Operator Theory and Harmonic Analysis: OTHA 2020, Part II – Probability-Analytical Models, Methods and Applications (Springer Proceedings in Mathematics & Statistics, 358) [1st ed. 2021] 3030768287, 9783030768287

Citation preview

Springer Proceedings in Mathematics & Statistics

Alexey N. Karapetyants Igor V. Pavlov Albert N. Shiryaev   Editors

Operator Theory and Harmonic Analysis OTHA 2020, Part II – Probability-Analytical Models, Methods and Applications

Springer Proceedings in Mathematics & Statistics Volume 358

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Alexey N. Karapetyants • Igor V. Pavlov • Albert N. Shiryaev Editors

Operator Theory and Harmonic Analysis OTHA 2020, Part II – Probability-Analytical Models, Methods and Applications

Editors Alexey N. Karapetyants Institute of Mathematics, Mechanic and Computer Sciences and Regional Mathematical Center Southern Federal University Rostov-on-Don, Russia

Igor V. Pavlov Department of Higher Mathematics Don State Technical University Rostov-on-Don, Russia

Albert N. Shiryaev Steklov Mathematical Institute Russian Academy of Sciences Moscow, Russia

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-030-76828-7 ISBN 978-3-030-76829-4 (eBook) https://doi.org/10.1007/978-3-030-76829-4 Mathematics Subject Classification: 26-02, 30-02, 32-02, 35-02, 42-02, 45-02, 46-02, 47-02, 60-02, 92-02 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is the second volume of the two-volume series entitled Operator Theory and Harmonic Analysis. Vol. 1:

New General Trends and Advances of the Theory

and Vol. 2:

Probability-Analytical Models, Methods, and Applications.

Volume 1 is devoted to harmonic analysis and its applications in general, while Volume 2 is focused on probabilistic and mathematical (statistical) methods in applied sciences, but still in the context of general harmonic analysis and its numerous applications. The volumes’ readership is the pool of researchers interested in various aspects of harmonic analysis and operator theory: real and complex variable methods, applications to PDE’s, mathematical modeling based on applied harmonic analysis and probability-analytical methods, and exploration of new themes and trends. The contributions to both volumes are based on the matter supposed to be presented at the annual International Scientific Conference on Modern Methods and Problems of Operator Theory and Harmonic Analysis and Their Applications (OTHA-2020, http://otha.sfedu.ru/), canceled due to Covid19 restrictions. The Editors are very grateful to all the authors for their valuable contributions and for a strong willingness to support mathematical activities and communications in the hope of the soonest resumption of regular conferences and safe mutual visits. The Editors express an immense sorrow on the occasion of the recent loss of remarkable scientists and brilliant persons, Hrachik Hayrapetyan (Armenia), who is one of the authors of the first volume, and Vladimir Pilidi (Russia), who was an active member of Program Committees of OTHA conferences, and Vladimir Nogin (Russia), who was colleague and teacher of quite a few participants of OTHA.

v

vi

Preface

The first volume contains words in memoriam of our dear friends Hrachik Hayrapetyan, Vladimir Pilidi, and Vladimir Nogin. Rostov-on-Don, Russia Moscow, Russia

A. Karapetyants I. Pavlov A. Shiryaev

Contents

A Conditional Functional Limit Theorem for a Decomposable Branching Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . V. I. Afanasyev A Probabilistic Interpretation of Conservation and Balance Laws. . . . . . . . . Ya. I. Belopolskaya

1 19

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Alexander Bendikov, Alexander Grigor’yan, and Stanislav Molchanov

43

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis

91

Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109 Sergey G. Bobkov, Maria A. Danshina, and Vladimir V. Ulyanov Combinatorial Identities with Binomial Coefficients . . . .. . . . . . . . . . . . . . . . . . . . 135 Ya. M. Erusalimsky Random Tempered Distributions on Locally Compact Separable Abelian Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 Manuel L. Esquível and Nadezhda P. Krasii On Solutions of Stochastic Equations with Current and Osmotic Velocities . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167 Yuri E. Gliklikh Stochastic Methods in Investigation of Modern Networks . . . . . . . . . . . . . . . . . . 185 Vladimir A. Gorlov and Alla V. Makarova

vii

viii

Contents

Double-Barrier Option Pricing under the Hyper-Exponential Jump Diffusion Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197 S. M. Grudsky and O. A. Mendez-Lara Single Jump Filtrations: Preservation of the Local Martingale Property with Respect to the Filtration Generated by the Local Martingale . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 219 Alexander A. Gushchin and Assylliya K. Zhunussova Local Time and Local Reflection of the Wiener Process .. . . . . . . . . . . . . . . . . . . . 233 I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev Random Harmonic Processes with New Properties . . . . . .. . . . . . . . . . . . . . . . . . . . 243 Elena Karachanskaya On Solvability of One Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253 A. Kh. Khachatryan A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 273 Oleg Kudryavtsev New Procedure for Applying the Cramér–von Mises Test for Parametric Families of Distributions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 Gennady Martynov CVaR Hedging in Defaultable Jump-Diffusion Markets . . . . . . . . . . . . . . . . . . . . 309 Alexander Melnikov and Hongxi Wan Out-of-Sample Utility Bounds for Empirically Optimal Portfolios in a Single-Period Investment Problem .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 335 Dmitry B. Rokhlin A Guaranteed Deterministic Approach to Superhedging: Optimal Mixed Strategies of the Market and Their Supports . . . . . . . . . . . . . . 355 Sergey N. Smirnov Some Properties of Regularly Varying Functions and Series in the Orthant .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 373 A. L. Yakymiv Influence of the Configuration of Particle Generation Sources on the Behavior of Branching Walks: A Case Study . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 387 E. B. Yarovaya Lanchester Model with the Random Coefficients . . . . . . . .. . . . . . . . . . . . . . . . . . . . 407 V. G. Zadorozhniy

A Conditional Functional Limit Theorem for a Decomposable Branching Process V. I. Afanasyev

Abstract A decomposable Galton–Watson branching process with two particle types is studied. It is assumed that a particle of the first type produces equal numbers of particles of the first and second types, while a particle of the second type produces only particles of their own type. Under the condition that the total number of particles of the second type is greater than N → ∞, a functional limit theorem for the process describing the number of particles of the second type in different generations is proved. Keywords Decomposable Galton–Watson branching process · Feller diffusion · Local time of a Brownian excursion · Functional limit theorems

1 Introduction and Statement of Main Result Consider a decomposable Galton–Watson branching process with two particle types. It is assumed that a particle of the first type produces equal numbers of particles of the first and second types, while a particle of the second type produces only particles of their own type. Let ϕ (s) and ψ (s), s ≥ 0, be the generating functions of nonnegative integervalued random variables ξ and η, respectively. Suppose that the maximal span of the distribution of the random variable ξ is equal to 1. Suppose also that Eξ = 1, Varξ : =σ12 ∈ (0, ∞) ; Eη = 1, Varη : =σ22 ∈ (0, ∞)

(1)

(note that, along with the symbols σ12 and σ22 , we use the notations b1 = σ12 /2 and b2 = σ22 /2). Let us introduce the generating functions for the descendants of one

V. I. Afanasyev () Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_1

1

2

V. I. Afanasyev

particle of the first and second types, respectively: for s1 , s2 ≥ 0, f1 (s1 , s2 ) = ϕ (s1 s2 ) , f2 (s1 , s2 ) = ψ (s2 ) . Let ξn and ηn denote the numbers of particles of the first and second types, respectively, in the nth generation of the branching process under consideration. It is assumed that ξ0 = 1 and η0 = 0. We set 1 =

∞ n=1

ξn , 2 =

∞ n=1

ηn .

Decomposable Galton–Watson branching processes with several particle types were first studied under the nonextinction condition (for a detailed bibliography, see [1]) or under the condition of extinction at a fixed moment of time (see [1] and [2]). In [3–7], decomposable Galton–Watson branching processes with two particle types were considered under conditions imposed on the total number of particles of the first or second types. For instance, various functional limit theorems for the process {(ξn , ηn ) , n ∈ N0 } considered under the condition that 1 = N, where N → ∞, proved in [3] with restrictions (1) and in [4] with more general restrictions for which, in particular, that the values σ12 and σ22 can be infinite. x We set 2 (x) = n=0 ηn for x ≥ 0. In [5, 6], decomposable Galton–Watson branching processes with two particle types were considered under the condition that 2 (aN) > N 2 , where a is a constant belonging (0, ∞] and N → ∞ (N ∈ N). In particular, in [6], under restrictions (1), a functional limit theorem was proved for the process {ξn , n ∈ N0 } considered under the condition that 2 > N, where N → ∞. In [7], under more general restrictions than (1), a one-dimensional version of this theorem was proved, and in addition, a similar statement is obtained when replacing the condition {2 > N} with the condition {2 = N}, where N → ∞. In the present paper, we study the process {ηn , n ∈ N0 } considered under the condition that 2 (aN) > N 2 , where N → ∞. Note that the random process {ξn , n ∈ N0 } is a critical Galton–Watson branching process in which the generating function of the descendants of one particle is ϕ (·). For a fixed process {ξn , n ∈ N0 }, the process {ηn , n ∈ N0 } is a critical Galton– Watson branching process with ξn immigrants in the nth generation for each n ∈ N. Note that here, the generating function of the descendants of one particle (including those of each immigrant) is ψ (·). Let {Y (t) , t ≥ 0} be a Feller diffusion, i.e., a homogeneous Markov random process with continuous trajectories starting from a positive state and having an absorbing state at 0 whose transition probability density p(t, x, y), where t, x, y > 0, satisfies the equality 

∞

e

−λy

0

for t, x > 0 and λ ≥ 0.

   x λx p (t, x, y) dy = exp − − exp − 1 + λt t

A Conditional Functional Limit Theorem for a Decomposable Branching Process

3

We set for a ∈ (0, ∞] 

a

Sa =

Y (b2 t) dt. 0

It is further assumed that the Feller diffusion starts from state 1 (i.e., Y (0) = 1 a.s.), if the opposite is not specified. It is proved in [8] that the random variable Sa is finite and positive and its distribution is absolutely continuous. Moreover, for λ ≥ 0, E exp (−λSa ) =



√

√ exp −x λ/b2 tanh a λ/b2 , a ∈ (0, ∞) ;

√  exp −x λ/b2 , a = ∞.

In the special case, when a = ∞, this random variable S∞ has absolutely continuous distribution with the probability density

2  1 f (z) = √ e−1/ 2σ2 z , z > 0. 2πσ2 z3/2

We set for a ∈ (0, ∞] 

∞

c (a) = 0

 P z2 Sa/z > 1 dz. z3/2

Note that for a ∈ (0, ∞], 

∞

0 < c (a) ≤ 0

  ∞   P z2 S∞ > 1 4 4 dz = 2 P S > y dy = 2E S∞ < ∞ ∞ z3/2 0

(the latter inequality is easy to obtain using the explicit representation for f (z), z > 0). We introduce for a ∈ (0, ∞] the probability density

 1 P z2 Sa/z > 1 pa (z) = , z > 0. c (a) z3/2 Let us assume that convergence of a sequence of random processes in distribution in the space D (0, ∞) means convergence in distribution in the space D[v, +∞) with the Skorokhod topology for each v > 0. Now, we formulate the main result. Theorem 1 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞,

η

t N

N

  D  , t > 0 2 (aN) > N 2 → { Y (b2 t) , t > 0| Sa > 1} ,

(2)

where Y (0) has absolutely continuous distribution with the probability density D

pa (·) and the symbol → means convergence in distribution in the space D (0, ∞).

4

V. I. Afanasyev

2 Auxiliary Statements We set for t ≥ 0 YN (t) =

ηt N . N

It is proved in [3, Theorem 3] that, if restrictions (1) are valid, then, as N → ∞, D

{ YN (t) , t > 0| 1 = N} → {Y (b2 t) , t > 0} ,

(3)

D

where the symbol → means convergence in distribution in the space D (0, ∞). Denote for a ∈ (0, ∞] ζN (a) =

2 (aN) . N2

It is proved in [5, Theorem 1] that, if restrictions (1) are valid and 0 < b < a ≤ ∞, then, as N → ∞, D

D

{ ζN (a) | 1 = N} → Sa , { ζN (a) − ζN (b) | 1 = N} → Sa − Sb .

(4)

In addition to relations (3) and (4), we prove the following result. Lemma 1 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞, D

{ YN (t) , t > 0; ζN (a) | 1 = N} → {Y (b2 t) , t > 0; Sa } ,

(5)

D

where the symbol → means convergence in distribution in the space D (0, ∞) × R. Proof We fix such v and u that 0 < v < u < ∞ and introduce random processes Y = {Y (b2 t) , t ∈ [v, u]} ; YN = {YN (t) , t ∈ [v, u]} , N ∈ N. Let f be a continuous bounded mapping of the space D [v, u] into R. It is required to show that, as N → ∞, D

{ f (YN ) , ζN (a) | 1 = N} → (f (Y ) , Sa ) . Let numbers x ∈ R and y ∈ (0, ∞) be fixed. Set P (N) = P ( f (YN ) ≤ x, ζN (a) ≥ y| 1 = N) .

(6)

A Conditional Functional Limit Theorem for a Decomposable Branching Process

5

Note that (1) 

2 (aN) = N

(2)

v + N

 (3)  v , u + N u ,

(7)

where 0 < v < u < a and (1)  N v

=

v N  

ηn ,

 (2)

N v , u

n=1

=

u N  

aN 

(3) 

n=v N+1

ηn , N

u =

ηn .

n=u N+1

In view of (7),





  P1 N, v , u ≤ P (N) ≤ P2 N, v , u , ε + P3 N, v , u , ε ,

(8)

where ε ∈ (0, y) and









P1 N, v , u = P f (YN ) ≤ x,

    ≥ y  1 = N , 

(2)  v ,u

N

N2

    (2)



 N v , u





P2 N, v , u , ε = P f (YN ) ≤ x, ≥ y − ε 1 = N ,  N2



P3 N, v , u , ε = P



    ≥ ε 1 = N . 

(1)  (3)  v + N u

N2

N

 



Set v

= min v, v and u

= max u, u . We consider two functionals on the space D v

, u

:  z → f ({z (t) , t ∈ [v, u]}) , z →

u

v

z (t) dt,

  



 where z ∈ D v

, u

. These functionals are continuous for 

such

z ∈ D v , u , which are themselves continuous functions on the segment v , u . Since trajectories of a Feller diffusion are continuous a.s., the mapping  z → f ({z (t) , t ∈ [v, u]}) ,



u

v

 z (t) dt ,

6

V. I. Afanasyev

  where z ∈ D v

, u

, is continuous on almost all trajectories of the Feller diffusion. Hence (see Theorem 5.1 in [9]), in view of (3), as N → ∞,       u

 u

 D  YN (t) dt  1 = N → f (Y ) , Y (b2 t) dt . (9) f (YN ) ,  v

v

Note that  (2)

   u

  Y v  + Y u 



N N  N v ,u  − YN (t) dt  ≤  2   N N v

(10)

and (see (3)), as n → ∞, 





 YN v + YN u  D  1 = N → 0.  N

(11)

It follows from (9)–(11) that, as N → ∞,  f (YN ) ,

     u

D  Y (b2 t) dt .  1 = N → f (Y ) ,  v

(2)   v ,u

N

N2

(12)

Since convergence in distribution implies complete convergence of corresponding distribution functions, we obtain from (12) that











lim P1 N, v , u = P f (Y ) ≤ x,

v

N→∞











lim P2 N, v , u , ε = P f (Y ) ≤ x,

N→∞

u



u

v

 Y (b2 t) dt ≥ y ,

(13) 

Y (b2 t) dt ≥ y − ε

(14)

and relations (13) and (14) are valid for all but a countable set of (x, y) ∈ R× (0, ∞). Further,

 P3 N, v , u , ε     (3)    (1)  N u

N v

ε  ε  ≥  1 = N + P ≥  1 = N . (15) ≤P 2 2 N2 N2

A Conditional Functional Limit Theorem for a Decomposable Branching Process

7

From (15) in view of relation (4), we obtain that

 lim sup P3 N, v , u , ε ≤ P



v

0

N→∞

ε Y (b2 t) dt ≥ 2



 +P

a u

ε Y (b2 t) dt ≥ 2



and, hence, lim

v →0,u →a



lim sup P3 N, v , u , ε = 0.

