URkXjdV
r?2`2 BN0 Bb i?2 i2bi 7mM+iBQM bT +2 Q7 i?2 M+2biQ` T`Q+2bb N0 X 1t KTH2 RkXjXk, " `iH2ii bT2+i`mK Q7 i?2 Q`B;BM H > rF2b T`Q+2bbX AM i?2 T `iB+mH ` + b2 r?2`2 h(t, Z) = h(t)- h(ν, Z) = h(ν)- M/ i?2`27Q`2
1 ρλ0 μN (dν) = dν . μ0 (dν) + 1−ρ |1 − h(ν)|2 d 3
(> rF2b- RNdR - RNdR#)X ("`ûK m/ M/ J bbQmHBû- kyyk)- ("`ûK m/- J bbQmHBû M/ _B/QH}- kyy8)X
9d3
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
A7 BM //BiBQM N0 Bb SQBbbQM T`Q+2bb rBi? p2` ;2 BMi2MbBiv λ0 - r2 Q#i BM i?2 Q`B;BM H 7Q`KmH Q7 > rF2b 7Q` i?2 bT2+i` H /2MbBiv, fN (ν) =
λ0 (1 − ρ)|1 − h(ν)|2
.
M /KBbbB#H2 bT +2 Q7 i2bi 7mM+iBQMb Bb BN = L1C (Rm ) ∩ L2C (Rm ) Ub22 M2ti 2t K@ TH2VX 1t KTH2 RkXjXj, > rF2b T`Q+2bb rBi? *QtB M M+2biQ` T`Q+2bbX q?2M N0 Bb *Qt T`Q+2bb b BM 1t KTH2 NXkXRj UrBi? biQ+? biB+ BMi2MbBiv T`Q+2bb {λ(t)}t∈Rm i? i Bb rB/2@b2Mb2 bi iBQM `v biQ+? biB+ T`Q+2bb rBi? K2 M λ M/ TQr2` bT2+i` H K2 bm`2 μλ V
1 λ0 dν . μN (dν) = & &2 μλ (dν) + & & 1−ρ &1 − E h(ν, Z) & q2 K v i F2 7Q` i2bi 7mM+iBQM bT +2 BN = L1C (Rm ) ∩ L2C (Rm )X AM/22/- B7 f ∈ L1C (Rm )∩L2C (Rm )- i?2 bQHmiBQM ϕ Q7 1[MX URkXjeV b iBb}2b i?2 +QM/BiBQM E [ϕ(·, Z)] ∈ L1C (Rm ) ∩ L2C (Rm ) = BN0 U1t KTH2 NXkXRjVX h?2 T i? iQr `/b i?2 T`QQ7 Q7 h?2Q`2K RkXjXR Bb T`2T `2/ #v i?2 M2ti irQ H2KK bX G2KK RkXjX9 6Q` ϕ ∈ L1C ( × Q) ∩ L2C ( × Q) Var ϕ(t, z) M (dt × dz) = λ Rm
r?2`2 λ (t) :=
Rm
K
Rm
E |ϕ(t, Z)|2 dt,
¯ (dt × dz) − λ (t) dt Q (dz) , M (dt × dz) := N ¯0 (dt × dz ) + ¯ (dt × dz ) h (t − s, z ) N h (t − s, z ) N K
Ui?2 biQ+? biB+ BMi2MbBiv Q7
Rm
K
n≥1 Nn V M/ λ = λ − λ0 X
S`QQ7X aBKTHB}2/ MQi iBQM Q7 i?2 FBM/ ϕ(t, z) M (dt × dz) = ϕ dM rBHH #2 mb2/X q2 ? p2 ϕ dM = ϕ dM n , n≥1
¯n (dt × dz) − λn (t)dt Q(dz)X :Bp2M Fn−1 - N ¯n Bb SQBbbQM r?2`2 M n (dt × dz) = N T`Q+2bb rBi? BMi2MbBiv K2 bm`2 λn (t)Q(dz)dt- M/ i?2`27Q`2- #v i?2 +Qp `B M+2 7Q`KmH UjX9V& & & o ` ϕ dM n & Fn−1 = E ϕ2 (t, Z) λn (t)dt, Rm
RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa
M/ E
9dN
&
& ϕ dM n && Fn−1 = 0,
M/ +QMb2[m2MiHv- #v i?2 +QM/BiBQM H p `B M+2 7Q`KmH & &
& & o ` ϕ dM n = E o ` ϕ dM n && Fn−1 + o ` E ϕ dM n && Fn−1 = λn E ϕ2 (t, Z) dt. Rm
HbQ 7Q` HH j, k ≥ 1&
& ϕ dM j+k =E ϕ dM j E ϕ dM j+k && Fj+k−1 = 0 . E ϕ dM j h?2`27Q`2- }M HHv o `
ϕ dM
=
o `
ϕ dM n
=
n≥1
= λ
n≥1
E ϕ2 (t, Z) dt
λn Rm
E ϕ2 (t, Z) dt . Rm
G2KK RkXjX8 A7 i?2 72`iBHBiv ` i2 h b iBb}2b +QM/BiBQMb URkXjjV M/ URkXj9V- i?2M UBV i?2`2 2tBbib- 7Q` Mv ;Bp2M F ∈ L1C ( × Q) ∩ L2C ( × Q)- mMB[m2 bQHmiBQM ϕ ∈ L1C ( × Q) ∩ L2C ( × Q) Q7 h(s − t, z)E [ϕ(s, Z)] ds = F (t, z), M/ ϕ(t, z) − Rm
L1C
UBBV 7Q` ;Bp2M f ∈ L1C (Rm ) ∩ L2C (Rm )- i?2`2 2tBbib mMB[m2 bQHmiBQM ϕ ∈ ( × Q) ∩ L2C ( × Q) Q7 URkXjeVX
S`QQ7X X 6Q` 7mM+iBQM v : R×K → R- /2MQi2 E [v (t, Z)] #v v¯ (t) M/ v (−t, z) #v vˇ (t, z)X *QM/BiBQM v ∈ L1C ( × Q) ∩ L2C ( × Q) BKTHB2b i? i v¯ ∈ L1C (Rm ) ∩ L2C (Rm )X HbQ- i?2 6Qm`B2` i` Mb7Q`K ν → v(ν, z) Q7 i?2 7mM+iBQM t → v(t, z) Bb mMB7Q`KHv ∗ #QmM/2/X 6BM HHv- #2+ mb2 ρ < 1- ν → 1 − E h(ν, Z) Bb mMB7Q`KHv #QmM/2/ r v 7`QK 0X ˇ¯ ∗ gX aBM+2 F¯ ∈ L1 (Rm ) ∩ L2 (Rm )*QMbB/2` i?2 `2M2r H 2[m iBQM g = F¯ + h C C
¯ˇ ∗n Bb i?2 mMB[m2 bQHmiBQM g ∈ M/ BM pB2r Q7 +QM/BiBQM URkXjjV- g := F¯ ∗ h n≥0
L1C (Rm ) ∩ L2C (Rm )XN h?2 6Qm`B2` i` Mb7Q`K Q7 g Bb
N h?2 +QMp2`;2M+2 Q7 i?2 b2`B2b BM L1C (Rm ) b r2HH b BM L2C (Rm ) Bb ;m ` Mi22/ #v i?2 BM2[m HBiB2b 'a ∗ b'L1 ≤ 'a'L1 'b'L1 M/ 'a ∗ b'L2 ≤ 'a'L1 'b'L2 c mMB[m2M2bb 7QHHQrb 7`QK ˇ¯ i?2 2[m HBiv =g − = g = h ∗ (g − g )- r?2`2 g Bb MQi?2` + M/B/ i2 bQHmiBQM- r?B+? BKTHB2b =¯ = 'g − g 'L1 ≤ h L1 'g − g 'L1 X >2M+2 mM/2` +QM/BiBQM URkXjjV- M2+2bb `BHv 'g − g 'L1 = 0X
93y
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a E F (ν, Z) . g (ν) = 1 − E h (ν, Z)∗
URkXj3V
6`QK i?2 Q#b2`p iBQMb K /2 BM i?2 }`bi HBM2b Q7 i?2 T`QQ7- r2 MQi2 i? i ν → g (ν) Bb mMB7Q`KHv #QmM/2/X .2}M2 MQr i?2 7mM+iBQM (t, z) → ϕ (t, z) #v ϕ (t, z) := h (s − t, z) g (s) ds + F (t, z) . Rm
q2 ? p2 i? i ϕ ∈ L1C ( × Q)- bBM+2
|ϕ (t, Z)| dt ≤ E |F (t, Z)| dt E Rm Rm
+E |h (t, Z)| dt Rm
Rm
URkXjNV
|g (t)| dt < ∞.
HbQ ϕ ∈ L2C ( × Q)X hQ T`Qp2 i?Bb- Bi bm{+2b iQ b?Qr i? i ϕ (ν, z) ∈ L2C ( × Q) #2+ mb2 i?2M- #v i?2 SH M+?2`2HĜS `b2p H B/2MiBiv
2 2 E |ϕ (t, Z)| dt = E |ϕ (ν, Z)| dν < ∞ . Rm
Rm
6Q` i?Bb Tm`TQb2- i F2 7Q` }t2/ z i?2 6Qm`B2` i` Mb7Q`K Q7 #Qi? bB/2b Q7 URkXjNV, ϕ (ν, z) = h (ν, z)∗ g (ν) + F (ν, z) , Q`- BM pB2r Q7 URkXj3V h (ν, z)∗ E F (ν, Z) . ϕ (ν, z) = F (ν, z) + 1 − E h (ν, Z)∗ aBM+2 F (t, z) ∈ L2C ( × Q)- Bi 7QHHQrb #v i?2 SH M+?2`2HĜS `b2p H B/2MiBiv i? i F (ν, z) ∈ L2C ( × Q)X Ai `2K BMb iQ b?Qr i? i h (ν, z) g(ν) ∈ L2C ( × Q) . h?Bb 7QHHQrb 7`QK i?2 7 +i i? i g Bb mMB7Q`KHv #QmM/2/- b r2 `2K `F2/ 2 `HB2` M/ i? i & & &2 &2 & &2
& & & & & & 2iπνt E &h (ν, Z)& = E && h (t, Z) e dt&& ≤ E && h (t, Z) dt&& , Rm
Rm
}MBi2 +QMbi Mi U#v ?vTQi?2bBb URkXj9VVX "X h?Bb Bb +H2 `Hv T `iB+mH ` + b2 Q7 X q2 MQi2 7Q` 7mim`2 `272`2M+2 i? i BM i?Bb + b2-
RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa
93R
⎛
⎞ ∗ (ν, z) h ⎠ . ϕ(ν, z) = f (ν) ⎝1 + 1 − E h∗ (ν, Z)
URkX9yV
q2 MQr im`M iQ i?2 T`QQ7 Q7 h?2Q`2K RkXjXRX S`QQ7X 6Q` ϕ- i?2 mMB[m2 bQHmiBQM Q7 URkXjeV ¯ (dt × dz) − λ (t)Q(dz)dt ϕ(t, z)M (dt × dz) = ϕ(t, z) N Rm K Rm K ¯ (dt × dz) − = ϕ(t, z)N ϕ(t, z)λ (t)Q(dz)dt. Rm
HbQ
Rm
K
Rm
K
K
ϕ(t, z)λ (t)Q(dz)dt ¯ (ds × dz ) Q(dz)dt = ϕ(t, z) h(t − s, z )N m Rm K R K ¯ (ds × dz ) = h(t − s, z )E [ϕ(t, Z)] dtN Rm ×Rm ×K ˇ − ·, z ) ∗ E [ϕ(·, Z)] (s) N ¯ (ds × dz ). h(s = Rm
K
¯ =N ¯ + N ¯0 h?2`27Q`2- bBM+2 N ϕ(t, z)M (dt × dz) Rm K ˇ − ·, z) ∗ E [ϕ(·, Z)] (t) N ¯ (dt × dz) = ϕ(t, z) − h(t Rm K ¯0 (dt × dz). − ϕ(t, z)N Rm
K
qBi? f ∈ BN UBM T `iB+mH ` f ∈ L1C (Rm ) ∩ L2C (Rm )V- r2 ? p2 ¯0 (dt × dz) = ϕ(t, z)M (dt × dz) + ϕ(t, z)N Rm
K
Rm
K
f (t)N (dt). Rm
HbQ- #v G2KK RkXjX9 ϕ(t, z)M (dt × dz) = λ E |ϕ(t, Z)|2 dt o ` m m R K R =λ E |ϕ(ν, Z)|2 dν. Rm
LQr-
93k
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
E
¯0 (dt × dz) ϕ(t, z)M (dt × dz) ϕ(t, z)N Rm K Rm K &
& ¯0 (dt × dz) = 0. =E E ϕ(t, z)M (dt × dz)&& F0 ϕ(t, z)N
Rm
Rm
K
K
h?2`27Q`2 ¯0 (dt × dz) = o ` ϕ(t, z)M (dt × dz) + ϕ(t, z)N m Rm K R K ¯ o ` ϕ(t, z)M (dt × dz) + o ` ϕ(t, z)N0 (dt × dz) = Rm K Rm K ¯0 (dt × dz) . λ E |ϕ(t, Z)|2 dt + o ` ϕ(t, z)N Rm
Rm
K
PM i?2 Qi?2` ? M/- #v i?2 mMBp2`b H +Qp `B M+2 7Q`KmH UNX9RV ¯ ϕ(t, z)N0 (dt × dz) o ` Rm K = |E [ϕ(ν, Z)]|2 μ0 (dν) + λ0 o ` (ϕ(ν, Z)) dν. Rm
*QK#BMBM; i?2 #Qp2- r2 ? p2 o ` f (t)N (dt) m R =λ E |ϕ(ν, Z)|2 dν + Rm
Rm
Rm
|E [ϕ(ν, Z)] |2 μ0 (dν)
+ λ0 Rm
o ` (ϕ(ν, Z)) dν
= A + B + C. "v 6Q`KmH URkX9yV-
1 + o ` h(ν, Z) dν, A = λ |f (ν)|2 Rm |1 − E h(ν, Z) |2 1 μ0 (dν), B= |f (ν)|2 Rm |1 − E h(ν, Z) |2 o ` h(ν, Z) dν. C= |f (ν)|2 λ0 Rm |1 − E h(ν, Z) |2
_2+ HHBM; i? i λ = ρλ0 /(1−ρ)- r2 Q#i BM }M HHv i? i 7Q` HH f ∈ L1C (Rm )∩L2C (Rm ) f (t)N (dt) o ` Rm ⎛ ⎞ 1 ⎠ μ0 (dν) + λρdν + λo ` h(ν, Z) dν , = |f (ν)|2 ⎝ Rm |1 − E h(ν, Z) |2 M/ i?Bb HHQrb mb iQ B/2MiB7v μN b URkXj8VX
RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a
RkX9
93j
1t +i a KTHBM; Q7 > rF2b S`Q+2bb2b
URy V aBM+2 i?2 > rF2b T`Q+2bb Bb +Hmbi2` TQBMi T`Q+2bb- Bi Bb M im` H iQ bF B7 i?2 2t +i b KTHBM; K2i?Q/ Q7 "`Bt M/ E2M/ HH TTHB2bX AM i?Bb bT2+B H + b2- r2 b22F iQ Q#i BM QM }MBi2 rBM/Qr W := [0, a] M 2t +i b KTH2 Q7 > rF2b T`Q+2bb N ≡ {Tn }n∈Z QM i?2 HBM2 rBi? ` M/QK 72`iBHBiv ` i2 h(t, Z)- r?2`2 Z Bb ` M/QK ∞ 2H2K2Mi BM bQK2 K2 bm` #H2 bT +2 (K, K)- M/ bm+? i? i ρ := E 0 h(t, Z) dt < 1X h?Bb T`Q+2bb Bb +Hmbi2` TQBMi T`Q+2bb r?Qb2 ;2`K TQBMi T`Q+2bb N0 Bb SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM μ M/ bm+? i? i Zn Un ≥ 1V Bb #` M+?BM; TQBMi T`Q+2bb Q7 ` M/QK 72`iBHBiv ` i2 h(t, Z) rBi? bBM;H2 M+2biQ` HQ+ i2/ i 0X AM T`BM+BTH2- i?2 "`BtĜE2M/ HH T2`72+i bBKmH iBQM K2i?Q/ Q7 N QM i?2 }MBi2 rBM/Qr W = [0, a] TTHB2bX Ai +QMbBbib Q7 irQ bi2Tb, UBV :2M2` i2 SQBbbQM TQBMi T`Q+2bb QM (−∞, 0] rBi? BMi2MbBiv μ (t)P (L ≥ −t) r?2`2 L Bb i?2 ivTB+ H H2M;i? U2tiBM+iBQM iBK2V Q7 #` M+?BM; TQBMi T`Q+2bb QM R+ rBi? ` M/QK 72`iBHBiv ` i2 h(t, Z) M/ rBi? bBM;H2 M+2biQ` HQ+ i2/ i 0X UBBV 6Q` 2 +? Q7 i?2 TQBMib Q7 i?Bb SQBbbQM T`Q+2bb- ;2M2` i2 #` M+?BM; TQBMi T`Q+2bb2b Q7 ` M/QK 72`iBHBiv ` i2 h(t, Z) rBi? bBM;H2 M+2biQ` HQ+ i2/ i Tn mMiBH vQm Q#i BM QM2 rBi? TQBMib BM (0, ∞)X Uh?2 ` iBQM H2 Bb i? i i?2 TQBMi Q7 N0 HQ+ i2/ i −t ? b T`Q# #BHBiv P (L ≥ −t) Q7 ? pBM; TQBMib BM [0, ∞)XRR a22 ?Qr2p2` i?2 7Q`K H DmbiB}+ iBQM BM am#b2+iBQM jXeXV 1t +i b KTHBM; `2[mB`2b i? i i?2 SQBbbQM T`Q+2bb QM (−∞, 0] Q7 BMi2MbBiv μ (t)P (L ≥ −t) #2 }MBi2X Bb ∞h?Bb BKTHB2b bQK2 `2bi`B+iBQMbX 6Q` BMbi M+2- B7 μ #QmM/2/- i?2 +QM/BiBQM 0 P (L > t) dt = E[L] < ∞ rBHH ;m ` Mi22 i? iX AM T `iB+mH `- L Kmbi #2 }MBi2- i? i Bb +QMp2`;2M+2 Q7 i?2 +Hmbi2` iQ i?2 2KTiv TQBMi T`Q+2bb Ui?2 bi iBQM `v bi i2V Kmbi i F2 TH +2 BM }MBi2 iBK2X Bb ∞ h?2 +QMp2`;2M+2 i?2M BM p `B iBQM- M/ bm{+B2Mi +QM/BiBQM 7Q` i?Bb Bb E 0 th(t, Z) dt < ∞X TTHB+ iBQM Q7 i?2 "`BtĜE2M/ HH K2i?Q/ `2[mB`2b i?2 +QMbi`m+iBQM Q7 SQBbbQM T`Q+2bb QM R+ Q7 BMi2MbBiv μ (t)P (L > t)XRk h?2 K BM T`Q#H2K Bb i? i bi2T UBV `2[mB`2b BM T`BM+BTH2 M 2t +i 2tT`2bbBQM 7Q` i?2 T`Q# #BHBiv i? i L > t- r?2`2 L Bb i?2 H2M;i? Q7 #` M+?BM; TQBMi T`Q+2bb N rBi? bBM;H2 M+2biQ` i 0 M/ rBi? ` M/QK 72`iBHBiv ` i2 h(t, Z)X *H2 `Hv P (L > t) = P (N ((t, ∞)) = 0) . M 2t +i 2tT`2bbBQM Q7 P (L > t) Bb MQi p BH #H2X >Qr2p2`- i?2 T`Q#H2K + M #2 +B`+mKp2Mi2/ BM i?2 7QHHQrBM; K MM2`X amTTQb2 i? i r2 FMQr 2tTHB+BiHv b2[m2M+2b Q7 MQM@M2; iBp2 7mM+iBQMb {ln }n≥1 M/ {un }n≥1 `2bT2+iBp2Hv MQM@/2+`2 bBM; M/ MQM@BM+`2 bBM;- M/ #Qi? +QMp2`;BM; TQBMirBb2 iQ P (L > t) BM bm+? t r v i? i ||un −ln ||∞ := supt≥0 |un (t)−ln (t)| dt → 0X amTTQb2 KQ`2Qp2` i? i 0 u0 (t) μ(t) dt < Ry
(JǠHH2` M/ _ bKmbb2M- kyy9)X P7 +Qm`b2 W = [0, a] ⊆ [0, ∞)- #mi HBiiH2 i?Qm;?i rBHH +QMpBM+2 i?2 `2 /2` i? i r2 + M T`2i2M/ i? i i?2 ;Q H Bb iQ b KTH2 N QM [0, ∞) rBi?Qmi //BiBQM H +Qbi bBM+2 i?2 TQBMib Q7 N0 7i2` a rBHH MQi #2 mb2/X Rk >2`2- iQ 7 +BHBi i2 MQi iBQM- iBK2 Bb `2p2`b2/- bQ i? i i?2 bm`pBpBM; TQBMib TT2 ` iQ #2 QM i?2 TQbBiBp2 ? H7@HBM2 ` i?2` i? M QM i?2 M2; iBp2 ? H7@HBM2X RR
939
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
∞X h?Bb Bb i?2 + b2 BM T `iB+mH ` B7 u0 (t) = 1−G(t) 7Q` bQK2 +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM QM R+ Q7 }MBi2 K2 M M/ supt≥0 μ (t) < ∞X h?2 +QMbi`m+iBQM i?2M ;Q2b b 7QHHQrbX 6B`bi ;2M2` i2 i?2 TQBMib UBM }MBi2 MmK@ #2`V Q7 SQBbbQM T`Q+2bb Q7 BMi2MbBiv u0 (t) μ(t)X G2i t1 - Ę- tk #2 i?2b2 TQBMibX :2M@ 2` i2 M BB/ b2[m2M+2 V1 - Ę- Vk - Q7 ` M/QK p `B #H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X lM/2` i?2 +m`p2 y = u0 (t) μ(t)- i?2 TQBMib (t1 , V1 u0 (t1 ) μ(t1 ))- Ę- (tk , V1 u0 (tk ) μ(tk )) Q7 R2+ 7Q`K SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X LQi2 i? i i?2 T`Q# #BHBiv i? i Mv Q7 i?2b2 TQBMib HB2 QM i?2 +m`p2 y = μ (t)P (L > t) Bb MmHH- M/ i?2`27Q`2- Bi rBHH ? TT2M 7Q` }MBi2 n i? i MQM2 Q7 i?2 TQBMib (t1 , V1 u0 (t1 ) μ(t1 ))- Ę- (tk , V1 u0 (tk ) μ(tk )) HB2b #2ir22M i?2 +m`p2b y = un (t) μ(t) M/ y = ln (t) μ(t)X 6Q` i?2 bBKmH iBQM- F22T QMHv i?2 TQBMib ti bm+? i? i Vi u0 (ti ) μ(ti ) ≤ n (ti ) μ(ti ) bBM+2 i?2b2 TQBMib `2 2t +iHv i?Qb2 HvBM; mM/2` i?2 +m`p2 y = P (L > t)X Ai `2K BMb iQ }M/ i?2 TT`QtBK iBM; 7mM+iBQMb un M/ ln X G2i Nk #2 N `2bi`B+i2/ iQ i?2 ;2M2` iBQMb 0- 1- Ę- kX AM T `iB+mH `- limk↑∞ Nk ((t, ∞)) = N ((t, ∞))X *QK@ k ((t,∞)) −θ N 7Q` Mv θ > 0- i F2 i?2 HBKBi b k ↑ ∞ iQ Q#i BM E e−θN ((t,∞)) Tmi2 E e M/ i?2M Q#i BM i?2 `2bmHi 7`QK i?2 Q#b2`p iBQM i? i lim E e−θN ((t,∞)) = P (N ((t, ∞)) = 0) . θ↑+∞
hQ +QKTmi2 E e−θNk ((t,∞)) - BMi`Q/m+2 i?2 ` M/QK 2H2K2Mib Yk rBi? p Hm2b BM (Mp (R+ ), Mp (R+ )) r?Qb2 /Bbi`B#miBQM Qk Bb i? i Q7 Nk X G2i MQr {Yk,n }n∈N #2 M BB/ b2[m2M+2 Q7 ` M/QK 2H2K2Mib /Bbi`B#mi2/ b Yk X h?2 /Bbi`B#miBQM Q7 Nk ((t, ∞)) Bb i?2 b K2 b i? i Q7 1{τn ≤t} Yk−1,n (t − τn ) n≥0
r?2`2 i?2 τn Ƕb `2 i?2 TQBMib Q7 i?2 }`bi ;2M2` iBQM Q7 N - 7Q`KBM; *Qt TQBMi T`Q+2bb Q7 ` M/QK BMi2MbBiv h(t, Z)X h?2`27Q`2 E e−θNk ((t,∞)) = E e−θ n≥0 1{τn ≤t} Yk−1,n (t−τn ) = E e−θ n≥0 1{τn ≤t} f (Yk−1,n ,t−τn ) , r?2`2 7Q` μ ∈ Mp (R+ ) M/ u ∈ R+ - f (μ, u) := μ((u, ∞))- bQ i? i E e−θ n≥0 1{τn ≤t} f (Yk−1,n ,t−τn ) t −θf (μ,t−s) E exp − 1 h(s, Z) Qk−1 (dμ) ds e
0
t
Mp (R+ )
0
Mp (R+ )
E exp LQr-
e
−θμ((t−s,∞))
− 1 h(s, Z) Qk−1 (dμ) ds
e−θμ((t−s,∞)) − 1 Qk−1 (dμ) = E e−θNk−1 ((t−s,∞))
Mp (R+ )
.
