Multidimensional Diffusion Processes (Classics in Mathematics) 3662222019, 9783662222010

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Classics in Mathematics Daniel W. Stroock S. R. Srinivasa Varadhan

Multidimensional Diffusion Processes

Daniel W.Stroock S. R. Srinivasa Varadhan

Multidimensional Diffusion Processes Reprint of the 1997 Edition

~ Springer

Daniel W. Stroock Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Ave Cambridge, MA 02139-4307 USA S. R. Srinivasa Varadhan New York University Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 10012 USA

O r i g i n a l l y published as Vol. 233 i n the series Grundlehren

der mathematischen

Wissenschaften

Mathematics Subject Classification (2000): 60J60,28A65

Library of Congress Control Number: 2005934787

ISSN 1431-0821 ISBN 978-3-662-22201-0 ISBN 978-3-540-28999-9 (eBook) D O I 10.1007/978-3-540-28999-9

This work is subject to copyright. A l l rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage i n data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, i n its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg, 2006 Originally published by Springer-Verlag Berlin Heidelberg New York in 2006 The use of general descriptive names, registered names, trademarks, etc. i n 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. Production: LE-TgX Jelonek, Schmidt & Vockler GbR, Leipzig Cover design: design & production Printed on acid-free paper

GmbH, Heidelberg

41/3142/YL-543 2 1 0

Grundlehren der mathematischen Wissenschaften 233 A Series of Comprehensive Studies in Mathematics

Series editors S.S. Chern J.1. Doob J. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M.M. Postnikov W. Schmidt D.S. Scott K. Stein J. Tits B.1. van der Waerden

Editor-in-Chief B. Eckmann

J.K. Moser

Springer-Verlag Berlin Heidelberg GmbH

Daniel w. Stroock S.R. Srinivasa Varadhan

Multidimensional Diffusion Processes

,

Springer

Daniel w. Stroock Massachusetts Institute of Technology Department of Mathematics 77 Massachusetts Ave Cambridge, MA 02139-4307 USA e-mail: [email protected]

S.R. Srinivasa Varadhan New York University Courant Institute of Mathematical Sciences 251 Mercer Street New York, NY 100U USA e-mail: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is available from the Llbrary of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 60J60, 28A65

ISSN 00]2-7830 ISBN 978-3-662-22201-0 ISBN 978-3-540-28999-9 (eBook) DOl 10.1007/978-3-540-28999-9

This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting. reuse of illustrations, recitation, broadcasting. reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 19650 in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BerteismannSpringer Science+Business Media GmbH http://www.springer.de

e Springer-Verlag Berlin Heidelberg 1979, 1997

Originally published by Springer-Verlag Berlin Heidelberg New York in 1997

The use of general descriptive names, registered names, trademarks, 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. Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

41/31421db - 5 4 3 :I. 1 0

To our parents: Katherine W. Stroock Alan M. Stroock S.R. Janaki S.V. Ranga Ayyangar

Contents

Frequently Used Notation . . . . . . . . . . . . . . . . . . . . . . . . . .

Xl

Chapter O. Introduction . . . . . . Chapter 1. Preliminary Material: Extension Theorems, Martingales, and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Weak Convergence, Conditional Probability Distributions and Extension Theorems . . . . . . . . . 1.2 Martingales . . . . . . . . . . . 1.3 The Space C([O, (0); Rd ) . . . . 1.4 Martingales and Compactness 1.5 Exercises . . . . . . . . . . . .

7 19 30 36 42

Chapter 2. Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure . . . . . . . . . . . . . . . . . . 2.1 Regularity of Paths . . . . . . . . . . . . . . . . 2.2 Markov Processes and Transition Probabilities 2.3 Wiener Measure . . . . . . . . . . . . . . . . . . 2.4 Exercises

46 46 51 56 60

Chapter 3. Parabolic Partial Differential Equations 3.1 The Maximum Principle 3.2 Existence Theorems . . . . . . . . . . . . . . . . 3.3 Exercises

7 7

65 65 71

79

Chapter 4. The Stochastic Calculus of Diffusion Theory 4.1 Brownian Motion . . . . . . . . . . . . . 4.2 Equivalence of Certain Martingales . . . 4.3 Ito Processes and Stochastic Integration. 4.4 Ito's Formula . . . . . . . . . . . . . . 4.5 Ito Processes as Stochastic Integrals .. 4.6 Exercises . . . . . . . . . . . . . . . . . .

82 82 85 92 104

Chapter 5. Stochastic Differential Equations 5.0 Introduction . . . . . . . . 5.1 Existence and Uniqueness

122 122 124

107

111

Contents

VIJI

131 132 134 136

5.2 On the Lipschitz Condition . . . . . . . . . . . . . . . 5.3 Equivalence of Different Choices of the Square Root 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 6. The Martingale Formulation 6.0 Introduction . . . . . . . . . . 6.1 Existence . . . . . . . . . . . . 6.2 Uniqueness: Markov Property 6.3 Uniqueness: Some Examples . 6.4 Cameron-Martin-Girsanov Formula 6.5 Uniqueness: Random Time Change. 6.6 Uniqueness: Localization 6.7 Exercises . . . . . .

145 149 152 157 161 165

Chapter 7. Uniqueness 7.0 Introduction . . . . 7.1 Uniqueness: Local Case 7.2 Uniqueness: Global Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

171 171 174 187 190

136

139

Chapter 8. Ito's Uniqueness and Uniqueness to the Martingale Problem 8.0 Introduction . . . . . . . . . . . . 8.1 Results of Yamada and Watanabe 8.2 More on Uniqueness. . . . . . 8.3 Exercises . . . . . . . . . . . . . .

195 195 195 . 204 . 207

Chapter 9. Some Estimates on the Transition Probability Functions 9.0 Introduction . . . . . . . . 9.1 The Inhomogeneous Case 9.2 The Homogeneous Case

. . . .

208 208 209 233

Chapter 10. Explosion . . . . . 10.0 Introduction . . . . . . . . 10.1 Locally Bounded Coefficients 10.2 Conditions for Explosion and Non-Explosion 10.3 Exercises . . . . . . . . . . . . . . . . . . . . .

. . . . .

248 248 249 254 259

Ito

Chapter 11. Limit Theorems . . . . . . . 11.0 Introduction . . . . . . . . . . . . . . . . . . . . 11.1 Convergence of Diffusion Process . . . . . . . 11.2 Convergence of Markov Chains to Diffusions 11.3 Convergence of Diffusion Processes: Elliptic Case. 11.4 Convergence of Transition Probability Densities 11.5 Exercises . . . . . . . . . . . . . Chapter 12. The Non-Unique Case . 12.0 Introduction . . . . . . . . . . . . 12.1 Existence of Measurable Choices 12.2 Markov Selections . . . . . . . .

261 261 262 266 272 279 283 285 285 286 . 290

ix

Contents

12.3 Reconstruction of All Solutions 12.4 Exercises . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . A.O Introduction . . . . . . . . . . . . . . . . . . . . . . . A.l Lp Estimates for Some Singular Integral Operators A.2 Proof of the Main Estimate A.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

296 302 . . . .

304 304 306 315 323

Bibliographical Remarks ..

328

Bibliography

331

........ .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Frequently Used Notation

I. Topological Notation. Let (X, p) be a separable metric space. 1) BO is the interior of B s; X. 2) B is the closure of B s; X. 3) oB is the boundary of B s; X.

4) 5) 6) 7) 8) 9)

PAx is the Borel field of subsets of X. Cb(X) is the set of bounded continuous functions f:X ..... R. B(X) is the set of bounded 31 x-measurable f:X ..... R. V p(X) is the set of bounded p-uniformly continuous f:X ..... R. M(X) is the set of probability measures on (X, 31 x ). Ilfll = suplf(x)1 for fE B(X). xeX

II. Special Notation for Euclidean Spaces

1) Rd is d-dimensional Euclidean space. d

3)

Ixl = (L>W /2 for x E Rd. 1 B(x, r) = {y E Rd: Ix - YI < r}.

4)

(x, y) = ~>jYj for x, Y E Rd.

2)

d

1

5) Sd-l ={xERd : Ixl= I}. 6) C(Rd) = {IE Cb(Rd): lim fix)

= O}.

7) C o( 0 there is a X such that

1.1.4 Theorem. Let

compact set K

~

inf Jl(K)

~

1-

f..

/lEr

Then

r

is pre compact in M(X) (i.e.,

r

is compact).

Proof We first recall that if X itself is compact, then by Riesz's theorem and standard results in elementary functional analysis, M(X) is compact. In general we proceed as follows. Choose a metric p on X equivalent to the original one such that (X, p) is totally bounded and denote by X its completion. Then X is compact and we can think of M(X) as being a subset of M(X). Thus it remains to show that if {Jln}f ~ rand Jln --+ j1 in M(X), then j1 can be restricted to X as a probability measure Jl, and Ji.n --+ Ji. in M(X). But, by our assumption on r, there is a sequence of compact sets {K/}, I ~ 1 in X such that Ji.n(K/) ~ 1 - 1/1 for n ~ 1. Since K/ is compact and therefore closed in X, it follows from Theorem 1.1.1 that j1(K/) ~ limn~oo Ji.n(K/) ~ 1 - 1/1. Thus

1.1. Weak Convergence, Conditional Probability Distributions, and Extension Theorems

and so we can restrict ji. to (X, J1.n --+ ji. in M(X) lim

n-co

for all cP

E

~x)

I

11

as a probability measure J1.. Finally since

cP dJ1.n =

I

cP dJ1.

U p(X), and by Theorem 1.1.1 this implies that J1.n --+ J1. in M(X).

Remark. Note that whereas Theorem 1.1.3 relies very heavily on the completeness of X, Theorem 1.1.4 does not use it at all. 1.1.5 Corollary. Let F £; Cb(X) be a uniformly bounded set of functions which are equicontinuous at each point of X. Given a sequence {J1.n}f £; M(X) such that J1.n --+ J1. in M(X), one has lim sup

n-oo cpeF

II

cP dJ1.n -

I

cP dJ1.1 =

o.

Proof Suppose there exists 8 > 0 such that lim sup sup n-+oo

II' E F

II

cP dJ1.n -

I

cP dJ1.1 >

8.

By choosing a subsequence if necessary, we can assume that for each n there is a CP1l in F such that

M = SUPrpEF Ilcpll and choose a compact set K in X such that J1.n(X\K) :s; 818M. It follows from this that J1.(X\K) :s; 818M. From the Ascoli-Arzela theorem and the Tietze extension theorem, we can find a t/I in Cb(X) such that It/I I :s; M, and a subsequence CP1li = t/l i of {CP1l} converging to t/I uniformly on K. But then

Let

sUP1l~ 1

IJ t/li dJ1.1Ii - Jt/li dJ1.1 :s; IJ (t/li -

t/I) dJ1.1Iil + IJ (t/li - t/I) dJ1.1

+ 11' t/I dJ1. ni =

f t/I dJ1.1

I'Kr (t/li - t/I) dJ1.ni l + IfX\K (t/li - t/I) dJ1.nil + IfK(t/li - t/I) dJ1.1 + IfX\K (t/li - t/I) dJ1.1

+ IJ t/I dJ1. ni -

Jt/I dJ1.l·

12

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Therefore 0< c :::;; lim )~OO

II I/Ij dJ1.nj - f I/Ij dJ1.,1

c < 0 + 2M· 8M

c

+ 0 + 2M· -8M + 0

2 which is a contradiction.

0

We now turn to the study of conditional probability distributions. Let (E, ff, P) be a probability space and I: s; ff a sub a-field. Then the conditional expectation of an integrable function f (') is another integrable function g (') which is I:-measurable and satisfies:

(1.2)

t

g(q)P(dq)

= tf(q)P(d q )

for all A

E

I:.

The function g(.) exists and is unique in the sense that any two choices agree almost surely with respect to P. (See Halmos [1950], for instance, to find a proof of the existence and elementary properties of conditional expectations.) The function 9 is denoted by E[J II:]. If we want to call attention to the measure P that is used, we use £P[J II:] in place of E[J II:]. In the special case when the function f(·) is the indicator function lB{q) of a set B in ff we refer to the conditional expectation as the conditional probability and denote it by P{B II:). It has some elementary properties inherited from the properties of conditional expectations. For instance if B j and B2 are disjoint sets in ff

Since P{B II:) can be altered on a set of measure zero for each B E ff it is important to know if one can choose a "nice" version of P{B II:) such that P{B II:) is a countably additive probability measure on ff for each q E E. Such a choice, if it can be made, will be called a conditional probability distribution and will be denoted by {Qq{B)}.

Definition. A conditional probability distribution of P given I: is a family Qq of probability measures on (E, .~) indexed by q E E such that (i) For each B E .~, Qq(B) is I:-measurable as a function of q (ii) For every A E I: and B E .~

P{A n B)

= I Qq(B)P(dq). 'A

1.1. Weak Convergence, Conditional Probability Distributions, and Extension Theorems

13

In general a conditional probability distribution need not exist However if we replace (E, ~) by a Polish space X and its Borel a-field fJ6, then for any P on (X, dB) and any sub a-field I: s; fJ6 a conditional probability distribution of P given I: exists. We state and prove this as a theorem.

1.1.6 Theorem. Let X be a Polish space and .JJ its Borel a-field. Let P be a probability distribution on (X, 26') and I: s; f!4 a sub a-field of fJ6. Then a conditional probability distribution {Qx} of P given I: always exists. Proof Since X is a Polish space, there exists on X an equivalent metric p such that

(X, p) is totally bounded. Therefore the space Up(X) of uniformly continuous bounded functions on (X, p) is separable. Let {Jj}f be a countable subset of U p(X) such that fl ( • ) == 1, {f}} f are linearly independent and the linear span W of {f}}f is dense in U p(X). We denote by gj some version of E[/; iI:). We can assume, without loss of generality, that g d .) == 1. For n ? 1 let An be the set of n-tuples h, r 2, ... , rn) with rational entries such that for all

x

E

X.