(16)

N→∞

It follows from (8), (13), and (14) that 



P f (Y ) ≤ x,

u

v



 Y (b2 t) dt ≥ y 

≤ P f (Y ) ≤ x,

u

v

≤ lim inf P (N) ≤ lim sup P (N) N→∞

N→∞



 Y (b2 t) dt ≥ y − ε + lim sup P3 N, v , u , ε . N→∞

Therefore, passing to the limit as v → 0, u → a and taking into account (16), we find that P (f (Y ) ≤ x, Sa ≥ y) ≤ lim inf P (N) N→∞

≤ lim sup P (N) ≤ P (f (Y ) ≤ x, Sa ≥ y − ε) . N→∞

Now, passing to the limit as ε → 0, we obtain lim P (N) = P (f (Y ) ≤ x, Sa ≥ y) .

N→∞

Thus, the required relation (6) is established. The lemma is proved. Set for z > 0 and a ∈ (0, ∞]  Sa(z)

2

= z Sa/z ; Y

(z)

(t) = zY

b2 t z

 , t ≥ 0.

It is known (see, for instance, [10]) that a Feller diffusion has the property of selfsimilarity, according to which

  D  Y (z) (t) , t ≥ 0 Y (0) = 1 = { Y (t) , t ≥ 0| Y (0) = z} .

(17)

8

V. I. Afanasyev

In particular,

  D  Sa(z)  Y (0) = 1 = { Sa | Y (0) = z} .

(18)

Moreover, relations (17) and (18) can be combined into one relation. Lemma 2 If restrictions (1) are valid and a ∈ (0, ∞], then for any z > 0, as N → ∞, D

{ YN (t) , t > 0; ζN (a) | 1 = zN} → { Y (b2 t) , t > 0; Sa | Y (0) = z} ,

(19)

D

where the symbol → means convergence in distribution in the space D (0, ∞) × R. Proof Set KN = zN; then, for N ≥ z−1 ,   t γ −1 , YN (t) = zYKN γN z N

(20)

  a γN−2 , γN z

(21)

2

ζN (a) = z ζKN where γN = zN/ zN. Note that

0 ≤ γN − 1 ≤ 1/KN .

(22)

By Lemma 1, as N → ∞, zYKN

    

 t a  D (z) (z)  , t > 0; z2ζKN = K → Y . , t > 0; S (t) 1 N a z z  (23)

We fix such v and u that 0 < v < u < ∞. Introduce a modulus of continuity for a function x ∈ D [v, u]: wx (δ; v, u) =

sup

t,s:|t −s|≤δ,

|x (t) − x (s)| , δ > 0.

t,s∈[v,u]

In view of (22),         t t  1 u v 2u  sup YKN γN ; , − YKN . ≤ wYKN z z  KN z z z t ∈[v,u]

(24)

Let number δ > 0 be fixed. Note that a functional that maps an element x of the space D [v, u] to its modulus of continuity wx (δ; v, u) is continuous for such x,

A Conditional Functional Limit Theorem for a Decomposable Branching Process

9

which are themselves continuous functions on the segment [v, u]. Since trajectories of a Feller diffusion are continuous a.s., then due to (23), as N → ∞,      v 2u  v 2u D  wY (z) δ; , , , (25) δ; = K → w (z) 1 N Y KN z z  z z where   

 t (z) YN = zYN , t ≥ 0 , Y (z) = Y (z) (t) , t ≥ 0 . z It follows from (25) that, as N → ∞,

 wY (z)

KN

  1 u v 2u  D 1 = KN → 0, ; ,  KN z z z

(26)

since limδ→0 wx (δ; v/z, 2u/z) = 0 for a continuous function x on the segment [v/z, 2u/z]. We obtain from (24) and (26) that, as N → ∞,           t t D   1 = KN → − YKN sup YKN γN 0. (27)  z z t ∈[v,u] Note that in view of (22), 0 ≤ ζK N

        a a a a − ζK N ≤ ζKN (1 + 1/KN ) − ζK N . γN z z z z

Therefore, if δ > 0 and N is large enough, then         a a+δ a a 0 ≤ ζK N γN − ζK N ≤ ζK N − ζK N . z z z z

(28)

By virtue of (4), as N → ∞,

        a+δ a D (z)  z ζK N − ζK N 1 = KN → Sa+δ − Sa(z) .  z z 2

(29)

Finally, in view of the continuity of almost all trajectories of a Feller diffusion,   (z) lim Sa+δ − Sa(z) = 0.

δ→0

It follows from (28)–(30) that, as N → ∞,         a   D ζ K γ N a − ζ K = K → 0.  1 N N  N z z 

(30)

(31)

10

V. I. Afanasyev

From relation (23), given (27) and (31), we find that, as N → ∞,

    t a  , t ∈ [v, u] ; z2 ζKN γN = K  1 N z z  

D → Y (z) (t) , t ∈ [v, u] ; Sa(z) . 

zYKN

γN

(32)

From (32), remembering relations (20)–(22), we obtain that, as N → ∞, 

D { YN (t) , t ∈ [v, u] ; ζN (a) | 1 = zN} → Y (z) (t) , t ∈ [v, u] ; Sa(z) . Therefore,

 D { YN (t) , t > 0; ζN (a) | 1 = zN} → Y (z) (t) , t > 0; Sa(z) , whence, in view of the relations (17) and (18), it follows the statement of the lemma. We also present the corrected formulations of Theorems 2 and 3 from [5]. Lemma 3 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞,   c (a) 1 P 2 (aN) > N 2 ∼ √ , N 

√ where c1 (a) = c (a) / 2 πb1 . Lemma 4 If restrictions (1) are valid and a ∈ (0, ∞], then, for any x > 0,    lim P 1 ≤ xN| 2 (aN) > N 2 =

N→∞

x

pa (z) dz, 0

where pa (·) is the function defined in Sect. 1.

3 Proof of Theorem 1 Let’s rewrite relation (2) in the equivalent form: as N → ∞, D

{ YN (t) , t > 0| ζN (a) > 1} → { Y (b2 t) , t > 0| Sa > 1} ,

(33)

where Y (0) has absolutely continuous distribution with the probability density pa (·). We fix such v and u that 0 < v < u < ∞. Let f be a continuous bounded and positive mapping of the space D [v, u] in R. For validity of (33), it is required

A Conditional Functional Limit Theorem for a Decomposable Branching Process

11

to show that 

∞

lim E ( f (YN ) | ζN (a) > 1) =

N→∞

E(z) ( f (Y ) | Sa > 1) pa (z) dz,

(34)

0

where, as before, YN = {YN (t) , t ∈ [v, u]} for N ∈ N, and Y = {Y (b2 t) , t ∈ [v, u]}; the notation E(z) , used instead of the notation E, indicates that Y (0) = z a.s. Note that for arbitrary positive numbers b and B (b < B), E ( f (YN ) | ζN (a) > 1) = E1 (N, b, B) + E2 (N, b, B) ,

(35)

where E1 (N, b, B) = E ( f (YN ) I (1 ∈ [bN, BN]) | ζN (a) > 1) , / [bN, BN]) | ζN (a) > 1) E2 (N, b, B) = E ( f (YN ) I (1 ∈ (here, I (A) is the indicator function of a random event A). It is clear that |E2 (N, b, B)| ≤ LP ( 1 ∈ / [bN, BN] | ζN (a) > 1) , where L = supx∈D[v,u] |f (x)|. Whence, in view of Lemma 4, we find that lim

lim sup |E2 (N, b, B)| = 0.

(36)

b→0,B→∞ N→∞

We show that  lim

∞

lim E1 (N, b, B) =

b→0,B→∞ N→∞

E(z) ( f (Y ) | Sa > 1) pa (z) dz.

(37)

0

Note that E1 (N, b, B) =

 K∈[bN,BN]

=



E (f (YN ) ; ζN (a) > 1, 1 = K) P (ζN (a) > 1) E ( f (YN ) I (ζN (a) > 1) | 1 = K)

K∈[bN,BN]

P (1 = K) P (ζN (a) > 1)

(the summation is performed using natural numbers K). It is known (see, for instance, [11, Chapter 2]) that, as n → ∞, 1 P (1 = n) ∼ . 2 πb1 n3

(38)

12

V. I. Afanasyev

Therefore, as N → ∞, 2 πb1 lim E1 (N, b, B) N→∞



= lim

N→∞

E ( f (YN ) I (ζN (a) > 1) | 1 = K) K 3/2P (ζN (a) > 1)

K∈[bN,BN]

 = lim

B

N→∞ b

E ( f (YN ) I (ζN (a) > 1) | 1 = zN) dz √ , z3/2 N P (ζN (a) > 1)

(39)

if at least one of these three limits exists. We consider a mapping that matches a pair (x, y), where x ∈ D [v, u], y ∈ R, the number f (x) I {y > 1}. This mapping is continuous for such (x, y) for which y = 1. Note that P(z) (Sa = 1) = 0. Therefore (see [9], Theorem 5.1]), we conclude using the Lemma 2 that lim E ( f (YN ) I (ζN (a) > 1) | 1 = zN) = E(z) (f (Y ) ; Sa > 1) .

N→∞

By Lemma 3, lim

N→∞

√ N P (ζN (a) > 1) = c1 (a) .

Hence, by the dominated convergence theorem, 

B

lim

N→∞ b



B

= b

E ( f (YN ) I (ζN (a) > 1) | 1 = zN) dz √ z3/2 NP (ζN (a) > 1) E(z) (f (Y ) ; Sa > 1) dz . c1 (a) z3/2

(40)

We obtain from (39) and (40) that 

B

lim E1 (N, b, B) =

N→∞

b

E(z) (f (Y ) ; Sa > 1) √ dz. 2 πb1 c1 (a) z3/2

(41)

Note, using the Lemma 3, relation (18), and the definition of pa (z), that E(z) (f (Y ) ; Sa > 1) √ = E(z) ( f (Y ) | Sa > 1) pa (z) . 2 πb1 c1 (a) z3/2

(42)

It follows from (41) and (42) that 

B

lim E1 (N, b, B) =

N→∞

b

E(z) ( f (Y ) | Sa > 1) pa (z) dz.

(43)

A Conditional Functional Limit Theorem for a Decomposable Branching Process

13

Passing in (43) to the limit as b → 0 and B → ∞, we get the required relation (37). From (35)–(37) follows relation (34). The theorem is proved. Remark 1 Note that Lemma 4 was not used to prove (43). For f ≡ 1, relation (43) means that 

B

lim P ( 1 ∈ [bN, BN] | ζN (a) > 1) =

N→∞

pa (z) dz. b

The latter relation allows us to establish Lemma 4 in a shorter way than in [5].

4 Conclusion The statement of Theorem 1 for a = ∞ can be given the following form: as N → ∞,    D  , t > 0 2 > N → { Y (b2 t) , t > 0| S∞ > 1} , √  N

 η √



t N

(44)

where Y (0) has absolutely continuous distribution with the probability density p∞ (·). It is interesting to compare (44) with the corresponding limit theorem for a particle in critical branching Galton–Watson process {Zn , n ∈ N0 }, starting with one  the zero generation and satisfying the condition that  > n, where  = ∞ n=0 Zn . Recall that, for a fixed process {ξn , n ∈ N0 }, the random process {ηn , n ∈ N0 } is also a critical branching Galton–Watson process; however, it is not created by one particle in the zero generation but ξn immigrants in the nth generation for each n ∈ N. Let {W(t) , t ∈ [0, 1]} be a standard Brownian motion. Recall that a Brownian excursion W0+ (t) , t ∈ [0, 1] is a limit in distribution in the space C [0, 1] as ε ↓ 0 of the Brownian motion considered under the condition that inft ∈[0,1] W (t) ≥ −ε and W (1) ≤ ε. The local time l0+ (y) at a level y ≥ 0 of the Brownian excursion   + W0 (t) , t ∈ [0, 1] is defined by 1 ε→0 ε

l0+ (y) = lim

 0

1

 I[y,y+ε] W0+ (t) dt,

where I[y,y+ε] (·) is the indicator function of the segment [y, y + ε]. Note  that there exists a continuous modification of the random process l0+ (y) , y ≥ 0 .

14

V. I. Afanasyev

It is proved in [12, Theorem 4.10] that, if the maximal span of the distribution of the random variable Z1 is equal to 1 and EZ1 = 1, VarZ1 : =σ 2 ∈ (0, ∞), then, as N → ∞, ⎫ ⎧     ⎬ ⎨ Z t √N 

σ σ   D l0+ t ,t ≥ 0 , (45) √ , t ≥ 0  = N → ⎭ ⎩ 2 2 N  D

where the symbol → means convergence in distribution in the space D [0, ∞) with the Skorokhod topology. As a corollary of relation (45), we establish the following limit theorem. Theorem 2 If the maximal span of the distribution of the random variable Z1 is equal to 1 and EZ1 = 1 and VarZ1 : =σ 2 ∈ (0, ∞), then, as N → ∞, ⎧   ⎫   ⎨ Z t √N ⎬ 

σ σ   D , t ≥ 0  > N → l0+ tα , t ≥ 0 , √ ⎩ ⎭ 2α 2 N 

(46)

where α is a random variable uniformly   distributed on the interval (0, 1) and independent of the process l0+ (t) , t ≥ 0 . Proof Note that for an arbitrary z > 0, as N → ∞,  ⎫ ⎧       √ ⎬ ⎨ Z t √N  σ σ z + D  zN l t , t ≥ 0 . → , t ≥ 0  = √ √  ⎭ ⎩ 2 0 2 z N 

(47)

The proof of relation (47) is based on (45) and is carried out similarly to the proof of Lemma 2 and is therefore not given here. Set for z > 0 and t ≥ 0 V

(z)

  Z √  √ t N σ σ z + l0 , N ∈ N. (t) = √ t ; UN (t) = √ 2 2 z N

Fix such v and u that 0 < v < u < +∞. Let f be a continuous bounded and positive mapping of the space D [v, u] in R. For validity (46), it is required to show that  −2   , (48) lim E ( f (UN ) |  > N) = Ef V α N→∞

  where UN = {UN (t) , t ≥ 0} and V (z) = V (z) (t) , t ≥ 0 . For B > 1, E ( f (UN ) |  > N) = E1 (N, B) + E2 (N, B),

(49)

A Conditional Functional Limit Theorem for a Decomposable Branching Process

15

where E1 (N, B) = E ( f (UN ) I ( ≤ BN) |  > N) , E2 (N, B) = E ( f (UN ) I ( > BN) |  > N) . Note that for B > 1, |E2 (N, B)| ≤ LP (  > BN|  > N) = L

P ( > BN) , P ( > N)

(50)

where L = supx∈D[v,u] |f (x)|. Since (see (38)), as N → ∞, 1 P ( = N) ∼ √ , 2 πbN 3

(51)

1 P ( > N) ∼ √ πbN

(52)

where b = σ 2 /2, then,

and lim

N→∞

1 P ( > BN) = √ . P ( > N) B

(53)

It follows from (50) and (53) that lim lim sup |E2 (N, B)| = 0.

(54)

B→∞ N→∞

Further, E1 (N, B) =

BN K=1

E ( f (UN ) |  = N + K)

P ( = N + K) . P ( > N)

(55)

In view of relations (51) and (52), we obtain from (55) that lim E1 (N, B) = lim

N→∞

BN K=1

N→∞

 = lim

N→∞ 1

B+1

√ N

E ( f (UN ) |  = N + K) 2 (N + K)3

E ( f (UN ) |  = zN)

dz , 2z3/2

(56)

16

V. I. Afanasyev

if at least one of these three limits exists. By virtue of (47),   lim E ( f (UN ) |  = zN) = Ef V (z) .