()
RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a
938
M/ i?2`27Q`2 t
E e−θNk ((t,∞)) = E exp E e−θNk−1 ((t−s,∞)) − 1 h(s, Z) ds . 0
h FBM; i?2 HBKBi b k ↑ ∞ vB2H/b t
((t,∞)) ((t−s,∞)) −θ N −θ N = E exp E e − 1 h(s, Z) ds E e 0
M/ i FBM; i?2 HBKBi b θ → +∞ ;Bp2b- rBi? f (t) = P (N ((t, ∞)) > 0)
t f (t) = E exp −ν(t, Z) + f (t − s)h(s, Z) ds , 0
t
r?2`2 ν(t, Z) := 0 h(s, Z) dsX h?2 `B;?i@? M/ bB/2 rBHH #2 /2MQi2/ #v Φ(f )X q2 `2 i?2`27Q`2 +QM+2`M2/ rBi? i?2 2[m iBQM f = Φ(f )
(f ∈ A),
r?2`2 A := {f : (R+ , B(R+ )) → ([0, 1]), B([0, 1]))}X q2 b?Qr i? i i?2 bQHmiBQM F (t) := P (N ((t, ∞)) > 0 Bb i?2 mMB[m2 bQHmiBQM- M/ i? i i?2`2 2tBbi b2[m2M+2b {gn }n≥1 M/ {hn }n≥1 - `2bT2+iBp2Hv MQM@/2+`2 bBM; M/ MQM@BM+`2 bBM;- rBi? i?2 +QKKQM HBKBi F M/ bm+? i? i ||fn − gn ||∞ → 0 . h?2 7QHHQrBM; bm++2bbBQM Q7 `2K `Fb rBHH #2 mb27mHX UBV h?2 b2[m2M+2 Q7 7mM+iBQMb {fn }n≥1 /2}M2/ #v fn (t) = P (Nn ((t, ∞)) > 0) Bb MQM@/2+`2 bBM;- ? b F 7Q` HBKBi- M/ r2 ? p2 b22M i? i Bi b iBb}2b i?2 `2+m``2M+2 fn = Φ(fn−1 )
(n ≥ 1) ,
rBi? f0 (t) ≡ 1X UBBV G2i f ∈ A M/ H2i Φ(n) #2 /2}M2/ `2+m`bBp2Hv #v Φ(0) (f ) := f - Φ(n) (f ) := Φ (Φ(f ))- M/ H2i fn := Φ(n) (f )X q2 ? p2 i? i (n−1)
f ≤ g ⇒ fn ≤ g n , f ≤ Φ(f ) ⇒ {fn }n≥1 Bb MQM@/2+`2 bBM;- M/ f ≥ Φ(f ) ⇒ {fn }n≥1 Bb MQM@BM+`2 bBM;. h?2 b2+QM/ M/ i?B`/ bb2`iBQMb #Qp2 `2 BKK2/B i2 +QMb2[m2M+2b Q7 i?2 }`bi QM2r?B+? Bb Bib2H7 M BKK2/B i2 +QMb2[m2M+2 Q7 i?2 /2}MBiBQM Q7 ΦX UBBBV Φ Bb +QMi` +iBQM QM A rBi? `2bT2+i iQ i?2 bmT MQ`KX JQ`2 T`2+Bb2Hv f, g ∈ A ⇒ ||Φ(f ) − Φ(g)||∞ ≤ ρ||f − g||∞ , r?2`2 ρ :=
R
E [h(t, Z)] dt U < 1VX
93e
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
S`QQ7X "v i?2 K2 M@p Hm2 i?2Q`2K- 7Q` Mv `2 H MmK#2`b a M/ b- ea −eb = (b−a)ec 7Q` bQK2 c ∈ [a, b]X AM T `iB+mH ` &
& t & & ||Φ(f ) − Φ(g)||∞ = sup &&E e−ν(t,Z) ec (f (t − s) − g(t − s))h(s, Z) ds && , t≥0
0
t r?2`2 i?2 [m MiBiv c = c(t, Z, f, g) HB2b #2ir22M 0 f (t − s)h(s, Z) ds M/ t g(t − s)h(s, Z) dsX AM T `iB+mH `- bBM+2 f, g ≤ 1- ec ≤ eν(t,Z) M/ i?2`27Q`2 0 & t
& & & ||Φ(f ) − Φ(g)||∞ ≤ sup &&E (f (t − s) − g(t − s))h(s, Z) ds && t≥0
∞ 0 ||f − g||∞ h(s, Z) ds = ρ||f − g||∞ . ≤E 0
UBpV F Bb i?2 mMB[m2 }t2/ TQBMi Q7 ΦX S`QQ7X h?2 bT +2 Q7 7mM+iBQMb A Bb +QKTH2i2 rBi? `2bT2+i iQ i?2 bmT MQ`K M/ Φ Bb +QMi` +iBQM QT2` iQ` QM i?Bb bT +2X h?2`27Q`2- #v i?2 }t2/ TQBMi i?2Q`2K- Bi ? b mMB[m2 }t2/ TQBMi- r?B+? r2 FMQr iQ #2 F X UpV qBi? fn := Φ(n) (f )-
||fn − F ||∞ → 0 .
S`QQ7X ||fn − F ||∞ = ||Φ(fn−1 ) − Φ(F )||∞ ≤ ρ||fn−1 − F ||∞ ≤ ρn ||f − F ||∞ . UpBV ||fn − F ||∞ ≤
ρn ||Φ(f ) 1−ρ
− f ||∞ X
S`QQ7X aBKBH `Hv iQ UpV- r2 ? p2 i? i ||fn − fn−1 )||∞ ≤ ρn−1 ||f1 − f )||∞ X HbQ #v UpV- ||F − f ||∞ = limm↑∞ ||fm − f ||∞ X h?2`27Q`2 ||F − f ||∞ ≤ lim (||f1 − f ||∞ + ||f2 − f1 ||∞ + · · · + ||fm − fm−1 ||∞ ) m↑∞
≤ lim ||f1 − f ||∞ (1 + ρ + · · · + ρm−1 ) ≤ m↑∞
||f1 − f ||∞ . 1−ρ
UpBBV A7 f ≤ Φ(f ) Q` f ≥ Φ(f )- i?2M fn → F 7`QK #2HQr Q` 7`QK #Qp2 `2bT2+iBp2HvX h?Bb 7QHHQrb 7`QK UBBVX h?2 7mM+iBQMb un M/ n `2 MQr /2}M2/X aBM+2 0 ≤ Φ(0)- fnu := Φ(n) (0) ↓ F X h F2 n (t) = 1 − fnu (t) bQ i? i n ↑ 1 − F X JǠHH2` M/ _ bKmbb2M 2t?B#Bi +mKmH iBp2 /Bbi`B#miBQM 7mM+iBQM G rBi? }MBi2 K2 M bm+? i? i G ≤ Φ(G)X h?2M fn := Φ(n) (G) ↑ F X h F2 un (t) = 1 − fn (t) bQ i? i un ↓ 1 − F X h?Bb Bb +`m+B H
RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a
93d
bBM+2 i?2 H;Q`Bi?K bi `ib #v +QMbi`m+iBM; Ub KTHBM;V SQBbbQM T`Q+2bb QM R+ Q7 BMi2MbBiv u0 (t) μ(t) = (1 − G(t)) μ(t) Q7 }MBi2 K bbXRj
h?2 * b2 Q7 MQM@SQBbbQMB M :2`K S`Q+2bb2b bHB;?i KQ/B}+ iBQM Q7 i?2 JǠHH2`Ĝ_ bKmbb2M i?BMMBM; T`Q+2/m`2 HHQrb mb iQ i`2 i i?2 + b2 Q7 ;2`K T`Q+2bb i? i Bb `2M2r H T`Q+2bbXR9 G2i N0 #2 M mM/2H v2/ `2M2r H b2[m2M+2 QM (0, +∞), 7Q` n ≥ 1- X0,n = S1 + · · · + Sn r?2`2 i?2 b2[m2M+2 Q7 MQM@M2; iBp2 ` M/QK p `B #H2b {Sn }n≥1 Bb BB/rBi? +QKKQM /Bbi`B#miBQM /KBiiBM; /2MbBiv f - rBi? +Q``2bTQM/BM; 7 BHm`2 ` i2 r(t) := 1− f f(t)(s) ds mMB7Q`KHv #QmM/2/ #v M < ∞X h?2 b2[m2M+2 Q7 TQBMib Q7 0 i?Bb `2M2r H T`Q+2bb K v #2 i?Qm;?i Q7 b #2BM; Q#i BM2/ #v i?BMMBM; SQBbbQM T`Q+2bb N 0 Q7 ` i2 M - rBi? b2[m2M+2 Q7 2p2Mi@iBK2b {X 0,n }n≥1 X h? i Bb {X0,n , n ≥ 1} ⊂ {X 0,n , n ≥ 1}X q2 ? p2 iQ i?BM N0 rBi? i?2 i?BMMBM; T`Q# #BHBiv p(x)X q2 TTHv i?Bb i?BMMBM; iQ i?2 /QKBM iBM; TQBMi T`Q+2bb N 0 - M/ r2 MQr FMQr ?Qr iQ /Q i?Bb #v i?2 TT`QtBK iBM; b2[m2M+2 K2i?Q/ /2b+`B#2/ #Qp2X h?2 MmK#2` Q7 TQBMib Q7 i?2 i?BMM2/ /QKBM i2/ SQBbbQM T`Q+2bb Bb }MBi2X h?2 TQBMib Q7 i?2 i?BMM2/ N0 `2 i?Qb2 TQBMib Q7 N0 i? i ? p2 bm`pBp2/ M/ `2 Q7 i?2 ivT2 X0,n X LQr- bi `i rBi? i?2 b KTH2 Q7 SQBbbQM T`Q+2bb Q7 BMi2MbBiv M P (L > t) Dmbi Q#i BM2/X G2i t1 - Ę- tk #2 Bib TQBMibX G2i mb HQQF i i?2 }`bi TQBMi- t1 M/ /` r ` M/QK MmK#2` V1 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X amTTQb2 r2 FM2r i?2 p Hm2 P (L > t1 )X h?2M r2 rQmH/ +?2+F B7 V1 M P (L > t1 ) Bb #2HQr Q` #Qp2 r(t1 )M P (L > t1 )- M/ /2+B/2 i? i t1 Bb Q` Bb MQi i?2 }`bi TQBMi Q7 i?2 i?BMM2/ `2M2r H T`Q+2bbX hQ /2+B/2 i?Bb- r2 +?2+F i?2 `2H iBp2 TQbBiBQMb Q7 V1 M un (t) M/ V1 M ln (t) rBi? `2bT2+i iQ i?2 i?`2b?QH/ r(t1 ) rBi? n bm{+B2MiHv H `;2 iQ ? p2 #Qi? TQBMib 2Bi?2` #Qp2 Q` #2HQr i?2 i?`2b?QH/X A7 #2HQr- i F2 t1 iQ #2 i?2 }`bi 2p2Mi iBK2 Q7 i?2 i?BMM2/ `2M2r H T`Q+2bbX A7 MQi- `2TH +2 t1 #v t2 M/ `2T2 i i?2 T`Q+2@ /m`2 b HQM; b vQm ? p2 M2; iBp2 Mbr2`X Ai K v ? TT2M i? i vQm 2M/ mT 7i2` k i`B Hb rBi? MQ TQBMiX amTTQb2 MQr i? i i?2 }`bi TQBMi i? i Bb `2i BM2/ b TQBMi Q7 i?2 i?BMM2/ `2M2r H T`Q+2bb KQM; i?2 HBbi t1 - Ę- tk Bb t - < kX q2 b? HH 2t KBM2 t+1 b r2 /B/ 7Q` t1 - i?Bb iBK2 rBi? i?2 i?`2b?QH/ r(t+1 − t )- M/ bQ QMX h?2 + b2 Q7 ;2`K TQBMi T`Q+2bb- i? i Bb- r?2M i?2 iBK2 tBb Bb `2p2`b2/ /2H v2/ TQBMi T`Q+2bb Bb bBKBH `- Kmi iBb Kmi M/BbX JQ`2 ;2M2` HHv- i?2 + b2 r?2`2 i?2 ;2`K T`Q+2bb Bb- r?2M i?2 iBK2 tBb Bb `2p2`b2/- TQBMi T`Q+2bb rBi? biQ+? biB+ BMi2MbBiv rBi? `2bT2+i iQ Bib BMi2`M H ?BbiQ`v λ(t) ≤ M `2+2Bp2b bBKBH ` i`2 iK2Mi- mbBM; i?2 `2;2M2` iBp2 7Q`K Q7 i?2 biQ+? biB+ BMi2MbBivX lM/2` //BiBQM H +QM/BiBQMb i? i rBHH MQi #2 /Bb+mbb2/ ?2`2- i?2 #QmM/2/M2bb bbmKTiBQM + M HbQ #2 bmTT`2bb2/X h?2 +?QB+2 fn := Φ(n) (1) rQmH/ H2 / iQ i?2 +Q``2+i HBKBi fn ↑ F - #mi BM i?Bb + b2 i?2 SQBbbQM T`Q+2bb Q7 BMi2MbBiv u0 (t) μ(t) = μ (t) +QmH/ ? p2 M BM}MBi2 MmK#2` Q7 TQBMib- 7Q` BMbi M+2 B7 μ (t) Bb TQbBiBp2 +QMbi MiX q2 `272` iQ (JǠHH2` M/ _ bKmbb2M- kyy9) 7Q` i?2 /2i BHb +QM+2`MBM; i?2 Q#i2MiBQM Q7 bm+? BMBiB H p Hm2 Q7 i?2 mTT2` #QmM/BM; T`Q+2bbX R9 ("`ûK m/- kyRe)X Rj
933
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
RkX8 "` M+?BM; SQBMi S`Q+2bb2b qBi?Qmi M+2biQ` h?Bb b2+iBQM 72 im`2b +`BiB+ H > rF2b TQBMi T`Q+2bb2b U?TTV rBi? }MBi2 p2` ;2 BM@ i2MbBiB2bX *`BiB+ HBiv K2 Mb i? i i?2 72`iBHBiv ` i2 BMi2;` i2b iQ 1 Ui?Bb +Q``2bTQM/b iQ i?2 mbm H +`BiB+ H #` M+?BM; T`Q+2bbV M/- BM i?2 +QMi2ti Q7 > rF2b TQBMi T`Q@ +2bb2b rBi? }MBi2 p2` ;2 BMi2MbBiv- Bi Bb 2[mBp H2Mi iQ i?2 #b2M+2 Q7 M+2biQ`bX h?2 }`bi [m2biBQM Bb- /Q bm+? T`Q+2bb2b 2tBbi\ h?2 p2` ;2 BMi2MbBiv λ Q7 HBM2 ` bi iBQM `v > rF2b TQBMi T`Q+2bb QM i?2 HBM2 b iBb}2b ∞ λ=ν+λ h(t)dt , URkX9RV 0
M/ i?2`27Q`2- r?2M ν > 0- i?2 bi #BHBiv +QM/BiBQM ρ < 1 Bb M2+2bb `v M/ bm{+B2Mi 7Q` N iQ ? p2 }MBi2 p2` ;2 BMi2MbBivX > rF2b #` M+?BM; TQBMi T`Q+2bb rBi?Qmi M+2biQ`b Bb bi iBQM `v TQBMi T`Q@ +2bb N rBi? i?2 biQ+? biB+ BMi2MbBiv h(t − s)N (ds) . URkX9kV λ(t) = (−∞,t)
A7 bm+? TQBMi T`Q+2bb 2tBbib- M2+2bb `BHv λ = λ h- M/ i?2`27Q`2- i?2 `2[mB`2K2Mi Q7 }MBi2 M/ MQM@MmHH p2` ;2 BMi2MbBiv BKTHB2b i? i ∞ h(t)dt = 1 , URkX9jV 0
r?B+? +Q``2bTQM/b iQ i?2 +`BiB+ H + b2 Q7 #` M+?BM; T`Q+2bb i?2Q`vX *`BiB+ H /vM KB+b Ui? i Bb- rBi? biQ+? biB+ BMi2MbBiv Q7 ivT2 URkX9kVV rBi? }MBi2 MQM@MmHH p2` ;2 BMi2MbBiv K v ? p2 MQ bQHmiBQM- b i?2 7QHHQrBM; `2bmHi b?QrbX h?2Q`2K RkX8XR UR8 V G2i N #2 bi iBQM `v 2`;Q/B+ > rF2b TQBMi T`Q+2bb rBi?Qmi M+2biQ`b Ui? i Bb- bm+? i? i ν = 0V- M/ bbmK2 i? i i?2 72`iBHBiv ` i2 7mM+iBQM h b iBb}2b i?2 b?Q`i@` M;2 +QM/BiBQM ∞ th(t)dt < ∞ . URkX99V 0
h?2M i?2 p2` ;2 BMi2MbBiv λ := EN ([0, 1) Bb 2Bi?2` 2[m H iQ y Q` +∞- i? i Bb- i?2`2 /Q MQi 2tBbi MQM@i`BpB H +`BiB+ H > rF2b T`Q+2bb2b rBi? b?Q`i@` M;2 72`iBHBiv ` i2X S`QQ7X q`Bi2 + P (N (R+ ) = 0) = E [P (N (R,) = 0 | F0 )] - ∞ = E exp − 0 dt (−∞,0) N (ds)h(t − s) , - ∞ ≥ exp −E 0 dt (−∞,0) N (ds)h(t − s) ∞ = exp −λ 0 th(t)dt , R8
("`ûK m/ M/ J bbQmHBû- kyyR)X
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
93N
r?2`2 i?2 b2+QM/ 2[m HBiv Bb /B`2+i TTHB+ iBQM Q7 i?2 `2bmHi Q7 1t KTH2 8XRXR9- i?2 BM2[m HBiv 7QHHQrb 7`QK C2Mb2MǶb BM2[m HBiv- M/ i?2 H bi bi2T Bb bBKTH2 TTHB+ iBQM Q7 6m#BMBǶb i?2Q`2KX bbmKBM; i?2 p2` ;2 BMi2MbBiv λ Bb }MBi2- mM/2` i?2 b?Q`i ` M;2 ?vTQi?2bBb URkX99V- i?Bb + H+mH iBQM vB2H/b P (N (R+ ) = 0) > 0. LQi2 i? i 2`;Q/B+Biv Q7 N BKTHB2b i? i i?2 2p2Mi {N (R+ ) = 0} ? b T`Q# #BHBiv 2Bi?2` y Q` Rc Bi i?mb 7QHHQrb i? i- rBi? T`Q# #BHBiv R- N (R+ ) = 0 M/- +QMb2[m2MiHvλ = 0X _2K `F RkX8Xk 6B`bi Q#b2`p2 i? i B7 r2 ? p2 irQ BM/2T2M/2Mi > rF2b T`Q+2bb2b N1 M/ N2 rBi?Qmi M+2biQ`b +Q``2bTQM/BM; iQ i?2 b K2 72`iBHBiv ` i2- i?2 TQBMi T`Q+2bb N = N1 + N2 Bb HbQ > rF2b T`Q+2bb rBi?Qmi M+2biQ`bX HbQ- B7 > rF2b T`Q+2bb rBi?Qmi M+2biQ` Q7 p2` ;2 BMi2MbBiv λ Bb BM/2T2M/2MiHv i?BMM2/ rBi? T`Q#@ #BHBiv p- i?2 `2bmHiBM; TQBMi T`Q+2bb Bb Q7 bBKBH ` M im`2- rBi? p2` ;2 BMi2MbBiv pλX AM T `iB+mH `- Mv > rF2b T`Q+2bb rBi?Qmi M+2biQ`b Bb M BM}MBi2Hv /BpBbB#H2 TQBMi T`Q+2bbX _2+ HH i? i i?2 TQr2` bT2+i` H K2 bm`2 μN Q7 i?2 TQBMi T`Q+2bb N /KBib /2MbBiv fN ν fN (ω) := . URkX98V 2π(1 − h)|1 − h(ω)|2 Uh?2 6Qm`B2` i` Mb7Q`Kb `2 7`QK MQr QM BM i2`Kb Q7 i?2 TmHb iBQM ω = 2πν 7Q` MQi iBQM H +QMp2MB2M+2XV h?2 bT2+i` H /2MbBiv i2M/b iQ }MBi2 +QMbi Mi fN (0) = ν/(2π(1 − h)3 ) b ω i2M/b iQ x2`QX h?Bb K v #2 pB2r2/ b M BM/B+ iBQM Q7 b?Q`i ` M;2 /2T2M/2M+2 T`QT2`iv, 7Q` BMbi M+2- B7 QM2 BMp2biB; i2b i?2 #2? pBQm` Q7 p `N (0, T ] b T → ∞- mbBM; i?2 2tT`2bbBQM (eiωT −1)/(iω) Q7 i?2 6Qm`B2` i` Mb7Q`K Q7 1[0,T ] - Bi Bb 2 bBHv b22M i? i p `N (0, T ] ∼ T fN (0)
R
|eiu − 1|2 du = 2πT fN (0) u2
Ui?2 BMi2;` H 2[m Hb 2π- #v i?2 SH M+?2`2HĜS `b2p H B/2MiBivV r?B+? Bb [m HBi iBp2Hv B/2MiB+ H iQ i?2 #2? pBQm` Q7 i?2 SQBbbQM T`Q+2bb- M/ i?2 H ii2` +2`i BMHv /2b2`p2b iQ #2 + HH2/ b?Q`i@` M;2 /2T2M/2Mi- #2+ mb2 Bi ? b BM/2T2M/2Mi BM+`2K2MibX h?mb > rF2b T`Q+2bb2b 7Q` r?B+? i?2 bi #BHBiv +QM/BiBQM h < 1 ?QH/b `2 b?Q`i@` M;2 /2T2M/2MiX >Qr2p2`- i ν = 0fN (0) :=
λ , 2π|1 − h|2
M/ i?2`27Q`2- b QM2 TT`Q +?2b i?2 +`BiB+ H + b2- fN (0) #2+QK2b BM+`2 bBM;Hv H `;2X q2 MQr T`Qp2 i?2 2tBbi2M+2 Q7 +`BiB+ H > rF2b T`Q+2bb2b rBi? }MBi2 M/ MQM@ MmHH p2` ;2 BMi2MbBivX h?2 bT2+B}+ +QM/BiBQMb i? i r2 BKTQb2 QM i?2 72`iBHBiv ` i2- #2bB/2b URkX9jV- `2
9Ny
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a sup t1+α h(t) ≤ R ,
URkX9eV
lim t1+α h(t) = r ,
URkX9dV
t≥0
t→∞
7Q` }MBi2 +QMbi Mib r, R > 0 M/ r?2`2 α ∈ (0, 12 )X h?2 7QHHQrBM; `2bmHi rBHH #2 T`Qp2/ i i?2 2M/ Q7 i?Bb bm#b2+iBQMX h?2Q`2K RkX8Xj h?2`2 2tBbib MQMi`BpB H bi iBQM `v > rF2b TQBMi T`Q+2bb rBi?@ Qmi M+2biQ`b rBi? }MBi2 BMi2MbBiv- T`QpB/2/ i?2 72`iBHBiv ` i2 BMi2;` i2b iQ 1 M/ b iBb}2b i?2 +QM/BiBQMb URkX9eV M/ URkX9dV- r?2`2 α ∈ (0, 12 )X h?2 T`QQ7 Q7 2tBbi2M+2 Bb MQM@+QMbi`m+iBp2 M/ mb2b r2 F +QMp2`;2M+2 `;m@ K2MibX 6Q` i?Bb- r2 b? HH T`Q+22/ b 7QHHQrb, 7Q` ε ∈ (0, 1)- +QMbB/2` i?2 bi iBQM `v > rF2b T`Q+2bb rBi? T ` K2i2`b hε = (1 − ε)h- M/ ν ε = ελ- 7Q` bQK2 }t2/ +QM@ bi Mi λ > 0X Ai Bb r2HH /2}M2/- bBM+2 i?2 bi #BHBiv +QM/BiBQM ?QH/b- M/ ? b p2` ;2 BMi2MbBiv λ 7Q` HH ε > 0X b h?2Q`2K RkX8XR b?Qrb- B7 r2 dzb2M/ i?2 M+2biQ`b iQ BM}MBivǴ- i?2B` 72`iBHBiv ` i2 Kmbi 2t?B#Bi bQK2 HQM;@i BH T`QT2`iv B7 QM2 r Mib MQMi`BpB H HBKBi T`Q+2bbX h?2 b2i Q7 +QM/BiBQMb +?Qb2M Bb QM2 Q7 i?2KX G2KK RkX8X9 (i) q?2M URkX9eV M/ URkX9dV `2 b iBb}2/ eiu − 1 −α lim (h(ω) − 1)ω = r du, 0 < α < 1 . α+1 ω→0 R+ u