The set An is clearly countable. From the properties of conditional expectations,

for almost all x. Thus if we let, for each (rl' ... , rn)

then Frr I .... roj

E

I: and P[ F rrl .

. .• roj]

oc

F=

U

E

An'

= O. Therefore if

U

Then F E I: and P(F) = O. Choose x E X\F and fix it. We now define the linear functional Lx on W by

where fEW is written uniquely as

for some n and real numbers t], (2' ... , tn' We want to show now that LAf) is a nonnegative linear functional on W. Suppose fEW is non-negative. Then r 1.11 + ... + tn In = f ? O. Given any rational number I: > 0 we can find rationals (r[> ... , rn) such that

14 and

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Irj - tjl

< e for 1 5,j 5, n. Therefore

From the definition of the set F, it follows that

By letting e -+ 0 over the rationals, we conclude that t1g1(X) + ... + tngn(x) ~ 0, or equivalently Lx(f) ~ O. Since LAid = 1 and W is dense in V p(X), Lx(f) defined on Wextends uniquely as a non-negative linear functional on V p(X). We continue to call this extension Lx(f). We can view the space VAX) as the space C(X), where X is the completion of the totally bounded space (X, p). Note that X is compact. By the Riesz Representation Theorem, there is a probability measure Qx on (X, ~x) such that

for all] E C(X). Thus we have shown that for all x (X, [}Ix) such that

E

X\F, there is a probability measure Qx on

(1.3) for all i. (Here we use the notation]to denote the extension of ani E V AX) to X.) This shows that the mapping

x

-+

JJ(y)QAdy)

on X\F is I:[ X\F]-measurable for all lEW, and therefore for all Moreover, it is easy to see that ( 1.4)

e

[J J(y)Q.(dy),

A n (X\F)] =

E

S

V p(X) so

V p(X).

e[j(·), A]

for alII E V p(X) and A E I:. Given a compact set K in X, choose { for all n. We will produce a point q E nn Bn' We can assume, without loss of generality, that Bn E .?n· For ~ n ~ m and B E .?n we define n m . n(q, B) to be XB(q). For n > m and BE.? n we define n m . n(q, B) inductively by

°

°

nm . n(q, B)

=

I nn(q', B)nm .

n-

I(q, dq').

18

I. Preliminary Material: Extension Theorems, Martingales, and Compactness

Clearly P(B) =

I nO, "(q, B)Po(dq)

for

BE.?",

We also have for n > m

for BE.?", Define n 2': 0,

Then F~+ 1 £: F~ and

Therefore Po(nn F~) 2': e/2 and we can find qo tJ- No such thatnO,n(qo, Bn) 2': e/2 for all n 2': 0, Suppose we have found qo, ql,,,, ,qm E E such that qk E Ak-I (qk-d\Nk-1 for 1 ~ k ~ m and nk,n(% Bn) 2': e/2 k+' for 1 ~ k ~ m and n 2': 0, Let

Then F~N £: F~+ 1 and for n> m

Hence

1

1) > e/2m+ 2,

nm+ (q m' n°OI Fnm+

"=0

--

nO'

We can therefore conclude that Am(qm) Il F~+ 1 =1= 0; and so there is a qn+1 E An(qn)\Nn with the property that nm+!,n(qm+I,Bn) 2': e/,i m+2 for all n 2': 0, By induction on m, we have now shown that a sequence {qm}O' exists with the for all n 2': 0. In properties that qm+! E Am(qm)\Nm and infn nm,n(qm, Bn) > particular XBm (qm) = nm,m(qm, Bm) > and therefore Am(qm) c Bm, since Bm E .?m. Thus Am(qm) c Bm. Finally, qN E n~ Am(qm) for all N 2': 0, and so (1.9) implies that n~ Am(qm) =1= 0, This completes the proof. 0

nm

nm

°

°

We will now establish Kolmogorov's extension theorem for product spaces, Let IX E I let X a be a Polish space with its Borel

I be an infinite index set and for each

19

1.2. Martingales

rIa.

a-field fJ6 a • For sets F c I we denote by X Fthe product space F X a and by fJI F the product a-field E F fJ6. on X F' For sets G ~ F =f let a~ denote the canonical projection from XG onto X;" We denote a~ by a F.

rIa

o

1.1.10 Theorem. Suppose thatfor eachfinite set F we are given a probability measure PF on (XF' 3B F) such thatJor any two finite sets 0 =f F c G, P F = PG(a~r I. Then there is a unique probability measure P on (XI' 311 ) such that P F = Pai 1 for all finite F =f 0. Proof Uniqueness is obvious from the fact that fJl l is the smallest a-field generated by d = UF ai 1 (fJ6 F). To prove existence, we observe that there exists a finitely additive P on d such that PF = Pai I. Now suppose that {An}O" Ed is a nondecreasing sequence such that An 10. Without loss of generality, we will assume that An E 3B Fn , n ~ 0 where 01= Fo c FI ... c Fn ... and {Fn} are strictly increasing finite sets. Define :Fn = ainl(fJ6 FJ for n ~ O. It is easy to check that {.'¥ n' n ~ O} satisfy (1.9). Next, let {Q~} be a regular conditional probability distribution of PFn given (a~:-lrl(.qeFn-J and define

for all q E X I and B E .'¥ n' It is easily checked that nn is a transition function from (XI' .'¥n- d to (XI' :Fn) and that it satisfies the condition of Theorem 1.1.9. Thus by that theorem there is a unique probability measure P on (X I, a(Uo :F n)) such that P equals PFoaiol on :Fo and

P(B) =

f nn(q, B)P(dq)

for all BE .'¥n. By induction, we see that P equals P on UO" .'¥n. In particular P(An) = P(A n), the countable additivity of P implies that P(An)lO, and the theorem is proved. D

1.2. Martingales Throughout this section, E will denote a non-empty set of points q. :F is a a-algebra of subsets of E, and {:F,: t ~ O} is a non-decreasing family of sub aalgebras of :F. Given s ~ 0 and a map lJ on [s, (0) x E into some separable metric space (X, D), we will say that lJ is (right-) continuous if lJ( " q) is (right-) continuous for all q E E. If P is a probability measure on (E,:F) and lJ: [s, (0) x E -+ X, we will say that () is P-almost surely (right-) continuous if there is a P-null set N E :F such that lJ(" q) is (right-) continuous for q ¢ N. Given s ~ 0 and lJ on [s, 00) x E into a measureable space, lJ is said to be progressively measurable with respect to {:F t : t ~ O} after time s if for each t ~ s the restriction of lJ to [s, t] x E is ~[s. 'I x :Ft-measurable. Usually there is no need to

20

I. Preliminary Material: Extension Theorems, Martingales, and Compactness

mention s or {ff t : t 20}, and one simply says that 0 is progressively measurable. Note that 0 is progressively measurable with respect to {ff t : t 2 O} after time s if and only if Os, defined by Os(t, q) = O(s + t, q), is progressively measurable with respect to {ff t + s : t 20} after time O. Thus any statement about a progressively measurable function after time s can be reduced to one about a progressively measurable function after time O. This remark makes it possible to provide proofs of statements under the assumption that s = 0, even though s need not be zero in the statement itself. Exercises 1.5.11-1.5.13 deal with progressive measurability and the reader should work them out. The following lemma is often useful.

1.2.1 Lemma. If 0 is a right-continuous function from [s, 00) x E into (X, D) and if O(t, .) is fft-measurable for all t 2 s, then 0 is progressively measurable.

Proof Assume that s = O. Let t 2 0 be given and for n 2 1 define 0n(u, q) = 0 (

[nu] + 1 n

1\

) t, q .

Clearly On is 81[0, tl x fft-measurable for all n. Moreover, as n ~ 00, On tends to 0 on [0, t] x E. Hence 0 restricted to [0, t] x E is 36[0. tl x fft-measurable. 0 Given a probability measure P on (E, ff), s 20, and a function 0 on [s, 00) x E into C, we will say that (O(t), ff" P) is a martingale after time s if 0 is a progressively measurable, P-almost surely right-continuous function such that O(t) = O(t, .) is P-integrable for all t 2 sand

(2.1)

(a.s., P),

The triple (O(t), ff" P) is called a submartingale after time s if 0 is a real-valued, progressively measurable, P-almost surely right continuous function such that O(t) is P-integrable for all t 2 sand

(2.2)

(a.s., P),

In keeping with the remarks following the definition of progressive measurability, we point out that s plays a rather trivial role here and that any statement proved for the case s = 0 can be proved in general simply by replacing 0 by Os and {ff t: t 2 O} by {ff t + s: t 2 O}. Thus, although theorems will be stated for general s, they will be proved under the assumption that s = O. Usually, it will not even be necessary to mention when the (sub-) martingale begins and we will simply say the (O(t), ff" P) is a (sub-) martingale. We begin our study of martingales and submartingales with the following lemma. Like nearly everything in this theory, the original version is due to J. L. Doob. Thus we will be somewhat lax in the assignment of credit before each theorem.

21

1.2. Martingales

1.2.2 Lemma. Let (O(t), !Ft , P) be a submartingale after time s with values in a closed interval I s R. Assume that g is a continuous, non-decreasing, convex function on I into [0, 00) such that g 0 O(t) is P-integrablefor all t ~ s. Then (g 0 O(t), !Ft> P) is a submartingale after time s. In particular, if (O(t), ''#'1> P) is a martingale or a non-negative submartingale after time s and r is a number greater than or equal one such that 1O(t) I' is P-integrable for all t ~ s, then (I O(t) 1', !F t , P) is a submartingale after time s.

Proof Assume that s = O. By the version of Jenssen's inequality for conditional expectation values:

Thus, if 0

~ tl

< t 2 , then E[g(O(t2)) 1 !FtJ ~ g(E[O(t2) 1 !FtJ) ~ g(O(t 1))

a.s.

This completes the proof of the first assertion. The second assertion is immediate in the case when (O(t), !F t , P) is a non-negative submartingale; simply take g(x) = x' on [0, 00). Thus the proof will be complete once we show that (I O(t) j, !Ft' P) is a submartingale if (O(t), !Ft , P) is a martingale. But if(O(t), !FI> P) is a martingale, then (a.s., P), and so ( 1O(t) I, !Ft> P) is a submartingale.

0

1.2.3 Theorem. If (O(t), !Ft> P) is a submartingale after time s, thenfor any;' > 0 and

all T> s:

(2.3) I n particular, if 0 is non-negative, then

(2.4)

p( sup O(t) ~ ;.) ~ ~ E[O(T)], S5:t5: T

and for all r > 1 (2.5)

E [( sup O(t))'] II' sSlsT

~ ~ 1 E [O(T)'J 'I' r

(in the sense that the right-hand side is infinite if the left-hand side is).

22

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Proof. Assume that s = O. Relation (2.4) is immediate from (2.3) and (2.5) follows from (2.3) by the nice real-variable theory lemma mentioned in Exercise 1.5.4. Since 8 is P-almost surely right-continuous, (2.3) will be proved once we show that for any n ~ 1 and 0 = to < ... < tn = T:

PC~~:n8(tk) ~ A) ~ 1E[8(T), 0~~:n8(tk) ~ A]. To this end, define Ao = {8(t o) ~ A}, and for 1 ~ k ~ n set max O( tJ < A}.

and

O:S;i:s;k

Clearly Ai

(i

Aj =

o ~ k ~ n. Hence

0

if i :1= j, {maxO~"S" 8(tk)~ A} =

Uo Ab

and A" E ~tk for

1 [ ~;t1 ~n E[O(T), A k ] =;tE 8(T), o":::nO(tk ) ~ A] ,

and the proof is complete.

0

We now want to introduce the important concept of a stopping time. A function t: E --+ [0, 00) U {oo} is called a stopping time (relative to {~t: t ~ O}) if, for all t ~ 0, {t ~ t} E ~t. Examples of stopping times are plentiful. Certainly if t == t for some t ~ 0, then t is a stopping time. A not quite so trivial example is obtained by considering the first time that a right-continuous, progressively measurable 0 comes close to a closed set c:

= inf{t ~ 0:

t

[O(u): 0 ~ u ~ t]

(i

C:j: 0}.

In this connection, observe that the last time that 0 leaves C is not a stopping time since one has to know the entire history ofthe path 0(·, q) in order to determine if it never visits C after time t. The question of when the entrance time tA

= {inf t ~ 0:

O(t)E A}

is a stopping time is a difficult but interesting problem. If the trajectories are continuous and A is a closed set, then tA is easily seen to be a stopping time. For the general case, see Dynkin [1965], Chapter IV, Section 1 for a discussion of the measurability ofvarious entrance times relative to the completed a-algebras. Since we will be working exclusively with processes that are right continuous and almost surely continuous for every closed set C, the contact time t

= {inf t ~ 0:

[8(u): 0

~

u ~ t]

(i

C:1= 0}

is a stopping time and agrees almost surely with the entrance time tc.

23

1.2. Martingales

Given a stopping time r, we define fF, = {A

E

fF: A

{r ~ t}

II

E

fF,

for all

t ~ O}.

It is easy to check that fF, is a sub a-algebra of fF and that fF, = fF, if r == t. Intuitively, fF, should be thought of as the set of events" before time r". (Lemma 1.3.2 in the next section makes this intuitive picture precise in the case of a path space.) The following lemma collects together some elementary facts about stopping times.