N→∞

(57)

We obtain from (56) and (57) by the dominated convergence theorem that 

B+1

lim E1 (N, B) =

N→∞

1

 dz  Ef V (z) 2z3/2

and, therefore,  lim

∞

lim E1 (N, B) =

B→∞ N→∞

1

  dz Ef V (z) . 2z3/2

(58)

It follows from (49), (54), and (58) that  lim E ( f (UN ) |  > N) =

N→∞

1

∞

 dz  Ef V (z) . 2z3/2

(59)

Note that  1

∞

 1   −2    dz  −2   u Ef V (z) = Ef V du = Ef V α . 2z3/2 0

(60)

Due to relations (59) and (60), we obtain (48). The theorem is proved. Comparing relations (44) and (46), we see a significant difference in the asymptotic behavior of the random processes     , t > 0 2 > N , √  N

 η √

t N



⎫ ⎧     ⎬ ⎨ Z t √N  , t ≥ 0  > N , √ ⎭ ⎩ N 

although, in each of these processes, the reproduction of particles is carried out according to probabilistic laws with the mathematical expectation 1 and a finite variance. This is due to the fact that the conditions for the forming of these processes differ. The first of them (with a fixed process {ξn }) is formed due to immigrants, and the second due to a single initial particle.

References 1. Vatutin, V.A., D’yakonova, E.E.: Decomposable branching processes with a fixed extinction moment. Proc. Steklov Inst. Math. 290, 103–124 (2015) 2. Vatutin, V.A., D’yakonova, E.E.: Extinction of decomposable branching processes. Discrete Math. Appl. 26(3), 183–192 (2016)

A Conditional Functional Limit Theorem for a Decomposable Branching Process

17

3. Afanasyev, V.I.: Functional limit theorems for the decomposable branching process with two types of particles. Discrete Math. Appl. 26(2), 71–88 (2016) 4. Vatutin, V.A.: A conditional functional limit theorem for decomposable branching processes with two types of particles. Math. Notes. 101(5), 778–789 (2017) 5. Afanasyev, V.I.: On a decomposable branching process with two types of particles. Proc. Steklov Inst. Math. 294, 1–12 (2016) 6. Afanasyev, V.I.: A functional limit theorem for decomposable branching processes with two particle types. Math. Notes. 103(3), 337–347 (2018) 7. Vatutin, V.A., D’yakonova, E.E.: Decomposable branching processes with two types of particles. Discrete Math. Appl. 28(2), 119–130 (2018) 8. Lindvall, T.: Limit theorems for some functionals of certain Galton-Watson branching processes. Adv. Appl. Probab. 6(2), 309–321 (1974) 9. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) 10. Afanasyev, V.I.: Invariance principle for the critical Galton–Watson process attaining a high level. Theory Probab. Appl. 55(4), 559–574 (2011) 11. Kolchin, V.F.: Random Mappings. Optimization Software, Publications Division, New York (1986) 12. Drmota, M.: Random Trees. Springer, Wien (2009)

A Probabilistic Interpretation of Conservation and Balance Laws Ya. I. Belopolskaya

Abstract We construct stochastic systems associated with parabolic conservation and balance laws written in the form of nonlinear parabolic systems and prove that solution of stochastic systems allows to construct weak and mild solutions of the Cauchy problem for original partial differential equations (PDE) partial differential equations systems. Keywords Stochastic differential equations · Hyperbolic and parabolic conservation laws · Balance laws · Diffusion processes

1 Introduction The laws of classical physics as a rule express conservation or balance of certain quantities such as mass, momentum, angular momentum, and energy. A conservation law is an equation in divergence form ∂u  + ∇yi f i (u) = 0, ∂t d

(1.1)

i=1

which expresses the fact that the quantity of u(t, y) ∈ R d does not evolve in time d ∂u dt u(t, y(t)) = 0. We use notations ∇yi u = ∂yi . A system of the form 1  ∂um  + ∇xi fmi (u) = cmk (u)uk ∂t

d

d

i=1

k=1

(1.2)

is called a system of balance laws. Ya. I. Belopolskaya () Saint Petersburg State University of Architecture and Civil Engineering, St Petersburg, Russia Sirius University, Sochi, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_2

19

20

Ya. I. Belopolskaya

Including into consideration dissipation mechanisms, one comes to parabolic systems of the form d1  d1 d d   ∂um  ij + ∇yi fmi (u) = ∇y2i ,yj [Bmk (u)uk ] + cmk (u)uk ∂t k=1 i,j =1

i=1

(1.3)

k=1

(systems with cross-diffusion). The aim of this article is to give a probabilistic interpretation to parabolic conservation and balance laws that is to systems of the form (1.3). Recall that a ij probabilistic interpretation to systems of nonlinear parabolic equations with Gmk ≡ Gij (systems with self-diffusion) was developed in [3, 4]. Here, we construct and analyze probabilistic representations of solutions of the Cauchy problem for the system (1.3). In other words, we derive stochastic problems associated with the Cauchy problem for these parabolic systems, investigate the stochastic problems, and show connections between their solutions and solutions of the Cauchy problem for (1.3). Connections between scalar nonlinear parabolic equations and stochastic differential equations have been investigated by many authors since the pioneer papers by McKean [9] and Freidlin [6]. Let us discuss a difference between these two approaches analyzing constructions suggested in these papers and the possibility to extend them to systems of nonlinear parabolic equations. In [9], there was a constructed stochastic model associated with the Cauchy problem for the Vlasov equation ∂u 1 = (A2 [u]u) − div(a[u]), ∂t 2

u(0, y) = u0 (y)

(1.4)

d ∂ 2 which arises in plasma physics. Here,  = i=1 ∂y 2 is the Laplace operator, i d div a[u](y) = i=1 ∇yi ai [u](y), and coefficients A[u] ∈ R d ⊗ R d , a[u] ∈ R d have the form   . . . ai (y, x1 , . . . , xd )u(x1 ) . . . u(xd )dx1 . . . dxd , i = 1, . . . , d. ai [u](y) = R

R

(1.5) The problem (1.4) was treated in [9] as a forward Kolmogorov equation for a distribution u(t, dy) = P {ξ(t) ∈ dy} of a stochastic process ξ(t) ∈ R d governed by a stochastic equation ξ(t) = ξ(0) +

 t  0

+

 t 

R

R

R



 ...

0

 a(ξ(s), x1 , . . . , xd )u(s, dx1 ) . . . u(s, dxd ) ds+

 ...

A(ξ(s), x1 , . . . , xd )u(s, dx1 ) . . . u(s, dxd ) dw(s), R

A Probabilistic Interpretation of Conservation and Balance Laws

21

where w(t) ∈ R d is the Wiener process defined on a probability space (, F, P ), ξ(0) does not depend on w(t), and P (ξ(0) ∈ dy) = u0 (y)dy. An alternative probabilistic approach to the Cauchy problem for semilinear parabolic equations was suggested in [6]. To explain this approach, let us consider the Cauchy problem 1 ∂u = T rA(x, u)∇ 2 uA(x, u) + a(x, u) · ∇u + c(x, u)u, ∂t 2

u(0, x) = u0 (x), (1.6)

where T rA(x, u)∇ 2 uA(x, u) =

d 

d 

Aik (x, u)∇x2i ,xj uAkj (x, u), a(x, u)·∇u =

i,j,k=1

ai (x, u)∇xi u.

i=1

If u(t, x) solves (1.6), then a function vT (t, x) = u(t, x) satisfies the Cauchy problem ∂vT 1 + T rA(x, vT )∇ 2 vT A∗ (x, vT ) + a(x, vT )∇vT + c(x, vT )vT = 0, ∂t 2

vT (T , x) = u0 (x).

(1.7) One can treat (1.7) as the backward Kolmogorov equation and hence to reduce its solution to solution of a stochastic system dξ(θ ) = a(ξ(θ ), vT (θ, ξ(θ )))dθ + A(ξ(θ ), vT (θ, ξ(θ )))dw(θ ), # $T vT (t, x) = E u0 (ξt,x (T ))e t

ξ(t) = x, (1.8) % c(ξt,x (τ ),vT (τ,ξt,x (τ )))dτ . (1.9)

Provided one can prove the existence and uniqueness of a solution to the systems (1.8), (1.9) and in addition prove that the function vT (t, x) is twice differentiable in x, we can verify that this function satisfies the Cauchy problem (1.7) in the classical sense. As a result, one obtains a probabilistic representation of a classical solution for the Cauchy problem (1.7), and hence, for a solution u(t, x) of the Cauchy problem (1.4), we get % # $T u(t, x) = vT (t, x) = E u0 (ξt,x (T ))e t c(ξt,x (τ ),vT (τ,ξt,x (τ )))dτ . It is important to underline that the Cauchy problem (1.4) admits a natural measure-valued solution that provided initial data u0 is a measure while (1.6) does not admit such a solution. On the other hand, in applications, it is natural to interpret a solution to (1.8) as a density of a probability measure. This goal requires an approach to the Cauchy problem (1.6) alternative to [6].

22

Ya. I. Belopolskaya

To derive a stochastic model which is compatible with the interpretation of a solution (1.6) as a density of a certain probability measure, one can apply an approach developed in [7, 8]. Within this approach, one considers a regularized Cauchy problem d d  1  2 ∂v = ∇xi xj ([Aik Akj ](x, ρ ∗v)v)+ ∇xi (ai (x, ρ ∗v)v)+c(x, ρ ∗v)v, ∂t 2 i,j,k=1

i=1

(1.10) v(0, x) = u0 (x), $ where ρ ∗ v(x) = R d ρ (x − y)v(y)dy with ρ (x) being a mollifier and u0 (x) is a density of a probability measure. To construct a stochastic interpretation of (1.10), one may apply the McKean approach to the regularized Eq. (1.10) and finally prove the existence of a limit of a solution to a stochastic system associated with this equation as ρ → δ. At the end, one has to verify that a solution to the limiting stochastic system allows to obtain the required probabilistic interpretation of a solution to the Cauchy problem (1.8). New phenomena arise when we turn to systems of (nonlinear) parabolic equations. While constructing stochastic models for these systems, we reveal that one can separate a group of systems of the form (1.3); namely, systems with diffusion ij coefficients Gmk ≡ Gij such that the Cauchy problem (1.3) can be reduced to the Cauchy problem for a system of backward Kolmogorov equations, and hence, its solution admits a probabilistic interpretation [3, 4]. The correspondent construction is an extension of the Freidlin approach. On the other hand, systems of the form (1.3) ij with general coefficients Gmk do not admit this reduction and require an alternative approach to construct a probabilistic interpretation of the Cauchy problem solution. The correspondent approach which extends the McKean approach to systems of nonlinear forward parabolic equations was developed in papers [1, 2, 5]. ij In this paper, we consider a class of systems (1.3) with Gmk (u) = d kj ik k=1 Mm (u)Mm (u) which allows to extend results obtained in [7] for scalar equations and construct stochastic counterparts to parabolic systems of this class. Besides, we omit an essential assumption assumed in [7] that cm (u) are bounded functions and prove that there exist both a solution of a stochastic system and weak and mild solutions of the Cauchy problem for an associated mollified parabolic system defined on a certain time interval. We postpone the proof of a possibility to go to a limit as the mollifier goes to delta function till ongoing paper because of a volume limitation. The remaining part of the article is organized as follows. In Sect. 2, we consider a particular case of a system similar to (1.3) and construct two equivalent stochastic systems associated with the system under consideration. One of them allows to obtain a probabilistic representations of a weak and a mild solution of the Cauchy problem for a system of this type, while the other allows to obtain a constructive way to prove the existence and uniqueness of a solution to stochastic system itself.

A Probabilistic Interpretation of Conservation and Balance Laws

23

The model system which motivates conditions on coefficients in this paper is the SKT system proposed by Shigesada, Kawashima, and Teramoto [10] ∂um = (A2m (y, u)um ) + cm (y, u)um , ∂t

(1.11)

um (0, y) = u0m (y), m = 1, 2, A2m (y, u) = αm +

2 

αmk uk (y), cm (y, u) = cm −

k=1

2 

cmk uk (y),

k=1

to model population dynamics. Here, αm , αmk , cm , andcmk are positive constants. In Sect. 3, we prove the existence and uniqueness theorem for solutions of the stochastic system associated with the original partial differential equations (PDE) system assuming that all coefficients of the PDE system are bounded. In the last section, we omit this assumption and prove that solutions um (t, y) to systems under investigation are bounded at least for some time interval [0, T1 ] with T1 depending on coefficients of the PDE system and initial data. This allows to apply the results of the previous section to systems extending (1.11).

2 Systems of Conservation and Balance Laws with Cross-Diffusion In this section, we consider a particular case of conservation and balance law systems of the form (1.4) and construct stochastic systems associated with their weak and mild solutions. Let us start with some preliminary notions and notations. Denote by C0∞ (R d ) (C0∞ ([0, T ] × R d )) the space of infinite differentiable functions defined on R d (on [0, T ] × R d ) with compact supports, and let Cb (R d ) be the space of bounded continuous functions f : R d → R with the norm f ∞ = supx |f (x)|. We use notations C k (R d ) for the space of k times differentiable real valued functions defined on R d . For functional spaces of a vector space X valued functions f : R d → X, we use notations of the form Cb (R d ; X), C k (R d ; X), and so on.

$ 1 Denote by Lp the space functions f with the norm f p = R d |f (x)|p dx p < ∞, and let W k,p (R d ) denote the Sobolev space of order k in (Lp (R d ),  · p ), 1 ≤ p ≤ ∞. Let P(R d ) denote the space of Borel probability measures and C([0, T ]; P(R d )) be the space of continuous functions defined on [0, T ] and valued in P(R d ).

24

Ya. I. Belopolskaya

Our aim is to construct a probabilistic representation of a solution to the Cauchy problem 1 ∂um = T r∇ 2 (Bm (y, u)um ) − div(bm (y, u)um ) + cm (y, u)um , ∂t 2

(2.1)

um (0, y) = u0m (y), m = 1, . . . , d1 , ∗, provided Bm = Mm Mm

Mm (y, u) = αm (y) +

d1 

bm (y, u) = bm (y) +

αmq (y)uq (y),

q=1

d1 

bmq (y)uq (y),

q=1

(2.2) cm (y, u) = cm (y) −

d1 

cmq (y)uq (y),

(2.3)

q=1

where αm (y), αmq (y) ∈ R d ⊗ R d , cm (y), cmq (y) ∈ R, and bm (y) and bmq (y) ∈ R d are bounded Lipschitz continuous functions. If cm ≡ 0, one can interpret the system (2.1) as a system of nonlinear forward Kolmogorov equations for densities of probability measures μm (t, dy) w.r.t the Lebesgue measure $ since it is easy to verify that its solution u (t, y) is positive if u (y) > 0 and m 0m R d um (t, y)dy = 1 if $ u (y)dy = 1. d 0m R Let [Avm ]∗ um = 12 T r∇ 2 (Bm (y, u)um ) − div(bm (y, u)um ) and Avm h = 1 2 2 T rBm (y, v)∇ h + bm (y, v) · ∇h. With this notations, we have  Rd

h(y)[Avm ]∗ um (t, y)dy =

 Rd

(Avm h(y))um (t, y)dy.