URkX93V
(ii) A7 BM //BiBQM α < 12 - i?2M i?2 p `B M+2b V ε = p `N ε [0, T ] `2 #QmM/2/ b ε → 0- M/ +QMp2`;2 b ε → 0 iQ i?2 }MBi2 HBKBi |eiωT − 1|2 λ V0 = dω . URkX9NV 2π R ω 2 |1 − h(ω)|2 S`QQ7X (i) 7i2` +? M;2 Q7 p `B #H2b- i?2 H27i@? M/ bB/2 Q7 URkX93V `2 /b ω −1−α (eiu − 1)h(u/ω)du . R+
h?2 KQ/mHmb Q7 i?2 BMi2;` M/ Bb- BM pB2r Q7 URkX9eV- H2bb i? M R|eiu −1|u−α−1 - r?B+? Bb BMi2;` #H2 r?2M α < 1X h?2`27Q`2 URkX93V 7QHHQrb 7`QK URkX9dV #v /QKBM i2/ +QMp2`;2M+2X (ii) h?2 bT2+i` H 7Q`KmH TTHB2/ iQ N ε rBi? i?2 7mM+iBQM f = 1[0,T ] vB2H/b |eiωT − 1|2 λ Vε = dω. 2π R ω 2 |1 − (1 − ε)h(ω)|2 h?2 BMi2;` M/ Bb K tBKBx2/ 7Q` 1 − ε = _2(h(ω))/|h(ω)|2 X h?Bb 2Mbm`2b i? i i?2 BMi2;` M/ Bb #QmM/2/ #Qp2 #v |eiωT − 1|2 |h(ω)|2 × , 2 ω |AKh(ω)|2
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
9NR
r?2`2 AKz /2MQi2b i?2 BK ;BM `v T `i Q7 i?2 +QKTH2t MmK#2` zX AM i?2 M2B;?#Qm`@ ?QQ/ Q7 y- b v BM bQK2 /QK BM |ω| ≤ ωc - i?Bb mTT2` #QmM/ Bb BMi2;` #H2- bBM+2 BM pB2r Q7 URkX93V i?2 b2+QM/ 7 +iQ` Bb Q7 Q`/2` |ω|−2α - M/ r2 ? p2 bbmK2/ α < 12 X AM i?2 +QKTH2K2Mi `v /QK BM |ω| > ωc - MQiB+2 }`bi i? i i?2 7mM+iBQM ω → |h(ω)| Bb +QMiBMmQmb- ;Q2b iQ x2`Q i BM}MBiv #v i?2 _B2K MMĜG2#2b;m2 H2KK - M/ Bb i?2`2@ 7Q`2 #QmM/2/ #Qp2 #v bQK2 +QMbi Mi θ(ωc ) > 0X Ai Bb 2 bBHv b22M i? i M2+2bb `BHvθ(ωc ) < 1X PM2 i?mb ? b |ω| > ωc ⇒ |1 − (1 − ε)h(ω)| ≥ 1 − (1 − ε)θ(ωc ) ≥ 1 − θ(ωc ) > 0 . h?2 BMi2;` M/ BM i?2 /2}MBiBQM Q7 V ε Bb i?mb #QmM/2/ #Qp2 #v bQK2 BMi2;` #H2 7mM+iBQM r?B+? /Q2b MQi /2T2M/ QM εX h?2 +QM+HmbBQMb Q7 i?2 H2KK i?2M 7QHHQr #v /QKBM i2/ +QMp2`;2M+2X _2K `F RkX8X8 LQi2 i? i- BM T `iB+mH `- i i?2 HBKBi- i?2 TQr2` bT2+i` H /2MbBiv Bb +QMbi Mi × |ω −2α | BM i?2 M2B;?#Qm`?QQ/ Q7 i?2 x2`Q 7`2[m2M+v- M/ i?2 HBKBi Q7 o `N [0, T ] = O(T 1+2α ) b T → ∞X AM i?Bb b2Mb2- i?2 HBKBi T`Q+2bb 2t?B#Bib HQM;@ ` M;2 /2T2M/2M+2X >Qr2p2`- BM Q`/2` iQ K F2 i?Bb `;mK2Mi `B;Q`Qmb- r2 ? p2 iQ T`Qp2 i? i i?2 HBKBi Q7 i?2 p `B M+2 Bb BM/22/ i?2 p `B M+2 Q7 i?2 HBKBiX h?Bb rBHH #2 /QM2 H i2` Uh?2Q`2K RkX8XRRVX S`QQ7X UQ7 h?2Q`2K RkX8XjV AM pB2r Q7 G2KK RkX8X9- i?2 7 KBHv Q7 ` M/QK p `B@ #H2b N ε [0, T ] Bb iB;?iX h?Bb 2Mbm`2b i? i i?2 }MBi2@/BK2MbBQM H /Bbi`B#miBQMb Q7 i?2 T`Q+2bb2b N ε HbQ 7Q`K iB;?i 7 KBHvX b +QMb2[m2M+2- i?2 H rb Q7 i?2 N ε `2 iB;?iXRe *QMbB/2` i?2M Mv HBKBi /Bbi`B#miBQM b ε → 0X q2 }`bi 2bi #HBb? i? i i?Bb /Bbi`B#miBQM Bb BM/22/ i? i Q7 > rF2b T`Q+2bb rBi? T ` K2i2`b ν = 0 M/ hX hQ b22 i?Bb- Bi Bb 2MQm;? iQ T`Qp2 i? i
b
E [1A N ((a, b])] = E 1A
λ(s) ds a
7Q` HH a < b- M/ HH 2p2Mib A BM bQK2 b2KB`BM; ;2M2` iBM; FaN X h F2 A Q7 i?2 7Q`K {N (C1 ) = k1 , . . . , N (Cn ) = kn } 7Q` BMi2;2`b k1 , . . . , kn M/ K2 bm` #H2 bm#@ b2ib C1 , . . . , Cn ∈ (−∞, a]X AM Q`/2` iQ b?Qr i? i i?Bb 7Q`KmH ?QH/b i`m2 r?2M N Bb i?2 +Q``2bTQM/BM; HBKBiBM; T`Q+2bb- QM2 b?QmH/ TTHv i?2 7Q`KmH iQ N ε M/ +?2+F i? i QM2 + M ;Q iQ i?2 HBKBi ε → 0 BM i?2 7Q`KmH X "mi N (a, b]1A Bb 2[m H iQ g(N (a, b], N (C1 ), . . . , N (Cn )) 7Q` bQK2 +QMiBMmQmb 7mM+iBQM g Ui FBM; /p Mi ;2 Q7 i?2 7 +i i? i i?2b2 ` M/QK p `B #H2b `2 BMi2;2`@p Hm2/VX h?2`27Q`2g(N ε (a, b], N ε (C1 ), . . . , N ε (Cn )) +QMp2`;2b r2 FHv iQ g(N (a, b], N (C1 ), . . . , N (Cn ))X hQ b?Qr i? i i?2 HBKBiBM; T`Q+2/m`2 Bb p HB/ 7Q` i?2 H27i@? M/ bB/2- Bi bm{+2b iQ b?Qr i? i i?2 p `B #H2b g(N ε (a, b], N ε (C1 ), . . . , N ε (Cn )) `2 mMB7Q`KHv BMi2;` #H2XRd h?Bb Bb i`m2 bBM+2 i?2v `2 #QmM/2/ #v i?2 ` M/QK p `B #H2b N ε (a, b] r?Qb2 b2+QM/ KQK2Mib `2 #QmM/2/- b b?QrM BM G2KK RkX8X9X Re Rd
(. H2v M/ o2`2@CQM2b- RN33)- h?2Q`2K NXRXoA- TX kd9X ("BHHBM;bH2v)- h?2Q`2K 8X9X
9Nk
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a G2i
b
ϕ(N ) :=
h(t − s)N (ds)1A .
dt a
(−∞,t)
hQ T`Qp2 i? i i?2 HBKBiBM; T`Q+2/m`2 Bb p HB/ 7Q` i?2 `B;?i@? M/ bB/2- QM2 ? b iQ b?Qr i? i i?2 2tT2+i iBQM Q7 ϕ(N ε ) i2M/b iQ i? i Q7 ϕ(N )X LQr ϕ(N ) Bb +QMiBMmQmb 7mM+iBQM Q7 N 7Q` i?2 TT`QT`B i2 iQTQHQ;v QM i?2 + MQMB+ H bT +2 Q7 HQ+ HHv }MBi2 TQBMi T`Q+2bb2bc b22 a2+iBQM RX9X h?2`27Q`2 ϕ(N ε ) +QMp2`;2b r2 FHv iQ ϕ(N )- bBM+2 N ε +QMp2`;2b r2 FHv iQ N X Ai `2K BMb iQ b?Qr i? i i?2 p `B #H2b ϕ(N ε ) `2 mMB7Q`KHv BMi2;` #H2X .2}MBM; b dt h(t − s)N ε (ds)1A , Y ε := a
(−∞,t)
Bi Bb 2MQm;? iQ b?Qr i? i i?2 Y ε ? p2 #QmM/2/ b2+QM/ KQK2MiX TTHvBM; i?2 /2}MBM; 7Q`KmH UNXR9V Q7 i?2 " `iH2ii bT2+i` H K2 bm`2- r2 Q#i BM E(Y ε )2 = [λ(b − a)]2 + |f (ω)|2 fN ε (ω)dω, R
r?2`2
b−s
f (s) :=
h(u)du. a−s
h?2 b2+QM/ KQK2Mi Q7 Y ε Bb mMB7Q`KHv #QmM/2/ r?2M i?2 7mM+iBQM |f (ω)|2 Bb BMi2@ ;` #H2 i BM}MBivX lbBM; i?2 7 +i i? i f Bb /Bz2`2MiB #H2 rBi? BMi2;` #H2 /2`Bp iBp2Bi 7QHHQrb 7`QK i?2 _B2K MMĜG2#2b;m2 H2KK i? i |f (ω)| = O(ω −1 ) b ω → ∞?2M+2 i?2 mMB7Q`K BMi2;` #BHBiv Q7 i?2 Y ε - rBi? i?2 b K2 T`QQ7 b i? i Q7 T `i UiiV Q7 G2KK RkX8X9X *QmTHBM; S`QT2`iB2b q2 MQr //`2bb irQ BMiBK i2Hv +QMM2+i2/ Bbbm2bX h?2 }`bi QM2 Bb i?2 #2? pBQm` Q7 i` MbB2Mi +`BiB+ H T`Q+2bb2b- M/ BM T `iB+mH `- i?2B` +QMp2`;2M+2 BM bQK2 b2Mb2 iQ bi iBQM `BivX h?2 b2+QM/ QM2 Bb i?2 [m2biBQM Q7 mMB[m2M2bbX LQi2 rBi? `2bT2+i iQ i?2 H ii2` Bbbm2 i? i i?2 bmK Q7 irQ BM/2T2M/2Mi +`BiB+ H > rF2b T`Q+2bb2b rBi? i?2 b K2 72`iBHBiv ` i2 h Bb +`BiB+ H > rF2b T`Q+2bb2b rBi? i?2 b K2 72`iBHBiv ` i2 hX h?mb mMB[m2M2bb + M #2 2tT2+i2/ iQ ?QH/ QMHv B7 QM2 //b +QMbi` BMi- b v- iQ ? p2 }t2/ p2` ;2 BMi2MbBivX h?2 Bbbm2 Q7 +QMp2`;2M+2 rBHH #2 i`2 i2/ }`bi- pB +QmTHBM;X *QMbB/2` irQ /Bb@ iBM+i TQBMi T`Q+2bb2b N1 - N2 M/ bbmK2 i? i 7Q` i = 1, 2- Ni /KBib QM R+ i?2 biQ+? biB+ BMi2MbBiv λi (t) = νi (t) + h(t − s)Ni (ds) , URkX8yV (0,t)
r?2`2 νi Bb bQK2 MQM@M2; iBp2 F0 @K2 bm` #H2 T`Q+2bbX 6Q` BMbi M+2- B7 i?2b2 T`Q@ +2bb2b `2 /2}M2/ QM R− b r2HH- i?2 T `iB+mH ` 7Q`K νi (t) = h(t − s)Ni (ds) URkX8RV (−∞,0]
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
9Nj
+Q``2bTQM/b iQ i?2 +H bbB+ H > rF2b T`Q+2bb /vM KB+bX "v :`B;2HBQMBbǶb BK#2//BM; i?2Q`2K- Bi K v #2 bbmK2/ i? i i?2`2 2tBbib M ?TT N QM R2 - BM/2T2M/2Mi Q7 F0 bm+? i? i 7Q` i = 1, 2- QM R+ Ni (dt) = N (dt × [0, λi (t)]), r?2`2 λi (t) = νi (t) +
(0,t)
URkX8kV
h(t − s)Ni (ds)X
LQi2 i? i i?Bb +QMbi`m+iBQM K v vB2H/ BKT`QT2` Ui? i Bb- MQi HQ+ HHv }MBi2V TQBMi T`Q+2bb2bX >Qr2p2` B7 i?2 biQ+? biB+ BMi2MbBiB2b `2 HQ+ HHv BMi2;` #H2- i?2 TQBMi T`Q+2bb2b `2 T`QT2`X hQ ;m ` Mi22 i?Bb- Bi bm{+2b i? i ν(t) U7Q` +QMp2MB2M+2 r2 KQK2Mi `BHv /`QT i?2 BM/B+2bV #2 HQ+ HHv BMi2;` #H2X AM/22/ λ(t) = ν(t) + h(t − s)N (ds). (0,t)
h FBM; 2tT2+i iBQMb M/ Bi2` iBM; i?Bb 2[m iBQM- QM2 Q#i BMb i?2 7QHHQrBM; #QmM/, ν ∗ h∗n (t) . URkX8jV E[λ(t) | F0 ] = ν(t) + n>0
AMi2;` i2/ #2ir22M y M/ T - i?2 `B;?i@? M/ bB/2 2[m Hb T ν(t)dt 0 , T 1 − 0 h(t)dt r?B+? Bb }MBi2 mM/2` Qm` bbmKTiBQMb QM i?2 7mM+iBQM hX >2M+2- E[λ(t) | F0 ] Bb XbX HQ+ HHv BMi2;` #H2X q2 MQr ;Bp2 +QM/BiBQMb mM/2` r?B+? i?2 irQ b?B7i2/ T`Q+2bb2b St N1 M/ St N2 #2+QK2 +HQb2 BM bQK2 b2Mb2 b t → ∞X G2KK RkX8Xe bbmK2 i? i #Qi? ν1 M/ ν2 `2 HQ+ HHv BMi2;` #H2- M/ i? i lim
t→∞
|ν1 (t) − ν2 (t)| ∞ = 1 XbX h(s)ds t
URkX89V
6Q` HH a < blim P [N1 = N2 QM (t + a, t + b] | F0 ] = 0 XbX
t→∞
URkX88V
S`QQ7X h?2 TQBMi T`Q+2bb Δ /2}M2/ #v Δ(dt) = |N1 (dt) − N2 (dt)| /KBib QM R+ i?2 biQ+? biB+ BMi2MbBiv δ(t) = |ν1 (t) − ν2 (t) + h(t − s)(N1 (ds) − N2 (ds))| . (0,t)
AM T `iB+mH `-
δ(t) ≤ |ν1 (t) − ν2 (t)| +
h(t − s)Δ(ds) . (0,t)
9N9
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
h FBM; 2tT2+i iBQMb M/ Bi2` iBM; i?Bb 2[m iBQM vB2H/b i?2 #QmM/, |ν1 − ν2 | ∗ h∗n (t) . E[δ(t) | F0 ] ≤ |ν1 − ν2 |(t) +
URkX8eV
n>0
∞ 6Bt ε > 0- M/ b2H2+i T > 0 bQ i? i 7Q` u ≥ T - |ν1 − ν2 |(u) ≤ ε u h(s)ds Ubm+? T 2tBbib XbX 7Q` HH ε > 0 #v bbmKTiBQM i URkX89VVX h?2M 7Q` u ≥ T - URkX8eV BKTHB2b ∞ u T E[δ(u) | F0 ] ≤ ε ds h∗n (u − s) h(v)dv + h∗n (u − s)|ν1 − ν2 |(s)ds. n≥0
0
s
n>0
0
b h BMi2;` i2b iQ 1- i?2 ;2M2`B+ i2`K BM i?2 }`bi bmKK iBQM `2 /b u u−v u h∗n (v)dv − h∗n (v)dv h(s)ds . 0
0
0
Q7 p `B #H2 `;mK2Mi b?Qrb i? i i?2 b2+QM/ i2`K i?2`2 2[m Hb u +? M;2 ∗(n+1) h (v)dvX h?Bb 2M #H2b mb iQ + M+2H i2`Kb BM i?Bb }`bi bmKK iBQM- r?B+? 0 i?2M `2/m+2b 2t +iHv iQ εX AMi2;` iBM; i?Bb BM2[m HBiv #2ir22M t + a M/ t + b vB2H/b t+b T t+b ∗n E[δ(u) | F0 ]du ≤ (b − a)ε + |ν1 − ν2 |(s) h (u − s) du ds . t+a
0
n>0
t+a
.2MQiBM; #v U i?2 `2M2r H K2 bm`2 bbQ+B i2/ rBi? i?2 T`Q# #BHBiv /2MbBiv h- i?2 b2+QM/ i2`K BM i?Bb `B;?i@? M/ bB/2 Bb #QmM/2/ #Qp2 #v T |ν1 − ν2 |(s)U ([t + a − s, t + b − s]) ds. 0
Ai 7QHHQrb #v /QKBM i2/ +QMp2`;2M+2 i? i (t+a,t+b) E[δ(u) | F0 ]du ;Q2b iQ y b t → ∞X AM/22/- U ([t + a − s, t + b − s] i2M/b iQ 0 = 1/ th(t)dt #v i?2 `2M2r H i?2Q`2K- M/ Bb #QmM/2/ #v i?2 }MBi2 [m MiBiv U ([0, b − a]) b bBKTH2 `;mK2Mi b?QrbX h?2 +QM+HmbBQM Q7 i?2 H2KK 7QHHQrb #v MQiB+BM; i? i E[|N1 − N2 | (t + a, t + b] | F0 ] = E[δ(u) | F0 ]du , (t+a,t+b)
r?B+? ;Q2b iQ x2`Q b t → ∞- #v i?2 T`2pBQmb M HvbBbX h?2`27Q`2 lim E [|N1 − N2 | (t + a, t + b] | F0 ] = 0 XbX
t→∞
7`QK r?B+? URkX88V 7QHHQrbX
G2KK RkX8Xd *QMbB/2` TQBMi T`Q+2bb2b Ni (i = 1, 2) QM R- /KBiiBM; QM R+ i?2 `2bT2+iBp2 biQ+? biB+ BMi2MbBiB2b h(t − s)Ni (ds) (i = 1, 2) . λi (t) = (−∞,t)
bbmK2 BM //BiBQM i? i 7Q` bQK2 λ > 0-
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
9N8
Ni [−t, 0] = λ a.s. (i = 1, 2) . URkX8dV t h?2M- 7Q` i = 1, 2- νi (t) := (−∞,0] h(t − s)Ni (ds) Bb HQ+ HHv BMi2;` #H2 M/ ∞ h(s)ds ∼ α−1 λrt−α b t → ∞ , νi (t) ∼ λ lim
t→∞
t
rBi? r b BM URkX9dVX AM T `iB+mH `- i?2 bbmKTiBQMb Q7 G2KK k `2 HH b iBb}2/X S`QQ7X 6Q` }t2/ t > 0- QM2 ? b t νi (s)ds = 0
t−u
Ni (du) (−∞,0]
h(s)ds . −u
aTHBi i?2 BMi2;` H Q7 i?2 `B;?i@? M/ bB/2 ++Q`/BM; iQ r?2i?2` u ≥ −1 Q` u < −1 Ui?2 +?QB+2 −1 ∞Bb `#Bi` `vVX h?2 }`bi BMi2;` H bQ Q#i BM2/ Bb i?2M #QmM/2/ #Qp2 #v Ni [−1, 0] 0 h(s)ds- r?B+? Bb }MBi2X AM pB2r Q7 URkX9eV- i?2 b2+QM/ BMi2;` H Bb bK HH2` i? M Ni (du)Rt(−u)−1−α . _2TH +2 (−u)−1−α #v i?Bb 2tT`2bbBQM 2[m Hb
∞
(−∞,−1)
(1 + α)z −2−α dz M/ TTHv 6m#BMBǶb i?2Q`2K iQ b22 i? i −u
∞
Rt(1 + α)
z −2−α Ni [−z, −1)dz .
1
AM pB2r Q7 URkX8dV- i?2 BMi2;` M/ Bb 2[mBp H2Mi iQ λz −1−α b z → ∞- bQ i? i i?2 BMi2;` H Bb }MBi2X h?mb νi Bb HQ+ HHv BMi2;` #H2X q`Bi2 νi (t) = h(t − s)Ni (ds) + h(t − s)Ni (ds) . (−∞,−1]
(−1,0]
h?2 b2+QM/ i2`K Q7 i?2 `B;?i@? M/ bB/2 Bb- 7Q` H `;2 t- Q7 Q`/2` t−1−α X 6Bt ε > 0 M/ +QMbB/2` T bm+? i? i 7Q` t ≥ T U+7X URkX9dVV- r−1 t1+α h(t) ∈ (1 − ε, 1 + ε)X 6Q` t ≥ T - i?2 }`bi i2`K Bb 2[m H iQ ∞ r(t − s)−1−α Ni (ds) = r(1 + α) u−2−α duNi [t − u, −1] , C t+1
(−∞,−1]
7Q` bQK2 +QMbi Mi C BM (1 − ε, 1 + ε)X "2+ mb2 Q7 i?2 bbmKTiBQM URkX8dV- i?Bb H bi i2`K Bb 2[mBp H2Mi iQ ∞ u−2−α (u − t)du , Cr(α + 1)λ t+1 −1
−α
Bib2H7 2[mBp H2Mi iQ Cα rλt X _2+ HHBM; i? i εBb `#Bi` `v- i?Bb H ii2` i2`K Bb ∞ 2[mBp H2Mi iQ νi (t) b t → ∞- M/ ?2M+2 Q7 Q`/2` t h(s)dsX h?2 #Qp2 H2KK b HHQr mb iQ 2bi #HBb? i?2 7QHHQrBM; `2bmHiX
9Ne
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
h?2Q`2K RkX8X3 amTTQb2 i? i i?2 72`iBHBiv ` i2 h b iBb}2b URkX9jV- URkX9eV M/ URkX9dVX h?2 7QHHQrBM; T`QT2`iB2b i?2M ?QH/, U V 6Q` HH λ > 0- i?2`2 2tBbib i KQbi QM2 /Bbi`B#miBQM 7Q` bi iBQM `v > rF2b T`Q+2bb rBi? p2` ;2 BMi2MbBiv λ M/ bm+? i? i limt→∞ N [−t, 0]/t = λ XbXc H2i πλ (·) /2MQi2 i?Bb /Bbi`B#miBQM r?2M Bi 2tBbibX U#V πλ Bb 2`;Q/B+ M/ KBtBM;X U+V Mv bi iBQM `v > rF2b T`Q+2bb bm+? i? i Λ := limt→∞ N [−t, 0]/t Bb XbX }MBi2 /KBib i?2 2`;Q/B+ /2+QKTQbBiBQM P (N ∈ ·) = πλ (·)P (Λ ∈ dλ) . U/V Mv TXTX N QM R+ rBi? biQ+? biB+ BMi2MbBiv λ(t) = νi (t) + h(t − s)N (ds) , (0,t)
∞
r?2`2 νi (t) ∼ Λ t h(s)ds b t → ∞ 7Q` M XbX }MBi2 `XpX Λ bm+? i? i i?2 /Bbi`B#miBQM π(Λ) 2tBbib XbX b iBb}2b i?2 r2 F +QMp2`;2M+2 T`QT2`iv w P (St N ∈ ·) → πλ (·)P (Λ ∈ dλ) .