1.2.4 Lemma. If r is a stopping time, then r is fF,-measurable. If r is a stopping time and 0 is a progressively measurable function, then O(r) == O(r(·), .) is fF,-measurable on {r < oo}. Finally, given stopping times 0' and r:

(i)

+ r, 0' v r, and 0' 1\ r

are stopping times, (ii) if A E fF", then A II {a < r} and A II {a ~ r} are in fF"A" (iii) if 0' ~ r, then fF" s:;; fF,. 0'

Proof The proof that r is fF,-measurable is left to the reader. Suppose that 0 is progressively measurable and let t ~ 0 be given. Definefr on ({r ~ t}, fF,[{r ~ t}]) by fr(q) = (r(q) At, q). Thenfr is measurable into ([0, t] x E, ~[O.'l x fF,). Since 0 restricted to [0, t] x E is ~[O"l x fF,-measurable, it follows that ~ 0 fr is

fF,[{r ~ t}]-measurable on {r ~ t}. But 0 fr is just the restriction of 0 to {r ~ t}, and so the assertion that O(r) is fF,-measurable on {r < oo} has been proved. The proof of (i) is easy and is left to the reader. To prove (ii), we first show that {a < r} and {r ~ a} are in fF" II fF T • By an obvious complementation argument, it suffices to prove that {a < r} E fF" II fF T' Given t ~ 0, let Q, denote the rational numbers in [0, t]. Then 0

{a < r}

II

{r ~ t} =

U{a ~ s} II {r > s} II {r ~ t}

seQ.

and the right-hand side is certainly a member of fF,. This proves that E fF,. To show that {a < r} E fF", note that

{a < r}

{a < r}

II

{a

~

t}

=

({a

~

t}

II

{r> t}) u ({a < r}

II

{r ~ t});

and ({a ~ t} II {r> t}) E fF, by definition, whereas ({a < r} II {r ~ t}) E fF" since we have just seen that {a < r} E fF,. We now know that both {a < r} and {a ~ r} are in fF" II fF r . To complete the proof of (ii), let A E fF" and t ~ be given. Then

°

(A

n {a < r}) n {a A r

~

t} = (A n {a

~

t}) n ({a < r} n {a

~

t})

E!Fe

because {a < r} E fF" and

(A

II

{a

~

r})

II

{a 1\ r

~

t} = (A

II

{a

~

t})

II

({a

~

r}

II

{r

~

t})

E

fF,

24

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

since {O' ~ r} of (ii). 0

E

!F,. Now (ii) is proved. Finally, (iii) is obviously just a special case

We now tum to Doob's optional stopping time theorem. Let 0' and r be stopping times with values in the finite set {to, ... , t N}, where 0 = to < ... < tN = T, and assume that 0' ~ r. Given a martingale (O{t), !F" P} and A E !Fa' we have:

E[O{r), A] =

N

L E[O{tk), A n {r =

k=O N

L E[O(T), A n

=

k=O

=

N

L E[O{t k ), A n

"=0

tk}]

{r = tk }] {O'

= t,,}] = E[O{O'), A),

since A E!F a £; !F, and therefore A n {r = t,,} and A n {O' = t,,} are in !F't. This proves that

(2.6)

£P[O{r) 1!Fa] = O{O')

a.s.

Next suppose that (O{t), !F" P) is a submartingale and define A{O) = 0 and A{t,,) - A{t,,_ d = (E[O{t,,)I!F'H] - O{t,,_ d) for 1 ~ k ~ N. Define ,,(T) = O{T) and

for 0 ~ k < N. Finally, set M{t) = ,,(t /\ T) - A{t /\ T). Then (M{t~ !F" P) is a martingale, and so, by (2.6~

E[M{r)l§a] = M{O') Since A{r)

(2.7)

~

a.s.

A{O'), it follows that E[O{r) 1!Fa]

~

O{O')

a.s.

1.2.5 Theorem. Let 0' and r be bounded stopping times such that s ~ 0' P) is a non-negative submartingale after time s, then

§"

(2.8)

E[O{r) 1§ a]

O{O')

a.s.

E[O{r)l§a] = O{O')

a.s.

~

and if (O{t~ !F" P) is a martingale, then (2.9)

~

r. If{O{t),

25

1.2. Martingales

Finally, if(O(t), ~" P) is a non-negative submartingale after time sand T> s, then {O(r v s): r is a stopping time bounded by T} is a uniformly P-integrable family. Proof Assume that s = O. Let (O(t), ~" P) be a non-negative submartingale. Choose T > 0 so that U :::; r :::; T - 1. Given n ~ 1, define _[nu]+l

Un -

n

an

d

_[nr]+l rn . n

Then Un and rn are stopping times and Un S rn s T. Since finite number of values, we have from (2.7) that:

(2.10)

E[O(rn)l~aJ ~

O(u n )

a.s.

(2.11 )

E[O(T) I~tJ ~ O(rn)

a.s.

(2.12)

E[O( T) I~ aJ ~ O(U n)

a.s.

Un

and rn take only a

From (2.11) and (2.12), respectively, we know that

and

Since O(T)

~

0, these imply:

E[O(rn)' O(rn)

~

).]

s E[O(T),

sup O(t) ~ ).] OSlsT

and

But, by (2.41

and so we have proved that {O(u n ): n ~ I} and {O(rn): n ~ 1} are uniformly Pintegrable. Since 0 is P-almost surely right continuous, we now know that O(un )-+ O(u) and O(rn) -+ O(r) in I!(P). Finally, if A E ~ a' then A E ~ an' since U S Un; and so by (2.10):

Letting n -+

00,

we now get (2.8).

26

I. Preliminary Material: Extension Theorems, Martingales, and Compactness

The proof that {O(r): r a stopping time bounded by T} is uniformly integrable when (O(t), fFp P) is a non-negative submartingale is accomplished in exactly the same way as we just proved that {O(rn): n ~ I} is uniformly integrable. The details are left to the reader. Finally, suppose that (O(t), fFr' P) is a martingale. Then (I O(t) I, fFp P) is a non-negative submartingale. Thus if an and rn are defined as in the preceding, O(rn) n ~ I} and O(a n ) n ~ I} are uniformly P-integrable families and so O(rn) -+ O(r) and O(a n) -+ O(a) in L1 (P). Since, by (2.6),

{I

I:

{I

I:

E[O(rn)lfFaJ = O(a n)

a.s.

for all n ~ 1, the rest of the argument is exactly like the one given at the end of the submartingale case. 0

1.2.6 Corollary. Let r: [0, (0) x E -+ [s, (0) be a right-continuous function such that r(t, .) is a bounded stopping timefor all t ~ 0 and r(" q) is a non-decreasingfunction for each q E E. If (O(t), fF" P) is a (non-negative sub-) martingale after time s, then (O(r(t)), fFt(r) , P) is a (non-negative sub-) martingale after time O. 1.2.7 Corollary. If r ~ s is a stopping time and (O(t), fF" P) is a (non-negative sub-) martingale after time s, then (O(t A r), fF" P) is a (non-negative sub-) martingale after time s. Proof By Corollary 1.2.6 (O(t then for t2 > t1:

A

r), fFr,\t, P) is a (sub- )martingale. Thus if A E fF r1 , (~)

E[0(t2 Ar), A (\ {r > tdl = E[0(t1 Ar), A (\ {r > td], since A (\ {r > td {r ~ td, and so:

E

fF rlAt . On the other hand, 0(t2 Ar)

= O(r) = 0(t1 Ar) on

(~)

E[ O( t 2 Ar), A (\ {r ~ t I}] = E[ O( t 1 Ar), A (\ {r ~ t d]' Combining these, we get our result.

0

The next theorem is extremely elementary but amazingly useful. It should be viewed as the" integration by parts" formula for martingale theory. 1.2.8 Theorem. Let (O(t), fFr' P) be a martingale after time sand '1: [s, (0) x E -+ C a continuous, progressively measurable function with the property that the variation 1'1I(t, q) of'1(', q) on [s, t] is finite for all t ~ sand q E E. Iffor all t ~ s

(2.13 )

then (O(t)'1(t) - J~ O(s)'1(ds), fF" P) is a martingale after time s.

27

1.2. Martingales

Proof Assume s = O. Using Exercise 1.5.5, one can easily see that J~ O(up,(du} can be defined as a progressively measurable function. Moreover, (2.13) certainly implies that O(t},,(t} - J~ O(up,(du} is P-integrable. Now suppose that 0 S tl < t2 and that A E :F'I' Then

since

r

But if I.l = t2 - t1> then

(0{t2) - O{u)),,(du) =

!~~ ktl (0{t 2) - o( t x (" ( t 1

+ ~ I.l ) -

1

+ ~ I.l ) )

,,( t 1

+k:

1 I.l ) )

a.s.,

and, by {2.13} and the Lebesgue dominated convergence theorem, the convergence is in I!{P). Finally,

for all n

~0

and 1 S k S n. Thus

E [( (0(t2) - O(u)),,(du), A] = 0 and the proof is complete.

0

The final topic of the present section is a theorem which will serve us well in what follows. Basically, this result shows that the martingale property is invariant under certain ways of conditioning a measure. Before we state the theorem, we need the following lemma, which is often useful. 1.2.9 Lemma. Let 0: [s, 00) x E -+ C be a progressively measurable, P-almost surely right continuous function such that O(t) is P-integrablefor all t ~ s. Let D £; [s, 00) be a countable dense set. If 0 is non-negative and

(2.14) for all t 1, t 2 E D such that t 1 < t 2, then (O(t), after time s. If

a.s. ~"

P) is a non-negative submartingale

(2.15) for all t 1, t2 ED such that tl < t 2, then (O(t),

a.s. ~I>

P) is a martingale after time s.

28

I. Preliminary Material: Extension Theorems, Martingales, and Compactness

Proof Assume that s = O. Clearly the proof boils down to showing that in either case the family {I O(t) I: t E [0, T] n D} is uniformly P-integrable for all TED. Since (2.15) implies (2.14) with 10(')1 replacing 0('), we need only show that non-negativity plus (2.14) implies that {O(t): t E [0, T] n D} is uniformly Pintegrable. To this end, we mimic the proof of (2.4), and thereby conclude that

p(

sup rE[D,rlnD

O(t)~A)slE[O(T)]'

A>O.

Combining this with

E[O(t), O(t)

~

A]

s

E[O(T), O(t)

s we conclude that {O(t): t

E

~

A]

E [O(T),

sup rE[O,rlnD

O(t)

~ ;.], t E [0, T]

[0, T] n D} is uniformly P-integrable.

n D,

0

1.2.10 Theorem. Assume that for all t ~ 0 the a-algebra :Fr is countably generated. Let r ~ s be a stopping time and assume that there exists a conditional probability distribution {Qq'} of P given :F,. Let 0: [s, (0) x E ~ Rl be a progressively measurable, P-almost surely right-continuous function such that O(t) is P-integrable for all t ~ s. Then(O(t),:F ro P) is a non-negative submartingale after time s if and only if (O(t 1\ r), :Fr, P) is a non-negative submartingale after time s and there exists a P-null set N E:F, such thatfor all q' ¢ N, (O(t)X[s,rl(r), :Fr, Qq') is a non-negative submartingale after time s. Next suppose that 0: [s, (0) x E ~ C is a progressively measurable, P-almost surely right-continuous function such that O(t) is P-integrable for all t ~ s. Then (O(t), :Fro P) is a martingale after time s if and only if(O(t 1\ r),:F p P) is and there is a P-null set N such that (O(t) - (J(t 1\ r), :Fro Qq') is a martingale after time s for all q' ¢ N.

Proof Assume that s = O. We suppose that ((J(t), :Fro P) is a martingale. Then by Corollary 1.2.7 so is (O(t I\r), :F" P). Let 0 S t1 < t 2 , BE:F, and A E ~rl be given. Then EP[EQ'[0(t 2 ), A], B n {r S t 1}] = e[(J(t 2 ), A n B n {r S t 1}] =

e[O(td, A n B n {r S t 1}]

=

EP[EQ'[O(td, A), B n {r S td)'

Here we have used the fact that A n B n {r S t 1} is in :Fri' Since B E :F, is arbitrary this implies that for P-almost all q' satisfying r(q') S t 1

29

1.2. Martingales

Taking a single null set for a countable subalgebra of sets A generating $l'1 we obtain a null set N 'I '12 such that, for q' ¢ N'l, 12 and r(q') ~ t I •

We now take a countable dense set D in [0, (0). We can then find a single null set N such that for q' ¢ N

EQ-IO(t 2) 1 $l,,]

=

O(td a.s. Qq'

provided t l , t2 ED and t2 ~ tl ~ r(q'). From Lemma 1.2.9 we can now conclude that for q' ¢ N, (O(t), $l" Qq.) is a martingale for t ~ r(q'). This can of course be restated as (O(t) - O(t A r(q')~ $l" Qq') is a martingale for t ~ O. Since

P[q': Qq,[q: r(q) = r(q')] = 1] = 1

(2.16)

we are done. Now suppose (O(t), $l" P) is a non-negative sub-martingale. Then by Corollary 1.2.7 so is (O(t A r), $l" Pl. By replacing equalities by the obvious inequalities we can conclude that there is a null set N in $lt such that (O(t), $lt, Qq') is a non-negative submartingale for t ~ r(q') provided q' ¢ N. We note that this is equivalent to X[O, t)(r(q'))O(t) being a non-negative submartingale for t ~ O. Again by (2.16) we are done. We now turn to the converse proposition. If o ~ tl < t2 and A E $l'1 are given, in the martingale case

e[0(t 2), A]

=

EP[EQ"[0(t 2),

= EP[~-'[0((t2

An A

r(q')) v td, A]]

=

e[EQ"[0((t2 A r) v t l , A]]

=

EP[O(t l ), A n {r

~ t l }]

+ e[O(r A( 2), A

= EP[O(td, An {r

~ t l }]

+ EP[O(r Atd, A n {r> t l }]

n {r > td]

= EP[O(t d; A]. The submartingale case is proved in the same manner by replacing the equalities by inequalities at the relevant steps. 0

1.2.11 Remark. It is hardly necessary to mention, but for the sake of completeness we point out that everything we have said about almost surely right-continuous martingales and submartingales is trivially true for discrete parameter martingales and submartingales. That is, if (E, $l, P) is a probability space, {$l,,: n ~ O} a non-decreasing family of sub a-algebras, {O,,: n ~ O} a sequence of P-integrable complex valued random variables, such that 0" is $l,,-measurable, then (0", $l", P) is a martingale (submartingale) if (0" is real-valued) and

E[O(n

(~)

+ 1)I$l,,] = O(n)

a.s.