(2.4)

To construct a stochastic counterpart of (2.1), we consider a fixed probability space (, F, P ), independent Wiener processes wm (t) ∈ R d , and independent random variables ξ0m defined on this probability space. We write ξ0m ∼ μ0m when ξ0m has a distribution μ0m that is P (ξ0m ∈ dy) = μ0m (dy) = u0m (y)dy and assume below that ξ0m do not depend on wm (t). We denote by L(ξm ) a probability measure γm in (, F) generated by the process ξm (t), t ∈ [0, T ]. Consider a stochastic system 

t

ξm (t) = ξ0m + 0



t

bm (ξm (s), u(s, ξm (s)))ds +

Mm (ξm (s), u(s, ξm (s)))dwm (s), 0

(2.5)

A Probabilistic Interpretation of Conservation and Balance Laws

25

where u = (u1 , . . . , um , . . . , ud1 ), P {ξ0,m ∈ dy} = u0m (y)dy, and um (t, y) = μm (t,dy) are a density of a measure μm (t, dy) defined by a relation dy   t  h(y)μm (t, dy) = E h(ξm (t)) exp cm (ξm (s), u(s, ξm (s)))ds

 Rd

(2.6)

0

(called the Feynman-Kac formula) valid for all h ∈ C0∞ (R d ). If the measure μm (t, dy) has a density um (t, y), the left-hand side of (2.6) can be rewritten as  Rd

  t  h(y)um (t, y)dy = E h(ξm (t)) exp cm (ξm (s), u(s, ξm (s)))ds . 0

Systems (2.5), (2.6) together make a closed system, but we need a more constructive system associated with (2.1). To derive a required system, we consider first a mollified system 

t

ξm (t) = ξ0m +

 bm (ξm (s), u(s, ξm (s)))ds +

0

t

Mm (ξm (s), u(s, ξm (s)))dwm (s),

ξ0 ∼ μ0 ,

0

(2.7)   t  um (t, y) = E ρ(y − ξm (t ))exp cm (ξm (s), u(s, ξm (s)))ds ,

(2.8)

0

where ρ : R d → R+ is a mollifier. Denote by γm ∈ P(Cd ) a distribution generated on  = Cd by the process ξm (t), L(ξm ) = γm . When ρ = δ, where δ is the Dirac delta function, then (2.7) and (2.8) can be reduced to (2.5) and (2.6) at least formally, and thus, applying the Ito formula, we can verify that um (t, y) is a weak solution to (2.1). Below, we prove that if μm (t, dy) = um (t, y)dy and um satisfy (2.8) with ρ = δ, then it is connected with a weak solution of the mollified system 1 ∂μm = T r∇ 2 (Bm (y, ρ∗v)vm )−div(bm (y, ρ∗μ)μm )+cm (y, ρ∗μ)μm , ∂t 2

(2.9)

μm (0, dy) = μ0m (dy), m = 1, . . . , d1 , $ by a relation um = ρ ∗ μm = R d ρ(· − x)μm (dx). To verify this, we need some conditions. We say that condition C 2.1 holds if coefficients bm : R d × R d1 → R d , Mm : R d × R d1 → R d ⊗ R d , cm : R d × R d1 → R are Lipschitz continuous and bounded (on each compact) with respect to space variables x ∈ R d and uniformly bounded in (x, u) ∈ R d × R d1 by constants Kb , KM , and Kc .

26

Ya. I. Belopolskaya

Denote by Cd = C([0, T ]; R d ) the space of bounded functions with sup&1 norm  ·  and by C = C([0, T ]; dm=1 Xm ) where Xm = R d . Let F (f )(y) = $ −2πix·y dx be the Fourier transform of a real valued function f defined on R d f (x)e Rd . γ

Lemma 1 Assume C 2.1 holds and there exists a solution (ξm (t), μm (t, dy)) of a system 



t

ξm (t) = ξ0m +

t

bm (ξm (s), (ρ ∗ μγ )(s, ξm (s)))ds +

0

Mm (ξm (s), (ρ ∗ μγ )(s, ξm (s)))dwm (s),

0

(2.10)

ξ0m ∼ μ0m , 



γ

h(y)μm (t, dy) =

Rd

Cd

h(ξm (t, ω))e

$t

0 cm (ξm (s,ω),ρ∗μ

γ (s,ξ

m (s,ω)))ds

γm (dω),

L(ξm ) = γm

(2.11) γ

Then, there exists a solution (ξm (t), um (t, y)) of a system 

t

ξm (t) = ξ0m +



t

b(ξm (s), uγ (s, ξm (s)))ds+

0

Mm (ξm (s), uγ (s, ξm (s)))dwm (s),

0

(2.12)

ξ0m ∼ μ0m , γ um (t, y)

 =

Cd

ρ(y −ξm (t, ω))e

$t

0 cm (ξm (s,ω),u

γ (s,ξ

m (s,ω)))ds

γm (dω),

L(ξm ) = γm . (2.13)

γ

γ

and um = ρ ∗ μm . If in addition the Lebesgue measure of the measurable set {x ∈ R d : F(ρ)(x) = 0} is equal to zero, then the measure μm (t) satisfying (2.11) is γ uniquely determined by the function um (t, ·) satisfying (2.13) and vice versa. Proof The proof of Lemma 1 is very close to the proof of Theorem 6.1 in [7]. Let γ γ (ξm , um ) satisfy (2.12) and (2.13); then, the Fourier transform of um has the form γ

F (um )(t, x) = F (ρ)(x)κ γ (t, x), where  κ (t, x) = γ

e Cd

−2πix·ξm (t,ω)

 exp

t

 γ

cm (ξm (s), u (s, ξm (s)))ds γm (dω). 0

A Probabilistic Interpretation of Conservation and Balance Laws

27

γ

Since by assumption cm are bounded, then κm are bounded, nonnegative definite functions. Choosing a sequence of complex numbers ak and points xk ∈ R d andk = 1, . . . , d,, we obtain that d  d 

ak a¯ j e −iy·(xk −xj ) =

 d 

k=1 j =1

⎞  d ⎛ d    −iy·xk ⎝ −iy·x −iy·x k j⎠ = | |2 , ak e aj e ak e j =1

k=1

k=1

γ

that yields κm ≥ 0. This by the Bochner theorem results that there exists a finite γ nonnegative Borel measure μm (t) on R d such that for all x ∈ R d , κm (t, x) = $ γ γ −2πix·y μm (t, dy). It remains to prove that μm (t, dy) satisfy (2.11). Rd e γ Note that μm (t, dy) are the Schwartz (tempered) distributions such that $ γ γ γ −1 F (κm ) = μm (t), and for any test function h, we have | R d h(y)μm (t, dy)| ≤ γ γ γ h∞ μm (t, R d ) < ∞. Thus, from F (um )(t) = F (ρ)F (μm (t)), we deduce that γ γ um (t) = ρ ∗ μm (t). On the other hand, for all test functions h,  Rd

 =

F −1 (h)(x)

 Cd

Rd



 =

Cd

Rd



 =



γ

h(y)μm (t, dy) =

Cd

Rd

Cd

γ

Rd

e−2πix·ξm (t,ω) exp



t



κm (x)F −1 (h)(x)dx = γ

Rd

  cm (ξm (s, ω), uγ (s, ξm (s, ω)))ds γm (dω) dx =

0

 $ t γ F −1 (h(x))e−2πix·ξm (t,ω) dx e 0 cm (ξm (s),u (s,ξm (s)))ds γm (dω) =

 $ t γ F −1 (h(x))e−2πix·ξm (t,ω) dx e 0 cm (ξm (s,ω),(ρ∗μ )(s,ξm (s,ω)))ds γm (dω) = 

 =

F −1 (κm )(x)h(x)dx =

h(ξm (t)) exp

t

 cm (ξm (s, ω), (ρ ∗ μγ )(s, ξm (s, ω)))ds γm (dω),

0

thus, we proved that if (ξm (t), um ) solve (2.12) and (2.13) , then (ξm (t), μm ) solve (2.10) and (2.11) . To prove the converse statement, we assume that (ξm (t), μm ) solve (2.10) and γ γ γ (2.11). Setting um (t, y) = ρ ∗ κm (t), we verify that (2.13) holds. Since μm (t) is a finite measure, we derive Eq. (2.13) choosing ρ(x − ·) for a test function. γ γ Finally, note that from F (um )(t) = F (ρ)F (μm (t)), we have that if the Lebesgue γ d measure  of the set {x ∈ R : F(ρ)(x) = 0} is equal to zero, then F(μm (t)) = γ γ γ F(um (t ))  F(ρ) a.e. for t ∈ [0, T ]. Hence, μm is uniquely defined by um and vice versa. 

28

Ya. I. Belopolskaya

Now, we describe connections between the stochastic system 

t

ξm (t ) = ξ0m +

 b(ξm (s), (ρ ∗ μ)(s, ξm (s)))ds +

0

t

Mm (ξm (s), ρ ∗ μ(s, ξm (s)))dwm(s),

0

(2.14)  Rd





γ

h(y)μm (t, dy) =

Cd

h(ξm (t, ω)) exp

t

 cm (ξm (s, ω), (ρ ∗ μm )(s, ξm (s, ω)))ds γm (dω),

0

(2.15) with P {ξm (0) ∈ dy} = u0m (y)dy,

L(ξm ) = γm

and the PDE system (2.9). We say that measures μm (t) ∈ M(R d ), m = 1, . . . , d1 are weak measure-valued solutions of the system (2.9) with initial data μm (0, dy) = u0m (y)dy if for any h ∈ C0∞ ([0, T ] × R d ) equalities 

T 0



 Rd

 ∂h 1 2 + T rBm (y, v)∇ h + bm (y, v) · ∇h + cm (y, v)h dt+ μm (t, dy) ∂t 2 (2.16)  + h(0, y)μm (0, dy) = 0 Rd

are valid. Theorem 1 Assume C 2.1 holds and there exists a unique solution (ξm (t), μm (t, dy)) of (2.14) and (2.15). Then, μ = (μ1 , . . . , μm , . . . , μd1 ) is the unique weak measure-valued solution of the system (2.9). Proof Consider processes ζ (t) = h(ξm (t))ηm (t), where h ∈ C0∞ (R d ) is a test function and ξm (t), ηm (t) satisfy (2.14) and (2.15) and apply the Ito formula to obtain  E[h(ξm (t))ηm (t)] = E[h(ξ0m )]+ E[h(ξm (s))cm (ξm (s), (ρ∗μm )(s, ξm (s)))ηm (s)]ds+ 0t



t

+

E[∇h(ξm (s)) · bm (ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds+

0

+

1 2



t 0

E[T rBm ∇ 2 h(ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds.

A Probabilistic Interpretation of Conservation and Balance Laws

29

Substituting it into the right-hand side of (2.15), we get 

 Rd

h(y)μm (t, dy) = 

Rd t

+

 h(x)μ0m (dx)+ E[h(ξm (s))cm (ξm (s), (ρ∗μm )(s, ξm (s)))ηm (s)]ds+ 0t

E[∇h(ξm (s)) · bm (ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds+

0

+

1 2



t

E[T rBm ∇ 2 h(ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds.

0

 

3 Existence and Uniqueness of a Solution to (2.12) and (2.13) Let for a fixed m the process ξm (t) be a canonical process on Cd = C([0, T ]; R d ), ξm (t, ω) = ω(t), Bt (Cd ) = σ {ξm (s), 0 ≤ s ≤ t}, and γm = L(ξm ) be a distribution of ξm . Denote by Pr (Cd ) the set of Borel probability measures on Cd with finite moments up to the order r and by Cb (Cd ) the space of bounded continuous real &1 valued functions on Cd . Let C = C([0, T ]; dm=1 Xm ) for Xm = R d . d Let the space P2 (R ) be equipped by the Wasserstein distance Wt of order 2 

 Wt2 (γm , γ˜m )

2

= infπ∈(γm ,γ˜m )

sup ξm (s, ω) − ξm (s, ω) ˜ π(dω, d ω)) ˜

Cd ×Cd 0≤s≤t

for all t ∈ [0, T ] where (γm , γ˜m ) denotes the set of Borel probability measures on Cd ×Cd , π ∈ P(Cd ×Cd ) with fixed marginals π(F ×Cd ) = γm (F )andπ(Cd ×F ) = γ˜m (F ), F ⊂ Cd . We use notations B∞ (0, K) and B(0, K) for centered balls of radius K in spaces L1 (R d ; R d1 ) and L∞&([0, τ ] × R d ; R d1 ), τ < T , respectively. Denote by  = m , m = Cd and Fm = σ {ξm (s); 0 ≤ s ≤ t}. Let ξm (t) be canonical processes on (m , Fm ) = C([0, T ]; R d ), that is, ξm : Cd → Cd and ξm (t) = ωm (t), t ≥ 0, ωm ∈ Cd . Consider the system 

t

ξm (t) = ξ0m +

 bm (ξm (s), uγ (s, ξm (s)))ds +

0

t

Mm (ξm (s), uγ (s, ξm (s)))dwm (s),

ξ 0 ∼ μ0 ,

0

(3.1) γ

um (t, y) =

 Cd

ρ(y − ξm (t, ω))ηm (t, ω)γm (dω).

(3.2)

30

Ya. I. Belopolskaya

In this section, we discuss the existence and uniqueness of a solution to this system. When cm ≡ 0, Eq. (3.2) can be treated as an equation linking the law of the process ξm (t) and the function um . If the mollifier ρ = δ is the Dirac delta function, then um (t) is the density of the marginal law of ξm (t). When cm = 0, then um (t) is implicitly connected to the law γm of the whole path of ξm (·) and (3.2) associates a probability law γm on Cd to a real valued function um (t). Given a probability measure γm ∈ P(Cd ), we consider the equation  t   γ γ um (t, y) = ρ(y − ξm (t, ω))exp cm (ξm (s), u (s, ξm (s)))ds γm (dω), Cd

0

(3.3) where ξm is a canonical process. Theorem 2 Assume that C 2.1 holds and sup ρ(x) ≤ K0 . Given probability measures γm ∈ P(Cd ), the system (3.2) admits a unique solution u = (u1 , . . . , um , . . . , ud1 ). Proof Denote by Z1 the linear space of real valued continuous processes gm (t), m = 1, . . . , d1 , t ∈ [0, T ] defined on the canonical space Cd such that  gm ∞,1 = E γm [ sup |gm (s)|] =

sup |gm (s, ω)|γm (dω) < ∞.

0≤t ≤T1

Cd 0≤t ≤T1

Let Z1,d1 be the linear space of R d1 -valued continuous processes g defined on the canonical space C such that  g∞,d1 = E γ [ sup g(s)] = 0≤t ≤T

sup g(s, ω)γ (dω) < ∞.

C 0≤t ≤T1

The spaces (C∞,1 ,  · ∞,1 ) and (C∞,d1 ,  · ∞,d1 ) are Banach spaces. For K ≥ 0, denote by  · K ∞,1 an equivalent norm in Z1 γm −Kt gm K |gm (t)|]. ∞,1 = E [ sup e 0≤t ≤T1

Denote by 

t

Gm (t, ξm (ω), u(ξm (ω))) = exp{

cm (ξm (τ ), u(τ, ξm (τ )))dτ }.

0 γ

Given γm (t) ∈ P(Cd ), we denote by m : C1,d1 → C([0, T ] × R d × R d1 ; R) a map defined by γ

m (v)(t, y) =

 Cd

ρ(y − ξm (t, ω))Gm (t, ξm (ω), v(ω))γm (dω)

(3.4)

A Probabilistic Interpretation of Conservation and Balance Laws

31

and by κ : C([0, T1 ] × R d × R d1 ; R) → C1,d1 a map given by κ(v)(t, ω) = vm (t, ω(t)).