S`QQ7X U V G2i N1 M/ N2 #2 irQ bi iBQM `v > rF2b T`Q+2bb2b bm+? i? i lim Ni [−t, 0]/t = λ .
t→∞
G2i N1 M/ N2 #2 b BM i?2 +QmTHBM; H2KK UG2KK kV lim P N1 = N2 QM (t + a, t + b] | F0 = 1 XbX t→∞
6`QK i?Bb Bi 7QHHQrb i? i i?2 }MBi2@/BK2MbBQM H /Bbi`B#miBQMb Q7 N1 M/ N2 `2 i?2 b K2- ?2M+2 i?2B` /Bbi`B#miBQMb +QBM+B/2, i?2`2 Bb i KQbi QM2 /Bbi`B#miBQM 7Q` bi iBQM `v > rF2b T`Q+2bb N bm+? i? i limt→∞ Ni [−t, 0]/t = λ XbX- M/ r2 /2MQi2 Bi #v πλ r?2M Bi 2tBbibX U#V πλ Bb KBtBM; U M/ ?2M+2 2`;Q/B+V, *QMbB/2` bQK2 `#Bi` `v }MBi2 BMi2`p H [a, b]- M/ H2i A- B #2 irQ K2 bm` #H2 bm#b2ib Q7 i?2 bT +2 N Q7 TQBMi T`Q+2bb2bBMpQHpBM; QMHv i?2 `2bi`B+iBQM Q7 i?2 TQBMi T`Q+2bb iQ i?2 BMi2`p H [a, b]X JBtBM; ?QH/b B7 7Q` HH bm+? a < b- A M/ Blim P (N ∈ A, St N ∈ B) = P (N ∈ A)P (N ∈ B) .
t→∞
*QMbB/2` irQ BM/2T2M/2Mi +QTB2b Ni Q7 N - M/ TTHv i?2 +QMbi`m+iBQM Q7 G2KK RkX8Xe iQ i?2K- bi `iBM; MQi 7`QK iBK2 y #mi 7`QK iBK2 bX Ai i?2M ?QH/b i? i
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
9Nd
P (N ∈ A, St N ∈ B)
, = P (N1 ∈ A, St N2 ∈ B) + P (N1 ∈ A, St N1 ∈ B) − P (N1 ∈ A, St N2 ∈ B) .
LQi2 i? i N1 +QBM+B/2b rBi? N1 QM (−∞, b]- r?BH2 N2 Bb +QMbi`m+i2/ 7`QK N2 M/ ¯ QMHv- M/ Bb i?mb BM/2T2M/2Mi Q7 i?2 2p2Mi {N1 ∈ A}X h?2 }`bi i2`K BM i?2 N T`2pBQmb 2[m iBQM Bb i?2M 2t +iHv 2[m H iQ P (N ∈ A)P (N ∈ B)X h?2 i2`K BM +m`Hv #` +F2ib ;Q2b iQ y b t → ∞- #2+ mb2 Bib KQ/mHmb Bb bK HH2` i? M i?2 p `B iBQM /Bbi M+2 #2ir22M i?2 /Bbi`B#miBQMb Q7 N1 M/ N2 r?2M `2bi`B+i2/ iQ [t + a, t + b] M/ #v G2KK RkX8Xe i?2 H ii2` ;Q2b iQ x2`Q b t → ∞X U+V 1`;Q/B+ /2+QKTQbBiBQM, "v bbmKTiBQM- limt→∞ N [−t, 0]/t = Λ- r?B+? Bb M XbX }MBi2 ` M/QK p `B #H2 U2tBbi2M+2 Q7 i?2 XbX HBKBi Λ Bb BKTHB2/ #v "B`F?QzǶb 2`;Q/B+ i?2Q`2Kc Bib }MBi2M2bb M22/b iQ #2 bbmK2/ i?Qm;?VX Ai Bb 2 bv iQ b?Qr i? i i?2 /Bbi`B#miBQM Q7 N +QM/BiBQM HHv QM Λ = l Bb biBHH i?2 /Bbi`B#miBQM Q7 bi iBQM `v ?TT- mM/2` r?B+? limt→∞ N [−t, 0]/t = l XbX Ai i?2M 7QHHQrb 7`QK U V i? i i?2 /Bbi`B#miBQM Q7 N +QM/BiBQM HHv QM Λ = l Kmbi +QBM+B/2 rBi? πl X ai i2K2Mi U/V 7QHHQrb 7`QK i?2 +QmTHBM; H2KK BM M Q#pBQmb r vX a2+QM/@Q`/2` S`QT2`iB2b h?2 2tBbi2M+2 Q7 bi iBQM `v +`BiB+ H > rF2b T`Q+2bb2b Nλ rBi? p2` ;2 BMi2MbBiv λ 7Q` Mv λ > 0 ? b #22M 2bi #HBb?2/- #mi i i?Bb TQBMi i?2B` b2+QM/@Q`/2` bi`m+im`2 Bb v2i iQ #2 M Hvb2/X b K2MiBQM2/ Dmbi 7i2` i?2 T`QQ7 Q7 G2KK RkX8X9- r2 M22/ iQ b?Qr i? i i?2 p `B M+2 Q7 i?2 HBKBi Bb 2[m H iQ i?2 HBKBi Q7 i?2 p `B M+2X K P#b2`p2 i? i 7Q` rBi? +QKT +i HH g ∈ C0 Ui?2 bT +2 Q7 +QMiBMmQmb 7mM+iBQMb bmTTQ`iV- pB2rBM; g(x)Nλ (dx) b i?2 r2 F HBKBi Q7 i?2 g(x)N ε (dx) r?2M ε 7QHHQrb b2[m2M+2 /2+`2 bBM; iQ y HQM; r?B+? i?2 N ε +QMp2`;2 r2 FHv- 6 iQmǶb H2KK BKTHB2b i? i |g(ω)|2 λ p ` g(x)Nλ (dx) ≤ dω . URkX83V 2π |1 − h(ω)|2
h?2 M2ti H2KK ;Bp2b M BM2[m HBiv BM i?2 +QMp2`b2 /B`2+iBQMX G2KK RkX8XN amTTQb2 i? i i?2 72`iBHBiv ` i2 h b iBb}2b URkX9jV- URkX9eV- M/ URkX9dVX 6Q` HH g ∈ L1C (R) ∩ L2C (R)- M/ 2p2`v bi iBQM `v > rF2b T`Q+2bb N bm+? i? i Λ := limt→∞ N [−t, 0]/t XbX }MBi2
p `
g(x)N (dx) ≥ p `(Λ)
2 g(x)dx
+
E(Λ) 2π
|g(ω)|2 |1 − h(ω)|2
dω.
URkX8NV
S`QQ7X Ai bm{+2b iQ 2bi #HBb? URkX8NV BM i?2 + b2 r?2`2 Λ `2/m+2b iQ +QMbi MibBM+2 i?2 ;2M2` H 7Q`KmH 7QHHQrb 7`QK i? i + b2 #v mbBM; i?2 +QM/BiBQM H p `B M+2 7Q`KmH X q2 i?mb bbmK2 Λ = λ XbX 7Q` bQK2 +QMbi Mi λ > 0X A7 g(x)N (dx) ? b BM}MBi2 p `B M+2- i?2 H2KK ?QH/b i`BpB HHvX >2M+2 bbmK2 i? i o ` g(x)N (dx)
9N3
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
Bb }MBi2X 6Q` bi iBQM `v > rF2b T`Q+2bb rBi? }MBi2 p2` ;2 BMi2MbBiv λ- M/ f ∈ L1C (R) ∩ L2C (R) ˜ ∗ f (x)]N (dx) = λ |f (ω)|2 dω, URkXeyV p ` [f (x) − h 2π ˜ = h(−t)- M/ h ˜ ∗ f (x) = h(x ˜ − t)f (t)dtX TTHvBM; 7Q`KmH URkXeyV iQ r?2`2 h(t) i?2 7mM+iBQM ˜ + ··· + g ∗ h ˜ ∗n f := g + g ∗ h vB2H/b
&2 & n & & λ ∗(n+1) 2& k& ˜ p ` [g(x) − h |g(ω)| & ∗ g(x)]N (dx) = h(ω) & dω . & & 2π k=0 ˜ ∗n (x)N (dx) → 0 b n → ∞X 1tT M/BM; i?2 H27i@? M/ b Bb b?QrM #2HQr- o ` g ∗ h bB/2 Q7 i?2 T`2pBQmb 2tT`2bbBQM- M/ H2iiBM; n → ∞- QM2 i?mb b22b i? i URkX8NV ?QH/b rBi? 2[m HBiv BMbi2 / Q7 BM2[m HBiv- mM/2` i?2 bbmKTiBQM i? i Λ = λ XbX M/ p ` g(x)N (dx) Bb }MBi2X
˜ ∗n (x)N (dx) +QMp2`;2b iQ x2`Q b n → ∞X *QMbB/2` q2 MQr b?Qr i? i o ` g ∗h ˜ ∗n ∗ g (x + t)N (dx)X LQi2 i? i QM2 T`Q+2bb Bb Q#i BM2/ h i?2 T`Q+2bb X n (t) := 7`QK i?2 T`2pBQmb QM2 #v +QMpQHmiBQM H }Hi2`BM;, n ds. X n+1 (t) = h(s)Xt+s
URkXeRV
G2i C 0 (u) /2MQi2 *Qp(X 0 (t), X 0 (t))X LQi2 i? i C 0 (u) Bb #QmM/2/ #Qp2 #v p `X 0 (0)- r?B+? Bb }MBi2 #v bbmKTiBQMX G2i mb MQr b?Qr i? i i?2 T`Q+2bb X 1 /KBib bT2+i` H K2 bm`2 μX "Q+?M2`Ƕb i?2Q`2K 2Mbm`2b Bib 2tBbi2M+2 T`QpB/2/ i?2 T`Q+2bb Bb rB/2@b2Mb2 bi iBQM `vX Ai Bb bi iBQM `v b2+QM/@Q`/2` T`Q+2bb- M/ Bi `2K BMb iQ T`Qp2 +QMiBMmBiv BM i?2 [m /` iB+ K2 MX q`Bi2 p ` X 1 (y) − X 1 (0) 0 0 = *Qp (h(s − y) − h(s))X (s) ds, (h(t − y) − h(t))X (t) dt = C 0 (t − s)(h(s − y) − h(s))(h(t − y) − h(t)) ds dt . 2 h?2 H bi i2`K Bb bK HH2` BM #bQHmi2 p Hm2 i? M C 0 (0) |h(t − y) − h(t)| dt r?B+? i2M/b iQ x2`Q b y → 0 bBM+2 h Bb BMi2;` #H2X AM pB2r Q7 URkXeRV- X n HbQ /KBib bT2+i` H K2 bm`2- ;Bp2M #v μ(dω)|h(ω)|2n−2 X Aib BMi2;` H Bb 2[m H iQ i?2 ˜ ∗n (x)N (dx)- r?B+? r2 r Mi iQ T`Qp2 ;Q2b iQ x2`QX aBM+2 h Bb p `B M+2 Q7 g ∗ h +QMiBMmQmb M/ |h(ω)| < 1 7Q` HH ω = 0- i?Bb +QMp2`;2M+2 ?QH/b B7 r2 + M T`Qp2 i? i μ({0}) = 0X h?2 7QHHQrBM; H2KK - r?Qb2 T`QQ7 Bb ;Bp2M H i2`- rBHH #2 mb27mHX G2KK RkX8XRy G2i {Xn }n>0 #2 b2[m2M+2 Q7 +2Mi2`2/ ` M/QK p `B #H2b- +QM@ p2`;BM; iQ y BM T`Q# #BHBiv b n → ∞X bbmK2 7m`i?2` i? i 7Q` HH n > 0- Xn ≤cx Y r?2`2 Y Bb ` M/QK p `B #H2 rBi? }MBi2 p `B M+2- M/ ≤cx /2MQi2b i?2 +QMp2t biQ+? biB+ Q`/2`BM;X h?2M limn→∞ p `(Xn ) = 0X
RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_
9NN
n G2i Xn := (1/n) 0 X 1 (t)dt−E(X01 )X "v "B`F?QzǶb 2`;Q/B+ i?2Q`2K- Xn +QMp2`;2b XbX- M/ i?mb 7Q`iBQ`B BM T`Q# #BHBiv- iQ y b n → ∞X HbQ- i FBM; Y := X 1 (0) − E(X 1 (0))- Bi ?QH/b i? i Xn ≤cx Y - M/ Y ? b }MBi2 p `B M+2X G2KK RkX8XRy i?mb TTHB2b- M/ ?2M+2 limn→∞ p `(Xn ) = 0X "v i?2 bT2+i` H 7Q`KmH - QM2 ? b & & iωn & e − 1 &2 & μ(dω) . & p `(Xn ) = & iωn & 6`QK i?Bb Bi 7QHHQrb i? i limn→∞ p `(Xn ) = μ({0}) M/ i?2`27Q`2 M2+2bb `BHv μ({0}) = 0X S`QQ7X amTTQb2 i? i p `(Xn ) /Q2b MQi +QMp2`;2 iQ yc #v +QMbB/2`BM; bm#b2[m2M+2 bbmK2 rBi?Qmi HQbb Q7 ;2M2` HBiv i? i p `(Xn ) Bb #QmM/2/ 7`QK #2HQr #v σ 2 > 0X lbBM; +QMp2`;2M+2 BM T`Q# #BHBiv Q7 Xn iQ y- 7Q` HH δ > 0- i?2`2 2tBbib M n(δ) bm+? i? i P Xn2 > 13 σ 2 ≤ δ, n ≥ n(δ) . ∞ q`Bi2 p `(Xn ) b 0 P (Xn2 > t)dt- M/ bTHBi i?Bb BMi2;` H Qp2` i?`22 BMi2`p Hb [0, σ 2 /3)- [ 13 σ 2 , 13 σ 2 (1 + δ −1 ))- [ 13 σ 2 (1 + δ −1 ), +∞)X "QmM/BM; P (Xn2 > t) 7`QK #Qp2 #v R QM i?2 }`bi BMi2`p H- M/ #v P (Xn2 > σ 2 /3) QM i?2 b2+QM/- QM2 Q#i BMb, ∞ P (Xn2 > t)dt ≥ σ 2 − 13 σ 2 − 13 σ 2 = 13 σ 2 , n ≥ n(δ) . (σ 2 /3)(1+δ −1 )
h?Bb BM im`M Bb 2[mBp H2Mi iQ + E Xn2 − 13 σ 2 (1 + δ −1 ) ≥ 13 σ 2 ,
n ≥ n(δ) .
>Qr2p2`- #v bbmKTiBQM- i?2 H27i@? M/ bB/2 Bb #QmM/2/ #Qp2 #v 1 E(Y 2 − σ 2 (1 + δ −1 ))+ , 3 bBM+2 i?2 7mM+iBQM x → (x2 − t)+ Bb +QMp2t 7Q` HH tX aBM+2 δ + M #2 i F2M `#Bi` `BHv bK HH- i?Bb BKTHB2b i? i limt→∞ E(Y 2 − t)+ ≥ 13 σ 2 > 0- r?B+? Bb MQi +QKT iB#H2 rBi? i?2 bbmKTiBQM i? i o `(Y ) Bb }MBi2- ?2M+2 +QMi` /B+iBQMX h?2 7QHHQrBM; `2bmHiR3 Bb Q#i BM2/ b +Q`QHH `v iQ G2KK RkX8XNX Ai +QKTH2@ K2Mib h?2Q`2K RkX8X3- r?B+? H27i QT2M i?2 [m2biBQM Q7 2tBbi2M+2 Q7 i?2 /Bbi`B#miBQMb πλ X h?2Q`2K RkX8XRR lM/2` i?2 bbmKTiBQMb Q7 G2KK RkX8XN- i?2`2 2tBbib M 2`@ ;Q/B+ /Bbi`B#miBQM πλ 7Q` 2p2`v λ > 0X HbQ- 7Q`KmH URkX8NV ?QH/b rBi? 2[m HBivBX2X 7Q` HH g ∈ L1C (R) ∩ L2C (R)- M/ 2p2`v bi iBQM `v > rF2b T`Q+2bb N bm+? i? i Λ := limt→∞ N [−t, 0]/t XbX }MBi2
p ` R3
g(x)N (dx) = p `(Λ)
("`ûK m/ M/ J bbQmHBû- kyyR)X
2 g(x)dx
+
E(Λ) 2π
|g(ω)|2 |1 − h(ω)|2
dω .
URkXekV
8yy
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
S`QQ7X *QK#BMBM; i?2 HQr2` #QmM/ URkX8NV Q7 G2KK RkX8XN rBi? i?2 mTT2` #QmM/ URkX83V r2 ? / 7Q` Nλ 2M #H2b mb iQ /2/m+2- 7Q` g ∈ C0K - i? i p `(Λ) = 0- r?2`2 Λ := limt→∞ Nλ ([−t, 0]/tX h?mb #v h?2Q`2K RkX8X3 U V- Nλ Kmbi #2 /Bbi`B#mi2/ b πλ - r?B+? i?mb 2tBbib UMQi2 i? i bBM+2 Mv /?2`2Mi /Bbi`B#miBQM iQ i?2 b2[m2M+2 {N ε } ? b iQ #2 /Bbi`B#mi2/ b πλ - {N ε } Bb MQi QMHv iB;?i #mi HbQ r2 FHv +QMp2`@ ;2MiVX h?Bb +QK#BM iBQM Q7 HQr2` M/ mTT2` #QmM/b HbQ b?Qrb i? i p `
g(x)Nλ (dx) =
λ 2π
|g(ω)|2 |1 − h(ω)|2
dω .
0 )c i?2 ;2M2` H + b2 h?mb URkXekV ?QH/b r?2M N Bb /Bbi`B#mi2/ b πλ ( M/ 7Q` g ∈ CK 0 7QHHQrb 7`QK i?2 +QM/BiBQM H p `B M+2 7Q`KmH U M/ #v /2MbBiv Q7 CK VX
RkXe E2`bi M SQBMi S`Q+2bb2b h?2 bBKTH2bi 2t KTH2 Bb TQBMi T`Q+2bb N QM R rBi? i?2 FtN @BMi2MbBiv λ(t) := ϕ h(t − s)N (ds) ,
URkXejV
(−∞,t)
r?2`2 ϕ : R → R Bb MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM M/ h : R → R Bb K2 bm` #H2 7mM+iBQM + HH2/ U #mbBp2Hv #mi +QMp2MB2MiHvV i?2 72`iBHBiv ` i2- bm+? i? i t < 0 ⇒ h(t) = 0 M/
R+
|h(t)|dt < ∞ .
am+? TQBMi T`Q+2bb Bb bQK2iBK2b + HH2/ MQM@HBM2 ` > rF2b T`Q+2bbXRN aiQ+? biB+ .Bz2`2MiB H avbi2K oB BK#2//BM;- MQM@HBM2 ` > rF2b T`Q+2bb + M #2 pB2r2/ b i?2 bQHmiBQM Q7 biQ+? biB+ /Bz2`2MiB H 2[m iBQM /`Bp2M #v SQBbbQM K2 bm`2X AM 7 +i- #v i?2 BMp2`b2 BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXeV- bm+? TQBMi T`Q+2bb + M #2 `2T`2b2Mi2/ b N (dt) = 1(0,ϕ(X(t))] (z)N (dt × dz) , R+ h(t − s)N (ds) , URkXe9V X(t) = (−∞,t)
r?2`2 N Bb ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R × R+ rBi? BMi2MbBiv RX h?Bb ivT2 Q7 `2T`2b2Mi iBQM rBHH TH v +`m+B H `QH2 BM i?2 7Q`i?+QKBM; M HvbBbX JQ`2 ;2M2` H KQ/2Hb Q7 bBKBH ` M im`2 rBHH 2p2Mim HHv #2 +QMbB/2`2/- bi `i@ BM; rBi? 1t KTH2 RkXeXk- rBi? Ft @BMi2MbBiB2b 7Q` ?BbiQ`B2b H `;2` i? M i?2 BMi2`M H ?BbiQ`vX RN q?2M ϕ Bb #QmM/2/- bm+? T`Q+2bb2b `2 BM 7 +i bT2+B H + b2b Q7 i?Qb2 +QMbB/2`2/ BM i?2 b2KBM H `iB+H2 (E2`bi M- RNe9)X
RkXeX E1_ahL SPALh S_P*1aa1a
8yR
1t KTH2 RkXeXR, M2m`QM 7B`BM; KQ/2HX >2`2 t λ(t) = λ0 exp − h(t − s) N (ds) , −∞
r?2`2 h : R+ → R+ U TQbBiBp2 7mM+iBQM QM (0, +∞)- MmHH QM (−∞, 0]V r?Qb2 bmT@ TQ`i Bb BM+Hm/2/ BM [0, a] 7Q` bQK2 TQbBiBp2 }MBi2 aX h?Bb }ib i?2 ;2M2` H 7` K2rQ`F Q7 i?2 T`2pBQmb 2t KTH2X >2`2 i?2 `2;2M2` iBQM TQBMib `2 i?2 TQBMib t Q7 M ?TT N0 Q7 BMi2MbBiv λ0 bm+? i? i N0 ([t − a, t)) = 0X h?Bb KQ/2H Bb Q7 BMi2`2bi BM i?2 M2m`Qb+B2M+2b #2+ mb2 Q7 Bib TiBim/2 iQ KQ/2H `27` +iQ`v T2`BQ/b BM M2m`QM H bTBF2 +iBpBivX 6Q` BMbi M+2- B7 h(t) = α1 1[0,1) (t) + α2 1[1,2) (t) + α3 1[2,3) (t) + α4 1[3,4) (t) r2 ? p2 λ(t) =λ0 exp{−α1 N ((t, t − 1]) + α2 N ((t − 1, t − 2]) + α3 N ((t − 2, t − 3]) + α4 N ((t − 3, t − 4])} , r?2`2 iBK2 Bb ?2`2 K2 bm`2/ BM bm{+B2MiHv bK HH mMBib +QKT iB#H2 rBi? M2m`Q#BQ@ HQ;B+ H TTHB+ iBQMbX h F2 7Q` BMbi M+2 λ0 = 10 , α1 = 100 , α2 = 2 , α3 = 0.5 , α4 = 0.1. A7 i iBK2 t- i?2`2 Q++m``2/ bTBF2 BM i?2 T`2pBQmb QM2 iBK2 mMBi BMi2`p H- i?2 M2m`QM Bb BM?B#Bi2/ BM i? i Bib }`BM; BMi2MbBiv Bb i?2M H2bb i? i 10e−100 - r?B+? Bb M2;HB;B#Hv bK HHX aTBF2b Q++m``BM; BM i?2 iBK2 BMi2`p H (t − 4, t − 1] BM i?2 #b2M+2 Q7 bTBF2 BM i?2 BMi2`p H (t − 1, t] HbQ ? p2 M BM?B#BiQ`v 2z2+i- #mi MQi bQ ` /B+ HX h?2 bi`B+i BM?B#BiQ`v T2`BQ/ i?2`27Q`2 ? b H2M;i? 1- r?2`2 b i?2 `2H iBp2 BM?B#BiQ`v T2`BQ/ ? b H2M;i? 3X h?2 # bB+ KQ/2H URkXejV rBHH #2 2M`B+?2/X 6Q` BMbi M+2- QM2 K v rBb? iQ +QMbB/2` KmHiBp `B i2 MQM@HBM2 ` > rF2b T`Q+2bb2bX 1t KTH2 RkXeXk, M2m` H M2irQ`F KQ/2HX >2`2 N := (Ni ; 1 ≤ i ≤ M ) Bb KmHiBp `B i2 TQBMi T`Q+2bbX h?2 BMi2MbBiv Q7 Ni Bb Q7 i?2 7Q`K λi (t) =
M
cji (t) exp −
j=1
t −∞
hii (t − s) Ni (ds)
t
cji (t) = αji (t)ϕji −∞
,
hji (t − s) Nj (ds)
.