30

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

for all n ~ O. The obvious analogues of (1.2.4) through (1.2.10) now hold in this context. Indeed, if one wishes, it is obviously possible to think of this set-up as a special case of the continuous parameter situation in which everything has been made constant on intervals of the form [n, n + 1).

1.3. The Space C([O, CX)); Rd ) In this section we want to see what the theorems in Section 1.1 say when the Polish space is C([O, 00); R d ). The notation used in this section will be used throughout the rest of this book. Let 0 = Od = C([O, 00); Rd ) be the space of continuous trajectories from [0, 00) into Rd. Given t ~ 0 and WE 0 let x(t, w) denote the position of win Rd at time t. Define

D(w,w')=

f

~

sUPo';f,;nlx(t,w)~x(t,w')I,_

n=12 1 + sUPo,;r,;nlx(t, w)-x(t,w)1

on 0 x O. Then it is easy to check that D is a metric on 0 and (0, D) is a Polish space. The convergence induced by D is uniform convergence on bounded tintervals. We will use A to denote the Borel a-field of subsets of (0, D). Clearly the map x(t) given by w -. x(t, w) is D continuous and therefore measurable, for each t ~ 0. Thus

a[x(t): t On the other hand, if WO

E

n,

~

t ~ 0 and

0]

I: >

O.

S n})

n) + Pk(A m n {Tk> n}).

Letting m and then k -> 00, we see that Fn(Am) -> 0 as m -> 00. A similar argument shows that Fn+ 1 equals Pn on ...I1n for n 2: O. Thus, by the preceding paragraph, there is a P on (0, ...11) such that P equals Fn on ...11 n for all n 2: O. Finally we must check that P equals P k on ...I1tk . Given A E ...I1tk n ...11 n' we have that

P(A) = Pn(A) = lim P/(A n {T/ > n}). /-00

But for I 2: k

IPk(A) -

P/(A n {T/ > n}) I = P/(A n {T/

S;

n}) -> 0

36

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

as 1-+ 00. Thus P(A) = Pk(A) for A E .At. (\ .An. But, by Lemma 1.2.2, .Atk is generated by the maps x(t 1\ Tk), for t ~ O. Therefore

Hence P equals Pk on .Atk , and we are done.

0

1.4. Martingales and Compactness In the preceding section we developed necessary and sufficient conditions for the compactness of measures on (n, .A). Like most general results, these conditions are not particularly useful when applied to special situations. It is the purpose of this section to develop a useful condition for compactness. The condition that we have in mind is ideally suited to the study of Markov processes and, more generally, processes for which there is a plentiful supply of associated martingales. Given p > 0 and WEn, define To(W) = 0 and for n ~ 1:

Tn(W) = inqt ~ Tn-I(W): Ix(t, w) - X(Tn-I(W), w)1 ~ pj4}. Here it is understood that Tn(W) = 00 if either Tn- 1(W) = 00 or there fails to exist a ~ Tn- 1(w) such that Ix(t, w) - X(T n_ 1 (W), W) I ~ pj4. Since W is a continuous path, it must always be true that either Tn-I(W) = 00 or Tn-I(W) < Tn(W) and Tn(W) -+ 00 as n -+ 00. Thus for T> 0 (this T is arbitrary but fixed throughout), we can define

t

N

= N(w) = min{n:

Tn+ I(W) > T}

and

(4.1) We need the following lemma. 1.4.1 Lemma. Let tl and t2 be any pair ojpoints in [0, T] such that It2 - tIl < b",(p). Then Ix(t2' w) - X(tI' w)1 ~ p and so

sup{ Ix(t2' w) - X(tI' w)l: 0 ~ tl < t2 ~ T and

It2 - tIl < bro(p)} ~ p.

Proof Consider the partition of [0, T] into the subintervals [To(W), T1 (w)), ... , [TN(ro)- 1 (w), TN(ro)(W)), and [TN(ro)(W), TJ. All of these subintervals, except possibly the last one, must have length greater than bro(p), Thus, either both t 1 and t 2 lie in the same subinterval, or they lie in adjacent subintervals. Since over any subinterval the distance of the path from its position at the left hand end never exceeds pj4, the oscillation of the path over any subinterval must be less than or

37

1.4. Martingales and Compactness

equal to p/2. Hence the oscillation over the union of two successive subintervals cannot exceed p. In particular, IX(t2' w) - X(tl' w)1 :s;; p. 0 The preceding lemma shows that the problem of estimating P(sup{ Ix(t 2) x(tdl: O:s;; tl < t 2 :s;; T and t2 - tl < J} ~ p) reduces to estimating

(4.2) The method that we are going to use to estimate the latter quantity depends on the following two hypotheses about P: 1.4.2 Hypothesis. For all non-negativefE C~(Rd) there is a constant Af ~ 0 such that (f(x(t)) + Aft, ..4(t, P) is a non-negative submartingale. 1.4.3 Hypothesis. Given a non-negative f E C~(Rd), the choice of A fin (1.4.2) can be made so that it works for all translates off Under these hypotheses, we are going to develop an estimate for the quantity in (4.2) which depends only on the constants A f . Let t: > 0 be given and choose I. E C~(Rd) so that 1.(0) = 1, J.(x) = 0 for Ix I ~ t:, and 0 :S;;J. :s;; 1. Given a E Rd , denote by f~ the function defined by f~(x) = f,(x - a). Now choose A, by Hypotheses 1.4.2 and 1.4.3 so that (f~(x(t)) + A, t, .ltt> P) is a non-negative submartingale for all a E Rd. Note that by Theorem 1.2.10. if r is a stopping time and {Qw'} is a r.c.p.d. of P given..4(" then there is a P-null set F E ..4(, such that (f~(x(t)) + A, t )X[o. tJ(r) is a non-negative Qw submartingale for all w' ¢ F. The null set F will of course depend, in general, on a E Rd. However, by taking a countable dense set D £; Rd and choosing F so that (f~(x(t)) + A, t)X[o. tJ(r) is a Qw' submartingale for all a E D and w' ¢ F, we see that it is possible to take F independent of a E Rd. We can now prove the following lemma. 1.4.4 Lemma. For any n

~

0 (a.s., P)

on

{r. < oo}.

Proof Let t: = p/4 in the preceding discussion and let {Qw'} be a r.c.p.d. of P given ..4(,., Then we can choose a P-null set FE .H,. so that

is a non-negative submartingale for all w' ¢ F, wheref~(x) = f,(x - x(r.(w'), w')) if r.(w') < 00 and f~·(·) == 1 otherwise. In particular, by Theorem 1.2.5,

38

I. Preliminary Material: Extension Theorems, Martingales, and Compactness

for w' ¢ F. In other words,

1':' ~ 1, and 'n+ 1 ~ 'n(W') + 0 and p > O. Because of Lemma 1.4.1, this will be done once we show that

(4.3)

lim sup P{{w: c5(l){p)

~

c5}) = O.

6"0 Pef¥

Note that, from the definition of 15",{p) in (4.1),

P{15.{p)

~ (5) ~ p( min rj -

rj-1

1 sjsk

k

~

L P{1:j -1:

i -1

~

1

~ kc5A p / 4

+ P{N >

~ (5) + P{N > k) (5) + P{N > k) k~

where we have used Lemma 1.4.4 to get the last line. Thus, the proof will be complete once we establish that

(4.4)

lim sup P{N > k) = O.

k .. oo P e f¥

But, by Lemma 1.4.4, we know that for any

to

> 0 and P

E~:

EP [e-(t i + 1 -t i )\vIt'.J ~ P{ri+ 1 - rj ~ to \ vlt't') + e-10P{rj+1 - rj

e- 1o + (I - e- 10 )P{1:j+ 1 - rj ~ e- 1o + {I - e- 10 )toAp/4 a.s.

~

~ to \ vlt't')

>

to \ vIt'•.)

40

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

Choosing to in a suitable manner, we can make

Thus, by Lemma 1.4.5, sup P(N ;;:: k)

~ eT;'k,

Pe~

and this certainly guarantees (4.4).

0

Although Theorem 1.4.6 is well-adapted to the study of continuous time Markov processes having continuous paths, it is not suitable, as it now stands, for the approximation of such processes by discrete time parameter processes. We will now make the necessary modifications to get a theorem which covers this situation. Given an h > 0, let nit stand for the subset men such that x( " m) is linear over each interval of the form Uh, (j + 1)h], j = 0, 1,.... Given men,., define t~(m) = 0 and for n ;;:: 1, t:(m) = inf{t;;:: t:-l(m): t = jh for somej;;:: 0 and Ix(t, m) - X(t:_l(m), ml ;;:: p/4}. We again adopt the convention that t:(m) = 00 if t:-l(m) = 00 or if Ix(t, m)X(t n -l(m), m)l < p/4 for all t;;:: t:_,(m). Once more, either t:-l(m) = 00 or t:(m) > t:- 1 (m) for all n ;;:: 1 and lim t:(m) =

00.

Define for T> 0 (arbitrary but fixed):

N*(m)

= inf{n;;::

0: t:+,(m) > T},

~:(p) = min{t:+ 1 (m)

and

- t:(m): 0 ~ n ~ N*(m)}

O:(m) = max{ Ix((j + l)h, m) - x(jh, m)l: 0

~jh ~

T}.

In place of Lemma 1.4.1, we now have the following lemma.

1.4.7 Lemma. Iftl> t2

E

[0, T] and 0

~

t2 - tl

~ ~:(p~

then

In particular, SUp{lX(t2' m) - X(tl' m)l: t" t2 ~

p + 20:(m).

E

[0, T]

and 0 ~ tl - t2 ~ ~:(p)}

41

1.4. Martingales and Compactness

Proof Given t 1 and t 2, let t1' and t~ be, respectively, the smallest and largest multiple of h such that t1' ~ t 1 and t~ :5 t 2 • Repeating, word for word, the argument used to prove Lemma 1.4.1, we see that Ix(t~, w) - x(t1', w)1 :5 p. On the other hand, neither Ix(ti, w) - X(tb w)1 nor Ix(t!, w) - X(t2' w)1 can exceed et(w), and so the proof is complete. D By analogy with hypotheses 1.4.2 and 1.4.3, we now state appropriate hypotheses for a P on (0, At) which is concentrated on 0h'

1.4.8 Hypothesis. For all non-negativefE CO'(Rd) there is a constant AI ~O such that (f(x(jh)) + AAjh), Atjh' P) is a non-negative submartingale. 1.4.9 Hypothesis. Given a non-negative f E CO'(R d), the choice of A I in 1.4.8 can be made so that it works for all translates off The analogue of Lemma 1.4.4 in this context is the following lemma, whose proof is identical to the proof of Lemma 1.4.4 when one takes remark 1.2.11 into account. The details are left to the reader.

1.4.10 Lemma. For any b which is a multiple of h and any n

~

0,

We are now ready to prove the analogue of Theorem 1.4.6.

1.4.11 Theorem. L!?t {h n: n ~ O} be a non-increasing sequence of positive numbers such that hn -+ 0 as n -+ 00. Let {Pn: n ~ O} be a sequence of probability measures on (0, .Jt) such that P n is concentrated on 0h.' Assume that each Pn satisfies hypotheses 1.4.8 and 1.4.9 (with h = hn) and that the choice of the constants A I can be made independent of n. If for each T > 0 and e > 0

(4.5)

lim

L

n-oo Os.jh,,'5:T

Pn(lx((j+ l)h n)-x(jhn)1 ~e)=O,

and lim sup Pn ( Ix(O) I ~

(4.6)

f?oo

then {P n: n

~

/) = 0,

n~O

O} is precompact.

Proof The proof here goes by analogy with the proof of Theorem 1.4.6. We must show that

(4.7)

lim Iimp n( sup Ix(t)-x(s)1 b"'O "-00 O:O:;s 0 and p > O.

t-st g;s where ff's is the completion of ffs in (E, ff, P). (That is A E ff's if and only if there is an A in ffs with A ~A c B where BE ff and P(B) = O.} Show that (O(t), ff't+ 0, P) is a martingale.

1.5.8. Suppose (O(t),

1.5.9. Use Theorem 1.2.8 to show that if (O(t), ffp P} is a continuous real valued martingale which is almost surely of bounded variation, then for almost all q, O(t) is a constant in t. Note that this conclusion is definitely false if one drops the assumption of continuity. 1.5.10. Suppose (O(t), fft, P) is a martingale on (E, ff, P) such that E(O(t))2 < 00. Show that E(O(t)f is an increasing function oft with a finite limit as t -+ 00. Use this to show that O(n) tends in mean square to a limit O( 00) as

SUPt~O

n -+

00.

Next, use Doob's inequality to prove

In particular 8(t) -+ 8(00) a.e. as t -+ (fj. Finally, show that if r is an extended stopping time relative to {~}, then EP[8(oo)I~] = 8(r) a.e. P. This is an especially easy case of Doob's Martingale Convergence Theorem.