(3.5)

It should be mentioned that κ ◦ γ : C1,d1 → C1,d1 , that is, g = κ(γ (g)). Let us prove the existence and uniqueness of a solution to the equation u = (γ ◦ κ)(u)

(3.6)

which is equivalent to the system (3.2). Assume first that the map κ ◦ γ has a fixed point g ∈ C1,d1 . Then, choose γ v = γ (g). Since g is a fixed point of the map κ ◦ γ , we have g = κ(γ (g)),

(3.7)

hence, v γ is a solution to (3.6). To prove uniqueness of a solution to (3.6), we assume on the contrary that there exist two solutions v and g of this equation such that v = (γ ◦ κ)(v) and g = (γ ◦ κ)(g). Set X = κ(v), Y = κ(g). Since v = γ (X), we have X = κ(v) = κ(γ (X)). In a similar way, we obtain Y = κ(g) = κ(γ (Y )). Recall that X and Y are fixed points of κ ◦ γ , and thus, the equality X = Y holds γ almost everywhere. It remains to prove that the map κ ◦ γ has a unique fixed point X ∈ C1,d1 as it is required. Given a pair (X, Y ) ∈ C1,d1 × C1,d1 for any pair (t, y) ∈ [0, T ] × R d , we have γ

γ

|m (X) − m (Y )|(t, y) =  =

Cd

ρ(y − ξm (t, ω))[Gm (t, ξm (ω), X(ω)) − Gm (t, ξm (ω), Y (ω))]γm (dω) ≤ 

t

≤ K0 etKc Lc E

  t  X(s) − Y (s)ds ≤ K0 etKc Lc E eKs e−Ks X(s) − Y (s)ds

0

0



t

≤ K0 et Kc Lc

+ eKs E

0

≤ K0 e

t Kc

, sup e−τ K X(τ ) − Y (τ ) ds ≤

0≤τ ≤T

eKt − 1 Lc K

,

+ sup X −

0≤τ ≤t

Y K ∞,1

.

32

Ya. I. Belopolskaya γ

γ

γ

Next, we consider κ(m (X))(t) = m (X)(t, ξm (t)) and κ(m (Y ))(t) = and deduce

γ m (Y )(t, ξm (t))

+ E

, sup e

−Kt

0≤t ≤T1

γ |κ(m (X))(t)

−

γ κ(m (Y ))(t)|

= ,

+ =E

sup e

−Kt

0≤t ≤T1

γ |m (X)(t, ξm (t))

≤ K0 eKc T Lc

γ − m (Y )(t, ξm (t))

|≤

1 X − Y ∞,1 . K

Taking sup in t ∈ [0, T ] and summing over m, we get γ (X) − γ (Y ) ≤

d1 

γ

γ

m (X) − m (Y ) ≤ d1 K0 eKc T Lc

m=1

1 X − Y ∞,1 . K

Thus, we can choose K large enough, namely, K > d1 K0 et Kc Lc , and verify that (κ ◦ γ ) is a contraction in (C1 ,  · K ∞,1 ). Hence, by fixed-point theorem, we obtain that there exists a unique solution of Eq. (3.6).   We have proved that given a probability measure γm , there exists a unique solution uγ of (3.6). Dependence of this solution on γm remains to be studied. γ

Lemma 2 Let assumptions of Theorem 2 be valid. Then, for the solution um of (3.6), an inequality |uγ (t, y) − uγ˜ (t, y)| ˜ 2 ≤ L[y − y ˜ 2+

d1 

Wt2 (γm , γ˜m )]

(3.8)

m=1

holds for any couple γm , γ˜m ∈ P2 (Cd ) × P2 (Cd ) for all (t, y, y) ˜ ∈ [0, T ] × Cd × Cd . Proof Given (γm , γ˜m ) ∈ P2 (Cd ) × P2 (Cd ), we evaluate γ

γ˜

γ

γ

γ

γ˜

˜ 2 ≤ 2[um (t, y) − um (t, y) ˜ 2 + um (t, y) ˜ − um (t, y) ˜ 2. um (t, y) − um (t, y) (3.9) γ

Lipschitz continuity of um (t, y) with respect to y follows immediately from Lipschitz continuity of the mollifier ρ and boundedness of the stochastic process γ Gm (t, ξm (ω), um (ω)). To estimate the second term on the right-hand side of (3.8),

A Probabilistic Interpretation of Conservation and Balance Laws

33

evaluate a difference 

γ˜

γ

˜ − um (t, y)| ˜ 2= |um (t, y)

Cd

 −

u0m (y˜ − ξm (t, ω))Gm (t, ξm (ω), uγ (ξm (ω)))γ (dω)− (3.10) γ˜

2

u0m (y˜ − ξm (t, ω))G ˜ ˜ u (ξm (ω))) ˜ γ˜ (d ω)| ˜ ≤ m (t, ξm (ω),

Cd

 ≤

Cd ×Cd

|ρ(y˜ − ξm (t, ω))Gm (t, ξm (ω), uγ (ξm (ω)))−

2 −ρ(y˜ − ξm (t, ω))G ˜ ˜ uγ˜ (ξm (ω)))| ˜ π(dω, d ω) ˜ m (t, ξm (ω),

for any π ∈ (γ , γ˜ ). Keeping in mind Lipschitz continuity of ρ and properties of cm , we can verify the estimate |ρ(y˜ − ξm (t))Gm (t, y, z) − |ρ(y˜ − ξ˜m (t))Gm (t, y, ˜ z˜ )|2 ≤ ≤ 2L2c e 2tKc ξm (t) − ξ˜m (t)2 + 4K02 L2c e 2tKc



t

[ξm (s) − ξ˜m (s)2 + z(s) − z˜ (s)2 ]ds ≤

0

   t 2 2 ˜ ≤ C (1 + t) sup ξm (s) − ξm (s) + z(s) − z˜ (s) ds . s≤t

0

Combining the last inequalities and assuming that um (t, y) is Lipschitz continuous in y (which will be proved below), we get by replacing y and y˜ by ξm (t) and ξ˜m (t) (3.11) |um (t, ξm (t)) − uγ˜ (t, ξ˜m (t))|2 ≤  2 ˜ sup ξm (s) − ξ˜m (s)2 dπ(ω, ω)]+ ˜ ≤ C[ξm (t) − ξm (t) + (1 + t) γ

Cd ×Cd s≤t



 +C

Cd ×Cd

t

|uγ˜ (s, ξm (s)) − uγ˜ (s, ξ˜m (s))|2 dsdπ(ω, ω)]. ˜

0

$ γ γ˜ ˜ Integrating Set qm (s) = Cd ×Cd |um (s, ξm (s)) − um (s, ξ˜m (s))|2 π(dω, d ω). (3.11) against π, we get  qm (t) ≤ C

d1 t  0 m=1

 qm (s)ds + M(t)

Cd ×Cd

| sup ξm (s, ω) − ξm (s, ω) ˜ 2 π(dω, d ω) ˜ s≤t

34

Ya. I. Belopolskaya

for all t ∈ [0, T1 ]. Summing the last inequality in m and applying the Gronwall 1 the following inequality lemma, we obtain for q(t) = dm=1  q(t) ≤ C

sup

d1 

Cd ×Cd s≤t m=1

ξm (s, ω) − ξm (s, ω) ˜ 2 π(dω, d ω). ˜

Keeping in mind (3.10), we deduce d1 

+ γ γ˜ |um (t, y) ˜ − um (t, y)| ˜ 2

m=1



≤ C y − y ˜ + 2

sup

d1 

Cd ×Cd s≤t m=1

, ξm (s, ω) − ξm (s, ω) ˜ π(dω, d ω) ˜ . 2

(3.12) The above inequality holds for any π ∈ (γ , γ˜ ); thus, we can take infimum over all π ∈ (γ , γ˜ ) that yields (3.8).   Now, we are ready to prove the following result. Theorem 3 Assume that C 2.1 holds. Then, there exists a unique solution (ξm (t), um (t, y)) of the system (3.1) and (3.2). The process ξm (t) ∈ R d is a Markov process, and um (t, y) are bounded Lipschitz continuous in y ∈ R d real valued functions. Proof Let us fix the measures γm ∈ P2 (Cd ) considered above. We deduce from γ Theorem 2 and lemma 2 that there exists a unique solution um (t, y) of a system (3.4) γ and the functions um (t, y) are bounded and Lipschitz continuous both in y ∈ R d and γm ∈ P2 (Cd ). This allows to assert that there exists a unique strong solution γ ξm (t) of the SDE (3.1) since coefficients of these equations are nonrandom and satisfy conditions of classical theorem about existence and uniqueness of a strong γ solution to an SDE. Moreover, the processes ξm (t) are Markov processes. Applying the Burkholder-Davis-Gundy theorem and Jensen inequality, we deduce that there exists a positive constant C depending on constants in C 2.1 and T such that γ E[sups≤T1 ξm (t2 ] ≤ C(1 + E[ξ0m 2 ]). This results that the law L(ξm ) belongs γ to P2 (Cd ). Set m (γm ) = L(ξm ) and note that m : P2 (Cd ) → P2 (Cd ). Our aim now is to prove that m is a contraction in P2 (Cd ) equipped with the Wasserstein metric. γ γ Let γm1 and γm2 belong to P2 (Cd ) and um1 andum2 be solutions of (3.2), associated 1 2 i with γm and γm , respectively. Besides, let γm = L(ξmi ), where ξmi , i = 1, 2, are solutions of  dξmi (t) = ξ0m +

t 0

 bm (ξmi (s), uγi (s, ξmi (s)))ds +

0

t

i Mm (ξmi (s), uγi (s, ξmi (s)))dwm (s).

A Probabilistic Interpretation of Conservation and Balance Laws

35

From the Wasserstein metric definition, we deduce |WT ((γm1 ), (γm2 ))|2 ≤ E[ sup ξm1 (t) − ξm2 (t)2 ].

(3.13)

0≤t ≤T

Keeping in mind estimates in C 2.1 and Lemma 2, we deduce that there exists a positive constant C such that + E[ sup 0≤t≤τ

ξm1 (t ) − ξm2 (t )2]

≤C



τ

E[ sup 0

0≤s≤t

ξm1 (s) − ξm2 (s)2 ]dt

+ 0

τ

, Wt (γm1 , γm2 )dt

,

(3.14) for any τ ∈ [0, T ]. By the Gronwall lemma, we obtain from (3.14)  E[ sup ξm1 (t) − ξm2 (t)2 ] ≤ eCT 0≤t ≤τ

0

τ

Wt (γm1 , γm2 )dt

and substituting it into the right-hand side of (3.12), we deduce WT2 ((γm1 ), (γm2 )) ≤ T eCT WT2 (γm1 , γm2 ).

(3.15)

As a result, we get that  is a contraction if T eCT < 1.

 

4 Existence and Uniqueness of a Stochastic System Solution with Growing Coefficients In this section, we omit the assumption that coefficients in (2.12) and (2.13) are bounded and prove that there exists the unique solution ξm (t), um (t, y) of this system such that um (t) are bounded over a certain interval [0, T1 ]. Let us preliminarily obtain some auxiliary results concerning the Cauchy problem solution of a linearized version of (2.1) and its stochastic counterparts. Assume that v(t, y) ∈ R d1 is a given vector function with bounded Lipschitz continuous in y ∈ R d positive components vm (t, y), m = 1, . . . d1 , and consider the Cauchy problem 1 ∂um = T r∇ 2 (Bm (y, v)um ) + div(bm (y, v)um ) + cm (y, v)um , ∂t 2 um (0, y) = u0 (y), m = 1, . . . , d1 .

(4.1)

36

Ya. I. Belopolskaya

We say that condition C 4.1 holds if there exist positive constants K0 , K, L0 , L, Lu,v , Lc , n0 , andn1 such that: 1. bm (y, v)2 + |Mm (y, v)|2 ≤ K[1 + v2 ], y ∈ R d , v ∈ R d1 . 2. bm (x, u) − bm (y, v)2 + |Mm (x, u) − Mm (y, v)|2 ≤ L[x − y2 + Lu,v u −  v2 ] x ∈ R d , where |Mm | = di,j =1 |M ij |2 is the Frobenius matrix norm. 3. supy u0m (y) ≤ K0 , u0m (y) − u0m (x) ≤ L0 x − y, x, y ∈ R d . 4. |cm (y, v)| ≤ n0 + n1 vand |cm (x, u) − cm (y, v)| ≤ Lc [x − y + Lu,v u − v]. To construct a stochastic representation of a solution to (4.1), we consider SDEs dξm (s) = bm (ξm (s), v(s, ξm (s)))ds + Mm (ξm (s), v(s, ξm (s)))dwm(s),

ξm (0) = x ∈ R d ,

(4.2) where wm (t) ∈ R d are independent Wiener processes defined on a fixed probability space (, F, P ). From general theory of SDEs, we have the following result [4]. Lemma 3 Assume that condition C 4.1 holds and vm (s, x) are positive functions, such that sup |vm (s, x)| ≤ Kv (s), |vm (s, x) − vm (s, y)| ≤ Lv (s)x − y, x

where Kv (s) and Lv (s) are positive functions bounded over an interval [0, T1 ]. Then, there exist unique solutions ξm (t) ∈ R d of (4.2), t ∈ [0, T1 ] possessing the Markov property. In addition, processes 



t

ηm (t) = exp

(4.3)

cm (ξm (s), v(s, ξm (s)))ds 0

$t satisfy estimates Eηm (t)2 ≤ exp{ 0 [n0 + n1 ]Kv (s)ds}. For the Markov process ξm (t) satisfying (4.2), we denote by Pmv (0, x, t, dy) = P {ξm (t) ∈ dy|ξm (0) = x} its transition probability. We say that measures μm (t) ∈ P(R d ) are mild measure-valued solutions of (4.1) with initial data μm (0, dy) = u0m (dy) if for any hm ∈ C0∞ (R d ), t ∈ [0, T ]  Rd



 h(y)μm (t, dy) = +

Rd

Rd

Pmv (0, x, t, dy)μm (0, dx)+ 

 t 0

h(y)

Rd

Pmv (θ, zm , t, dy)cm (z, v(θ, , z))μm (θ, dz)dθ

.

(4.4)

A Probabilistic Interpretation of Conservation and Balance Laws

37

Lemma 4 Let vm be bounded Lipschitz continuous positive functions, m = 1, . . . , d1 , and condition C 4.1 holds. Then, the process ξm (t) satisfying (4.2) and the process ηm (t) of the form (4.3) together give rise to measures μm (t) such that  Rd

h(y)μm (t, dy) = E [h(ξm (t))ηm (t)]

(4.5)

holds for any h ∈ C0∞ (R d ). In addition, measures μm (t, dy) are unique measurevalued mild solutions of the Cauchy problem (4.1). Proof Since lemma conditions ensure the existence and uniqueness of processes ξm (t) and ηm (t) satisfying (4.2) and (4.3), we can observe that for each fixed m, the expression lm (h) = E[h(ξm (t))ηm (t)] is a bounded linear functional over the space C0∞ (R d ). Approximating bounded functions on R d by C ∞ functions, we can define a probability measure μm (t) considering lm (h) as a bounded linear functional over Cb (R d ) which by the Riesz theorem defines a unique measure μm (t) on R d satisfying (4.5). To prove that the measure μm (t) defined by (4.5) is a mild measure-valued solution of (4.1), we note that properties of the conditional expectations allow to rewrite the right-hand side of (4.5) in the form E [h(ξm (t))ηm (t)] = E[h(ξm (t))]+ 

t

+E

(4.6) 

cm (ξm (s), v(s, ξm (s)))ηm (s)E[h(ξm (t)|ξm (s))]ds =

0

 = +E

 t 

Rd

Rd

h(y)Pmv (0, x, t, dy)u0m(dx)+ 

 Rd

0



Rd

h(y)Pmv (s, ξm (s), t, dy)cm (z, v(s, ξm (s)))ηm (s)ds

.

$ Since R d h(y)Pmv (s, z, t, dy)cm (z, v(s, , z)) is a bounded function, we choose it for a test function and obtain an equality  t  E 0



 Rd

=

Rd

h(y)Pmv (s, ξm (s), t, dy)cm (z, v(s, ξm (s)))ηm (s)ds

 t 0

Rd

=

Pmv (s, zm , t, dy)cm (z, v(s, , z))μm (s, dz)ds.