h?2 hii 7mM+iBQMb `2 MQM@M2; iBp2 BMi2;` #H2 M/ Q7 #QmM/2/ bmTTQ`i- i?2 hij 7mM+iBQMb `2 7Q` i = j BMi2;` #H2 M/ Q7 #QmM/2/ bmTTQ`i Q7 `#Bi` `v bB;MX h?2 7mM+iBQMb ϕij `2 #QmM/2/ MQM@M2; iBp2 MQM@/2+`2 bBM; 7mM+iBQMbX h?2B` +?QB+2 M/ i?2 +?QB+2 Q7 i?2 bB;Mb Q7 i?2 hij b /2i2`KBM2 i?2 bi`2M;i? M/ 2t+Bi iQ`vfBM?B#BiQ`v bi imb Q7 i?2 bvM Tb2 i → jX h?2 biQ+? biB+ T`Q+2bb2b {αij }t≥0 - 1 ≤ i, j ≤ M - `2
8yk
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
MQM@M2; iBp2 M/ #QmM/2/- iQ r?B+? + M #2 bbB;M2/ p `BQmb `QH2b- BM T `iB+mH ` i?2 BM~m2M+2 Q7 2ti2`M H biBKmHBX PM2 K v HbQ rBb? iQ +QMbB/2` MQM@HBM2 ` > rF2b T`Q+2bb2b rBi? ` M/QK 72`iBHBiv ` i2X 1t KTH2 RkXeXj, /2H v bvbi2KX i 2p2Mi@iBK2 Tn Q7 TQBMi T`Q+2bb N QM R +mbiQK2` b?Qrb mT i b2`pB+2 7 +BHBiv `2[mB`BM; íb2`pB+2 iBK2í σn X h?2 b2[m2M+2 {σn }n∈Z Bb BB/ M/ BM/2T2M/2Mi Q7 N X +mbiQK2` ``BpBM; i iBK2 Tn /2T `ib 7`QK i?2 bvbi2K i iBK2 Tn + σn X h?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t Bb 1{Tn t X(t) = n∈Z
Q`- 2[mBp H2MiHvX(t) =
1{Tn 2`2 & & & 1 , . . . , N M ) − ϕi (N 1 , . . . , N M )&& &ϕi (N & &M M 0 & & 0 & j (ds × dz) − j (ds × dz)&& hji (−s, z)N hji (−s, z)N =& & & −∞ K −∞ K j=1
≤
M 0 j=1
j=1
−∞
& & & & hji (−s, z) &N j − N j & (ds × dz) , K
M/ i?2`27Q`2 E2`bi MǶb +QM/BiBQM Bb b iBb}2/X *QM/BiBQM ρ(A) < 1 ;m ` Mi22b i?2 2tBbi2M+2 Q7 bi iBQM `v KmHiB@ivT2 > rF2b T`Q+2bb rBi? }MBi2 p2` ;2 BMi2MbBiB2bX
1t KTH2 RkXeXe, h?2 hvTB+ H 1t KTH2X h?2 T`2pBQmb 2t KTH2b M/ BM 7 +i i?2 K DQ`Biv Q7 2t KTH2b +QM+2`M TQBMi T`Q+2bb2b N rBi? biQ+? biB+ BMi2MbBiv Q7 i?2 7Q`K UmMBp `B i2 + b2- 7Q` bBKTHB+BivV
(ds × dz h(t − s, z) N
λ(t) = ϕ (−∞,t)
,
K
r?2`2 ϕ Bb 1@GBTb+?BixX E2`bi MǶb +QM/BiBQM Bb i?2M b iBb}2/ bBM+2 ky
(E2`bi M- RNe9)X
8y9
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
& & &ϕ &
& & h(t − s, z) μ (ds × dz) − ϕ h(t − s, z) μ (ds × dz) && (−∞,t) K (−∞,t) K & & & & & h(t − s, z) μ (ds × dz) − h(t − s, z) μ (ds × dz)&& ≤& (−∞,t) K (−∞,t) K & & & & h(t − s, z) | μ−μ | (ds × dz)&& = && (−∞,t) K |h(t − s, z)| | μ−μ | (ds × dz) . ≤
(−∞,t)
K
HBiiH2 i?Qm;?i rBHH +QMpBM+2 i?2 `2 /2` i? i ϕ B7 α@GBTb+?Bix 7Q` `#Bi` `v TQbBiBp2 α- E2`bi MǶb +QM/BiBQM rQmH/ biBHH #2 b iBb}2/ UDmbi K ii2` Q7 MQi iBQMVX h?2 7QHHQrBM; [m2biBQMb `Bb2 M im` HHv, M ) rBi? i?2 #Qp2 1 , . . . , N X .Q2b i?2`2 2tBbi DQBMiHv bi iBQM `v 7 KBHv (N /vM KB+b\ "X Ab Bi mMB[m2\ *X Ab Bi bvKTiQiB+ HHv bi #H2\ BM r? i b2Mb2\ mM/2` r? i BMBiB H +QM/BiBQMb\ bvKTiQiB+ bi #BHBiv Bb /2}M2/ BM i?2 7QHHQrBM; r vX 6B`bi r2 /2}M2 i` MbB2Mi bQHmiBQMX M ) rBi? i?2 /v@ 1 , . . . , N .2}MBiBQM RkXeXd i` MbB2Mi bQHmiBQM Bb Mv 7 KBHv (N M KB+b UDV QM (0, ∞)X M BMBiB H +QM/BiBQM Bb Mv T`QT2`iv (P) `2H iBp2 iQ i?2 1 , . . . , N M ) iQ (−∞, 0]X `2bi`B+iBQM Q7 (N 1 , . . . , N M QM (−∞, 0] M/ r2 i?2M h?2`27Q`2 r2 `2 ;Bp2M i?2 TQBMi T`Q+2bb2b N +QMbi`m+i N1 , . . . , NM QM (0, +∞] rBi? i?2 biQ+? biB+ Ft @F2`M2Hb /2b+`B#2/ #Qp2X h?2 /vM KB+b UDV `2 b B/ iQ #2 bvKTiQiB+ HHv bi #H2 mM/2` i?2 BMBiB H +QM@ /BiBQM (P) B7 i?2`2 2tBbib mMB[m2 bi iBQM `v bQHmiBQM M/ B7 i?2 +Q``2bTQM/BM; )+ Ui?2 `2bi`B+iBQM iQ (0, +∞) Q7 St N V +QM@ i` MbB2Mi bQHmiBQM N Bb bm+? i? i (St N p2`;2b b t → ∞ iQ i?2 bi iBQM `v /Bbi`B#miBQM QM (0, +∞)X "v +QMp2`;2M+2 r2 K2 M +QMp2`;2M+2 BM /Bbi`B#miBQM Q` +QMp2`;2M+2 BM p `B iBQM UiQ #2 K /2 T`2+Bb2 7Q` 2 +? `2bmHiVX "QmM/2/ 62`iBHBiv _ i2 rBi? "QmM/2/ amTTQ`i h?2 [m2biBQMb - " M/ * }M/ 2 bv M/ TQbBiBp2 Mbr2`b BM i?2 bT2+B H + b2 r?2`2 h : R → R ? b bmTTQ`i {t ∈ R; h(t) = 0} BM+Hm/2/ BM [0, a] 7Q` bQK2 a ∈ R+ - M/ r?2M KQ`2Qp2` ϕ Bb #QmM/2/ Ub v- #v Λ < ∞VX AM 7 +i- HQ+ H }MBi2M2bb Bb ;m ` Mi22/ ˜ bBM+2 7`QK URkXe9V- i?2 TQBMib Q7 N `2 M2+2bb `BHv TQBMib Q7 i?2 TQBMi T`Q+2bb N ˜ QM R /2}M2/ #v N (C) = N (C × [0, Λ])- r?B+? Bb BM/22/ HQ+ HHv #QmM/2/ bBKTH2 T`Q+2bbX h?2 2tBbi2M+2 Q7 bi iBQM `v TQBMi T`Q+2bb N rBi? i?2 /vM KB+b URkXe9V + M ˜ a #2 i?2 TQBMi T`Q+2bb 7Q`K2/ #v i?2 TQBMib T˜n Q7 N ˜ #2 b?QrM b 7QHHQrbX G2i N
RkXeX E1_ahL SPALh S_P*1aa1a
8y8
˜ a X h?2b2 `2 bm+? i? i T˜n − T˜n−1 > a- M/ + HH {T˜na }n∈Z i?2 b2[m2M+2 Q7 TQBMib Q7 N a ˜a ˜ dz`2;2M2` iBQM TQBMibǴ- #2+ mb2 7Q` HH t ∈ [Tn , Tn+1 ) λ(t) = ϕ h(t − s)N (ds) (T˜na ,t)
/Q2b MQi /2T2M/ QM N #27Q`2 T˜na X h?mb r2 ? p2 M 2tTHB+Bi 7Q`K 7Q` λ(t) 7Q` HH t i? i /Q2b MQi `2[mB`2 FMQrH2/;2 Q7 i?2 r?QH2 T bi Q7 N X lMB[m2M2bb Q7 i?2 bi iBQM `v bQHmiBQM Bb +H2 ` 7`QK i?Bb `;mK2MiX AM/22/- H2i N #2 bi iBQM `v bQHmiBQM- M/ +QMbi`m+i 7`QK Bi N b BM i?2 BMp2`bBQM i?2Q`2K- M/ MQr +QMbi`m+i N b #Qp2mbBM; N X h?2 BMi2MbBiv Q7 N Bb HbQ Q7 i?2 7Q`K h(t − s)N (ds) λ (t) = ϕ (T˜nα ,t)
a a 7Q` t ∈ [T˜na , T˜n+1 )X Ai Bb i?2M +H2 ` i? i N = N QM HH (T˜na , T˜n+1 ]- n ∈ Z- i? i Bb N ≡NX
:Bp2M Mv BMBiB H +QM/BiBQM U TQBMi T`Q+2bb QM (−∞, 0]V- i` MbB2Mi bQHmiBQM N rBHH +QMp2`;2 BM p `B iBQM iQ i?2 bi iBQM `v QM2X AM/22/- H2i N #2 i?2 i` MbB2Mi bQHmiBQM bQ +?Qb2M- M/ +QMbi`m+i 7`QK N i?2 SQBbbQM T`Q+2bb N rBi? i?2 p2` ;2 BMi2MbBiv R QM R+ × R+ - b BM i?2 BMp2`bBQM i?2Q`2KX 6`QK N QM R × R+ - +QMbi`m+i i?2 bi iBQM `v bQHmiBQM N X h?2M N M/ N +QmTH2 i iBK2 T˜1a X G2i mb #2 KQ`2 bT2+B}+ #Qmi i?2 bBKmH iBQMX h?2 ;Q H Bb iQ bBKmH i2 QM i?2 TQbBiBp2 ? H7@HBM2 bi iBQM `v TQBMi T`Q+2bb rBi? i?2 `2[mB`2/ /vM KB+bX 6Q` i?Bb QM2 }`bi +QMbi`m+ib QM i?2 M2; iBp2 ? H7@HBM2 SQBbbQM T`Q+2bb rBi? BMi2MbBiv Λ, t(0) < 0- t(−i) < 0- XXX- mMiBH i?2 }`bi TQBMi t(−L) < 0 bm+? i? i t(−L)−t(−L−1) > aX ai `i i?2 +QMbi`m+iBQM 7`QK t(−L) B;MQ`BM; i?2 TQBMib #27Q`2 bBM+2 i?2v /Q MQi z2+i i?2 b?Q`i ` M;2 biQ+? biB+ BMi2MbBivX PM2 i?2`27Q`2 ? b b2[m2M+2 t(−L)XXX- t(0)- iQ r?B+? QM2 //b i?2 TQBMib t(1) > 0- t(2) > 0 Q7 SQBbbQM BMi2MbBiv 1 QM i?2 TQbBiBp2 ? H7@HBM2X PM2 Kmbi MQr b2H2+i i?2 TQBMib Q7 i?2 #Qp2 b2[m2M+2 i? i rBHH #2 `2i BM2/ b TQBMib Q7 i?2 TQBMi T`Q+2bb Q7 BMi2`2biX amTTQb2 i? i i?Bb b2H2+iBQM ? b #22M T2`7Q`K2/ 7Q` i?2 TQBMib t(−L)- XXX- t(n)X PM2 i?2M FMQrb i?2 TQBMi T`Q+2bb rBi? i?2 /2bB`2/ /vM KB+b #27Q`2 USh1_ RkX >qE1a SPALh S_P*1aa1a
r?2`2 a > 0X Uh?Bb [m MiBiv /2T2M/b QMHv QM i?2 `2bi`B+iBQM N− Q7 N iQ (−∞, 0]XV 6Q` 7mim`2 `272`2M+2- r2 ;Bp2 H #2Hb iQ i?`22 T `iB+mH ` BMBiB H +QM/BiBQMb, − ) < ∞ M/ lim εa (t, N− ) = 0 , sup εa (t, N
(Pd,1 )
− ) < ∞ M/ lim E [εa (t, N− )] = 0 , sup E εa (t, N
(Pd,2 )
t↑∞
t≥0
t↑∞
t≥0
M/
M i,j=1
∞ 0
j (ds × dz) hji (t − s, z)N
dt < ∞ .
(Pv )
(−∞,0]×K
h?2 K BM `2bmHi Q7 i?Bb b2+iBQM Bb, h?2Q`2K RkXeXN UkR V A7 ρ(A) < 1- i?2`2 2tBbib mMB[m2 bi iBQM `v KmHiBp `B i2 1 , . . . , N M ) rBi? i?2 /vM KB+b UDV b iBb7vBM; E2`bi MǶb K `F2/ TQBMi T`Q+2bb (N +QM/BiBQM URkXe8V M/ rBi? }MBi2 p2` ;2 BMi2MbBiB2b (λ1 , · · · , λM )X JQ`2Qp2`- mM/2` 2Bi?2` QM2 Q7 +QM/BiBQMb UPd,1 V Q` UPd,2 V- i?2 /vM KB+b `2 bvKTiQiB+ HHv bi #H2 BM /Bbi`B#miBQMX A7 KQ`2Qp2` 7Q` HH i, j- (1 ≤ i, j ≤ M ) ∞ t hji (t, z) Qj (dz) dt , URkXedV 0
K
mM/2` i?2 BMBiB H +QM/BiBQM UPv V- i?2 /vM KB+b `2 bvKTiQiB+ HHv bi #H2 BM p `B @ iBQMX #2 i?2 TQBMi T`Q+2bb QM R×K 1t KTH2 RkXeXRy, h?2 mMBp `B i2 + b2X G2i N rBi? biQ+? biB+ Ft @F2`M2H (ds × dz) Q(dz) , h(t − s, z)N λ(t, dz) = ϕ (−∞,t)
K
r?2`2 ϕ Bb α@GBTb+?Bix M/ ∞h : R × K → R Bb 7mM+iBQM bm+? i? i h(t, z) > 0 BKTHB2b i? i t ≥ 0- M/ 0 K |h(t, z)| dt Q(dz) < ∞X h?2 K i`Bt A `2/m+2b iQ ∞ i?2 b+ H ` α 0 K |h(t, z)| dt Q(dz)X h?2`27Q`2 B7 i?Bb [m MiBiv Bb bi`B+iHv bK HH2` i? M 1- i?2`2 2tBbib mMB[m2 bi iBQM `v /Bbi`B#miBQM rBi? i?2 ;Bp2M /vM KB+bX h?2 BMBiB H +QM/BiBQMb `2 bQK2iBK2b ;Bp2M BM /Bz2`2Mi r v- MQi BMpQHpBM; i?2 T`2?BbiQ`v Q7 i?2 TQBMi T`Q+2bbX q2 K v HQQF i i?2 7QHHQrBM; dzi` MbB2MiǴ /vM KB+b (ds × dz) Q(dz) . λ(t, dz) = ϕ ν(t) + h(t − s, z)N (0,t)
K
QM (−∞, 0] h?Bb + M #2 `2/m+2/ iQ i?2 bi M/ `/ + b2 /2b+`B#2/ #Qp2- #v ;BpBM; N b 7QHHQrbX h?2`2 Bb QMHv QM2 TQBMi i T0 ≡ 0 M/ h(t, Z0 ) = ν(t) UBM 7 +i- i?2`2 − ) = t |ν(s)| ds M/ i?2`27Q`2 +QM/BiBQMb Bb MQ K `F i T0 VX AM i?Bb + b2- εa (t, N t−a UPd,1 V M/ UPd,2 V `2 i?2 b K2 M/ `2 / kR
(J bbQmHBû- RNN8 M/ RNN3)- ("`ûK m/ M/ J bbQmHBû- RNNe)X
RkXeX E1_ahL SPALh S_P*1aa1a
t
t
|ν(s)| ds < ∞ M/ lim
sup t≥0
8yd
t↑∞
t−a
r?2`2 b +QM/BiBQM UPv V Bb
∞
|ν(s)| ds = 0 , t−a
|ν(s)| ds < ∞ ,
0
M/ URkXedV Bb
∞
t|h(t, z)| Q(dz)
0
dt < ∞ .
K
q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K RkXeXNX S`QQ7X ai2T R, 1tBbi2M+2 Q7 bi iBQM `v bQHmiBQMX G2i N i U1 ≤ i ≤ M V #2 BM@ /2T2M/2Mi SQBbbQM T`Q+2bb2b QM R × K × R- Q7 `2bT2+iBp2 BMi2MbBiv K2 bm`2b dt×Qi (dz)×du U1 ≤ i ≤ M VX G2i {Ft }t∈R #2 i?2 BMi2`M H ?BbiQ`v Q7 (N 1 , . . . , N M )X Ai K v #2 bbmK2/ i? i i?2b2 TQBMi T`Q+2bb2b `2 /2}M2/ QM T`Q# #BHBiv bT +2 (Ω, F, P ) 2M/Qr2/ rBi? K2 bm` #H2 ~Qr {θt }t∈R - i? i (P, θt ) Bb 2`;Q/B+ M/ N i U1 ≤ i ≤ M V Bb θt @+QKT iB#H2X 6Q` i?Bb- QM2 K v +?QQb2 iQ rQ`F QM bQK2 + MQMB+ H bT +2 Q7 TQBMi T`Q+2bb2bX .m2 iQ i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b(P, θt ) rBHH MQi Dmbi #2 2`;Q/B+- #mi HbQ KBtBM;X (n) U1 ≤ i ≤ M V QM R × K× M/ *QMbi`m+i `2+m`bBp2Hv TQBMi T`Q+2bb2b N i (n) MQM@M2; iBp2 biQ+? biB+ T`Q+2bb2b {λi (t)}t∈R U1 ≤ i ≤ M V- n ≥ 0V #v (n) (n) (dt × dz) = N (n) N i (dt × dz × [0, λi (t)]) i
M/ (n+1)
λi
1(n) , . . . , St N (n) , (t) = ϕi St N M
URkXe3V
URkXeNV
(0)
bi `iBM; rBi? λi (t) ≡ 0 U1 ≤ i ≤ M VX "v E2`bi MǶb +QM/BiBQM & & & (n+1) & (n+1) (0)& − λi (0) E &λi
M & & & (n) (n−1) && (dt × dz) . E hji (−t, z) &N − N ≤ i i i=1
(−∞,0)
K
& & & (n) (n−1) && +QmMib i?2 TQBMib (t, z, u) Q7 N i bm+? i? i u 7 HHb h?2 TQBMi T`Q+2bb &N −N i i (n)
(n−1)
#2ir22M λi (t) M/ λi (t)X "v `2+m``2M+2- Bi Bb 2 bBHv b?QrM i? i i?2 T`Q+2bb2b (n) {λi (t)}t∈R U1 ≤ i ≤ M V `2 Ft @T`2/B+i #H2X "v h?2Q`2K 8XdXj- i?2 F2`M2H & & & & (n) (n−1) (t)& Qi (dz) &λi (t) − λi & & & (n) (n−1) &&X AM T `iB+mH `- #v i?2 bKQQi?@ Bb biQ+? biB+ Ft @BMi2MbBiv F2`M2H Q7 &N − N i i BM; i?2Q`2K Uh?2Q`2K 8XRXkyV-
8y3
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a & & & (n+1) & (n) E &λi (0)& − λi (0)
M & & & (n) & (n−1) E hji (−t, z) &λi (t) − λi (t)& dt × Qi (dz) . ≤ (−∞,0)
i=1
K
h?Bb `2+m`bBp2 +QMbi`m+iBQM Bb θt @+QKT iB#H2 M/ (P, θt ) Bb 2`;Q/B+ UKBtBM;V- i?2 (n) biQ+? biB+ T`Q+2bb2b {λi (t)}t∈R U1 ≤ i ≤ M V- n ≥ 0V `2 DQBMiHv bi iBQM `v- M/ i?2`27Q`2 & & & (n+1) & (n) E &λi (0)& − λi (0) M & & & (n) & (n−1) E &λi (0) − λi (0)& hji (−t, z) dt × Qi (dz) , ≤ (−∞,0)
i=1
i? i Bb-
K
& & & & & (n+1) & & (n) & (n) (n−1) E &λi (0) − λi (0)& ≤ AE &λi (0) − λi (0)& .
6Q` HH k ≥ 0- i?2 MQ`K Q7 An Bb 2[mBp H2Mi iQ nk ρ(A)n b n ↑ ∞X aBM+2 #v ?vTQi?2bBb ρ(A) < 1- Bi 7QHHQrb i? i & && (n) & (n−1) E &λi (0) − λi (0)& < ∞ (1 ≤ i ≤ M ) n≥0
Q`- 2[mBp H2MiHv #v bi iBQM `Biv& && (n) & (n−1) E &λi (t) − λi (t)& < ∞
(t ∈ R) .
n≥0 (n)
h?Bb BKTHB2b i? i 7Q` HH t ∈ R- λi (t) +QMp2`;2b HKQbi bm`2Hv M/ BM L1 b n → ∞X JQ`2Qp2`- 7Q` HH #QmM/2/ C ∈ B(R) & && (n) (n−1) & P − N N (dt × dz) = 0 & i & i C
n≥0
≤
K
E
n≥0
= (C)
&
& & (n) (n−1) && (dt × dz) &Ni − N i C
K
& & & (n) & (n−1) E &λi (0) − λi (0)& < ∞ .
n≥0
"v i?2 "Q`2HĜ* Mi2HHB H2KK & & & (n) (n−1) && (dt × dz) = 0 BXQX = 0 . P &Ni − N i C
K
(n) (C × K) `2K BMb +QMbi Mi 7i2` AM T `iB+mH `- 7Q` HH #QmM/2/ C ∈ B(R)- N i i }MBi2 ` M/QK iBK2X h?Bb K2 Mb i? i i?2`2 2tBbib HQ+ HHv }MBi2 TQBMi T`Q+2bb N (n) ( C × K) 7Q` HH #QmM/2/ C ∈ B(R)X (C × K) = N bm+? i? i limn↑∞ N i i i +QmMib i?2 TQBMib (t, z) bm+? i? i |N | ({t, z} × [0, λi (t)]) > 0X q2 b?Qr i? i N i AM 7 +i- #v 6 iQmǶb H2KK -
RkXeX E1_ahL SPALh S_P*1aa1a E
8yN
& &
& & &Ni (dt × dz) − N i (dt × dz × [0, λi (t)])& C K & &
& & (n) ≤ lim E &N i (dt × dz × [0, λi (t)]) − N i (dt × dz × [0, λi (t)])& n↑∞ C K & & & (n) & = (C) lim E &λi (0) − λi (0)& = 0 . n↑∞
i /KBib i?2 biQ+? biB+ Ft @BMi2MbBiv F2`M2H λi (t)Qi (dz)X Ai `2K BMb iQ h?2`27Q`2- N 1 , . . . , St N M , p2`B7v i? i λi (t) = ϕi St N & & & 1 , . . . , St N M && E &λi (t) − ϕi St N & & & & (n) ≤ E &λi (t) − λi (t)&
& & & (n−1) && (ds × dz) + E hji (t − s, z) &N j − Nj j
& & & & (n) = E &λi (t) − λi (t)& + E
(−∞,t)
∞
0
j
L
& & & & (n−1) hji (s, z) ds Qi (dz) E &λi (0) − λi (0)& . L
& & & 1 , . . . , St N M && = 0- i?mb +QM+Hm/BM; i?2 G2iiBM; n → ∞ vB2H/b E &λi (t) − ϕi St N T`QQ7X Bb θt @+QKT iB#H2- M/ i?2`27Q`2 2`;Q/B+X LQi2 i? i i?2 HBKBiBM; T`Q+2bb N ai2T kX q2 MQr b?Qr +QMp2`;2M+2 BM /Bbi`B#miBQM iQ i?2 #Qp2 bi iBQM `v bQHmiBQM }`bi rBi? BMBiB H +QM/BiBQM UPd,1 VX h?Bb rBHH ;m ` Mi22 Q7 i` MbB2Mi bQHmiBQM N mMB[m2M2bb Q7 bi iBQM `v /Bbi`B#miBQM rBi? }MBi2 p2` ;2 BMi2MbBiB2b b iBb7vBM; Bb bm+? bi iBQM `v bQHmiBQM rBi? }MBi2 BMi2MbBiB2b +QM/BiBQM UPd,1 V #2+ mb2 B7 N λ1 , . . . , λM t εa (t, N− ) = λj hji (s − u, z) du Qi (dz) ds i,j
=
λj
i,j
t
λj a
(−∞,0]
K
hji (u, z) du Qi (dz) ds t−a
i,j
≤
t−a
(s,∞)
K
hji (u, z) du Qi (dz) , (s,∞)
K
b iBb}2b +QM/BiBQM UPd,1 VX 7`QK r?B+? r2 b22 i? i N QM (−∞, 0] b iBb7vBM; +QM/BiBQM UPd,1 VX h?2M G2i #2 ;Bp2M TQBMi T`Q+2bb N QM (0, ∞) rBi? i?2 /2bB`2/ /vM KB+b b 7QHHQrbX G2i N i U1 ≤ i ≤ M V +QMbi`m+i N #2 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM (0, ∞)×K rBi? `2bT2+iBp2 BMi2MbBiv K2 bm`2b QM (−∞, 0]X *QMbi`m+i
× Qi U1 ≤ i ≤ M V M/ BM/2T2M/2Mi Q7 i?2 #Qp2 ;Bp2M N (n) `2+m`bBp2Hv i?2 TQBMi T`Q+2bb2b N U1 ≤ i ≤ M V QM R × K #v BKTQbBM; i? i i?2v i +QBM+B/2 rBi? Ni U1 ≤ i ≤ M V QM (−∞, 0] × K- M/ i? i QM (0, +∞)-
8Ry
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a (n) (n) (dt × dz) = N (n) (t)]) N i (dt × dz × [0, λi i
M/
(n+1)
λi
1(n) , . . . , St N (n) , (t) = ϕi St N M
URkXdyV
(0)
bi `iBM; rBi? λi (t) ≡ 0 U1 ≤ i ≤ M VX AM pB2r Q7 E2`bi MǶb +QM/BiBQM 0
T
& & & (n+1) & (n) E &λi (t) − λi (t)& | F0N dt
∞ hji (t, z) dt Qj (dz) ≤ j
0
T
& & & (n) & (n−1) E &λi (t) − λi (t)& | F0N dt .