1.5, Exercises

°

45

1.5.11. Let (E, ~) be a measurable space and ~I' t 2:: a non-decreasing family of u-fields such that ~ = U(UI ~I)' Given A S; [0, (0) x E, we say that A is progressively measurable if XA(', .) is a progressively measurable map from [0, (0) x E into R. Show that the class of progressively measurable sets constitute au-field and that a functionf: [0, (0) x E --+ (X, aJ) is progressively measurable if and only if it is a measurable map relative to the u-field of progressive measurable sets. 1.5.12. With the same notation as above, let r: E -. [0,00] be an extended non-negative real valued function such that for each t > 0, {q: r(q) s; t} E !Fe. If f: [0, (0) x E --+ (X, aJ) is progressively measurable show that f(t, q) defined by f(t, q) = f(t /\ r(q), q) is again a progressively measurable function. In fact show that if 1: is the u-field of progressively measurable sets in [0, (0) x E, then the map .: [0, (0) x E --+ [0,(0) x E defined by .(t, q) = (t /\ ,(q), q) is a measurable map of([O, (0) x E, 1:) into itself. (Hint: it is enough to verify that for each t > 0, the map . restricted to [0, t] x E is a measurable map of ([0, t] x E, aJ[O. II X ~I) into itself. Since the second component of . is the identity map it is enough to check that r(q) 1\ s: [0, t) x E -. [0, t] is aJ[o.t) x !Fe measurable. Since r(q) 1\ s cannot exceed't, we need only check that for each a < t the set {(s, q): ,(q) /\ S ~ a} is in aJ[o. II X ~I' Clearly

{r(q) /\ S ~ a} = ([0, t] x {q: r(q)

°

~

a}) u [0, a] x E.)

1.5.13. Let A(t), t 2:: be a non-increasing family of subsets of E such that A(t) E ~I for each t 2:: 0. Consider the set

A=

U (t, A(t)) =

{(t, q): q E A(t)}.

12:0

ns progressively measurable then

B=

U (t, B(t)) =

°

and B(O) = A(O). Show that if A is

{(t, q): q E B(t)}

12:0

and

B\A are progressively measurable too. (Hint: Consider the functionf(t, q) = XA(t, q). From the fact that f(t, q) is progressively measurable show that J defined by

1(t, q) =

f(t - 0, q) = ~~~ f(t( 1 -~). q)

is again progressively measurable. Identify J as XB')

Chapter 2

Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure

2.1. Regularity of Paths Suppose that for each n 2': 1 and 0 S tl < .. , < tn we are given a probability distribution Plio .... I. on the Borel subsets of (Rd)". Assume that the family {PII . ... , IJ is consistent in the sense that if {SI' ... , Sn- d is obtained from {t l' ... , tn} by deleting the kth element t k , then PSI • .... 8.-1 coincides with the marginal distribution of Plio .... 1. obtained by removing the kth coordinate. Then it is obvious that the Kolmogorovextension theorem (cf. Theorem 1.1.10) applies and proves the existence of a unique probability measure P on (Rd)[o. (0) such that the distribution of ("'(td, ... , "'(tn)) under P is PII, .... I•. (Here, and throughout this chapter, '" stands for an element of (Rd)[o. 0, then for

r 1 EBlRd:

QII(rd = P(t/I(t 1 ) E rd = e[p(O, t/I(O); =

f P(O, y; t

1,

rl)Qo(dy)

=

t 1,

rdl

f P(O, y; t

1,

rdpo(dy)

= PII(r t )· Now assume that QIJ, .... 1. = PII . .... 1•• Then for any bounded measurable

f: (R")" -.... R, we have

54

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

In particular, ifr l ,

r n+1

... ,

E f!iRd

and

f(YI' ... , Yn) = Xrl(YI) ... XrJYn)P(tn, Yn; tn+ I' rn+ d, then, by (2.3),

Q,lo .... '.+ l(r 1 x =

... x

r n+ d = EP(f(t/J(t d,

... , t/J(t n ))]

I· .. I f(YI' ... , Yn)P'l ..... ,.(dYI dYn) If P(tn' Yn; tn+ dYn )P'l ..... ,.(dYI X ...

I'

+1

x

X ...

x dYn)

fl x ", x f n +l

= P tl • ...• tn + 1 (r 1

x ... x

r n + 1 )•

Thus the proof can be completed by another application of Exercise 1.5.1.

0

Of course, there is no reason why a Markov process should always have to be realized on ((Rd)[O. (0), .s1J(Rd)[o.OO))' In fact, we want the following definition.

2.2.3 Definition. Let (E, !F, P) be a probability space and {!F,: t 2': O} a nondecreasing family of sub a-algebras of !F. Let P(s, x; t, .) be a transition probability function and J1 a probability measure on (Rd, f!iRd). Given a function ~: [0, (0) x E-+R d, the triple (~(t), !F" P) is called a Markov process on (E,!F) with transition probability function P(s, x; t, .) and initial distribution J1 if ~(t) is !F,-measurable for all t 2': 0 and (2.5)

P(~(O) E

r) = J1(r),

and

(2.6)

P(~(t)

for all 0

~

s < t and r

E

E

r I!Fs ) = P(s, ~(s); t, r)

(a.s., P)

.s1JR d.

Notice that if E = (Rd)[o. OO),!F = (.s1J Rd)[O. (0), and!F, = a[t/J(u): 0 ~ u ~ tl, then the preceding definition is consistent with the one given in 2.2.1. The case in which we will be most interested is when E = n, !F = .At, !F, = .At" and ~(t) = x(t). In fact, if (x(t), .At" P) is a Markov process on (0, .At), we will call it a continuous

Markov process.

2.2.4 Theorem. Let P(s, x; t, .) be a transition probability function such that for each T > 0 there exist IX = IXT > 0, r = rT 2': 1 + IXT, and C = CT for which

(2.7)

sup

Yl eRd

I IY-Ytlrp(tl'Yl;t2,dY)~Clt2-tll1+", 0~tl1' ..• , rJ>n E Cb(Rd), then for 0 < tl < ... < tn:

To this end, let u ~ sand rJ>

EP[rJ>(fJu(v)), A]

=

=

Cb(R d ) be given; Then for v > u and A

f EP[rJ>(fJ(v + u) A

=

E

t(f

.

E

!Fu:

fJ(u, q)) I!Fu](q)P(dq)

rJ>(y - fJ(u, q))gd(V, y - fJ(u, q)) dY)P(dq)

(J rJ>(y)giv, y) dY)P(A).

Thus

In particular, (3.4) proves (3.3) when n = 1. Next suppose that (3.3) holds for n. Let 0 < tl < ... < tn+1 and rJ>1, ... , rJ>n+l E Cb(Rd) be given. Applying (3.4) to u = tn + to, v = t n + 1 - tn' and rJ>(y) = rJ>n+I(Y + z), we have:

59

2.3. Wiener Measure

Thus, since (3.3) holds for n:

e[¢t(Pro(td) ... ¢n+ t(Pro(t n+d), A]

f

= EP[¢n+t(Prn+ro(tn+t - tn) + Pro(tn, q))l§rn+ro](q) A

X

¢1(Pro(tt, q)) ... ¢n(Protn' q))P(dq)

f

= ¢1 (Pro(t d) ... ¢n(Pro(tn)) A

x

f ¢n+ t(y)gd(tn+t - tn' Y - Pro(tn)) dyP(dq)

= E'/Y(d) [¢I(X(t d ) ... ¢n(X(tn)) X

f ¢n+ (y)gd(tn+ 1

1 -

tn' Y - X(tn)) ]P(A)

= E'/Y(dl¢t(x(td) ... ¢n(X(tn))¢n+l(X(tn+1 ))]P(A) Thus the induction is complete.

0

2.3.3 Theorem. If (P(t), § " P) is an s-Brownian motion and r is a stopping time satisfying r 2 s define Pr( . ) by:

,P(t + r(q), q) - p(r(q), q) fir(t. q) = \ jo Then for A

E

(3.5)

if r(q)
a bounded .It-measurable function on EP[(J>

0

Q

into R, we have

Pro A n {r < oo}] = ElI'O(d)[(J>]P(A n {r < oo}).

In particular, (a.s., P).

(3.6)

Proof It is certainly enough to check (3.5) when function on Q. But in that case,

(J>

is a bounded continuous

60

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

t+

where

'. =

:::...[n....:.(r_:_s~)]

if r 0 and lpM E Cg'(R4) satisfies: 0 ~ lpM ~ 1, lpM == 1 on B(O, M), and lpM == 0 on R4\B(0, M + 1), then the unique

T..,

69

3.1. The Maximum Principle

°

1M E C~' 2([0, t) X Rd ) n Cb([O, t] X Rd ) satisfying (aIMlas) + Ls 1M = 0, Os s < t, with/M(t, .) = CPM' has the property thatIM(s, .) E C(Rd) for all 5 s < t. To this end, note that we can find positive numbers A and B such that for all Xo E Rd the function: (s, x)

E

[0, t)

X

+ Ls 1/1 xo sO, 0 5 s < t. Thus if p == 11 Xo 1 ;:::: M p2cpM(') sl/lxo(t, .), we see from Theorem 3.1.1 that

Rd ,

+ 1, then, since

satisfies (01/1 xo las)

Os s < t.

In particular,

< -A

0
E[w,,] a.s. Hence the quadratic variation on [0, 1] of f3h q) over the dyadics is equal to one a.s., and therefore f3(., q) is a.s. not of bounded variation. This means that we cannot take an entirely naive approach to the definition of (1.5). To see how one can get around this sticky point, suppose that (J is a smooth function on [0, 00) and think of (1.5) as being defined through an integration by parts. Then J~ (J(s) df3(s) == f3(t)lJ(t) - f3(0)lJ(0) - J~ f3(s)lJ'(s) ds. An elementary calculation shows that Em (J(s) df3(s)] = 0 and E[(J~ (J(s) df3(sW] = J~ (J2(S) ds. Hence (J -> J~ (J(s) df3(s) establishes an isometry from r,2([O, t], A.) into r,2(E, P): where A. is Lebesgue measure. One can therefore extend the definition of J~ (J(s) df3(s) to cover all (J E r,2([O, t], A). This procedure was first carried out, and used with great success, by N. Wiener. The situation is somewhat more complicated when (J depends on q as well as s. To illustrate the sort of phenomenon encountered here, consider (J(s, q) = f3(s, q). Suppose first that we attempt to define

(1.6)

fo f3(s) df3(s) = lim 1

2"- 1

L

n-co k=O

f3(t k • n}(f3(tH

1.

n) - f3(t k • n))·

85

4.2. Equivalence of Certain Martingales

By elementary manipulation, we have that 2" - 1

L

fJ(t k , n}(fJ(tk+ 1, n) - fJ(t k , n))

k~O

=

2"f 1

k~O

=

(fJ(tk+ 1, n) + fJ(t k , n))(fJ(tk+ 1, n) - fJ(t k , n)) _ ~ W 2 2 n

!fJ2( 1) - !fJ 2(O) -

! w,. -+ !fJ2( 1) -

!fJ 2 (O)

-!

a,s,

as n -+ 00, Thus the lack of bounded variation manifests itself here in the appearance of the term -!- We will see later on in this chapter that -! results from the fact that dfJ(s) = " (ds)-t" and therefore dfJ2(S) = 2fJ(s) dfJ(s) + (dfJ(s)f = 2fJ(s) dfJ(s) + ds (cf. Ito's formula Theorem 4.4.1). It is important to notice that putting the increments of Brownian motion" in the future" as in (1.4) makes a difference (this is another manifestation of the absence of bounded variation ). For instance, one can easily check that 2"- 1

(1.7)

lim

L fJ(tk+ l.n}(fJ(tk+1.n) -

n-+oo k=O

fJ(tk,n)) = !fJ2(1) -

!fJ 2 (O) + 1-

Although the difference here seems to be small, it turns out that (1.6) is a far fJ(s) dfJ(s) than the one given by (1.7). Suffice it to say preferable way to define that the advantage with (1.6) is that the following relations hold:

g

E [( fJ(s) dfJ(S)]

=0

and

E [(( fJ(s) dfJ(s)

r]

=

E [( fJ2(S) dS].

These equations allow one to carry out a completion procedure analogous to that in the case where () depends only on s.

4.2. Equivalence of Certain Martingales Let (E, ff, P) be a probability space and {ff t : t ~ O} a nondecreasing family of sub a-fields of ff. Let s ~ 0 be arbitrary and a: [s, (0) x E -+ Sd and b: [s, 00) x E -+ Rd be bounded progressively measurable functions. For each (t, q) E [s, (0) x E the components of a(t, q) will be denoted by {dj(t, q)} and those of b(t, q) by {bi(t, q)}.

86

4. The Stochastic Calculus of Diffusion Theory

For any functionf(x) in the space C 2 (Rd) of twice continuously differentiable functions we define Lr f for t z s by

We note that if ~(.,.) is any progressively measurable function from [s, (0) x E -+ Rd then (Lt(q)f)(~(t, q)) defines another progressively measurable function of t and q for t z sand q E E. Usually we will suppress the variable q and write ~(t), a(t), b(t~ Lr for our objects. We shall suppose that we are given a function ~(t, q) mapping [s, (0) x E into Rd which is progressively measurable, right continuous in t, and almost surely continuous in t. The main result of this section is the following theorem which proves that various types of relations between L t and ~(t) are equivalent.