Substituting this equality into (4.6), we deduce that the right-hand side of (4.5) coincides with the right-hand side of (4.4). It remains to prove that the measure-valued mild solution to (4.1) is unique. Assume on the contrary that there exist two measure-valued mild solutions

38

Ya. I. Belopolskaya

μm (t)andνm (t) ∈ P(R d ) of (4.1) and set κm (t) = μm (t) − νm (t). Since cm (x, u) are bounded, we deduce that κm (t)T V < ∞. In addition to ∀h ∈ Cb (R d ), we have  t   h(y)κm (t, dy) = cm (z, v(θ, z))κm (θ, dz) h(y)Pmv (θ, z, t, dy)dθ. Rd

0

Rd

Rd

and hence, evaluating supremum over all h such that h∞ = supx |h(x)| ≤ 1, we obtain  t κm (θ, ·)T V dθ. κm (t, ·)T V ≤ Kc 0

Finally, by the Gronwall lemma, we deduce that κm (t, ·) = 0 and hence μm (t) = γm (t).   To construct a solution of the systems (2.12) and (2.13), we consider its successive approximations ξm(0) = x,

u(1) m (t, y) = u0m (y),

dξm(1) (t ) = −bm (ξm(1) (t ), u0 (ξm(1) (t )))dt + Mm (ξm(1) (t ), u0 (ξm(1) (t )))dwm(t ),

ξm(1) (0) = ξ0m ,

  t  (1) (1) (1) u(2) (t, y) = E u (y − ξ (t)) exp c (ξ (t), u (ξ (τ )))dτ , 0m m m 0 m m m 0

and dξm(k) (t) = −bm (ξm(k) (t), u(k) (t, ξm(k) (t)))dt + Mm (ξm(k) (t), u(k) (t, ξm(k) (t)))dwm (t),

ξm(k) (0) = ξ0m ,

(4.7)   t  (k+1) (k) (k) (k) u(k+1) (t, y) = E u (y − ξ (t)) exp c (ξ (τ ), u (τ, ξ (τ )))dτ 0m m m m m m 0

(4.8) and prove that (ξmk (t), uk+1 m (t, y)) converge as k → ∞ to a solution of the systems (2.12) and (2.13) if C 4.1 holds. To prove the convergence of successive approximations (4.7) and (4.8) to processes ξm (t) and bounded Lipschitz continuous functions um (t, y) as k → ∞ on some time interval [0, T1 ], we study first a linearized stochastic system. Denote by Kv (t) = supy |v(t, y)| = v(t)∞ and by K0m = u0m ∞ . Given a Lipschitz continuous bounded function v(t) ∈ R d1 , we consider a system including (4.2) and an equation   t  um (t, y) = E ρ(y − ξm (t)) exp cm (ξm (τ ), v(τ, ξm (τ )))dτ . 0

(4.9)

A Probabilistic Interpretation of Conservation and Balance Laws

39

Lemma 5 Assume that C 4.1 holds. Then, functions um (t) defined by (4.7) possess the following properties: there exists an interval [0, T1 ], such that T1
0 is symmetric and admits a continuous heat kernel p(t, x, y).2 The function p(t, x, y) can be estimated as follows: p(t, x, y) 

t .3 [t 1/α + d(x, y)]1+α

(1.2)

2 The function (x, y) → p(t, x, y) is continuous (and even locally Lipschitz continuous) w.r.t. the ultrametric d(x, y), but it is discontinuous w.r.t. the Euclidean metric |x − y|. 3 We write f  g if the ratio f/g is bounded from above and from below by positive constants for a specified range of variables. We write f ∼ g if the ratio f/g tends to identity.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

47

The function p(t) := p(t, x, x) does not depend on x. By [31, Proposition 2.3], it can be represented in the form p(t) = t −1/α A(logp t),

(1.3)

where A(τ ) is a continuous nonconstant α-periodic function; see also an extended version of this result in [8]. In particular, in contrary to the classical case (symmetric stable densities), the function t → p(t) does not vary regularly. There are already several publications on the hierarchical Laplacian acting on a general ultrametric measure space (X, d, m); see [1, 3–7, 28, 29]. By the general theory developed in [4, 5] and [6], any hierarchical Laplacian L acts in L2 (X, m) as essentially self-adjoint operator having a pure point spectrum. This operator can be represented in the form  (f (x) − f (y))J (x, y)dm(y).

Lf (x) =

(1.4)

X

The Markov semigroup (e−t L )t >0 admits with respect to m a continuous transition density p(t, x, y). It turns out that in terms of certain (intrinsically related to L) ultrametric d∗ , 1/d∗ (x,y)

J (x, y) =

N(x, τ )dτ,

(1.5)

0 1/d∗ (x,y)

p(t, x, y) = t

N(x, τ ) exp(−tτ )dτ,

(1.6)

0

where N(x, τ ) is the so-called spectral function related to L (which will be defined later).

1.2 Outline Let us describe the main body of the paper. In Sect. 2, we introduce the notion of homogeneous hierarchical Laplacian L and list its basic properties: the spectrum of the operator L is pure point; all eigenvalues of L have infinite multiplicity and compactly supported eigenfunctions; the heat kernel p(t, x, y) exists, and it is a continuous function having certain asymptotic properties; etc. As a special example,

48

A. Bendikov et al.

we consider the case X = Qp , the ring of p-adic numbers endowed with its standard ultrametric d(x, y) = |x − y|p , and the normed Haar measure m. The hierarchical Laplacian L in our example coincides with the Taibleson-Vladimirov operator Dα , the operator of fractional derivative of order α; see [40, 42] and [22]. The most complete source for the basic definitions and facts related to the p-adic analysis is [21] and [39]. The Schrödinger type operator H = L + V with hierarchical Laplacian L was studied in [11, 15, 26, 27, 29, 31, 32] (the hierarchical lattice of Dyson) and in [22, 41, 42] (the field of p-adic numbers). In the next sections, we consider the Schrödinger type operator acting on a homogeneous  ultrametric space X. We assume that the potential V is of the form V = σi 1Bi , where Bi are balls which belong to a fixed horocycle H (i.e., all Bi have the same diameter). The main aim here is to study the set Spec(H ). Under certain assumptions on V (e.g., V (x) → 0 at infinity  , etc.), we conclude that the set Spec(H ) is pure point (with possibly infinite number of limit points). We split the set Spec(H ) in two disjoint parts: the first part consists of the point λ = 0 and the eigenvalues of the operator L which correspond to the horocycle H (with compactly supported eigenfunctions) and the second part is the closure of a countably infinite set  of eigenvalues of the operator H (with non-compactly supported eigenfunctions). In the case of sparse potential V , i.e., when d(Bi , Bj ) → ∞ fast enough, we specify the structure of the set . In this connection, we would like to mention here pioneering works of S. Molchanov [29], D. Krutikov [24, 25], and A. N. Kochubei [22].  In the last section, we consider the potential V of the form V = σi (ω)1Bi , where σi (ω), ω ∈ (, , P ) are i.i.d. random variables, and embark on the localization theory. More precisely, we show that if the sequence of (nonrandom) distances d(Bi , Bj ) between locations tends to infinity fast enough, then the spectrum of H is pure point for P -a.a. ω ∈ . In the case when X is discrete, L is the Dyson Laplacian, Bi are Singletons, and V is ergodic; the localization theorem appeared first in the paper of Molchanov [29] (σi (ω) are Cauchy random variables) and later (under more general assumptions on σi (ω)) in the papers of Kritchevski [27] and [26]. The proof of this theorem is based on the self-similarity of H . This approach is not applicable to the case of (random) sparse potentials. The proof of the localization theorem for (random) sparse potentials presented in this paper is based on the abstract form of Simon-Wolff criterion [38] for pure point spectrum, the technique of fractional moments, the decoupling lemma of Molchanov, and Borel-Cantelli type arguments; see [1, 28].

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

49

2 Preliminaries 2.1 Homogeneous Ultrametric Space Let (X, d) be an ultrametric space. Recall that a metric d is called an ultrametric if it satisfies the ultrametric inequality d(x, y) ≤ max{d(x, z), d(z, y)}, which is stronger than the usual triangle inequality. Henceforth, we assume that the ultrametric space (X, d) is separable, non-compact, and proper, that is, each d-ball is a compact set. For any x ∈ X and r ≥ 0, let Br (x) = {y ∈ X : d (x, y) ≤ r} be a closed ball. The basic consequence of the ultrametric property is that Br (x) is an open set for any r > 0. Moreover, each point y ∈ Br (x) can be regarded as its center; any two balls of the same radius are either disjoint or identical; etc. See a survey part in paper [6, Section 1] and references therein. To any ultrametric space (X, d), one can associate in a standard fashion a tree T. The vertices of the tree are metric balls; the boundary ∂T can be identified with the one-point compactification X ∪ { }. We refer to [6, Section 1] for a treatment of the association between ultrametric space (X, d) and the tree T of its metric balls. Definition 2.1 Let m be a Radon measure on X. The triple (X, d, m) we call a homogeneous ultrametric measure space if the group of isometries of (X, d) acts transitively on X and if the measure m is invariant w.r.t. the action of this group. The following remarkable result is due to M. Del Muto and A. Figà-Talamanca; see paper [12, Section 2]. Theorem 2.2 Any homogeneous ultrametric measure space (X, d, m) can be identified with a certain locally compact Abelian group G equipped with a translation invariant ultrametric d and Haar measure m. For example, the set X = [0, +∞[ equipped with the ultrametric structure generated by p-adic intervals can be identified with Qp , the ring of p-adic numbers. The identification in Theorem 2.2 is not unique. One possible way to define the identification is to choose the sequence {an } of forward degrees associated with the tree of balls T. This sequence is two-sided if X is non-compact and perfect (has no isolated points), it is one-sided if X is compact and perfect, or if X is discrete. In the first case, we identify X with a , the ring of a-adic numbers; in the 2nd case with a ⊂ a , the ring of a-adic integers; and in the 3rd case with the discrete group [a : a ]  Za1 ⊕ Za2 ⊕ . . ., the weak sum of cyclic groups Zan (i.e., the set of all eventually finite sequences z = (z1 , z2 , . . . , zl , 0, 0, . . .), ∃l = l(z)). We refer the reader to [18] for the comprehensive treatment of the special groups a , a and Za1 ⊕ Za2 ⊕ . . ..

50

A. Bendikov et al.

2.2 Homogeneous Hierarchical Laplacian Let (X, d, m) be a non-compact homogeneous ultrametric measure space. Let B be the set of all open balls, B (x) ⊂ B the set of all balls centered at x, and C : B → (0, ∞) a function satisfying the following conditions: 1. m(A) = m(B)  ⇒ C(A) = C(B). 2. λ(B) := C(T ) < ∞. T ∈B:B⊆T

3. supB∈B(x) λ(B) = ∞ for any non-isolated x. The class of functions C(B) satisfying these conditions is rich enough. For example, let us fix α > 0, and for any two nearest neighboring balls B ⊂ B , set C(B) = m(B)−α − m(B )−α and then λ(B) = m(B)−α . Definition 2.3 Let C : B → (0, ∞) be as above; we define the homogeneous hierarchical Laplacian L pointwise as ⎛ ⎞   1 Lf (x) := C(B) ⎝f (x) − f dm⎠ . (2.1) m(B) B∈B(x)

B

Let D be the set of all compactly supported locally constant functions.4 The series in (2.1) diverges in general ,but for f ∈ D, it belongs to C∞ (X) ∩ L2 (X, m). Moreover, the operator L admits a complete in L2 (X, m) system of eigenfunctions fB =

1B

1B − , m(B) m(B )

(2.2)

where the couple B ⊂ B runs over all nearest neighboring balls. The eigenvalue corresponding to fB is the number λ(B ) defined at condition 2. In particular, we conclude that L : D → L2 (X, m) is an essentially self-adjoint operator. Each eigenvalue λ(B) has infinite multiplicity, so Spec(L) is pure point and coincides with its essential part.

4 The

set D is a dense subset in each of the Banach spaces C∞ (X) and Lp (X, m), 1 ≤ p < ∞.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

51

The intrinsic ultrametric d∗ (x, y) is defined as follows: d∗ (x, y) :=

0 when x = y , 1/λ(x  y) when x = y

(2.3)

where x  y is the minimal ball containing both x and y. In particular, for any open ball B, we have λ(B) =

1 . diam∗ (B)

(2.4)

The spectral function τ → N(τ ), see Eq. (1.5), is defined as a left-continuous stepfunction having jumps at the points λ(B), and N(λ(B)) = 1/m(B). The volume function V (r) is defined by setting V (r) = m(B), where the ball B has d∗ -radius r. It is easy to see that N(τ ) = 1/V (1/τ ).

(2.5)

The Markov semigroup Pt = e−t L admits a continuous density p(t, x, y) with respect to m; we call it the heat kernel. The function p(t, x, y) can be represented in the form given by Eq. (1.6). Respectively, the Markov generator L admits the representation given by Eqs. (1.4) and (1.5). The resolvent operator (L + λI)−1 , λ > 0, admits a continuous strictly positive kernel R(λ, x, y) with respect to the measure m. The resolvent operator is well defined for λ = 0, i.e., the Markov semigroup (Pt )t >0 is transient, if and only if for some (equivalently, for all) x ∈ X the function τ → 1/V (τ ) is integrable at ∞. Its kernel R(x, y) := R(0, x, y), called also the Green function, is of the form +∞

dτ ,r = d∗ (x, y). V (τ )

R(x, y) =

(2.6)

r

Under certain Tauberian conditions, Eq. (2.6) takes the form R(x, y) 

r ,r = d∗ (x, y). V (r)

For all these facts, we refer the reader to [4, 5], and [6].

(2.7)

52

A. Bendikov et al.

2.3 Subordination Let  : R+ → R+ be an increasing homeomorphism. For any two nearest neighboring balls B ⊂ B , we define

 C(B) =  (1/m(B)) −  1/m(B ) .

(2.8)

The following properties hold true: (i) λ(B) =  (1/m(B)). In particular, the hierarchical Laplacians L and LI d are related by the equation L = (LI d ).5 (ii) d∗ (x, y) = 1/ (1/m(x  y)). (iii) V (r) ≤ 1/−1 (1/r). Moreover, V (r)  1/−1 (1/r) whenever both  and −1 are doubling and m(B ) ≤ Cm(B) for some constant C > 0 and all neighboring balls B ⊂ B . In turn, this yields the heat kernel estimates

1 −1  p(t, x, y)  t · min t

    1 1 1  , , t m(x  y) m(x  y)

(2.9)

2.4 L2 -Multipliers As a special case of the general construction, consider X = Qp , the ring of padic numbers equipped with its standard ultrametric d(x, y) = |x − y|p . Remind that the ultrametric space (Qp , d) and the ultrametric space ([0, ∞), d) with nonEuclidean d (the Dyson’s model) are isometric. Let m be the normed Haar measure on the Abelian group Qp and F : f → f0the Fourier transform acting in L2 (Qp , m). It is known, see [22, 39, 42], that F : D → D is a bijection. Let  : R+ → R+ be an increasing homeomorphism. The self-adjoint operator (D) we define as L2 -multiplier, that is,  (ξ ) = (|ξ |p )f0(ξ ), ξ ∈ Qp . (D)f

(2.10)

By [5, Theorem 3.1], (D) is a homogeneous hierarchical Laplacian. The eigenvalues λ(B) of the operator (D) are of the form  λ(B) = 

 p . m(B)

(2.11)

the case (τ ) is a Bernstein function the association L = (LI d ) has been studied in the well-known Bochner’s subordination theory [16].