0
K
URkXdRV E2`bi MǶb +QM/BiBQM HbQ ;m ` Mi22b i? i (1) hji (t − s, z)tNj (ds × dz) , λi (t) ≤ ϕi (∅) + (−∞,0]
j
K (1)
M BM2[m HBiv r?B+?- BM pB2r Q7 i?2 BMBiB H +QM/BiBQM UPd,1 V- BKTHB2b i? i λi (t) Bb (n) HQ+ HHv BMi2;` #H2X AM2[m HBiB2b URkXdRV i?2M b?Qr i? i λi (t) Bb HQ+ HHv BMi2;` #H2 7Q` HH n ≥ 1 U1 ≤ i ≤ M VX *QM/BiBQM ρ(A) < 1 M/ BM2[m HBiB2b URkXdRV ;m ` Mi22 (n) (n) /KBi `2bT2+iBp2 HBKBib λ (t) M/ N bm+? i? i 7Q` i? i i?2 λi (t)Ƕb M/ i?2 N i i i HH C ∈ B((0, ∞)) M/ HH L ∈ K (C × L) = N N i (dt × dz × [0, λi (t)]) . i C
L
)- M/ qBi? i?2 b K2 `;mK2Mib b BM ai2T R- QM2 b?Qrb i? i λi (t) = ϕi (St (N ? b i?2 `2[mB`2/ /vM KB+bX h?Bb BKTHB2b i? i i?2`27Q`2 i?2 HBKBiBM; T`Q+2bb N T N dt < ∞ M/ i?2`27Q`2 E λi (t) | FtN Bb HQ+ HHv BMi2;` #H2 Ui?Bb E λi (t) | Ft 0 rBHH #2 M22/2/ H i2` QMVX #2 i?2 bi iBQM `v bQHmiBQM +QMbi`m+i2/ 7`QK i?2 SQBbbQM T`Q+2bb N G2i MQr N b BM ai2T RX 6Q` HH i U1 ≤ i ≤ M V H2i fi (t) := E |λi (t) − λi (t)| | FtN 1(0,∞) (t) , )X h FBM; E2`bi MǶb +QM/BiBQM BMiQ ++QmMir?2`2 λi (t) = ϕi (St (N (ds × dz) hji (t − s, z)N fi (t) ≤ j j
+
(−∞,0)
j
+
K
∞
λj t
t j
0
hji (s, z) ds Qj (dz)
URkXdkV
hji (t − s, z)fj (s) ds Qj (dz) ,
URkXdjV
K
K
RkXeX E1_ahL SPALh S_P*1aa1a
8RR
r?2`2 λj Bb i?2 p2` ;2 BMi2MbBiv Q7 Nj X G2i a > 0 #2 }t2/ M/ H2i Fi (t) := t f (s) dsX AMi2;` iBM; i?2 T`2pBQmb BM2[m HBiv #2ir22M t − a M/ t- r2 Q#i BM t−a i Fi (t) ≤ εi (t) +
t j
0
hji (t − s, z)Fj (s) ds Qj (dz) ,
URkXd9V
K
r?2`2 εi (t) :=
j
t
t−a
(−∞,0)
(ds × dz) hji (u − s, z)N j K
+
j
du
∞
λj a
hji (s, z) ds Qj (dz) . t−a
h?2 fi Ƕb `2 HQ+ HHv BMi2;` #H2- #2+ mb2 r2 ? p2 b22M 72r HBM2b #Qp2 i? i N N E λi (t) | Ft Bb HQ+ HHv BMi2;` #H2- M/ bQ Bb E λi (t) | Ft #2+ mb2 E [λi (t)] = λi < ∞X h?2`27Q`2 i?2 7mM+iBQMb Fi M/ εi U1 ≤ i ≤ M V `2 #QmM/2/ QM }MBi2 BMi2`p HbX h?2`27Q`2- #v Bi2` iBM; BM2[m HBiv URkXd9V M/ T bbBM; iQ i?2 HBKBi Fi (t) ≤
n≥0
j
t (n)
εj (t − s)gij (s) ds ,
URkXd8V
0
(0)
r?2`2 gij (t) = 1{i=j} 1{t=0} M/ (n+1) gij (t)
=
t k
0
hki (t − s, z) ds Qk (dz) . K
h?2 `B;?i@? M/ bB/2 Q7 URkXd8V 72 im`2b i?2 KmHiBp `B i2 `2M2r H K2 bm`2 bbQ+B i2/ rBi? i?2 /Bbi`B#miBQM 7mM+iBQMb K hji (t, z) Qj (dz)X aBM+2 ρ(A) < 1- i?2 `B;?i@? M/ bB/2 Q7 URkXd8V Bb }MBi2X aBM+2 εj (t) Bb #QmM/2/ M/ i2M/b iQ 0 b t → +∞- #v /QKBM i2/ +QMp2`;2M+2 TTHB2/ iQ URkXd8V- limt↑∞ Fi (t) = 0X P#b2`p2 i? i
& & & & N Fi (t) = E &Ni − Ni & (ds × dz) | F0 t−a,t]
K
QM [t − a, t] × K | F N ) . i = N ≥ 1 − P (N i 0 +QMp2`;2 BM /Bbi`B#m@ h?Bb BKTHB2b i? i i?2 }MBi2@/BK2MbBQM H /Bbi`B#miBQMb Q7 St N +QMp2`;2b BM /Bbi`B#miBQM iBQM iQ i?Qb2 Q7 N - r?B+? Bb 2MQm;? iQ ;m ` Mi22 i? i St N iQ N Uh?2Q`2K RX9XdVX h?2 b K2 +QM+HmbBQM ?QH/b B7 i?2 i` MbB2Mi bQHmiBQM ? b i?2 BMBiB H +QM/BiBQM UPd,2 V BMbi2 / Q7 UPd,1 VX Ai bm{+2b iQ TTHv i?2 `;mK2Mib #Qp2 rBi? fi (t) M/ Fi (t) `2TH +2/ #v i?2B` 2tT2+i iBQMbX ai2T jX q2 MQr b?Qr +QMp2`;2M+2 BM p `B iBQM iQ i?2 bi iBQM `v bQHmiBQM Ur?Qb2 rBi? BMBiB H +QM/BiBQMb 2tBbi2M+2 r b T`Qp2/ BM ai2T RV Q7 i` MbB2Mi bQHmiBQM N UPv V M/ mM/2` +QM/BiBQM URkXedVX Ai bm{+2b iQ b?Qr i? i i?2 i` MbB2Mi bQHmiBQM M/
8Rk
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
i?2 bi iBQM `v bQHmiBQM +QmTH2 BM }MBi2 iBK2X 6Q` i?Bb- BMi2;` i2 BM2[m HBiv URkXdjV 7`QK 0 iQ T 7Q` }MBi2 TQbBiBp2 T iQ Q#i BM T ∞ T fi (t) dt ≤ εi + fi (t) dt × hji (t, z) dt Qj (dz) , 0
0
j
r?2`2 εi =
j
+
∞
0
j
(−∞,0]
∞
0
K
(ds × dz) hji (t − s, z) N j
dt
K
thji (t, z) dt Qj (dz) 0
K
T ++Q`/BM; iQ UPv V M/ URkXedV- 0 fi (t) dt M/ εi `2 }MBi2 7Q` HH T > 0X Ai2` iBM; i?2 H bi BM2[m HBiv vB2H/b- BM p2+iQ` 7Q`K T f (t) dt ≤ An ε . 0
n≥0
∞ h?2`27Q`2- bBM+2 i?2 #QmM/ Bb BM/2T2M/2Mi Q7 T - 0 fi (t) dt < ∞X "mi
∞ & & & & N fi (t) dt = E &Ni − Ni & (dt × dz) | F0 (0,+∞]
0
K
i M/ N i Bb }MBi2X a B/ Qi?@ M/ i?2`27Q`2 i?2 MmK#2` Q7 TQBMib MQi +QKKQM iQ N +QMp2`;2b iQ N 2`rBb2- Ni M/ Ni +QmTH2 BM }MBi2 iBK2- r?B+? BKTHB2b i? i St N BM p `B iBQMX _ i2 Q7 1tiBM+iBQM *QMbB/2` MQM@HBM2 ` > rF2b TQBMi T`Q+2bb N r?B+? Bb 2KTiv QM (−∞, 0]- rBi? /vM KB+b QM R+ λ(t) = ν(t) + ϕ h(t − s, z)NZ (ds × dz) , URkXdeV (0,t)×K
r?2`2 ν: R+ →[0, ∞)- ν = 0- Bb HQ+ HHv BMi2;` #H2 7mM+iBQM- ϕ: R+ → [0, ∞)ϕ(0) = 0- Bb GBTb+?Bix 7mM+iBQM rBi? GBTb+?Bix +QMbi Mi R- M/ h : R × L → R Bb K2 bm` #H2 7mM+iBQM UMQM@M2+2bb `BHv MQM@M2; iBp2V r?B+? b iBb}2b URkX9VX bbmKBM; ∞
E[|h(t, Z1 )|]dt < 1 , 0
i?2 mMB[m2 bi iBQM `v T`Q+2bb N (0) +Q``2bTQM/BM; iQ i?2 /vM KB+b (0) h(t − s, z)NZ (ds × dz) ϕ (0,t)×K
Bb i?2 2KTiv T`Q+2bbX A7 KQ`2Qp2` r2 bmTTQb2 URkXeV M/
URkXddV
RkXeX E1_ahL SPALh S_P*1aa1a
∞
tE[|h(t, Z1 )|]dt < ∞
8Rj URkXd3V
0
2 +? T`Q+2bb N rBi? /vM KB+b URkXdeV Bb bm+? i? i St N i2M/b BM p `B iBQM iQ i?2 2KTiv T`Q+2bbX >2`2 ; BM- +QMp2`;2M+2 BM p `B iBQM i F2b TH +2 pB +QmTHBM;- M/ +QM/BiBQMb URkXddV M/ URkXeV `2 bm{+B2Mi 7Q` i?2 BMi2;` #BHBiv Q7 E[λ(t)]X AM i?2 2tTQM2MiB H + b2- i? i Bb r?2M bQK2 β > 0 2tBbib bm+? i? i ∞ eβt E[|h(t, Z1 )|]dt = 1 , URkXdNV 0
r2 ? p2 i?2 7QHHQrBM; `2bmHi, h?2Q`2K RkXeXRR lM/2` bbmKTiBQMb (12.79) M/ URkXNV- M/ B7 i?2 7mM+iBQM eβt ν(t) Bb /B`2+iHv _B2K MM@BMi2;` #H2 QM R+ - i?2M 7Q` HH K bm+? i? i ∞ βt e ν(t)dt URkX3yV K > ∞ 0 βt β 0 te E[|h(t, Z1 )|]dt i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt . AM i?2 #Qp2 BM2[m HBiv (12.80) i?2 `B;?i@? M/ bB/2 Bb /2}M2/ b 0 B7 i?2 /2MQKBM iQ` Bb BM}MBivX S`QQ7X h?2 T`QQ7 Bb bBKBH ` iQ i? i Q7 h?2Q`2K RkXkXRX h?2 mMB[m2 /Bz2`2M+2 HB2b BM i? i- BM i?2 MQM@HBM2 ` + b2- r2 ? p2- #v i?2 GBTb+?Bix T`QT2`iv Q7 ϕ- i?2 /272+iBp2 `2M2r H BM2[m HBiv t λ(t) ≤ ν(t) + λ(t − s)E[|h(s, Z1 )|]ds , URkX3RV 0
r?B+? vB2H/b- #v Bi2` iBM; M/ i FBM; BMiQ ++QmMi i? i λ(t) Bb HQ+ HHv #QmM/2/ QM R+ λ(t) ≤ ξ(t) , r?2`2 ξ(t) b iBb}2b i?2 /272+iBp2 `2M2r H 2[m iBQM t ξ(t − s)E[|h(s, Z1 )|]ds. ξ(t) = ν(t) + 0
h?2 `2bmHi 7QHHQrb #v TTHvBM; i?2 `2bmHib Q7 am#b2+iBQM 9XkX
AM i?2 bm#2tTQM2MiB H + b2, h?2Q`2K RkXeXRk lM/2` bbmKTiBQMb URkXddV- URkXd3V M/ URkXNV- B7 KQ`2Qp2` i?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv E[|h(t, Z1 )|] g(t) = ∞ E[|h(t, Z1 )|]dt 0
8R9
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
Bb bm#2tTQM2MiB H M/ i?2 7mM+iBQM ν Bb #QmM/2/ M/ bm+? i? i B = lim sup t→∞
ν(t) < ∞, G(t)
i?2M 7Q` HH K bm+? i? i K>
1−
∞ 0
B E[|h(t, Z1 )|]dt
i?2`2 2tBbib t0 = t0 (K) bm+? i? i 7Q` HH t ≥ t0 ∞ P (T > t) ≤ K G(s)ds . t
h?2 T`QQ7 Bb M HQ;Qmb iQ i? i Q7 h?2Q`2K RkXkXjX Ai bm{+2b iQ MQi2 i? i- BM i?2 T`2b2Mi + b2- BM2[m HBiv URkX3RV biBHH ?QH/bX "QmM/2/ GBTb+?Bix .vM KB+b h?2Q`2K RkXeXRj Ukk V G2i ϕ #2 α@GBTb+?Bix 7Q` bQK2 α > 0 M/ #QmM/2/ #v Λ > 0- M/ H2i h #2 BMi2;` #H2 QM R+ M/ bm+? i? i ∞ t|h(t)| dt > ∞ . URkX3kV 0
h?2M i?2`2 2tBbib mMB[m2 bi iBQM `v /Bbi`B#miBQM Q7 N rBi? /vM KB+b t λ(t) = ϕ h(t − s) N (ds) .
URkX3jV
0
JQ`2Qp2`- i?2 /vM KB+b URkX3jV `2 bi #H2 BM p `B iBQM rBi? `2bT2+i iQ i?2 BMBiB H +QM/BiBQM t+s lim |h(s − u)| N (du) ds = 0 a.s. URkX39V t↑∞
t
(−∞,0]
S`QQ7X G2i N #2 M ?TT QM R × R Q7 BMi2MbBiv 1X G2i {Ft }t∈R #2 i?2 BMi2`M H ?BbiQ`v Q7 N - i? i BbFt = σ(N (C) ; C ⊆ (−∞, t] × R) . Ai K v #2 bbmK2/ i? i N Bb /2}M2/ QM T`Q# #BHBiv bT +2 (Ω, F, P ) 2M/Qr2/ rBi? K2 bm` #H2 ~Qr {θt }t∈R - i? i (P, θt ) Bb 2`;Q/B+ M/ N Bb θt @+QKT iB#H2X 6Q` i?Bb- QM2 K v +?QQb2 iQ rQ`F QM bQK2 + MQMB+ H bT +2 Q7 TQBMi T`Q+2bb2bX .m2 iQ i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b- (P, θt ) rBHH MQi Dmbi #2 2`;Q/B+#mi HbQ KBtBM;X ai2T R, 1tBbi2M+2 Q7 bi iBQM `v bQHmiBQMX *QMbi`m+i `2+m`bBp2Hv i?2 FtN @T`2/B+i #H2 T`Q+2bb2b {λn (t)}t∈R M/ i?2 TQBMi T`Q+2bb2b Nn ++Q`/BM; iQ kk
(E2`bi M- RNe9)X
RkXeX E1_ahL SPALh S_P*1aa1a
8R8
λn (t) (C ∈ B(R)) , ds × 0, Λ C λn+1 (t) = ϕ h(t − s) Nn (ds)
Nn (C) =
N
URkX38V URkX3eV
(−∞,t)
M/ λ0 (t) ≡ 0X amTTQb2 Ui?Bb rBHH #2 T`Qp2/ H i2`V i? i- 7Q` HH #QmM/2/ C ∈ B(R)i?2 T`Q+2bb2b Nn XbX `2K BM 2p2Mim HHv +QMbi Mi QM C b n BM+`2 b2b- M/ H2i N /2MQi2 i?2 HBKBiBM; T`Q+2bbX h?Bb T`Q+2bb- r?B+? Bb θt @+QKT iB#H2- Bb i?2`27Q`2 bi iBQM `v U M/ KBtBM;V- M/ rBHH ? p2 i?2 2tT2+i2/ /vM KB+b B7 Bi +QmMib i?2 TQBMib Q7 N #2HQr i?2 +m`p2 y(t) = λ(t) = ϕ (−∞,t) h(t − s) N (ds) X G2i C #2 bQK2 #QmM/2/ b2i BM RX "v 6 iQmǶb H2KK &
&
& & &N (ds) − N (ds × 0, λ(s) )& E & & Λ C
&
& & λn (s) λ(s) && & ≤ lim E &N (ds × 0, Λ ) − N (ds × 0, Λ )& n↑∞
C |λn (0) − λ(0)| . URkX3dV ds lim E n↑∞ Λ C "v i?2 GBTb+?Bix T`QT2`iv Q7 ϕ |λn (0) − λ(0)| ≤ α
|h(−s)| |Nn − N | (ds) . (−∞,0)
h?2 `B;?i@? M/ bB/2 i2M/b iQ0 b n → ∞- M/ i?2`27Q`2- #v /QKBM i2/ +QMp2`;2M+2 Ui?2 /QKBM iBM; p `B #H2 Bb (−∞,0) |h(−s)| N (ds × [0, Λ]V- i?2 }`bi i2`K BM URkX3dV Bb MmHH- bQ i? i N ? b i?2 `2[mB`2/ BMi2MbBivX h?2 b2[m2M+2 {Nn }n≥1 +QMp2`;2b B7 /2}M2/ #v i?2 TQBMi T`Q+2bb N ({t}) := lim sup Nn ({t}) − lim inf Nn ({t}) (t ∈ R) N n↑∞
n↑∞
Bb Bb 2KTivX P#b2`p2 i? i i?2 FtN @BMi2MbBiv Q7 N := lim sup λn (t) − lim inf λn (t) λ(t) n↑∞
n↑∞
(t ∈ R)
+QmMib i?2 TQBMib Q7 N #2ir22M i?2 T`2/B+i #H2 +m`p2b t → lim supn↑∞ λn (t) bBM+2 N b M/ t → lim inf n↑∞ λn (t)X q`BiBM; λ(t) lim sup (λi (t) − λj (t)) ,
n↑∞ i,j≥n
Bi 7QHHQrb 7`QK i?2 GBTb+?Bix bbmKTiBQM i? i ≤ α lim sup λ(t) |h(t − s)| Ni (ds) − inf n
αA − αB ,
i≥n
(−∞,t)
j≥n
|h(t − s)| Nj (ds) (−∞,t)
8Re
*>Sh1_ RkX >qE1a SPALh S_P*1aa1a
r?2`2- rBi? `#Bi` `v TQbBiBp2 a A = lim sup |h(t − s)| Ni (ds) − inf n
M/
i≥n
B = lim sup n
j≥n
(−∞,t−a)
i≥n
|h(t − s)| Nj (ds) (−∞,t−a)
|h(t − s)| Ni (ds) − inf
j≥n
[t−a,t)
|h(t − s)| Nj (ds)
.
[t−a,t)
aBM+2 i?2`2 `2 QMHv }MBi2 MmK#2` Q7 TQBMib Q7 N BMpQHp2/ BM B- r2 ? p2 (ds) . B≤ |h(t − s)| N [t−a,t)
b 7Q` A- r2 ? p2 i?2 BKK2/B i2 #QmM/ |h(t − s)| N (ds × [0, Λ]) . A≤ (−∞,t−a)
h?2`27Q`2- 7Q` `#Bi` `v a > 0 ≤α |h(t − s)| N (ds × [0, Λ]) − λ(t) (−∞,t−a)
(ds) |h(t − s)| N
.
[t−a,t)
G2iiBM; a → ∞ ≤α λ(t)
(ds) . |h(t − s)| N
URkX33V
(−∞,t)
rBHH #2 HKQbi bm`2Hv 2KTiv B7 P (N (0, +∞) = 0) = 1X aBM+2 h?2 TQBMi T`Q+2bb N {N (t, +∞) = 0} ⊆ {N (0, +∞) = 0}- #v 2`;Q/B+Biv Q7 (P, θt )- i?Bb rBHH ?QH/ b (0, +∞) = 0) > 0X 6Q` i?Bb Bi bm{+2b i? i rBi? TQbBiBp2 T`Q# #BHBiv bQQM b P (N (0, +∞) = 0 | F N ) > 0X h?2 F N BMi2MbBiv Q7 N - r?B+? Bb Q7 i?2 7Q`K v(t, N − )P (N t 0 | F N M/ i?2`27Q`2 b iBb}2b K DQ` iBQM URkX33VX Ai 7QHHQrb i? i 2[m Hb E λ(t) t
−) ≤ v(t, N
|h(t − s)| N (ds × [0, Λ]) . (−∞,0)
h?2`27Q`2
(0, +∞) | F N ) = 0) ≥ exp − P (N 0
∞
0
|h(t − s)| N (ds × [0, Λ]) ds
.
(−∞,0)
h?2 `;mK2Mi Q7 i?2 2tTQM2MiB H Bb }MBi2 BM pB2r Q7 bbmKTiBQM URkX3kV- M/ i?2`2@ (0, +∞) | F N ) = 0) > 0- r?B+? +QM+Hm/2b i?2 T`QQ7 Q7 2tBbi2M+2 Q7 7Q`2 P (N 0 bi iBQM `v bQHmiBQMX G2i N #2 TQBMi T`Q+2bb rBi? BMBiB H +QM/BiBQM URkX39VX lb2 h?2Q`2K 8XdX3 iQ +QMbi`m+i M ?TT N bm+? i? i N +QmMib- QM R+ - i?2 TQBMib Q7 N #2HQr i?2 +m`p2 h(t − s) N (ds) . t → λ (t) = ϕ (−∞,t]
RkXdX 1s1_*Aa1a
8Rd
*QMbi`m+i i?2 bi iBQM `v TQBMi T`Q+2bb N 7`QK N b BM i?2 }`bi T `i Q7 i?2 T`QQ7X G2i Ft := FtN ∧ FtN Ut ∈ RVX AM T `iB+mH `- {Ft }t∈R Bb ?BbiQ`v Q7 #Qi? N M/ N X 6Q` HH s ∈ R M/ HH t > 0- /2}M2 fs (t) =:= P (|N − N |(s, s + t] = 0 | Fs ) M/ `2+ HH i? i i?2 TQBMi T`Q+2bb |N − N | /KBib i?2 Ft @BMi2MbBiv |λ(t) − λ (t)| QM R+ X "v i?2 GBTb+?Bix bbmKTiBQM QM ϕ |h(t − s)| N (ds) + |h(t − s)| N (ds) |λ(t) − λ (t)| ≤ α (−∞,0] (−∞,0] |h(t − s)||N − N |(ds) . +α (0,t)
`;mBM; 2t +iHv b BM 1t KTH2 8XRXR9 vB2H/b i?2 KBMQ` iBQM ∞ gs (u) du , fs (∞) ≥ exp − s
r?2`2
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|h(u − v)| N (dv)
.