4.2.1 Theorem. For any ~(. ), a(· ), b(· ) satisfying the conditions described above, the following are equivalent: (i) f(~(t)) - f~ (Luf)(~(u)) du is a martingale relative to (E,!Fro P)for t z s,for alifE CO'(Rd). (ii) f(t, e(t)) - f~ ((%u) + Lu)f(u, ~(u)) du is a martingale relative to (E, !Fro P) for t z s, for all fE q. 2([0, (0) X Rd ). (iii) f(t, ~(t)) exp[ - f~((%u) + Lu)f / f)(u, ~(u) du] is a martingale relative to (E, !Fro P) for t z s and for all f E q. :l([0, (0) X Rd ) which are uniformly positive. (iv) If e E Rd and g E q. 2([0, (0) X Rd) then XO,g(t) = exp 1

[(e,~(t) - ~(s)

-1t

b(u)du)

+ g(t,~(t»

f• (e + Vg, a(u)(e + Vg)(u, ~(u)) du

-2

r

- {' (:u + Lu )g(u,

~(u)) dU]

is a martingale relative to (E, !Fro P)for times t (v) If 0 E Rd then X 9 (t) = exp [ (e, ~(t) - ~(s) -

f. b(u) du) - 2f. (e, a(u)e) du 1

r

is a martingale relative to (E, !Fro P)for t (Vi) If e E Rd , then X i9 (t) = exp [i(e, ~(t) - ~(s) -

z s. r

]

z s.

It ] f.t b(u) du) + 2f. (e, a(u)O) du

is a martingale relative to (E, !Fro P)for t

~

s.

87

4.2. Equivalence of Certain Martingales

Moreover if any of the above equivalent relations holds, then for each t > s

where A =

SUPt~ •. qeE SUPI61=1

(2.2)

E

(0, a(t, q)O). In particular,for any r > 0,

[exp k~~~t I~(u) - ~(s) I]] ~ C

and the constant C depends only on B

s, r, A and B where

t -

= sup Ib(t, q)l. t2:S

qeE

Proof Assume (i). Let f E[J(t 2 , ~(t2)) - f(t 1 ,

= E[J(t 2, = E [f,:z =

E [(

E

Cg'([O, (0) x Rd ). Then for s ~ t 1 ~

~tl

(:~)(u, ~(t2)) du; A] + E [f,:z (Lv f)(t ~(v)) dv; A] 1,

(:~)(u, ~(u)) du; A]

~(t2)) - (:~)(u, ~(u)) ] du; A]

+ E [(2(L v f)(V,

~(v)) dv; A]

+ E[C(L v f)(t 1,

~(v)) -

(Lvf)(v,

= E [( (:~ + Luf )(u, ~(u)) du;

[(2 du

r(iu Lv

)(u,

tl

tl

U

~(v)) dv; A]

A]

~(v)) dv; A]

- E[{2 dvf(: Lvf)(U,

= E [(

E

- f(t l' ~(t2)) + f(t l' ~(t2)) - f(t l' e(t d); A]

+ E [(Z [(:~)(U,

+E

and A

A]

~(td);

~(t2))

t2

~(v)) du; A]

(fu + Luf )(U, ~(u)) du; A].

The last two terms in the preceding step cancel each other because Lv(oflou) = (Ojou)L v f and the two repeated integrals are double integrals over the same

88

4. The Stochastic Calculus of Diffusion Theory

region, namely: t 1 ~ U ~ v ~ t 2' It is obvious that the validity of the last equality for all f E CO'([O, (0) x R d ) implies the validity of the same equality for all f E q. 2([0, (0) X Rd ). Hence (i) implies (ii). Next assume (ii) and let f E ct· 2([0, (0) X Rd ) be uniformly positive. Take

tx(t) = f(t,

~(t)) -

((iu +

Lu f )(u,

~(u)) du

and

Then

Applying Theorem 1.2.8, we see that (ii) implies (iii). Assume (iii) and assume for the moment that ~(s) = O. Obviously (iv) would follow if we were allowed to take f(t, x) = exp[ (0, x) + g(t, x)). We cannot do this becausefis unbounded and not uniformly positive. To circumvent this problem, we choose for each M ~ 1 a uniformly positive function fM in ct· 2{[0, (0) X R d) such that fM{t, x) = exp[(O, x) + g(t, x)] for x ~ M and define LM=(inf[t~s: sup 1~(u)1 ~M])/\(Mvs).

Then by (iii) and Corollary 1.2.7, (Xe, g(t /\ LM)' :F t , P) is a martingale for t ~ s. Clearly Xe. g{t /\ LM) -+ X e, //(t) a.e. Pas M / 00 for each t ~ s. Thus we can establish (iv) if we can show that {X 6 • g(t /\ LM): M ~ I} is uniformly integrable for each t ~ s. But

Xi,//(t/\LM) = X 26. 2//(t /\ LM) exp

[t'\tM (0 + Vg, a{u)(O + Vg){u, ~(u)) dU]

~ CX 26, 2//{t /\ LM)

and E[X 26. 2//{t /\ LM)] = E[X 26. 2g{S)] = exp[2g{s, 0)]. This completes the proof when ~(s) = O. To remove this restriction, we define ~'(t) = ~(t) - ~(s). From (iii) and Exercise 1.5.7 it follows that for uniformly positive functions f in Ct,2([O, (0) x Rd )

89

4.2. Equivalence of Certain Martingales

is an (£, ffr' P) martingale for t ;:::: s. From what we proved earlier, it follows that for all () E Wand g E q' 2([0, 00) x Rd ), X~, it) is an (£, §'n P) martingale for t;:::: s, where X~,g(t) is the expression defined in the same way as XB,g(t) with ~'(t) replacing ~(t). If one defines gx.(t, x) = g(t, x' + x) for x, x' E Rd we have for s ..:; t 1 ..:; t2

Again using Exercise 1.5.7

Clearly X~,g(()t) = XB,g(t) for all t;:::: s, and we have therefore proved (iv). Obviously (v) is just the special case of (iv) when g == O. To see that (v) implies (vi) it is enough to show that (v) implies (2.1) and (2.2). Indeed, if the estimates (2.1) and (2.2) hold, then both sides of the equality

for S..:; tl ..:; t2 and A E §'rl' determine entire functions of () E Cd. Therefore the validity of the equation for all () E Rd implies its validity for all i() with () E Rd. To see that (v) implies (2.1), let I() I = 1 and p > 0 be given. Then by Theorem 1.2.3 and (v) we obtain

pt~~~r IAt, q) is jointly continuous, (iv) for almost all q E E,

J IAt, q) dx =

(6.3)

o

r

for all

t :2:

0 and

1

-2

II a(u)Xr(~(u)) du 0

r E f!lR'

In the case a(') == 1, (6.3) shows that 21.(t, q) is identifiable as the density, with respect to Lebesgue measure, of the occupation time functional, up until time t, of the path ~(" q). For this reason, Ix is called the local time at x. The existence of a local time was first observed by P. Levy [1948], but it was H. Trotter who first provided a rigorous proof. The tack which we adopt here is due to H. Tanaka. (See McKean [1975]') Although it is somewhat obscure in the present proof, what underlies the existence of a local time for a I-dimensional process is the fact that points have positive capacity in one dimension. The way in which this fact is used here is hidden in the bounded ness at its pole of the fundamental solution for 1- d2 /dx 2 . To be precise, the functional for which we are looking is given, at least formally, by 1 I "2 a(u) bg(u) - x) du,"

Ito

f

o

118

4. The Stochastic Calculus of Diffusion Theory

where b(') stands for the Dirac b-function. Hf(x) = x yO for x b( . ), and then Ito's formula predicts that 1 -f a(u) b(~(u) I

x) du

=

E

R then/"(') =

f XIx. OO)(~(u)) d~(u). I

f(~(t) - x) - f( -x) -

2 0 0

The terms on the right of this expression make sense and therefore indicate how we should go about realizing the left hand side. This observation is the basis of Tanaka's approach. We now outline some of the details. For n ~ 1, let!,. be defined on R by:

fn(x)

=

0

if

1 x (111) comes from the subspace of "chaos of mth order." For each n 2 1, let L\n be the set ~ = {s ERn; 0:$ Sl :$ S2 ... :$ Sn < oo}. Given 1 E L2(L\n) (with respect to the Lebesgue measure) we define

(6.6)

f I(s) d(n)x(s) f ""dx(sn) (dX(Sn_ d'" =

6.

0

and show that

(6.7)

E[

(f

6.

0

f (S) a(s) X",(s) dx(s) . •0

By induction, show that

X",(t) = 1 +

Lf III

j= 1

6J(sd'" 4>(Sj) d(j)x(s) + Rm(t),

121

4.6. Exercises

where

Then estimate E[R~(t)] and show that E[R~(t)] ~ 0 as m ~ 00. Conclude from this that X",(t) E t9~ £(m) for all t. Finally, prove Wiener's theorem by observing that the only function in L2("II') that is orthogonal to X",(t) for allljJ and t is the zero function. Now if X(t) is a martingale on the Wiener space with sUPtE[X2(t)] < 00, then by Exercise 1.5.10, X{t) = E[X IAt] and use the representation X = ~o(n). in terms of homogeneous chaos to conclude that X{t) is given by a stochastic integral

I 8{s) dx{s), .1

X{t) =

·0

where

E

I

.00

• 0

82 {s) ds
0: ~(t

+ b) -

~(t)

'" at(t, ~(t))(P(t + b) - P(t)) + b(t, ~(t)) 15

123

5.0. Introduction

where P(t) is a Brownian motion with the property that {P(t + c5) - P(t): c5 > O} is independent of e(t). In other words, e{') ought to solve the stochastic integral

equation: (0.2)

e(t) - e(s) =

l'

a!(u,e(u»dp(u) +

l' b(u,~(u»du.

There is a second, more formally acceptable but less intuitively appealing, way to arrive at (0.2). Namely, suppose that a and b are amenable to the techniques of Chapter 3, and let {P•. x: (s, x) E [0, 00) X Rd} be the associated Markov family of measures on (0, vH). Then, for f E CO'(Rd) and s ::;; t1 < t2 and x E Rd:

where L, is the operator having a(t, .) and b(t, .) as its coefficients. By the Markov property, one therefore has:

e

s •x

[J(x(t 2 ))

-

f(x(t d) IvH,J = EP'I.%lItI[J(x(t 2)) - f(x{t 1))] = EP').X('I)

=

[(2Lu f(x(u)) dU]

EPs.x [(Luf(X(U)) du IvH,I]'

But this means that

f(x(t)) -

,

f• Luf(x(u)) du

is an (0, vH,. p •. x) martingale after time s. Because this is true for anyf E CO'(R d), we conclude that x(· ) '" J"d(a(', x(· )), b(" x(·))) on (0, vH,. Pl. If one sets ~(. ) = x(·). Equation (0.2) can therefore be seen as a consequence of Theorem 4.5.2. In this connection, note that, by Theorem 4.3.8. if ~ satisfies (0.2), then conversely ~

'" J"d(a(·.

~(. )),

b(·.

~(. ))).

Regardless of the manner in which one arrives at (0.2), the question remains as to how one can exploit this equation to construct a diffusion process. Following Ito, what we are going to do in this chapter is to treat (0.2) as a equation for the trajectories of the diffusion in much the same way as one would approach an ordinary differential equation. In fact, it has been recently discovered by H. Sussman [1977] that, in special cases, Ito's method can be interpreted as a natural extension of the theory of ordinary differential equations.

124

5. Stochastic Differential Equations

5.1. Existence and Uniqueness Let (E, !F, P) be a probability space, {!F,: t ~ O} a non-decreasing family of sub O'-algebras of !F, and fJ( • ) is d-dimensional Brownian motion on (E, !F" P). Suppose that 0': [0, (0) X Rd -+ Rd ® Rd are measurable functions which satisfy the conditions: sup 1100(t,

(1.1 )

'2:0

xeRd

and for all x, y

E

where A
s:

(1. 7) ~

3E[ 1'1 - '1' 12]exp[3A2(4 + (T - s))(T - s)]

for all square integrable g;s-measurable '1 and '1' on E into Rd. Proof Let '1 and '1' be given and suppose and '1', respectively. Then:

~( . ) and

n· )are solutions to (1.3) for '1

Els!~~TI~(t) - ~'(tW J ~ 3E[I'1- '1'1 2] + 3E lS!~~T I((U(U,

~(u)) -

u(u,

~'(u))) dP(u)

+ 3E [s!~~ T If(b(U,

~(u)) -

b(u,

~'(u))) du rJ

~ 3E[I'1- '1'1 2] + 12E[!((U(U, ~(u)) -

~'(u))) dP(U)n

b(u,

~'(u)) 12 dU]

~ 3E[ 1'1- '1' 12] + 3A2(4 + (T - s))

f.. E[ I~(t) -

+ 3(T - s)E [( Ib(u,

~(u)) -

u(u,

n

T

~'(t) 12] dt .

127

5.1. Existence and Uniqueness

Putting d(T)

=

E[sUPssrsT I~(t) - ~'(tW], we have:

d(T) ~ 3E[ 11] - 1]' 12]

+ 3A 2(4 + (T - s))

f

T

d(t) dt.

s

An application of Gromwall's inequality now yields:

This proves, in particular, that ~(.) = ~'(.) almost surely if 1] = 1]'. Obviously, since we now know that ~(.) and ~'(.) must be ~(.; s, 1]) and ~(. ; s, 1]') respectively, it also proves (1.7). 0

5.1.3 Corollary. Let a and b be as before. For (s, x) E [0, 00) X Rd , let ~s. A. ) == ~(. v s; s, 1]), where 1] == x. Then ~s. A·) can be chosen to be a(f3(t)f3(s): t ~ s)-measurable. Moreover, if f3'(.) is a second Brownian motion on a second space (E', $';, PI) and if ~~. A. )is defined accordingly for the primed system, then the distribution of ~s. A· ) under P coincides with the distribution of ~~. A·) under P'. Proof To prove the first part, simply note that if 1] == x in Theorem 5.1.1, then, by induction, ~n(·; S, 1]) can be chosen to be a(f3(t) - f3(s): t ~ s)-measurable for each n ~ 0. Thus the assertion follows from (1.4). To prove the second part, define ~~(.; s, 1]) with 1] == x for the primed systems as in Theorem 5.1.1. By induction, the distribution of ~n( . ; s, 1]) under P and ~~(. ; s, 1]) under P' coincides for all n ~ 0. Therefore the same must be true of the distributions of ~s. A· ) and ~~. A. ). 0 Corollaries 5.1.2 and 5.1.3 tell us that when a and b satisfy (1.1) and (1.2), we can unequivocally talk about the distribution of the solution to (1.3) when 1] is constant. Since ~s. x( • ) is P-almost surely continuous, we think of its distribution as a probability measure p s • x on (Q, vIt). We urge the reader to bear in mind that, because of Corollary 5.1.3, the measure p s • x is a function of a and b alone; and not of the underlying Brownian motion f3(.) or the space (E, $'r' P).