5 In

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

53

Let p(t, x, y) be the heat kernel associated with the operator (D). Assume that both  and −1 are doubling; Then, Eq. (2.9) applies. Since for any x, y ∈ Qp , m(x  y) = |x − y|p , we obtain 

1 −1  p(t, x, y)  t · min t

    1 1 1  . , |x − y|p |x − y|p t

(2.12)

The Taibleson-Vladimirov operator Dα is L2 -multiplier; it can be written as a hypersingular integral operator Dα f (x) =

1 p (−α)



f (y) − f (x)

Qp

|y − x|1+α p

dm(y),

(2.13)

where p (z) = (1 − pz−1 )(1 − p−z )−1 is the p-adic gamma function; see [42, VIII.2]. The heat kernel pα (t, x, y) of the operator Dα admits two-sided bounds pα (t, x, y) 

t . (t 1/α + |x − y|p )1+α

(2.14)

In particular, the Markov semigroup (e−t D )t >0 is transient if and only if α < 1. In the transient case, the Green function Rα (x, y) is of the form α

Rα (x, y) =

1 1 . p (α) |x − y|1−α p

(2.15)

For all facts listed above, we refer the reader to [4, 5] and [6].

2.5 The Symbol of the Hierarchical Laplacian Identifying X with a locally compact Abelian group, we can regard −L as an isotropic Lévy generator. By (1.4), the operator L on D takes the form  (f (x) − f (y))J (x − y)dm(y),

Lf (x) =

(2.16)

X

or equivalently, in terms of the Fourier transform, 1 (θ ) = 0 0 Lf L(θ ) · f0(θ ), θ ∈ X,

(2.17)

54

A. Bendikov et al.

1p can be identified with Qp ) and 0 is the dual Abelian group (e.g., Q where X 0 )= L(θ

 [1 − Re $h, θ %]J (h)dm(h).

(2.18)

X

0 ) ≥ 0, the symbol of the Lévy generator −L, is a continuous The function L(θ negative definite function [10]. In particular, the function 0 L(θ ) is subadditive. By the subordination property [5, Theorem 3.1], the function 0 L(θ )2 is the symbol of 2 0 symmetric Lévy generator −L , so the function L(θ ) = 0 L(θ )2 is subadditive as well, i.e., it satisfies the triangle inequality 0 L(θ1 + θ2 ) ≤ 0 L(θ1 ) + 0 L(θ2 ).

(2.19)

Since −L is an isotropic Lévy generator [5, Sec. 5.2], a stronger property holds true. Theorem 2.4 The function 0 L(θ ) satisfies the ultrametric inequality 0 0 2 )}. L(θ1 + θ2 ) ≤ max{0 L(θ1 ), L(θ

(2.20)

Proof In order to simplify notation, we assume that X = Qp , the ring of p-adic numbers. Let B ⊂ B be two nearest neighboring balls centered at the neutral element. Notice that both B and B are compact subgroups of the group Qp , say B = p−k Zp and B = p−k−1 Zp . Applying the Fourier transform to the both sides of equation LfB = λ(B )fB , we get

1 0 L(θ )f1 B (θ ) = λ(B )f B (θ ).

(2.21)

The measure ωB = (1B m)/m(B) is the normalized Haar measure of the compact subgroup B, similarly for ωB . Since for any locally compact Abelian group, the Fourier transform of the normalized Haar measure of any compact subgroup A is the indicator of its annihilator group A⊥ , and in our particular case B ⊥ = pk Zp and (B )⊥ = pk+1 Zp , we obtain f1 B (θ ) = 1B ⊥ (θ ) − 1(B )⊥ (θ ) = 1∂B ⊥ (θ ),

(2.22)

where ∂B ⊥ is the sphere B ⊥ \ (B )⊥ . 0 ) takes constant value Equations (2.22) and (2.21) imply that the function L(θ

⊥ 0 λ(B ) on the sphere ∂B , i.e., L(θ ) = ψ(θ p ) for some function ψ(τ ) such that ψ(0) = 0 and ψ(+∞) = +∞. Since C ⊂ D implies λ(C) > λ(D), the function ψ(τ ) can be chosen to be continuous and increasing, so 0 L(θ ) = ψ(θ p ) satisfies the ultrametric Inequality (2.20) as claimed. 

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

55

3 The Schrödinger Type Operator 3.1 General Properties In what follows, we assume that (X, d, m) is a non-compact homogeneous ultrametric measure space. Let V be a real measurable function. Consider the Schrödinger type operator H u = Lu + V u,u ∈ D.

(3.1)

Our goal is to show that under certain mild conditions on V , the operator H defined by Eq. (3.1) is essentially self-adjoint. Theorem 3.1 Assume that V is locally bounded. Then, the following is true: (i) The operator H = L + V is essentially self-adjoint.6 (ii) Assume that V (x) → +∞ as x →  . Then the operator H has a compact resolvent. Consequently, the spectrum of H is discrete. (iii) Assume that V (x) → 0 as x →  . Then, the essential spectrum of H coincides with the spectrum of L. Thus, the spectrum of H is pure point, and the negative part of the spectrum consists of isolated eigenvalues of finite multiplicity. Proof (i) When X = Qp , the field of p-adic numbers, and L is the Taibleson-Vladimirov operator, the essential self-adjointness of H = L + V has been proved in [22, Sec. 3.2 and 3.4]; see also [2]. We provide a proof; in general case, (X, d, m) is a homogeneous ultrametric measure space and L a homogeneous hierarchical Laplacian acting on it. As the potential V is locally bounded, H : D → L2 (X, m) is a well-defined symmetric operator. Let us choose an open ball O, which contains the neutral element, and write Eq. (2.16) in the form ⎞ ⎛   Lf (x) = ⎝ + ⎠ [f (x) − f (x + y)]J (y)dm(y) O

Oc

= LO f (x) + LO c f (x). We have Hf = LO f + LO c f + Vf , where the operator V is the operator of multiplication by the function V (x). The operator LO c f = J (O c )(f − a ∗ f ), where a(y) = J (y)1O c (y)/J (O c ), is a bounded symmetric operator that, for the classical Schrödinger operator H = − + V in Rn , this statement is not true, unless V satisfies a certain lower bound; see [9, Chapter II, Theorem 1.1, and Example 1.1].

6 Recall

56

A. Bendikov et al.

in L2 (X, m) (as f → a ∗ f is the operator of convolution with probability measure a(y)dm(y)) and thus does not influence self-adjointness. As LO is minus Lévy generator, it is essentially self-adjoint (one more way to make this conclusion is that the matrix of the operator LO is diagonal in the basis {fB } of eigenfunctions of the operator L; see [23]). For any ball B which belongs to the same horocycle H as O, we denote HB the subspace of L2 (X, m), which consists of all functions f having support in B. Since O is a subgroup of the Abelian group X and each ball B ∈ H is a coset (i.e., belongs to the quotient group [X : O]), we conclude that HB is an invariant subspace of the symmetric operator HO = LO + V . Moreover, by symmetry HB reduces HO . The ultrametric space X can be covered by a sequence of nonintersecting balls Bn (recall that due to the ultrametric property, two balls of the same diameter either coincide or do not intersect). This leads to the orthogonal decomposition L2 (X, m) =

/

H Bn

n

where each HBn reduces HO . The restriction of the essentially self-adjoint operator LO to its invariant subspace HBn is an essentially self-adjoint operator, while the restriction of the operator V is bounded. Thus, HO is essentially self-adjoint as orthogonal sum of essentially self-adjoint operators HO,n , the restriction of HO to HBn . (ii) The proof is similar to the one for the Schrödinger operators given in [42, Theorem X.3]; the main tools are boundedness from below of the operator H and the Riesz-Rellich compactness criteria for subsets of L2 (X, m). (iii) Let us show that the operator V is L−compact. Then, by [19, Theorem IV.5.35], the essential spectrums of the operators H and L coincide. Recall that L−compactness means that if a sequence {un } is such that both {un } and {Lun } are bounded, then there exists a subsequence {u n } ⊂ {un } such that the sequence {V u n } converges. 1. Denote vn = Lun + un . By assumption, the sequence {vn } is bounded and un = R1 vn = r1 ∗ vn . It follows that the quantity 1/2

 |un (x + h) − un (x)|2 dm(x)

 ≤ vn L2

|r1 (z + h) − r1 (z)| dm(z)

tends to zero uniformly in n as h tends to the neutral element. Thus, the sequence {un } consists of equicontinuous on the whole in L2 (X, m) functions. The same is true for the sequence {V un }. Indeed, for any ball B which contains the neutral

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

57

element, we write 1/2

 |V (x + h)un (x + h) − V (x)un (x)| dm(x) 2

≤ I + I I + I I I,

where 1/2

 |un (x + h) − un (x)|2 dm(x)

I = V L∞  I I = un L2

, 1/2

|V (x + h) − V (x)|2 dm(x)

,

B

I I I = un L2 sup |V (x + h) − V (x)| . x∈B c

Clearly, I, I I , and I I I tend to zero uniformly in n as h tends to the neutral element and B ' X. 2. The sequence {V un } consists of functions with equicontinuous L2 (X, m) integrals at infinity. Indeed, for any ball B which contains the neutral element, we have  |V un (x)|2 dm(x) ≤ un L2 sup |V (x)| → 0 Bc

x∈B c

uniformly in n as B ' X. Thus, the sequence {V un } is bounded in L2 (X, m), consisting of equicontinuous on whole in L2 (X, m) functions with equicontinuous L2 (X, m) integrals at infinity. By the Riesz-Kolmogorov criterion of compactness in L2 (X, m), the set {V un } is compact, whence it contains a convergent subsequence {V u n }, as claimed.  The Case (X, d, m) Is countably infinite In the case when the homogeneous ultrametric measure space (X, d, m) is countably infinite, the statement (ii) of Theorem 3.1 can be complemented as follows. Theorem 3.2 Assume that (X, d, m) is countably infinite. Then, the following statements are equivalent: (i) The operator H has a discrete spectrum. (ii) |V (x)| tend to infinity as x → . Proof (ii) ⇒ (i) : Since X is discrete and L is a bounded symmetric operator, let us set d := L. Suppose that |V (x)| tend to infinity as x → . Then, for every given interval I = [a, b] and its neighborhood I = [a − d − 1, b + d + 1], there exist a finite set A of points x such that V (x) ∈ I . Let us choose v ∈ / I and define

58

A. Bendikov et al.

the operator H = L + V where



V (x) :=

V (x) if x ∈ /A . v if x ∈ A

The resolvent of the operator V : u(x) → V (x)u(x) is analytic inside of I , and, as a result, the resolvent of H is analytic inside of I . Indeed, it is straightforward to show that 2 2 2 2 2 2 2 2 2L(V − λI)−1 2 = 2(V − λI)−1 L2 ≤

d < 1, d +1

for any λ ∈ I . It follows that the operator   H − λI = (V − λI) E + L(V − λI)−1 is invertible. This in turn implies that the operator H has no spectrum inside the interval I . But the difference H −H is an operator of finite rank. Hence, the operator H has (in the same interval I ) not more than finite number of eigenvalues; see Lemma 3.12. Thus, we have already proved that the spectrum of H is discrete. (i) ⇒ (ii) : Suppose that the operator H has a discrete spectrum. Then, clearly the spectrum of H 2 is also discrete. Let E1 ≤ E2 ≤ · · · be the eigenvalues of H 2 . Then, by Courant’s min − max principle En = min max{(ψ, H 2 ψ) : ψ ∈ span(ψ1 , . . . , ψn ), ψ = 1}. ψ1 ,...,ψn

(3.2)

Assume that |V (x)| does not tend to +∞ as x →  . Then, there exists a sequence {xn } ⊂ X such that |V (xn )| ≤ C for some C > 0 and all n ≥ 1. It follows that (ψ, H 2 ψ) ≤ 2(d 2 + C 2 ), ∀ψ ∈ span(δx1 , δx2 , δx3 , . . .), ψ = 1.

(3.3)

Equations (3.2) and (3.3) imply that the interval [0, 2(d 2 + C 2 )] contains at list one limit point of the sequence {En }, i.e., the essential spectrum of H 2 (equivalently of H ) is not empty. This fact contradicts the discreetness of the spectrum of H 2 (or H ). This proves the second part of the theorem.  The Case (X, d, m) Is Perfect In the case when (X, d, m) is perfect (has no isolated points), the situation with the equivalence of statements (i) and (ii) of Theorem 3.2 is not so obvious. Consider a class K of potentials V of the form V =

 B∈H

σ (B)1B ,

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

59

where H is any fixed horocycle in the tree of balls. For a potential V ∈ K, let us select the following Hilbert subspaces of L2 (X, m) : • L+ = span{1B : B ∈ H}. • LB = span{fT : T  B}. / LB • L− = L2 (X, m) ( L+ = B∈H

The proofs of the following three lemmas are left to the reader. Lemma 3.3 The linear spaces L+ , LB , and L− are invariant subspaces for both operators H and L. Let H+ , HB and H− (resp. L+ , LB , and L− ) be the restriction of the operator H (resp. L) to L+ , LB , and L− , respectively. The following properties hold true: (i) H = H+ ⊕ H− . (ii) HB = L B + σ (B). / (LB + σ (B)). (iii) H− = B∈H

Remind that LfB = λ(B )fB for any open ball B. As B converges to a singleton, λ(B ) → +∞, whence LB has discrete spectrum. By the homogeneity property, Spec(LA ) is the same for all A ∈ H. Let us set • SH := Spec(LA ). • RV : =Range(V ). Lemma 3.4 In the introduced notation Spec(H− ) = SH + RV . In particular, the operator H− has a pure point (not necessary discrete) spectrum. Let us choose in each ball B ∈ H an element aB and consider a discrete ultrametric space (X , m , d ) with X = {aB : B ∈ H} induced by (X, m, d). Lemma 3.5 The operator L+ can be identified with certain hierarchical Laplacian L acting on (X , m , d ); respectively, the operator H+ can be identified with cer tain Schrödinger type operator H = L + V with potential V = a∈X V (a)δa . Theorem 3.3 For a potential V ∈ K, the statements (i) and (ii) of Theorem 3.2 are related by the implication (i) ⇒ (ii). The inverse implication (ii) ⇒ (i) holds true if and only if the set SH + RV has no accumulating points. Proof If we assume that Spec(H ) is discrete, then the operator H+ (whence the operator H ) has a discrete spectrum. Applying Theorem 3.2, we conclude that |V (x)| → +∞, i.e., (i) ⇒ (ii) as claimed.

60

A. Bendikov et al.

If the sequence {σ (B) : B ∈ H} contains a subsequence σ (Bk ) → −∞, then it may well happen that the set Spec(H− ) = SH +RV will contain a number of accumulating points, i.e., Spec(H ) in this case is not discrete. In particular, (ii) ⇒ (i) if and only if the set SH +RV has no accumulating points. 