(−∞,0]
∞
|h(u − v)| N (dv) du
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gs (u) = α
(−∞,s]
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∞
gs (u) du
,
s
i?2 H bi BM2[m HBiv `2 /b P (N = N QM (s, +∞) | Fs ) ≥ Z(s) − ε(s) . h?2 +QM+HmbBQM U+QmTHBM; Q7 N M/ N V i?2M 7QHHQrb 7`QK h?2Q`2K RX9XR8X AM/22/ Z Bb 2`;Q/B+ bBM+2 N Bbc Bi b iBb}2b URX9XR8V #2+ mb2 i?2 `;mK2Mi Q7 i?2 2tTQM2MiB H /2}MBM; Z(s) Bb }MBi2- #v bbmKTiBQM URkX3kVc M/ }M HHv- BM pB2r Q7 bbmKTiBQM ∞ URkX39V- s (−∞,0] |h(u − v)| N (dv) du → 0 b → ∞X
RkXd
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1t2`+Bb2 RkXdXRX 1tiBM+iBQM *QMbB/2` i?2 > rF2b #` M+?BM; T`Q+2bb QM [0, ∞) rBi? QM2 M+2biQ` ∞ i i?2 Q`B;BM M/ rBi? 72`iBHBiv ` i2 7mM+iBQM h(t) = ce−αt rBi? c < α UbQ i? i 0 h(t) dt < 1VX TTHv h?2Q`2K RkXeXRR iQ }M/ β M/ M mTT2` #QmM/ 7Q` K bm+? i? i P (T > t) ≤ Ke−βt X
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1t2`+Bb2 RkXdXkX o ++BM iBQM M/ +QM7BM2K2Mi a K2 [m2biBQM b BM 1t2`+Bb2 RkXdXR BM i?2 + b2 Q7 p ++BM iBQM Q` +QM}M2K2Mi Ub22 1t KTH2 RkXkXkc 7Q` +QM}M2K2Mi- i F2 7Q` Z1 ` M/QK p `B #H2 mMB7Q`KHv /Bbi`B#mi2/ BM [0, T ] 7Q` bQK2 T > 0VX 1t2`+Bb2 RkXdXjX AKT`Qp2/ ` i2 Q7 2tiBM+iBQM lb2 i?2 #QmM/b ;Bp2M BM 1t KTH2 8XRXR8 iQ ;Bp2 mTT2` M/ HQr2` #QmM/b Q7 i?2 i BH /Bbi`B#miBQM Q7 i?2 iBK2 iQ 2tiBM+iBQM- i?mb BKT`QpBM; M/ +QKTH2K2MiBM; h?2Q`2K RkXkXRX 1t2`+Bb2 RkXdX9X AM7Q`K iBQM i` Mb72` #v M2m`QMb *QMbB/2` i?2 KQ/2H Q7 1t KTH2 RkXeXR- bHB;?iHv KQ/B}2/ t λ(t) = Λ exp − h(t − s) N (ds) , −∞
r?2`2 h : R+ → R+ U TQbBiBp2 7mM+iBQM QM (0, +∞- MmHH QM (−∞, 0]V r?Qb2 bmTTQ`i Bb BM+Hm/2/ BM [0, a] 7Q` bQK2 TQbBiBp2 }MBi2 a- M/ r?2`2 Bb TQbBiBp2 ` M/QK p `B #H2 rBi? +/7 F X h?Bb ` M/QK p `B #H2 K v #2 +QMbB/2`2/ i?2 BMi2MbBiv Q7 i?2 bB;M H TTHB2/ iQ i?2 M2m`QMX :Bp2 M 2tT`2bbBQM 7Q` U V E Λ | FtN - M/ U#V i?2 Kmim H BM7Q`K iBQM #2ir22M Λ M/ N X
TT2M/Bt XR
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.vMFBMǶb h?2Q`2K h?2 i2+?MB+ H `2bmHi FMQrM b .vMFBMǶb i?2Q`2K Qr2b Bib BKTQ`i M+2 iQ i?2 7 +i i? i σ@}2H/b `2 Q7i2M /2}M2/ BM i2`Kb Q7 i?2B` ;2M2` iQ`b- M/ i?2`27Q`2- QM2 M22/b bQK2 iQQHb iQ /2i2`KBM2 r?2M ;Bp2M T`QT2`iv- T`Qp2/ 7Q` bQK2 +H bb Q7 b2ib- 2ti2M/b iQ HH i?2 b2ib BM i?2 σ@}2H/ ;2M2` i2/ #v i?Bb T `iB+mH ` +H bbX .2}MBiBQM XRXR G2i X #2 M `#Bi` `v b2iX h?2 +QHH2+iBQM S ⊆ P(X) Bb + HH2/ π@bvbi2K UQ7 b2ibV B7 Bi Bb +HQb2/ mM/2` }MBi2 BMi2`b2+iBQMbX .2}MBiBQM XRXk G2i X #2 M `#Bi` `v b2iX MQM2KTiv +QHH2+iBQM Q7 b2ib S ∈ P(X) Bb + HH2/ /@bvbi2K UQ7 b2ibV B7 U V X, ∅ ∈ SX U#V S Bb bi #H2 mM/2` bi`B+i /Bz2`2M+2 Ui? i Bb- B7 A, B ∈ S M/ A ⊆ B- i?2M B − A ∈ SVX U+V S Bb bi #H2 mM/2` b2[m2MiB H BM+`2 bBM; HBKBib Ui? i Bb- i?2 HBKBi Q7 MQM@ /2+`2 bBM; b2[m2M+2 Q7 b2ib BM S Bb BM SVX AM i?2 HBi2` im`2- /@bvbi2K Bb bQK2iBK2b + HH2/ .vMFBM bvbi2KX h?2Q`2K XRXj G2i X #2 M `#Bi` `v b2iX A7 i?2 +QHH2+iBQM S ∈ P(X) Bb π@ bvbi2K M/ /@bvbi2K- Bi Bb σ@}2H/X h?2 bK HH2bi /@bvbi2K +QMi BMBM; MQM@2KTiv +QHH2+iBQM Q7 bm#b2ib C ⊆ P(X) Bb /2MQi2/ #v d(C)X q2 MQr bi i2 .vMFBMǶb i?2Q`2KX h?2Q`2K XRX9 G2i S #2 π@bvbi2K /2}M2/ QM XX h?2M d(S) = σ(S)X h?2 M2ti `2bmHi Bb + HH2/ i?2 7mM+iBQM H .vMFBMǶb i?2Q`2KX Ai Bb mb27mH BM T`QpBM; i? i ;Bp2M +H bb Q7 7mM+iBQMb +QMi BMb HH i?2 7mM+iBQMb K2 bm` #H2 rBi? `2bT2+i iQ +2`i BM σ@}2H/X
© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9
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X k−1
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f (x)μ(dx) = X
X
f (T x)(μ ◦ T )(dx) ,
r2 ? p2 iQ T`Qp2 i? i i?2 K2 bm`2 ν := μ ◦ T Bb Q7 i?2 7Q`K ν(dx1 × · · · × dxk ) = dx1 × μ(dx2 × · · · × dxk ) 7Q` bQK2 HQ+ HHv }MBi2 K2 bm`2 μ QM X k−1 X 6Q` HH MQM@M2; iBp2 K2 bm` #H2 7mM+@ iBQMb ϕ : X k → R ϕ(x1 , . . . , xk )ν(dx1 × · · · × dxk ) = ϕ(x)(μ ◦ T )(dx) = ϕ(T −1 x)μ(dx) Xk Xk Xk = ϕ(x1 , x2 − x1 , . . . , xk − x1 )μ(dx1 × · · · × dxk ) k X = ϕ(x1 + h, x2 − x1 , . . . , xk − x1 )μ(dx1 × · · · × dxk ), Xk
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h?2Q`2K XRXRe q2 ? p2 i? i dV (P1 , P2 ) =
1 2
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dV (P1 , P2 ) = dV (P1 , P2 ) .
h?2Q`2K XRXR3 G2i h : Ω → Ω #2 K2 bm` #H2 TTHB+ iBQM Q7 Ω QMiQ Bib2H7 M/ H2i P1 M/ P2 #2 i?2 BK ;2b #v h Q7- `2bT2+iBp2Hv- i?2 T`Q# #BHBiB2b P1 M/ P2 QM (Ω, F)X h?2M dV (P1 , P2 ) ≤ dV (P1 , P2 ) , rBi? 2[m HBiv B7 KQ`2Qp2` h /KBib K2 bm` #H2 BMp2`b2 h−1 X .2}MBiBQM XRXRN G2i X M/ Y #2 irQ ` M/QK 2H2K2Mib rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (U, U)X h?2 p `B iBQM /Bbi M+2 dV (X, Y ) #2ir22M X M/ Y Bb- #v /2}MBiBQM- i?2 p `B iBQM /Bbi M+2 #2ir22M i?2B` /Bbi`B#miBQMb, dV (X, Y ) := sup (P (X ∈ A) − P (Y ∈ A)) . A∈U
h?2Q`2K XRXky 6Q` Mv T B` (X, Y ) ∈ D(α, β)- r2 ? p2 i?2 7mM/ K2Mi H +Qm@ THBM; BM2[m HBiv dV (α, β) ≤ P (X = Y ) , URkXNRV r?2`2 2[m HBiv Bb ii BM2/ #v bQK2 T B` (X, Y ) ∈ D(α, β)- r?B+? Bb i?2M b B/ iQ `2 HBx2 K tBK H +QBM+B/2M+2X
.2}MBiBQM XRXkR X h?2 b2[m2M+2 {Pn }n≥1 Q7 T`Q# #BHBiv K2 bm`2b QM (X, X ) Bb b B/ iQ +QMp2`;2 BM p `B iBQM iQ i?2 T`Q# #BHBiv P QM (X, X ) B7 lim dV (Pn , P ) = 0 .
n↑∞ V ar.
h?Bb Bb /2MQi2/ Pn → P X "X G2i {Xn }n≥1 #2 ` M/QK 2H2K2Mib rBi? p Hm2b BM i?2 K2 bm` #H2 bT +2 (U, U)- rBi? `2bT2+iBp2 /Bbi`B#miBQMb α M/ {αn }n≥1 X h?2 b2[m2M+2 {Xn }n≥1 Bb b B/ iQ +QMp2`;2 V ar. BM p `B iBQM iQ X B7 limn↑∞ dV (Xn , X) = 0X h?Bb Bb /2MQi2/ #v Xn → XX k9
A#B/2K- TX R3y- Kmi iBb Kmi M/BbX
XRX J1al_"AGAhu L. J1al_1
8k8
G2i Q #2 T`Q# #BHBiv K2 bm`2 bm+? i? i 7Q` HH n ≥ 1- Pn Q- 7Q` BMbi M+2 Q=
1 Pn . 2n n≥1
.2MQi2 #v fn U`2bTX- f V i?2 _ /QMĜLBFQ/ɷK /2`Bp iBp2 Q7 Pn U`2bTX- P V rBi? `2bT2+i L1
V ar.
iQ QX "v h?2Q`2K XRXRe- Pn → P B7 M/ QMHv B7 fn → f - r?2`2 L1 = L1C (Q)X LQi2 HbQ i? i B7 ϕ : X → C Bb #QmM/2/ 7mM+iBQM- i?2M ϕ dPn → ϕ dP , X
X
b 7QHHQrb 7`QK i?2 7 +i i? i ϕ dPn − ϕ dP = ϕ × (fn − f ) dQ X
X
X
M/ /QKBM i2/ +QMp2`;2M+2X h?2Q`2K XRXkk G2i Pn - Q M/ fn #2 /2}M2/ b #Qp2X amTTQb2 i? i i?2`2 2tBbib MQM@M2; iBp2 K2 bm` #H2 7mM+iBQM f 7`QK (X, X ) iQ (R, B) bm+? i? i Q@ X2X V ar. fn → f X h?2M Pn → P - r?2`2 P Bb i?2 T`Q# #BHBiv /2}M2/ #v P (A) = A f dQA ∈ XX h?2 T`QQ7 Bb /B`2+i +QMb2[m2M+2 Q7 a+?2zûǶb H2KK , G2KK XRXkj G2i f M/ fn - n ≥ 1- #2 Q@BMi2;` #H2 MQM@M2; iBp2 `2 H 7mM+iBQMb 7`QK (X, X ) iQ (R, B)rBi? lim f = f Q@ X2X M/ lim f dQ = X f dQX n↑∞ n n↑∞ n X h?2M limn↑∞ X |fn − f | dQ = 0X w2`Q@QM2 G rb h?2 x2`Q@QM2 H rb T`2b2Mi2/ BM i?Bb b2+iBQM +QM+2`M bvKTiQiB+ 2p2Mib M/ ;Bp2 +QM/BiBQMb mM/2` r?B+? bm+? 2p2Mib `2 i`BpB H Ui? i Bb- rBi? T`Q# #BHBiv 0 Q` 1VX .2}MBiBQM XRXk9 G2i {Xn }n≥1 #2 b2[m2M+2 Q7 ` M/QK p `B #H2b- M/ /2}M2 FnX := σ(X1 , . . . , Xn )X h?2 σ@}2H/ T X := ∩n≥1 σ(Xn , Xn+1 , . . .) Bb + HH2/ i?2 i BH σ@}2H/ Q7 i?Bb b2[m2M+2X h?2Q`2K XRXk8 h?2 i BH σ@}2H/ Q7 b2[m2M+2 {Xn }n≥1 Q7 BM/2T2M/2Mi ` M/QK p `B #H2b Bb i`BpB H- i? i Bb- B7 A ∈ T X - i?2M P (A) = 0 Q` 1X G2i (S, S) #2 bQK2 K2 bm` #H2 bT +2 M/ H2i μ #2 T`Q# #BHBiv K2 bm`2 QM BiX q2 b? HH rQ`F QM i?2 + MQMB+ H K2 bm` #H2 bT +2 Q7 S@p Hm2/ ` M/QK b2[m2M+2b(Ω, F) := (S N , S ⊗N ) , 2M/Qr2/ rBi? i?2 T`Q# #BHBiv K2 bm`2
8ke
SS1L.As P := μ⊗∞ .
AM T `iB+mH `- M 2H2K2Mi Q7 Ω ? b i?2 7Q`K ω = {x = (x1 , x2 , . . .) ∈ S N } . 6Q` 2 +? n ≥ 1- i?2 b2[m2M+2 {Xn }n≥1 /2}M2/ #v Xn (ω) := xn Bb BB/- M/ i?2 +QKKQM T`Q# #BHBiv /Bbi`B#miBQM Q7 i?2 Xn Ƕb Bb μX LQi2 i? i Xn (πω) = Xπ(n) (ω)X .2}MBiBQM XRXke U V }MBi2 T2`Kmi iBQM Q7 N Bb T2`Kmi iBQM π bm+? i? i π(i) = i 7Q` HH #mi }MBi2 MmK#2` Q7 BM/B+2b i ≥ 1X U#V M 2p2Mi A ∈ F bm+? π −1 A = A Bb + HH2/ 2t+? M;2 #H2X U+V h?2 σ@}2H/ E +QMbBbiBM; Q7 i?2 +QHH2+iBQM Q7 2t+? M;2 #H2 2p2Mib Bb + HH2/ i?2 2t+? M;2 #H2 σ@}2H/X h?2Q`2K XRXkd h?2 2p2Mib Q7 i?2 2t+? M;2 #H2 σ@}2H/ E `2 i`BpB H- i? i Bb- 7Q` Mv A ∈ E- P (A) = 0 Q` 1X
Xk aiQ+? biB+ S`Q+2bb2b >BbiQ`B2b AM i?2 7QHHQrBM;- i?2 BM/2t b2i T Bb 2Bi?2` QM2 Q7 i?2 7QHHQrBM;, R- R+ - N- ZX .2}MBiBQM XkXR G2i (Ω, F) #2 K2 bm` #H2 bT +2X h?2 7 KBHv {Ft }t∈T Q7 bm#@ σ@}2H/b Q7 F Bb + HH2/ ?BbiQ`v QM (Ω, F) B7 7Q` HH s, t ∈ T bm+? i? i s ≤ tFs ⊆ Ft . AM Qi?2` rQ`/b- ?BbiQ`v Bb MQM@/2+`2 bBM; 7 KBHv Q7 bm#@σ@}2H/b Q7 F BM/2t2/ #v TX Ai Bb HbQ + HH2/ }Hi` iBQMX h?2 σ@}2H/ F∞ := ∨t∈T Ft Bb- #v /2}MBiBQM- i?2 bK HH2bi σ@}2H/ i? i +QMi BMb Ft 7Q` HH t ∈ TX G2i {X(t)}t∈T #2 biQ+? biB+ T`Q+2bb /2}M2/ QM (Ω, F)X h?2 ?BbiQ`v {FtX }t∈T /2@ }M2/ #v FtX = σ(X(s) ; s ≤ t) Bb + HH2/ i?2 BMi2`M H ?BbiQ`v Q7 {X(t)}t∈T X Mv ?BbiQ`v {Ft }t∈T bm+? i? i Ft ⊇ FtX ,
t ∈ T,
Bb + HH2/ ?BbiQ`v Q7 {X(t)}t∈T X PM2 HbQ b vb, i?2 biQ+? biB+ T`Q+2bb {X(t)}t∈T Bb / Ti2/ iQ i?2 ?BbiQ`v {Ft }t∈T - Q` Ft @ / Ti2/X .2}MBiBQM XkXk G2i T = R Q` R+ X .2}M2 7Q` HH t ∈ T Ft+ := ∩s>t Fs . h?2 ?BbiQ`v {Ft }t∈T Bb + HH2/ `B;?i@+QMiBMmQmb B7 7Q` HH t ∈ T- Ft = Ft+ X
XkX ahP*>ahA* S_P*1aa1a
8kd
J2 bm` #BHBiv Ai Bb Q7i2M +QMp2MB2Mi iQ pB2r biQ+? biB+ T`Q+2bb b K TTBM; X : T×Ω → E/2}M2/ #v (t, ω) → X(t, ω)X h?Bb QT2Mb i?2 r v iQ p `BQmb K2 bm` #BHBiv +QM+2TibX .2}MBiBQM XkXj h?2 biQ+? biB+ T`Q+2bb {X(t)}t∈R Bb b B/ iQ #2 K2 bm` #H2 Bz i?2 K TTBM; 7`QK R × Ω BMiQ E /2}M2/ #v (t, ω) → X(t, ω) Bb K2 bm` #H2 rBi? `2bT2+i iQ B ⊗ F M/ EX h?Bb BKTHB2b BM T `iB+mH ` i? i 7Q` Mv ω ∈ Ω i?2 K TTBM; t → X(t, ω) Bb K2 bm` #H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(R) M/ EX HbQ- B7 E = R M/ B7 X(t) Bb MQM@M2; iBp2- QM2 + M /2}M2 i?2 G2#2b;m2 BMi2;` H X(t, ω)dt R
7Q` 2 +? ω ∈ Ω- M/ HbQ TTHv hQM2HHBǶb i?2Q`2K iQ Q#i BM
E X(t)dt = E [X(t)] dt . R
R
"v 6m#BMBǶb i?2Q`2K- i?2 H bi 2[m HBiv HbQ ?QH/b i`m2 7Q` K2 bm` #H2 biQ+? biB+ T`Q+2bb2b Q7 `#Bi` `v bB;M b iBb7vBM; E [|X(t)|] dt < ∞ . R
.2}MBiBQM XkX9 biQ+? biB+ T`Q+2bb {X(t)}t∈R+ i FBM; Bib p Hm2b BM i?2 K2 @ bm` #H2 bT +2 (E, E) Bb b B/ iQ #2 Ft @T`Q;`2bbBp2Hv K2 bm` #H2 Bz 7Q` HH t ∈ R+ i?2 K TTBM; (s, ω) → X(s, ω) 7`QK [0, t] × Ω BMiQ E Bb K2 bm` #H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B([0, t]) ⊗ Ft M/ EX *H2 `Hv- {X(t)}t∈R+ Bb i?2M Ft @ / Ti2/ M/ K2 bm` #H2X h?2Q`2K XkX8 G2i {X(t)}t∈R+ #2 biQ+? biB+ T`Q+2bb- i FBM; Bib p Hm2b BM iQTQHQ;B+ H bT +2 E U2M/Qr2/ rBi? Bib "Q`2H σ@}2H/ E = B(E)V- / Ti2/ iQ {Ft }t∈R+ M/ `B;?i@+QMiBMmQmb U`2bTX H27i@+QMiBMmQmbV- i?2M {X(t)}t∈R+ Bb Ft @T`Q;`2bbBp2Hv K2 bm` #H2X h?2Q`2K XkXe A7 i?2 MQM@M2; iBp2 biQ+? biB+ T`Q+2bb {X(t)}t∈R + Bb Ft @T`Q;`2bb@ Bp2Hv K2 bm` #H2- i?2M- 7Q` 2 +? t ∈ R+ - i?2 ` M/QK p `B #H2 (0,t] X(s, ω) ds Bb Ft @K2 bm` #H2X aiQTTBM; hBK2b T`BM+BT H MQiBQM BM i?2 i?2Q`v Q7 biQ+? biB+ T`Q+2bb2b Bb i? i Q7 biQTTBM; U`2bTX QTiBQM HV iBK2X AM i?Bb bm#b2+iBQM- i?2 BM/2t b2i Bb T Bb N- Q` R+ - M/ r2 /2}M2 T = T ∪ {+∞}X ¯ ` M/QK p `B #H2 τ Bb .2}MBiBQM XkXd G2i {Ft }t∈T #2 ?BbiQ`vX T@p Hm2/ + HH2/ M Ft @biQTTBM; U`2bTX QTiBQM HV iBK2 Bz 7Q` HH t ∈ T{τ ≤ t} ⊂ Ft
(`2bTX- {τ < t} ⊂ Ft ) .
8k3
SS1L.As
_2K `F XkX3 AM i?Bb `2K `F- i?2 BM/2t b2i Bb R+ X biQTTBM; iBK2 Bb M QTiBQM H iBK2 #2+ mb2 1 {T < t} = ∪n {T ≤ t − } ∈ Ft . n A7 i?2 }Hi` iBQM {Ft }t∈T Bb `B;?i@+QMiBMmQmb- M QTiBQM H iBK2 Bb biQTTBM; iBK2#2+ mb2 1 {T ≤ t} = ∩n {T < t + } ∈ Ft+ . n h?2Q`2K XkXN G2i {Tn }n∈N #2 b2[m2M+2 Q7 Ft @QTiBQM H iBK2b /2+`2 bBM; iQ i?2 ` M/QK p `B #H2 T X h?2M T Bb M Ft @QTiBQM H iBK2X h?2Q`2K XkXRy G2i {Ft }t∈R+ #2 `B;?i@+QMiBMmQmb ?BbiQ`v- M/ H2i τ #2 M Ft @ biQTTBM; iBK2X .2}M2 7Q` 2 +? n ≥ 1 M TT`QtBK iBQM τ (n) Q7 i?2 biQTTBM; iBK2 τ #v, ⎧ ⎪ B7 τ (ω) = 0 ⎨0 k τ (n, ω) = k+1 B7 ≤ τ (ω) < k+1 n 2 2n 2n ⎪ ⎩ +∞ B7 τ (ω) = ∞. h?2M τ (n) Bb M Ft @biQTTBM;@iBK2 /2+`2 bBM; iQ τ b n ↑ ∞X h?2Q`2K XkXRR G2i {X(t)}t∈R+ #2 `B;?i@+QMiBMmQmb U `2bTX H27i@+QMiBMmQmbV `2 H@p Hm2/ T`Q+2bb / Ti2/ iQ {Ft }t≥0 - M/ c #2 ;Bp2M `2 H MmK#2`X h?2 ` M/QK iBK2 τ := inf{t | X(t) ≥ c} Bb M Ft @biQTTBM; iBK2X AM i?2 H27i@+QMiBMmQmb + b2X(τ ) ≤ cX h?2Q`2K XkXRk G2i T = N Q` R+ X G2i {Ft }t∈T #2 ?BbiQ`vX G2i τ #2 M Ft @ biQTTBM; iBK2X h?2 +QHH2+iBQM Q7 2p2Mib Fτ = {A ∈ F∞ | A ∩ {τ ≤ t} ∈ Ft , 7Q` HH t ∈ T} Bb σ@}2H/- M/ τ Bb Fτ @K2 bm` #H2X G2i {X(t)}t∈T #2 M 1@p Hm2/ Ft @ / Ti2/ biQ+? biB+ T`Q+2bb- M/ H2i τ #2 }MBi2 Ft @biQTTBM; iBK2X A7 r2 /2}M2 X(τ ) #v X(τ )(ω) = X(τ (ω), ω)X h?2M- B7 T = N U `2bTX- T = R+ M/ {X(t)}t∈R+ Bb Ft @ T`Q;`2bbBp2Hv K2 bm` #H2V X(τ )1{τ t} UA ∈ Ft - t ∈ R+ V Bb /2MQi2/ #v Fτ − X .2}MBiBQM XkXR9 biQ+? biB+ T`Q+2bb {Y (t)}t≥0 rBi? p Hm2b BM M `#Bi` `v K2 bm` #H2 bT +2 (E, E) bm+? i? i 7Q` HH t ≥ 0 M/ HH ω ∈ Ω- i?2`2 2tBbib TQbBiBp2 MmK#2` ε(t, ω) bm+? i? i Y (t + h, ω) = Y (t, ω) 7Q` HH h ∈ [t, t + ε(t, ω)]- Bb b B/ iQ ? p2 HQ+ H bi2T bi`m+im`2X
h?2Q`2K XkXR8 G2i {Y (t)}t≥0 #2 biQ+? biB+ T`Q+2bb rBi? p Hm2b BM M `#Bi` `v K2 bm` #H2 bT +2 (E, E) M/ rBi? HQ+ H bi2T bi`m+im`2X G2i FtY := σ(Y (s) ; 0 ≤ s ≤ t) .