5.1.4 Theorem. Let {P s • x: (s, x) E [0, 00) X Rd} be thefamity of measures on (Q, vIt) associated with functions a and b satisfying (J.l) and (J .2). Thenfor bounded continuous cI>: Q -+ C, (s, x) -+ EP "1cI>] is a continuous function. In particular, {P s • x : (s, x) E [0, 00) X Rd} is measurable in the sense that (s, x) -+ £P '·1cI>] is measurable for all bounded vIt -measurable cI>: n -+ c. Proof Clearly it suffices to prove that

128

5. Stochastic Differential Equations

°

for each T> 0. Let ~ s' ~ s < T and x', x ~(t; s, r() almost surely, where 1]'

=

x'

E

Rd be given. For t 2': s, ~s'. x,(t) =

+ f a(u, ~s" x,(u)) dfJ(u) + f b(u, ~s'. x'(u)) du, s

s

s'

s'

Thus, by (1.7),

E [S!~YT 1~s. x(t) -

~s'. x,(t) 12] ~ 3E[ 1'1' -

x 12]exp[3A2(4 + (T - s))(T - s)),

and

If s'

~

t

~

s, then

~s"At) - ~s.At)

= x' - x + J~ a(u, ~s',x,(u)) dfJ(u) + J~b(u, ~s'.x,(u)) du,

and so

Since ~s. At) - ~s'. x,(t) = x - x' for

°t ~

~ s', this completes the proof.

0

We are now ready to prove the main result of this section.

°

5.1.5 Theorem. Let a and b satisfy (1.1) and (1.2) and define {P s • x : (s, x) [0, (0) X Rd} accordingly. For ~ s < t, x E Rd, and r E !14Rd, define

P(s, x; t, r)

= ps.Ax(t) E r).

Then (s, x) --+ P(s, x; t, r) is measurable on [0, t)

r

E

E

X

R d .. and for s ~ t 1 < t 2 and

£!#Rd :

(a.s., Ps, x)·

In particular, {ps. x: (s, x) E [0, (0) X Rd} is a continuous Markov family and P(s, x; t, .) is its transition probability function. Finally,for f E c~· 2([0, 00) x R d ), ~ s < t, and x E Rd:

°

f f(t, y)P(s, x; t, dy) - f(s, x) = { du f (:u

+ Lu )i(u, y)P(s, x; u, dy),

129

5.1. Existence and Uniqueness

where

Proof To prove the Markov property, let f E Cb(R d ) and s :-:; tl < t2 be given. Note that ~s.At) = ~(t; t l , ~s,Atd), t?: t l , almost surely. Thus if 0:-:; Sl < ... < Sn :-:; t 1 and f l' ... , f n E fJl Rd, then

e"'[J(x(t 2 )), {x(sd E fl' ... , x(sn) E f n}]

= E[J(~(t2; t l , ~s,x(tl)))' gs,AS1) E fl'

... , ~s,Asn) E fn}].

Hence, if we can show that

then the proof will be complete. To this end, note that for any Z E Rd, ~/1' .(t 2 ) is a(p(t) - P(td: t?: td-measurable and is therefore independent of !F/l' Hence Elf(~tIoZ(t2))I~I]

= Elf(~tIoZ(t2))]

f f(y)P(t

=

b

z; t 2 , dy)

a.s.

Next, suppose that '1: E --+ Rd is !F/l-measurable and takes on only a finite number of values Z l' ... , ZN • Then it is easy to see, from uniqueness and Exercise 4.6.8, that N

W2;

tl,

'1) =

L X(~=Zj}~/loZj(t2) 1

a.s.

Thus, for such an '1, we have:

Finally, choose a sequence {'1nH" of such rt's so that E[ I'1n - ~s, x(t d 12] --+ 0 as 00. Then ~(t2; t l , '1n)--+~(t2; t l , ~s,Atd) in probability, and so

n--+

n .... oo

= lim n .... 00

f f(y)P(t

b

'1n; t 2 , dy) =

f f(y)P(t

l,

~s,x(td; t 2 , dy)

130

5. Stochastic Differential Equations

almost surely. We have used here the fact that

and is therefore a bounded continuous function. The final assertion is an immediate consequence of the fact that x(·) '" fd(aa*(·, x(·)), b(·, x(· ))) on (0, .1(" p •. x). 0

5.1.6 Remark. If (1 and b satisfy (1.1) and (1.2), then we have seen that the unique solution to

fo

f

r

(1.8)

r

~(t) = x + a(u, ~(u)) dP(u) + b(u, ~(u)) du 0

is a measurable functional of P( .). This brings up a rather subtle point which we will have occasion to discuss in greater detail in Chapter 8. Suffice it to say now that when (1 and b fail to satisfy (1.1) and (1.2), there are situations in which ~(.) can satisfy (1.8) without it being true that ~(.) is a measurable functional of P(·). See Exercise 5.4.2.

5.1.7 Remark. We have already observed that if a and b satisfy (1.1) and (1.2), then

the measures p •. x depend only on (1 and b. On the other hand, (1 has no intrinsic meaning, since it is a == (1a* which is important and there may be a continuum of choices of (1 such that a = aa*. Thus we would like to know that p •. x really only depends on (1(1* and b. That this is indeed the case will be shown in Section 5.3.

5.1.8 Remark. The condition (1.1) can be considerably weakened. A good discussion of what one can say when (1.1) is abandoned can be found in the book of H. P. McKean [1969].

5.1.9 Remark. Suppose that a and b satisfy (1.1) and (1.2) and that, in addition, they are independent of time. Let ~ •. ,,(.), (s, x) E [0, (0) X R d, be defined accordingly relative to P(·) on (E, ~" P). Given s ;;:: 0, set ~; = ~r+., t;;:: s, and P'(t) = P(t + s) - P(s). Then P'(·) is a Brownian motion on (E, ~;, P). Moreover, if ~(.) = ~ •. x(· + s), then

~(t) = x +

fo (1(~(u)) dP'(u) + f b(~(u)) du, r

r

0

t ;;:: s.

Hence ~(.) = ~O. ,,( .), where ~O. ,,( .) is defined relative to P'(·) on (E, ~;, P). From this we see that the distribution of ~ •. ,,(. + s) coincides with that of ~o. ,,( • ). In other words, if e.: 0 -+ 0 is the map given by x(t, e.w) = x(t + s, w~ t ;;:: 0, then po." = p •. " e;l. In particular, for 0 ~ s < t: 0

P(s, x; t, r) = p •. Ax(t) E r) = p s • Ae; 1({x(t - s) E r})) = po. Ax(t - s) E

r) = P(O, x; t - s, r).

131

5.2. On the Lipschitz Condition

Hence, if P(t, x, .) == P(O, x, t, .), then

P(s, x; t, .) = P(t - s, x, .),

o ~ s < t.

The Markov process is therefore homogeneous in time.

5.2. On the Lipschitz Condition We saw in Section 5.1 that if we are given coefficients a and b such that b satisfies (1.2) and a can be written as a = uu· with u satisfying (1.2) then we can construct a diffusion corresponding to a and b. Unfortunately the assumptions on u are not directly transferable into assumptions on a. However we have the following theorems giving a wide class of coefficients a for which u exists with uu* = a and u satisfying (1.2).

st

5.2.1 Lemma. Let 0 < ex < A be given, and suppose that a E has all its eigenvalues in [ex, A]. Then the non-negative definite symmetric square root at of a is gi!Jen by the absolutely convergent series:

where cn is the nth coefficient in the power series expansion of(1 - x)t around x = O. In particular, the map a -+ at is analytic on into

st

st.

Proof Simply write a = A(I - (I - a/A)) and note that

5.2.2 Theorem. Let a: [0, (0) X Rd -+ Sd be a function such that O. If there is a C < 00 such that

Ila(s, x) - a(s, y)11 ~ Clx -

yl,

then

Proof Clearly it is enough to show that if x -+ a(x) is a Cqunction on Rl into Sd 2, x E Rl and 6 E Rd , then such that 0, there is a constant A p, T < 00 such that

gs.

E[

sup

O,;t,;1'

I~SI'Xl(t)-~S2'X2(t)IP] ~Ap,T(lsl-S21

+

Ix 1 - x21 2)P1 2

for all Sj, S2 E [0, T] and Xj, X2 E Rd. Next, apply Exercise 2.4.1, and the technique used in choosing a nice version of a local time (cf. Exercise 4.6.13) to find a new family gs, x('): (s, x) E [0, 00) X Rd} of right-continuous progressively measurable functions ~s. A') such that P(~s, x(') = ~s, A')) = 1 for all (s, x) E [0, 00) X Rd and for P-almost all q E E the map (s, x, t) -+ ~s, x(t, q) is continuous.

5.4.2. There is a distinction between the type of" path-wise" uniqueness proved in Corollary 5.1.2 and the "distribution" uniqueness proved in Theorem 5.3.2. In fact, this distinction will be the topic of Chapter 8. To prove that there really is a difference, let fJ(·) be a I-dimensional Brownian motion on some space (E, ff't, P). Define if x ~ 0 if x < 0 and

fo a(fJ(s)) dfJ(s), t

lJ(t) =

Note that lJ(· ) is again a Brownian motion on (E,

fo t

fJ(t)

= a(fJ(u)) dlJ(u),

t

~ 0. ff't,

P). Next observe that

t ~ 0,

and

fo t

-fJ(t)

= a(-fJ(u))dlJ(u),

t~O.

135

5.4. Exercises

(The latter equation turns on the fact that

p(( x(oM3(u)) du = 0) = 1 for all t :2: 0.) Thus we have here a u with the property that there is more than one solution to the corresponding Ito stochastic integral equation; and yet it is clear that all solutions have the same distribution, namely "If"!l.lo' This example was discovered by H. Tanaka (cf. Yamada-Watanabe [1971]). Next observe that lJ(t) = IP(t) I - 21o(t), where 10(') is the local time of P(·) at o(cf. Exercise 4.6.13). Thus P(·) is u( IP(t)l: t :2: O)-measurable. On the other hand, P(·) certainly is not u( IP(t) I: t :2: O)-measurable. Thus, in spite of the fact that

P(t) =

fo u(P(u)) d'P(u), t

t :2: 0

P(·) is not u(lJ(t): t :2: O)-measurable. Here is the example promised in Remark 5.1.1.

Chapter 6

The Martingale Formulation

6.0. Introduction We have now seen two approaches to the construction of diffusion processes corresponding to given coefficients a and b. Let us quickly recapitulate the essential steps involved in these different methods and see if we cannot extract their common features. The approach presented in Chapter 3 involved solving certain differential equations in which a and b appear as coefficients. If enough of these equations can be solved, then we are able to interpret the" fundamental solution" as a transition probability function P(s, x; t, .). Moreover, P(s, x; t, .) automatically satisfies estimates which permit us to apply the results of Chapter 2 and thereby construct, for each (s, x) E [0, (0) X R d , a probability measure Ps,x on n = C([O, (0), R d ) with the properties that:

(0.1)

p s . Ax(t) = x, 0 S t S s) = 1

and (a.s., ps. x)

(0.2)

for all sst 1 < t 2 and r E 91 Rd. The relationship between ps. x and the coefficient.; a and b is via P(s, x; t, .) and can be most concisely summarized by the equation:

(0.3)

f P(s, x; t, dy)f(y) - f(x) Rd

where

(0.4)

= ( s

du

f P(s, x; u, dy)Luf(y), Rd

137

6.0. Introduction

and Equation (0.3) holds for f E CO' (R d ). Combining (0.2) and (0.4), we can eliminate P(s, x; t, .) and state the connection between p s• x and the coefficients a and b by the equation:

or, more succinctly,

f Luf(x(u)) du t

(0.5)

f(x(t)) -

s

is a p s • x-martingale for allf E CO'(Rd). The second approach that we have discussed for constructing diffusions is Ito's method of stochastic integral equations. Here again there are intervening quantities between the coefficients a and b and the measures p s • x (described in the paragraph preceding Theorem 5.1.4). Namely, there is the underlying Brownian motion with respect to which the stochastic integral equation is defined and there is the choice of a satisfying a = aa*. That neither of these quantities is canonical can be seen most dramatically when one tries to carry out Ito's method on a differentiable manifold (cf. McKean [1969]). But, if one focuses on the direct relationship between p s • x and the coefficients a and b, matters simplify; and, in spite of the apparent differences between Ito's method and the construction via partial differential equations, one is led once again to Equations (0.1) and (0.5) as being the most concise description of the connection between p s• x and a and b (cf. Theorem 5.3.1). With the preceding discussion in mind, it seems only natural to ask if, in fact, (0.1) and (0.5) do not characterize what we should call "the diffusion with coefficients a and b starting from x at time s." Indeed, any measure with a legitimate claim to that title satisfies (0.1) and (0.5). Thus the real question is whether (0.1) and (0.5) are sufficient to uniquely determine a measure. That this may not be too much to hope for is already indicated in Theorem 4.1.1, from which it is easy to show that Wiener measure is characterized by (0.1) and (0.5) with a == I and b == O. On the other hand, one must not allow oneself to be over-optimistic because it is easy to find coefficients a and b for which (0.1) and (0.5) do not uniquely determine a measure. For instance, let a == 0 and suppose that b is a continuous vector field which admits several integral curves through a given point. Then it is obvious that each integral curve determines a different measure satisfying (0.1) and (0.5). Thus we should guess that there will be cases in which (0.1) and (0.5) together determine a unique measure, but that it cannot be entirely trivial to recognize such cases. Obviously, an interesting class of questions come out of these and related considerations. Before investigating them, as we will be throughout the remainder of this book, we will formalize the statement of the problem around which these questions revolve.