XkX ahP*>ahA* S_P*1aa1a
8kN
h?2MUBV i?2 ?BbiQ`v {FtY }t≥0 Bb `B;?i@+QMiBMmQmb- M/ UBBV 7Q` Mv Ft @biQTBM; iBK2 T FTY = σ(Y (s ∧ T ) ; s ≥ 0) . Y Y S`QQ7X UBV Ai bm{+2b iQ T`Qp2 i? i B7 A ∈ Ft+2 −k 7Q` HH k ∈ N- i?2M A ∈ Ft X LQr #v i?2 bbmK2/ HQ+ H bi2T bi`m+im`2- /2MQiBM; #v Q i?2 b2i Q7 ` iBQM H MmK#2`b,
1A = Φk (Y (s), s ∈ (Q ∩ [0, t + 2−k ]) ∪ {t + 2−k }) . .2}M2 Φk (Y (s), s ∈ Q ∩ [0, t]) := Φk (Y (s), s ∈ (Q ∩ [0, t + 2−k ]) ∪ {t + 2−k }) , r?2`2 G2i M/
Y (s) = Y (s)1[0,t) (s) + Y (t)1[t,t+2−k ] (s) . ξk := Φk (Y (s), s ∈ Q ∩ [0, t]) , Ωk := {ω ; Y (t + δ, ω) = Y (t, ω) 7Q` HH δ ∈ [0, 2−k ]} .
h?2M ξk Bb FtY @K2 bm` #H2- Ωk ↑ Ω M/ 1A∪Ωk = ξk 1Ωk X Ai 7QHHQrb i? i 1A = lim inf k ξk - M/ i?2`27Q`2 A ∈ FtY X UBBV h?2 BM+HmbBQM ⊇ Bb +H2 `X h?2 `2p2`b2 BM+HmbBQM rBHH }`bi #2 T`Qp2/ 7Q` M FtY @ biQTTBM; iBK2 i FBM; /2MmK2` #H2 b2i Q7 p Hm2b 0 ≤ a0 < a1 < . . . ≤ +∞X Mv 2p2Mi A ∈ FTY + M #2 /2+QKTQb2/ b A = ∪p≥0 Ap r?2`2 Ap := A ∩ {T = ap } Bb BM FaYp X b #Qp2- r2 + M r`Bi2 1Ap = ϕp (Y (t) ; t ∈ Q ∩ [0, ap ]) . LQr- bBM+2 T = ap QM Ap - t ∧ T (ω) = t B7 t ∈ [0, ap ]X h?2`27Q`2- 7Q` HH p ≥ 01Ap = ϕp (Y (t ∧ T ) ; t ∈ Q ∩ [0, ap ]) M/ BM T `iB+mH ` Ap ∈ σ(Y (s ∧ T ) ; s ≥ 0)X *QMbB/2` MQr i?2 ;2M2` H + b2X 6Q` HH k ≥ 1- i?2 iBK2 p2−k 1{(p−1)2−k ≤T t} ∈ ∨n≥1 Fa− 1 = Fa− ⊆ Fa . n n
XjX J_hAL:G1a
8jR
h?2Q`2K XkXky G2i X = {X(t)}t≥0 M/ Y = {Y (t)}t≥0 #2 irQ biQ+? biB+ T`Q@ +2bb2b / Ti2/ iQ i?2 `B;?i@+QMiBMmQmb ?BbiQ`v {Ft }t≥0 - M/ bm+? i? i 7Q` Mv Ft @ biQTTBM; iBK2 S E X(S)1{S t} = {Tn > t} ∩ {Z0 ∈ L0 } ∩
n−1
1{[0,s]} (Ti ) 1L (Zi ) = k
i=1
Bb +H2 `Hv BM σ (T0 , Z0 , T1 , Z1 , . . . , Tn−1 , Zn−1 , Tn )X U+V FT∞ − = ∨n FTn − = σ (Tn , Zn ; n ≥ 0) = ∨n FTn = FT∞ X
h?2 7QHHQrBM; `2bmHib 72 im`2 i?2 bvK#QHB+ +QMp2MiBQM T∞+1 := +∞X h?2Q`2K X9Xk G2i S #2 }MBi2 Ft @biQTTBM; iBK2X h?2M- 7Q` HH n ∈ N ∪ {+∞}FS ∩ {Tn ≤ S < Tn+1 } = FTn ∩ {Tn ≤ S < Tn+1 } , M/ i?2`2 2tBbib b2[m2M+2 {Rn }n∈N∪{+∞} Q7 MQM@M2; iBp2 FTn @K2 bm` #H2 ` M/QK p `B #H2b bm+? i? i 7Q` HH n ∈ N ∪ {+∞}S ∧ Tn+1 = (Tn + Rn ) ∧ Tn+1 QM {S ≥ Tn } . S`QQ7X Ai bm{+2b iQ b?Qr i? i QM2 + M bbQ+B i2 rBi? Mv ;2M2` iQ` U Q7 FS ;2M2` iQ` V Q7 FTn bm+? i? i U ∩ {Tn ≤ S < Tn+1 } = V ∩ {Tn ≤ S < Tn+1 } , M/ pB+2 p2`b X h?2 Q#b2`p iBQM i? i 7Q` HH k ∈ N- HH t ≥ 0- HH L ∈ K{N ((0, t ∧ S] × L) = k}∩{Tn ≤ S < Tn+1 } = {N ((0, t ∧ Tn ] × L) = k} ∩ {Tn ≤ S < Tn+1 } T2`KBib iQ /Q i?Bb- mbBM; UBBV Q7 h?2Q`2K X9XRX h?2 ` M/QK p `B #H2 S Bb FS @K2 bm` #H2 M/ FS = σ(Z0 ) ∨ σ (NZ ((0, t ∧ S] × L) ; t ≥ 0, L ∈ K) . "v i?2 `2bmHi Dmbi T`Qp2/- i?2`2 2tBbib K TTBM;b ψn Un ≥ 0V M/ ψ∞ bm+? i? i S 1{Tn ≤S a M/ HH ω ∈ Ω Tna (ω) := a1A (ω) + n1A (ω) . h?Bb Bb biQTTBM; iBK2X HbQ {(t, ω) ; t ≤ Tna (ω)} = A × [a, ∞) + A × [n, ∞) M/ i?2`27Q`2- B7 a ≤ b < n{(t, ω) ; t ≤ Tna (ω)} − {(t, ω) ; t ≤ Tnb (ω)} = A × [a, b) . h?Bb #2BM; i`m2 7Q` HH n- b M22/ MQi #2 +QMbi` BM2/ iQ #2 }MBi2X
h?2Q`2K X9X9 h?2 T`Q+2bb {X(t)}t≥0 Bb Ft @T`2/B+i #H2 B7 M/ QMHv X(0) Bb F0 @ K2 bm` #H2 M/ QM R+ \{0} f (n) (t, ω) 1{Tn (ω)T∞ } , URkXNdV X(t, ω) := n≥0
r?2`2 7Q` 2 +? n ∈ N ∪ {+∞}- i?2 K TTBM; (t, ω) → f (n) (t, ω) Bb FTn ⊗ B(R+ )@ K2 bm` #H2X S`QQ7X am{+B2M+v Bb H27i b M 2t2`+Bb2X 6Q` M2+2bbBiv- Bi bm{+2b #v h?2Q`2K X9Xj iQ T`Qp2 i?2 `2T`2b2Mi iBQM 7Q` T`Q+2bb2b Q7 i?2 7Q`K X(t) = 1{t≤S} - r?2`2 S Bb }MBi2 Ft @biQTTBM; iBK2X "mi i?Bb BM im`M Bb +QMb2[m2M+2 Q7 h?2Q`2K X9XjX h?2Q`2K X9X8 G2i A ∈ FTn − 7Q` bQK2 n ≥ 1X h?2`2 2tBbib M Ft @T`2/B+i #H2 T`Q+2bb {X(t)}t≥0 bm+? i? i X(Tn ) = 1A B7 Tn < ∞- M/ X(t) = 0 B7 t ∈ (Tn−1 , Tn ]X S`QQ7X "v UBBB- V Q7 h?2Q`2K X9XR- i?2`2 2tBbib K2 bm` #H2 7mM+iBQM f bm+? i? i 1A (ω) = f (T0 (ω), Z0 (ω), . . . , Tn−1 (ω), Zn−1 (ω), Tn (ω)) . h F2 X(t) := 1{Tn−1 0 ,
loc. r?2`2 n≥1 qn = ∞ M/ n≥1 pn < ∞X h?2M- #v i?2 TQbBiBpBiv +QM/BiBQM- Q P X >Qr2p2` Q M/ P `2 Kmim HHv bBM;mH ` bBM+2 Q(Xn → 0) = 0 M/ P (Xn → 0) = 1X h?2Q`2K X8Xk {Ln }n≥1 +QMp2`;2b Q@ HKQbi@bm`2Hv M/ P @ HKQbi@bm`2Hv iQ bQK2 ` M/QK p `B #H2 L∞ M/ dQ = 1{L∞ =∞} dQ + L∞ dP , r?2`2 P (L∞ = ∞) = 0X h?2Q`2K X8Xj Q P ⇔ EP [L∞ ] = 1 ⇔ Q(L∞ < ∞) = 1 , Q ⊥ P ⇔ EP [L∞ ] = 0 ⇔ Q(L∞ = ∞) = 1 .
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n
(n ≥ 1)
Xi
i=1
Bb MQM@M2; iBp2 K `iBM; H2 M/ Bi +QMp2`;2b HKQbi bm`2Hv iQ }MBi2 ` M/QK 1 ! p `B #H2 M∞ X G2i an := E[Xn2 ]X ULQi2 i? i ∞ n=1 an ≤ 1XV h?2Q`2K X8X9 h?2 7QHHQrBM; +QM/BiBQMb `2 2[mBp H2Mi, 1 !∞ ! 2 UBV ∞ n=1 an > 0- r?2`2 an = E[Xn ]X ULQi2 i? i n=1 an ≤ 1XV UBBV E[M∞ ] = 1X UBBBV Mn → M∞ BM L1 X UBpV {Mn }n≥0 Bb mMB7Q`KHv BMi2;` #H2X 1t KTH2 X8X8, E Fmi MBǰb .B+?QiQKv h?2Q`2KX G2i {Xn }n≥1 #2 b2@ [m2M+2 Q7 ` M/QK 2H2K2Mib rBi? p Hm2b QM i?2 K2 bm` #H2 bT +2 (E, E)X q2 K v bmTTQb2 i? i Bi Bb i?2 +QQ`/BM i2 b2[m2M+2 Q7 i?2 + MQMB+ H bT +2 (Ω, F) := (E N , E ⊗N )X G2i Q M/ P #2 irQ T`Q# #BHBiv K2 bm`2b QM (Ω, F) bm+? i? i i?2 b2@ [m2M+2 Bb BB/ `2H iBp2 iQ #Qi?X G2i QXn M/ PXn #2 i?2 `2bi`B+iBQMb Q7 Q M/ P `2bT2+iBp2Hv iQ σ(Xn ) M/ H2i Qn M/ Pn #2 i?2 `2bi`B+iBQMb Q7 Q M/ P `2bT2+iBp2Hv iQ Fn := σ(X1 , . . . , Xn )X q2 bbmK2 i? i 7Q` HH n ≥ 1- QXn PXn M/ /2MQi2 dQ i?2 +Q``2bTQM/BM; _ /QMĜLBFQ/ɷK /2`Bp iBp2 dPXXn #v fn (Xn )X h?2M 7Q` HH n ≥ 1Qn Pn M/ Ln =
dQn dPn
= Πni=1 fi (Xi )X aBM+2
n
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n
log fi (Xi ) < ∞
i=1
Bb i BH@2p2Mi Q7 i?2 b2[m2M+2- Bib T`Q# #BHBiv Bb 0 Q` 1X h?2`27Q`2- i?2`2 `2 QMHv irQ TQbbB#BHBiB2b- 2Bi?2` Q P Q` Q ⊥ P X 1t KTH2 X8Xe, E Fmi MBǰb *QM/BiBQMX E Fmi MBǶb i?2Q`2K Uh?2Q`2K X8X9V + M #2 TTHB2/ iQ i?2 bBim iBQM U M HQ;Qmb iQ i? i Q7 1t KTH2 X8X8 #Qp2V r?2`2 Ln =
n
Zi ,
i=1
r?2`2 {Zn }n≥1 Bb b2[m2M+2 Q7 BB/ MQM@M2; iBp2 ` M/QK p `B #H2b Q7 K2 M 1X "v i?Bb i?2Q`2K- i?2 +`Bi2`BQM Q7 #bQHmi2 +QMiBMmBiv Q7 Q rBi? `2bT2+i iQ P EP [L∞ ] = 1 ! 2 1 Q7 h?2Q`2K X8Xj Bb ∞ n=1 E[Zn ] > 0X "v i?2 b K2 `;mK2Mi b BM 1t KTH2 X8X8i?2 QMHv Hi2`M iBp2 iQ P Bb Q ⊥ P - M/ i?2`27Q`2 M2+2bb `v M/ bm{+B2Mi 1 ! 2 +QM/BiBQM 7Q` i?2 H ii2` Bb ∞ n=1 E[Zn ] = 0X
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© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9
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898
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89e
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89d
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893
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HX+X/X#X- R G TH +2 7mM+iBQM H ě Q7 SQBbbQM T`Q+2bb- 3k ě Q7 +Hmbi2` TQBMi T`Q+2bb- j9 ě Q7 TQBMi T`Q+2bb- Rd ;2M2` iQ`- ek ě Q7 ` M/QK K2 bm`2- Rd BM}MBi2bBK H ě- ek HB7Q9yR bi #H2 M/ +QMb2`p iBp2 ě- ek GBM/H2v T`Q+2bb- 9kN :B##b HQ+ H ě /Bbi`B#miBQM- k9N ě +? ` +i2`BbiB+b Q7 J `FQp }2H/- k9N ě TQi2MiB H- k9N ě 2M2`;v- k8R ;`B/- 8 " `iH2ii bT2+i`mK Q7 i?2 bi iBQM `v ě- HQ+ H +? ` +i2`BbiB+b- Rd9 HQ+ HBxBM; b2[m2M+2- 8jR j9e HQ+ HHv i?BMMBM; Q7 i?2 ě- Ryy ě }MBi2 K2 bm`2- R ě BMi2;` #H2- Rdk ? `/@+Q`2 KQ/2H- kjj HQM;@` M;2 /2T2M/2M+2- j8k > rF2b T`Q+2bb GQiF ĜoQHi2`` KQ/2H- Rkj HBM2 ` ě- 9dd GQvM2b KmHiBp `B i2 ě- 9e9 ěǶ 2[m iBQM- 9jy MQM@HBM2 ` ě- 8yy ěǶ b2[m2M+2- 9jy ? x `/ ěǶ p `B #H2- 9jy ě K2 bm`2- 9y ě ` i2- 9R M (E)- R ?BbiQ`v- 88 Mp (E)- k +QKT iB#H2 ě- kNy M(E)- k BMi2`M H Ĝ- Rdk Mp (E)- k ?TT- 8y μ(ϕ)- 3 bi M/ `/ ě- RNe Jf:AfRf∞- 99d- 98R JfJfEfy- 9yy BK#2//BM; i?2Q`2K K `F /B`2+i ě- kRR ě T`Q+2bb- k83 }`bi ě- kRR +QKT iB#H2 ě- k83 b2+QM/ ě- kRe- kke ě b2[m2M+2- 39 BMp2`b2 ě- kR9 K `F2/ TQBMi T`Q+2bb- e BMi2MbBiv ě +QKT iB#H2 rBi? ~Qr- k83 ě K2 bm`2- 9- dd ě rBi? BM/2T2M/2Mi BB/ K `Fb- d T`2/B+i #H2 ě- R3k bi iBQM `v ě- k8N biQ+? biB+ Ĝ- Ree J `FQp }2H/- k9N biQ+? biB+ ě F2`M2H- Rdk K `iBM; H2- 8jR BMp2`bBQM 7Q`KmH ě +QMp2`;2M+2 i?2Q`2K- 8jk S HK ě- kdk- kNN ě `2T`2b2Mi iBQM i?2Q`2K- RN8 HQ+ H ě- 8jR C MQbbv /2MbBiv- kjy bm#ě- 8jR E2`bi M bmT2`ě- 8jR ě TQBMi T`Q+2bb- 8yj K2 bm`2 ěǶb +QM/BiBQM- 8yj /Bzmb2 ě- Ry E2bi2MǶb #QmM/- R9k MQM@ iQKB+ ě- Ry
AL.1s ` M/QK ě- k J2+F2 ěǶb n@i? Q`/2` K2 bm`2- jkj ěǶb 7Q`KmH - ke9 ěǶb K2 bm`2- jRy ěǶb `2/m+2/ K2 bm`2- jk9 K2KQ`v BMi2`K2/B i2 ě- j8k HQM; ě- j8k KQK2Mi K2 bm`2- RR 7 +iQ`B H ě- Rk }`bi ě Q7 TQBMi T`Q+2bb- RR b2+QM/ ě Q7 TQBMi T`Q+2bb- RR- jj8 Kmim H BM7Q`K iBQM 7Q`KmH - jd3 N − εx - k8N N − x- k8N N L - Rdk NZ - d (N, Z)- d N (ϕ)- 3 M2 `2bi M2B;?#Qm` /Bbi M+2- jy8 M2B;?#Qm`b- k9y MQM@H iiB+2- Rjy QTiBQM H b KTHBM;- 8jk S HK ě /Bbi`B#miBQM- kej ě F2`M2H- jRR mi? Q`/2` ě- jkj ě T`Q# #BHBiv- ke9- jRR- jkj T2M2i` #H2 bT?2`2b ě KQ/2H- kj8 T2`K M2Mi H TQBMi T`Q+2bb- Rj T2`im`# iBQM M HvbBb- d3 T? b2 i` MbBiBQM- j8j TQBMi K2 bm`2- k TQBMi T`Q+2bb- k- 9N # b2 ě- e /Bbi`B#miBQM Q7 ě- R9 }`bi@Q`/2` ě- e- RR HB7i2/ ě- Rdj K `F2/ ě- e # b2 Q7 ě- 39 }MBi2 ě- kjj bi iBQM `v ě- k8N KmHiBp `B i2 ě- Rd8 M@i? Q`/2` ě- RR b2+QM/@Q`/2` ě- RR b2H7@2t+BiBM; ě- R38
888 b2KB@*Qt ě- R38 bBKTH2 ě- j bi iBQM `v ě- 9- k8N rB/2@b2Mb2 bi iBQM `v ě- jje SQBbbQM T`Q+2bb +QKTQmM/ ě- 8y /Qm#Hv biQ+? biB+ ě- 8 ?QKQ;2M2Qmb ě- 8- 9N K `F2/ ě- 39 J2+F2Ƕb +? ` +i2`Bx iBQM Q7 ě- kNe KBt2/ ě- 8 bi M/ `/ ě- 8 SQHH +x2FĜE?BMi+?BM2 7Q`KmH - 99j TQr2` bT2+i` H K2 bm`2 ě Q7 TQBMi T`Q+2bb- j9y ě Q7 biQ+? biB+ T`Q+2bb- jj9 ě Q7 i?2 b KTH2/ #`mb?- j3N T`2/B+i #H2 ě σ@}2H/- Red ě BMi2MbBiv- j39 ě T`Q+2bb- Red ě BM/2t2/ #v- Red ě T`QD2+iBQM- R3j ě bi`m+im`2- kNR T`22KTiBp2 `2bmK2- 9yR T`Q# #BHBiv ;2M2` iBM; 7mM+iBQM H- kk T`Q+2bbQ` b? `BM;- 98j T`Q;`2bbBp2Hv K2 bm` #H2- 8kd _ /QMĜLBFQ/ɷK /2`Bp iBp2- kdN ` M/QK ě }2H/- k93 ě K2 bm`2- k `2;2M2` iBp2 T`Q+2bb- R8R `2H iBp2Hv +QKT +i- R `2M2r H ě 2[m iBQM- RRN /272+iBp2 ě- RRN 7mM/ K2Mi H ě- Rky KmHiBp `B i2 ě- R8d ě 7mM+iBQM- RRd r2B;?i2/ ě- R9R ě K2 bm`2- RRd ě T`Q+2bb- RR8- jjd ě i?2Q`2K 2H2K2Mi `v ě- RR3 ě iBK2- RR8 KmHiBp `B i2 ě i?2Q`v- R8d `2+m``2Mi ě T`Q+2bb- RR3 i` MbB2Mi ě T`Q+2bb- RR3
88e `2bB/m H M HvbBb- j38 `2r `/ ě 7mM+iBQM- Rjd ě T`Q+2bb- Rjd _B2K MM Ƕb ě BMi2;` H- Rjy ě BMi2;` #H2- Rjy /B`2+i ěǶb BMi2;` H- Rjk /B`2+iHv ě BMi2;` #H2- Rjy- Rjk _BTH2vǶb 7mM+iBQM- jjN _BTH2vĜE2HHvǶb i?2Q`2K- k9R _vHH@L `/x2rbFBǶb 7Q`KmH - ke9 Sx (N )- k8N b KTH2 ě #`mb?- j3N ě +QK#- j33 ě Q7 TQBMi T`Q+2bb /Bbi`B#miBQM- dd ě b2[m2M+2- j33 ` M/QKě- j33 b KTHBM;- R8 ě SQBbbQM T`Q+2bb- dd 2t +i ě- N3 ` M/QK ě- j33 b/2- 9N b2H7@2t+BiBM; TQBMi T`Q+2bb- R38 b2KB@*Qt TQBMi T`Q+2bb- R38 b2KB@J `FQp ě F2`M2H- R83 ě T`Q+2bb- R83 b2KB@K `iBM; H2 K2i?Q/- jee b2`pB+2 /Bb+BTHBM2- 9jj b?B7i- k8e b?Qi MQBb2 ě TQr2` bT2+i` H K2 bm`2- j8e SQBbbQM ě- Ny bBM`- jR3 aHBpMv FǶb i?2Q`2K- RyN- RRy- kNj- jR8 aKBi?Ƕb `2;2M2` iBp2 7Q`KmH - R8e bKQQi?BM; 7Q`KmH - ReN- kjd bi #H2 bvKTiQiB+ HHv ě- 8y9 bi iBQM `v ě 7` K2rQ`F- k8e- k8N ě ;`B/- e ě K `F2/ TQBMi T`Q+2bb- k8N ě TQBMi T`Q+2bb- 9- k8N ě biQ+? biB+ T`Q+2bb- key rB/2@b2Mb2 ě ě TQBMi T`Q+2bb- jje
AL.1s biQ+? biB+ ;2QK2i`v- RyR aiQM2Ƕb /2+QKTQbBiBQM- R9N bi`2bb `2H2 b2 KQ/2H- Rd8 m;K2Mi2/ ě- j33- 9e8 bi`QM; J `FQp T`QT2`iv Q7 ?TTb- 8k- kk9 bm#2tTQM2MiB H /Bbi`B#miBQM- R9k i?BMMBM;- j9- j8e i` {+ BMi2MbBiv- 9y8- 9kN o M GB2b?QmiĜ" //2H2v BM/2t- jy8 oQ`QMQś +2HH- kNN q i M #2 ěǶb SQBbbQM +? ` +i2`Bx iBQM- eR 2ti2MbBQM Q7 ě- ky3 bT iB H ěǶb i?2Q`2K- k9y r2 F +QMp2`;2M+2- ke r?Bi2 MQBb2 : mbbB M ě- kR3 SQBbbQM ě- j9k rB/2@b2Mb2 bi iBQM `v ě TQBMi T`Q+2bb- jje ě biQ+? biB+ T`Q+2bb- jjj