138

6. The Martingale Formulation

Given bounded measurable functions a: [0, 00) X Rd ~ Sd and b: [0, (0) X Rd -+ Rd, define L, by (0.4). Given (s, x) E [0, (0) X Rd, a solution to the martingale problemfor L, (or a and b) startingfrom (s, x) is a probability measure P on (0, At) satisfying P(x(t) =x, O:S; t:S; s)

=1

such that

f(x(t)) -

j

"

L. f(x(u)) du

s

is a P-martingale after time s for allf E Cg'(Rd). The basic questions which arise in connection with the martingale problem for given coefficients are the following:

(i) Does there exist at least one solution starting from a specified point (s, x)? (ii) Is there at most one solution starting from (s, x)? (iii) What conclusions can be drawn if there is exactly one solution for each (s, x)? Questions (i) and (ii) are, respectively, those of existence and uniqueness; while the answers to (iii) presumably contain the justification for our interest in (i) and (ii). By analogy with Hadamard's codification of problems in the theory of partial differential equations, we will say that the martingale problem for a and b is well-posed if, for each (s, x), there is exactly one solution to that martingale problem starting from (s, x). To the student familiar with the modern theory of Markov processes via semigroups and resolvents the preceding formulation may seem somewhat odd. In particular when the coefficients do not depend explicitly on t, the modern theories describe the diffusion process generated by L as a Markov family of measures {P x: X E Rd} whose transition probability function comes from a semigroup which is generated by an extension of L. On the other hand in our formulation each measure Px or, in the time dependent case, each measure p s•x is described separately, and no mention is made about the connection between one P s , x and the others. There is cause for concern here, because unless one proves uniqueness, there will in general be no nice relation among the solutions starting from various points. (Although, as we will see in Chapter 12, under quite general conditions one can make a "Markov Selection" from among the multiplicity of solutions available for each starting point.) Thus, it should be clear that uniqueness is very vital for our formulation to yield a completely satisfactory theory. Once established, uniqueness will lead to many important consequences. For these reasons, after quickly establishing a reasonably general existence result, we are going to devote much of this chapter to the development of some techniques that will play an important role in establishing uniqueness results.

139

6.1. Existence

6.1. Existence We open this section with an easy construction which will serve us well in the sequel.

6.1.1 Lemma. Let s 2 0 be given and suppose that P is a probability measure on (n, AS), where AS == CT(X(t): t 2 s). If" E C([O, s], Rd) and P(x(s) = ,,(s)) = 1, then there is a unique probability measure b~ ® sP on (n, A) such that b~ ®sP(x(t) = ,,(t), 0 S t s s) = 1 and b~ ®sP(A) = P(A)for all A E AS. Proof The uniqueness is obvious. As for existence, let b~ be the point-mass on C([O, s], Rd) at " (i.e., b~({(X E C([O, s], Rd): (X(t) = ,,(t), 0 s t S s}) = 1) and let cI>: n --+ C([s, (0), Rd ) be the map defined by cI>(ro)(t) = x(t, ro), t 2 s. Clearly cI> is measurable on (n, AS), and therefore P a cI>- 1 is well defined. Define P = b~ x (P cI>-I) on X == C([O, s], Rd ) X C([s, (0), Rd ) and set X = {((X, 13) E C([O, s], Rd) X C([s, (0), Rd): (X(s) = f3(s)}. Then X is obviously a measurable subset of X and P(X) z b~({(X E C([O, s], Rd): (X(s) = ,,(s)})P a cI>- l({f3 E C([s, (0), Rd): f3(s) = ,,(s)}) = 1. Thus P can be restricted to X. Finally, '1': X --+ C([O, (0), Rd ) defined 0

by

'I'(((X, f3))(t)

=

J(X(t) ~f 0 s t < s \f3(t) if t> s

is clearly a measurable map of X onto C([O, (0), Rd ), and therefore the restriction of P to X determines, via '1', a probability measure on (n, A). It is easy to check that this is the desired measure b~ ® s P. D

6.1.2 Theorem. Let r be a finite stopping time on n. Suppose that w mapping ofn into probability measures on (n, A) such that

--+

Q", is a

(i) ro --+ Q",(A) is .ilr-measurable for all A E A, (ii) Q",(x(r(ro), .) = x(r(ro), ro)) = 1 for all ro E n.

Given a probability measure P on (n, A), there is a unique probability measure P ®r(.) Q. on (n, A) such that P ®r(.) Q. equals P on (n, Ar) and {b", (8)t(",) Q",} is a r.c.p.d. of p®T(.)Q.IA t • In particular, suppose that rzs and that 0: [s, 00) x n --+ C is a right-continuous, progressively measurable function after time s such that O(t) is P®t(.)Q.-integrable for all t 2 s, (O(t I\r), A" P) is a martingale after time s, and (O(t) - O(t 1\ r(w)), A" Q",) is a martingale after time s for each ro E n. Then (O(t), A r , P ®r(.)Q.) is a martingale after time s. Proof The second assertion is an immediate consequence of Theorem 1.2.10 once we have proved the first part. Moreover, the uniqueness assertion of the first part is obvious. Finally, to prove the existence of P. ®r(.) Q., it is enough to check that ro --+ bID ®r(w) Qw(A) is Ar-measurable for all A E A and then set A Ejl.

140

6. The Martingale Formulation

Once it is known that w -+ 15", ®,(",)Q",(A) is ..It,-measurable, the proof that R has the desired properties of P. ®,(.)Q. is easy, But if A = {x(td E r l , " ' , x(t n} Ern}, where n ~ 1,0::; tl < ", < tn' and r l , " ' , rn E fJl Rd , then

n-I

+ L X[t •. tk+tl(r(w}}Xrt(x(t l , w}} .. , Xr.(x(t k , w)) k=1

x Qw(x(tk+ dE

rk+ I'

.. "

x(tn} Ern)

+ X[t •. OO)(r(w))xdx(tl' w)) ... xdx(tn' w)), and this is clearly JI,-measurable,

0

We will soon use Theorem 6.1.2 to prove a quite general existence result. But before we do, we want to establish a very useful theorem which complements Theorem 6.1.2.

6.1.3 Theorem. Let a: [s, 00) x n-+Sdandb: [s, 00) x 0.-+ Rdbeboundedprogressively measurable functions. Suppose that P is a probability measure on (0., JI) and that ~: [s, (0) x 0. -+ Rd is a progressively measurable, right continuous P-almost surely continuous function for which ~(.) '" J:;(a(·), b(·)) with respect to (0., JI" P). Given a stopping time r ~ s and a r.c.p.d. {Q",} of PI JI" there exists a P-null set N E JI, such that ~(.) '" 5~(W)(a(·), b(·)) with respect to (0., A" Qw) for each W ~ N.

Proof Let

®

C;; CO'(Rd} be a countable, dense subset. For each f apply Theorem 1.2.10 to find a P-null set N J E A, such that

Xj(t)

=

f(~(t, w)) - f(~(r(w), w)) -

,

f

E

®, we can

(Luf)(~(u, w)) du

,(w)

is a martingale after r(w) with respect to (0., At, Qro) for each w ~ N r . Define N = [email protected] P-null set and N E A,. Moreover, for any fE CO'(Rd) and w ~ N, we can find {f,.}1' C;; such that fn -+ fin CO'(Rd) and therefore Xjjt}-+ Xj(t) boundedly and point-wise. Hence (Xj(t), A" Qw} is a martingale after r(w) for all f E C(f(Rd) and w f{. N. By Theorem 4.2.1, this completes the proof. 0

®

6.1.4 Lemma. Let c: [0, oo)-+Sd and b: [0, OO)-+Rd be bounded measurablefunctions and set L,

1

= ..

2

0 L c'}(t)-~· oxiox d

i ,j=1

..

2

j

+

0 L b'(t)-. oXi d.

i=1

141

6.1. Existence

Then the martingale problem for Lr is well-posed. Moreover, ~r {P s• x : (s, x) E [0, (0) X Rd} denotes the family of solutions determined by L" then (s, x) --+ ps. x(A) is measurable for all A E .It. Proof To prove the existence of a measurable family of solutions, we could invoke Theorems 5.1.4 and 5.3.2. Moreover, uniqueness could be proved by an application of Theorem 4.5.2 and Corollary 5.1.3. However, we prefer to take a different route. Define C(s, t) = J~ c(u) du and B(s, t) = J~ b(u) dufor 0 s sst. Then for all o s sst and x E Rd the function ({J(s, x; t,

e) = exp[i P. Clearly

f f(Y)J1{dy) =

lim e"u{x(O))] = e[!{x{O))] n' -+

00

144

6. The Martingale Formulation

for all f E Cb(R d ), and so P satisfies the correct initial condition. Moreover, if f E Cg'(Rd), then

fo L~ny(x(u)) du I

f(x(t)) is a Pn-martingale, where

Since for each t :::::: 0, the functions

fo L~ny(x(u)) du I

are equicontinuous at every point of 0, bounded independently of n, and tend to J~ Lu!(x(u)) du as n -> 00, we can apply Corollary 1.1.5 to prove that

EP[(f(X(t))- (Luf(x(U))dU)cI>l =

!~~ EPn [( f(x(t)) - (L~ny(x(u)) du )cI>].

for all t :::::: 0 and bounded continuous cI>. In particular, if 0 :s;; t 1 < t z and cI> is a bounded continuous ..illl-measurable function, then we have:

e[(f(x(t z))- (L uf(X(U))dU)cI>l =

e [( f(x(t d) -

.c

Lu f(x(u)) du )cI> J,

since this equality certainly holds when P and Lu are replaced by Pn' and L~n'), respectively. It is now easy to see that this equality persists when cI> is simply bounded and ..it/I-measurable, and so we have shown that x(·) ~ .'~(a(·), b(')) with respect to (0, ..itl , P), 0 6.1.7 Theorem. Let a: [0, (0)

X

Rd

->

Sd and b: [0, (0)

X

Rd -> Rd be bounded meas-

urable functions such that a(t, .) and b(t, .) are continuous for all t :::::: O. For each

s :::::: 0 and each probability measure J1 on R d , there is a probability measure P on

(n, ..it) such that

P{x{t)

E

r, 0 :s;; t :s;; s) = J1(r),

145

6.2. Uniqueness: Markov Property

and x(·) ~ ,j'd(a(', x(·)), b(', x(·))) with respect to (Q,.itl' P), In particular, the martingale problem for a and b has at least one solution starting from each (s, x) E [0, 00) X R d , Proof Clearly we need only prove the first assertion. To this end, let sand J.l be given and define as and bs by as(t, x) = a(t + s, x) and bs(t, x) = b(t + s, x). By Theorem 6,1,6, we can find a Q on (n,.it) with the properties that Q{x{O)

E

r)

=

J.l(r),

and x{·) ~ ,j'~(ak, x(·)), bA', x{·))) with respect to (n,.itl' Q). Therefore, if we now define s:n-+n by x(t, sw)=x((t-s)vO,w), t;;::O, and set P = Q ; 1, then it is easily seen that P has the desired properties, 0 u

6.2. Uniqueness: Markov Property We now have our basic existence result. Unfortunately, the corresponding uniqueness theorem is not so easy to prove; in fact, it is not even true. The purpose of the present section is to start laying the groundwork for a reduction procedure with which we will eventually be able to obtain a reasonably satisfactory uniqueness theorem. At the same time, the preparations that we make here will serve us well when it comes time to discuss the consequences of uniqueness. In order to appreciate what is going on in this section, it is helpful to think about how one might try proving uniqueness. Given two probability measures P and Q on (n, .it), one knows that they are equal if and only if

for all n 2': 1,0 S t 1 < ... < tn' and r l' ... , r n E !!4 Rd. That is, if the finite dimensional marginal distributions determined by P and Q agree. If one knows a priori that P and Q are Markovian in the sense that for any 0 S t1 < t2 the conditional distribution of X(t2) given .it tl is a function of t 1, t 2 , and x(td alone and if this function is the same for P and Q, then P = Q if x(O) has the same distribution under P and Q. Of course this line of reasoning cannot be applied without the a priori knowledge that P and Q are Markovian, Nonetheless, a variant of it is applicable to the study of martingale problems; and it is this variant which we now want to develop. 6.2.1 Theorem. Let a: [0, 00)

X Rd -+ Sd and b: [0, 00) X Rd -+ Rd be bounded measurable functions and suppose that P on (n, .it) has the properties that P{x{IT) = y)= 1 and X(')~Yd(a(" x(·)), b{"x('))) under P for some IT 2':0 and YER d , Thenfor any" E C([O, 00); R d ) satisfying ",(IT) = y, x(·) ~ Yd(a(-, x(· »), b(·, x(·))) under bn ® .. P. In particular, if x(·) ~ Yd(a(', x(· )), b(" x(·))) under P and if {P"'} is a r,c.p.d. of P i.it t where r 2': s is a finite stopping time, then there is a P-null set

146

6. The Martingale Formulation

N E .AT such that bx(t(w), CD)