3.2 Rank One Perturbations In this section, we assume that the homogeneous ultrametric measure space (X, d, m) is countably infinite. In this case, X can be identified with a countable Abelian group G equipped with an increasing sequence {Gn }n∈N of finite subgroups such that ∩Gn = {0} and ∪Gn = G. Each ball in G is a set of the form g + Gn for some g and n. As an example, one can consider the group G = Z(p1 )⊕ Z(p2 ) ⊕ . . ., the weak sum of cyclic groups, equipped with the sequence of its subgroups Gn = Z(p1 ) ⊕ Z(p2 ) ⊕ . . . ⊕ Z(pn ) ⊕ {0} ⊕ . . . . Let L be a homogeneous hierarchical Laplacian. We study spectral properties of the Schrödinger type operator H = L + V with potential V (x) = −σ δa (x), σ > 0. Clearly, H can be written in the form Hf (x) = Lf (x) − σ (f, δa )δa (x), that is, H can be regarded as a rank one perturbation of the operator L. In this connection, let us recall an abstract form of the Simon-Wolff theorem [38, Theorems 2 and 2’] about pure point spectrum of rank one perturbations. The Simon-Wolff Criterion Let A be a self-adjoint operator with simple spectrum on a Hilbert space H, and let ϕ be a cyclic vector for A, that is, {(A − λ)−1 ϕ | Im λ >0} is a total set for H. By the spectral theorem, H is unitary equivalent to L2 (R, μ0 ) in such a way that A is multiplication by x with cyclic vector ϕ ≡ 1. Here, μ0 is the spectral measure of ϕ for A. Let H = A + σ (ϕ, ·)ϕ be a rank one perturbation of the operator A. Set  F (x) :=

2 22 2 2 (x − y)−2 dμ0 (y) = lim 2(A − (x + i)I)−1 ϕ 2 . →0

Theorem 3.7 Fix an open interval ]a, b[. The following are equivalent: (i) For a.e. σ , H has only pure point spectrum in ]a, b[. (ii) For a.e. x ∈]a, b[, F (x) < ∞.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

61

In general, if H0 is the closed subspace generated by vectors {(A − λI)−1 ϕ |Im λ >0}, then its orthogonal complement (H0 )⊥ is an invariant space for H and H = A on (H0 )⊥ . Thus, the extension from the cyclic to general case is clear. The function ϕ = δa is not a cyclic vector for L because the operator L has many compactly supported eigenfunctions φ having support outside of a. Indeed, for any such φ, for all λ ∈ C with Im λ >0 and for some eigenvalue λk , we will have ((L − λI)−1 δa , φ) = (δa , (L − λI)−1 φ) = (δa , (λk − λ)−1 φ) = 0. We use the Krein-type identity below to show that the spectrum of the operator H = L − σ δa is pure point for all σ . Let ψ(x) = R(λ, x, y) be the solution of the equation Lψ(x) − λψ(x) = δy (x). Let ψV (x) = RV (λ, x, y) be the solution of the equation H ψV (x) − λψV (x) = δy (x). Notice that L and H are symmetric operators, whence both (x, y) → R(λ, x, y) and (x, y) → RV (λ, x, y) are symmetric functions. Theorem 3.8 In the notation introduced above RV (λ, x, y) = R(λ, x, y) +

σ R(λ, x, a)R(λ, a, y) , 1 − σ R(λ, a, a)

(3.4)

RV (λ, a, y) =

R(λ, a, y) 1 − σ R(λ, a, a)

(3.5)

RV (λ, a, a) =

R(λ, a, a) . 1 − σ R(λ, a, a)

(3.6)

and

Proof We have LψV (x) − λψV (x) = δy (x) + σ δa (x)ψV (x) = δy (x) + σ δa (x)ψV (a). It follows that ψV (x) = R(λ, x, y) + σ ψV (a)R(λ, x, a).

(3.7)

62

A. Bendikov et al.

Setting x = a in the above equation, we obtain ψV (a) = R(λ, a, y) + σ ψV (a)R(λ, a, a) or ψV (a)(1 − σ R(λ, a, a)) = R(λ, a, y). Since ψV (a) = RV (λ, a, y), we obtain Eq. (3.6). In turn, Eqs. (3.6) and (3.7) imply (3.4) and (3.5).  Theorem 3.9 The operator H = L −σ δa has a pure point Spectrum, which consists of at most one negative eigenvalue and countably many positive eigenvalues with accumulating point 0. The operator H has precisely one negative eigenvalue λσ− if and only if σ > 0 and one of the following two conditions holds: (i) the semigroup (e−t L)t >0 is recurrent, and (ii) the semigroup (e−t L )t >0 is transient and R(0, a, a) > 1/σ . If it is the case, then Spec(H ) consists of numbers λσ− < 0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 . Otherwise, Spec(H ) consists of numbers 0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 . If σ < 0, then Spec(H ) consists of numbers 0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 < λσ+ . The eigenvalues λk are at the same time eigenvalues of the operator L. All λk have infinite multiplicity and compactly supported eigenfunctions, the eigenfunctions of the operator L, whose supports do not contain a. The eigenvalue λσk (resp. λσ− , λσ+ ) is the unique solution of the equation R(λ, a, a) = 1/σ in the interval ]λk+1 , λk [ (resp. ] − ∞, 0[, ]λ1 , +∞[). Each λσk (resp. λσ− , λσ+ ) has multiplicity one and non-compactly supported eigenfunction ψk (x) = R(λσk , x, a) (resp. ψ− (x) = R(λσ− , x, a), ψ+ (x) = R(λσ+ , x, a)) (Fig. 2).

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

63

  Fig. 2 The roots λσ∗ of the equation R (λ, a, a) = 1/σ. The dashed graph corresponds to a recurrent case, the solid graph—to the transient case

Proof Let ϒ(X) be the tree of balls associated with the ultrametric space (X, d). Consider in ϒ(X) the infinite geodesic path from a to  : {a} = B0  B1  . . .  Bk  . . . . The series below converges uniformly and in L2 ,  δa =

1 B1 1 B0 − m(B0 ) m(B1 )



 +

1 B2 1 B1 − m(B1 ) m(B2 )

 +... =

∞ 

fBk .

(3.8)

k=0

Notice that all fBk are eigenfunctions of the operator L, i.e., LfBk = λ(Bk+1 )fBk = λk+1 fBk . By definition, R(λ, x, y) = (L − λ)−1 δy (x), whence we obtain 1 1 fB (a) + fB (a) + . . . λ1 − λ 0 λ2 − λ 1   1 1 1 − = λ1 − λ m(B0 ) m(B1 )   1 1 1 + − + ..., λ2 − λ m(B1 ) m(B2 )

R(λ, a, a) =

or in the final form   ∞  1 Ak 1 , Ak = − R(λ, a, a) = . λk − λ m(Bk−1 ) m(Bk )

(3.9)

k=1

Since λ → R(λ, a, a) is an increasing function, the equation 1 − σ R(λ, a, a) = 0, σ = 0,

(3.10)

64

A. Bendikov et al.

has precisely one solution λσk lying in each open interval ]λk+1 , λk [ , λk+1 < λσk < λk , k = 1, 2, . . . . Claim 1 All numbers λσk are eigenvalues of the operator H . Indeed, the function ψ(x) = R(λ, x, a) with λ = λσk satisfies the equation H ψ(x) − λψ(x) = Lψ(x) − λψ(x) − σ δa (x)ψ(x) = Lψ(x) − λψ(x) − σ δa (x)ψ(a) = Lψ(x) − λψ(x) − δa (x) = 0. Claim 2 All numbers λk are eigenvalues of the operator H . Indeed, for any ball B which does not contain a but belongs to the horocycle Hk−1 , we have HfB = LfB = λk fB . When σ > 0, there may exist one more eigenvalue λσ− < 0, a solution of the Eq. (3.10). Indeed, λ → R(λ, a, a) is an increasing function, continuous on the interval ] − ∞, 0]. Since R(λ, a, a) → 0 as λ → −∞ and R(λ, a, a) → R(0, a, a) ≤ +∞ as λ → −0, Eq. (3.10) has unique solution λ = λσ− < 0 in the cases (i) and (ii). The proof of the theorem is finished.  Example 3.10 The Dyson’s Laplacian. Consider the set X={0, 1, 2, . . .} equipped with the counting measure m and with the set of partitions {r : r = 0, 1, . . .}, each of which consists of all rank r intervals Ir = {x ∈ X : kpr ≤ x < (k + 1)pr }. The set of partitions {r } generates the ultrametric structure on X and the hierarchical Laplacian ⎛ +∞  ⎜ Dα f (x) = (1 − κ)κ r−1 ⎝f (x) − r=1

1 m(Ir (x))



⎞ ⎟ f dm⎠ , κ = p−α ,

Ir (x)

where the sum is taken over all rank r intervals Ir (x) containing x. The operator Dα admits a complete system of compactly supported eigenfunctions. Indeed, let I be an interval of rank r and I1 , I2 , . . . , Ip be its subintervals of rank r − 1. Let us consider p functions fIi =

1I 1Ii − , i = 1, 2, . . . , p. m(Ii ) m(I )

Each function fIi belongs to the domain of the operator Dα and Dα fIi = κ r−1 fIi .

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

65

When I runs over the set all p-adic intervals, the set of eigenfunctions fIi forms a complete system in L2 (X, m). In particular, Dα is essentially self-adjoint operator having pure point spectrum Spec(Dα ) = {0} ∪ {κ r−1 : r ∈ N}. Clearly, each eigenvalue λr = κ r−1 has infinite multiplicity. Let us compute the value R(λ) := R(λ, 0, 0) of the resolvent kernel for Dα . By Eq. (3.9), we have R(λ) =

  Ak 1 = (p − 1) . k λk − λ p (λk − λ) k≥1

k≥1

In particular, R(0) = +∞ if and only if α ≥ 1; Otherwise, R(0) =

p − 1 1 p−1 = . k(1−α) p p − pα p k≥0

Consider the operator H = Dα − σ δ0 , σ > 0. Let us compute the number Neg(H ) of negative eigenvalues of the operator H counted with their multiplicity. By Theorem 3.9, the operator H has at most one negative eigenvalue. It has exactly one negative eigenvalue if and only if either α ≥ 1 or 0 < α < 1 and σ > (p − pα )(p − 1)−1 . If we denote the set of pairs (α, σ ) which satisfy the above conditions by  and by 0 = R2+ \  its complement, we obtain Neg(H ) =

1 if(α, σ ) ∈  0 if(α, σ ) ∈ 0

which is shown on the picture below (Fig. 3).

Fig. 3 Sets 0 and 

66

A. Bendikov et al.

3.3 Finite Rank Perturbations As in the previous section, the ultrametric measure space (X, d, m) is countably infinite and homogeneous. For convenience, we assume that m(B) = diam(B) for any non-singleton ball B. Let L be a homogeneous hierarchical Laplacian. We study spectral properties of the Schrödinger type operator H = L + V with potential V (x) = − N i=1 σi δai (x), σi > 0. Clearly, H can be written in the form Hf (x) = Lf (x) −

N 

σi (f, δai )δai (x),

i=1

that is, H can be regarded as rank N perturbation of the operator L. Throughout this section, we use the following notation: • R(λ, x, y) is the solution of the equation Lψ(x) − λψ(x) = δy (x). We set → − → − → N R(λ, x, − a ) := (R(λ, x, ai ))N i=1 , and R(λ, a , a ) := (R(λ, aj , ai ))i,j =1 . • RV (λ, x, y) is the solution of the equation H ψ(x) − λψ(x) = δy (x). We set → → − − → N RV (λ, x, − a ) := (RV (λ, x, ai ))N i=1 , and RV (λ, a , a ) := (RV (λ, aj , ai ))i,j =1 . •  := diag(σi : i = 1, . . . , N). Theorem 3.11 The following properties hold true: 1. The set Spec(H ) is pure point; its essential part Specess (H ) coincides with the set Spec(L) = {0} ∪ {λk }, and its discrete part Specd (H ) in each open interval lying in the complement of Spec(L) consists of at most N distinct points, solutions of the equation → → det( −1 − R(λ, − a ,− a )) = 0.

(3.11)

2. For each k ∈ N, there exists δ > 1 such that mini =j d(ai , aj ) > δ implies that the operator H has precisely N distinct eigenvalues in each open interval (λs+1 , λs ): 1 ≤ s ≤ k. Moreover, there exists precisely N distinct negative eigenvalues of the operator H , provided one of the following two conditions is satisfied: (2.1) The semigroup (e−t L )t >0 is recurrent. (2.2) The semigroup (e−t L )t >0 is transient and all 1/σi < R(0, a, a).7 The proof of the first part of Theorem 3.11 is based on the Weyl’s theorem on the essential spectrum of compactly perturbed symmetric operators, see [19, Theorem IV.5.35], and on the following lemma.

7 Thanks

to the homogenuity assumption R(λ, a, a) does not depend on a.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

67

Lemma 3.12 Let A and B be two symmetric bounded operators and H = A + B. Assume that B is an operator of rank N operator. Let (a, b) be an interval lying in the complement of the set Spec(A). Then, the set Spec(H ) ∩ (a, b) consists of at most N distinct points. Proof By the Weyl’s essential spectrum theorem, Specess (H ) coincides with the set Specess (L) = {0} ∪ {λk }. Hence, the set Spec(H ) ∩ (a, b) may contain only finite number of eigenvalues each of which has finite multiplicity. Consider the case N = 1, that is, the operator B is of the form Bf = σ1 (f, f1 )f1 . Let λ ∈ (a, b) and let f be a nontrivial solution of the equation Hf −λf = 0. Then, f can be written in the form f = −σ1 (f, f1 )Rλ f1

(3.12)

where Rλ = (A − λ)−1 is the resolvent operator. It follows that (f, f1 ) = 0 and (f, f1 ) = −σ1 (f, f1 )(Rλ f1 , f1 ) or σ1 (Rλ f1 , f1 ) + 1 = 0.

(3.13)

The function φ(λ) = (Rλ f1 , f1 ) is strictly increasing on the interval (a, b). Indeed, applying the resolvent identity, we get dφ(λ) = (Rλ2 f1 , f1 ) = Rλ f1 2 > 0. dλ It follows that Eq. (3.13) has at most one solution lying in the interval (a, b). Assuming that Eq. (3.13) 2 has a2solution, denote it λ∗ . Then, (3.12) implies that the vector f∗ := Rλ∗ f1 / 2Rλ∗ f1 2 satisfies the equation Hf∗ − λ∗ f∗ = 0. Thus, the operator H has at most one eigenvalue in the interval (a, b). Without loss of generality, we may provide the induction from N = 1 to N = 2. Thus, assuming that the perturbation operator B is of the form Bf = σ1 (f, f1 )f1 + σ2 (f, f2 )f2 we set A f := Af + σ1 (f, f1 )f1

68

A. Bendikov et al.

and Hf := A f + σ2 (f, f2 )f2 . Observe that the operator A may have in the interval (a, b) at most one eigenvalue λ∗ . The2corresponding eigenspace is one-dimensional; call it $f∗ %, where f∗ := 2 Rλ∗ f1 / 2Rλ∗ f1 2. Let us consider two cases. First Case Assume that f2 ⊥ f∗ . Then, Hf∗ = A f∗ = λ∗ f∗ , i.e., λ∗ is an eigenvalue of the operator H . It follows that the orthogonal complement $f∗ %⊥ is a joint invariant subspace of the operators H and A and that these operators being restricted to $f∗ %⊥ , call them H⊥ and A ⊥ , satisfy H⊥ f = A ⊥ f + σ2 (f, f2 )f2 . The operator A ⊥ has no eigenvalues in the interval (a, b). Hence, by what we have already shown in the first part of the proof, the operator H⊥ has at most one eigenvalue in the interval (a, b). It follows that the operator H has at most two eigenvalues in the interval (a, b). Second Case Assume that f2 and f∗ are not orthogonal. Let Rλ := (H − λI)−1 and Rλ := (A − λI)−1 be the resolvent operators. The following identity holds true: (Rλ f, g) = (Rλ f, g) −

σ2 (Rλ f, f2 )(Rλ f2 , g) 1 + σ2 (Rλ f2 , f2 )

(3.14)

for any f, g, and λ = λ∗ lying in (a, b). Using the spectral resolution formula for the operator A , the fact that its spectral function Eλ in (a, b) has the only jump at λ = λ∗ and that the value of the jump Eλ∗ is the operator of orthogonal projection on the subspace $f∗ %, we get (f∗ , f )2 + O1 (1) λ − λ∗

(3.15)

(f∗ , f )(f∗ , f2 ) + O2 (1) λ − λ∗

(3.16)

(Rλ f, f ) = and (Rλ f, f2 ) =

where Oi (1) are analytic functions. Substituting asymptotic equations (3.15) and (3.16) in Eq. (3.14), we get analyticity of the function λ → (Rλ f, f ) at λ = λ∗ . In particular, this shows that λ = λ∗ is not an eigenvalue of H . On the other hand, λ∗ splits the interval (a, b) in two parts: (a, λ∗ ) and (λ∗ , b), each of which does not contain eigenvalues of the operator A . Then, as we have

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

69

already shown, each of these intervals contains at most one eigenvalue of the operator H . Since λ∗ is not an eigenvalue of the operator H , the number of distinct eigenvalues of H in the interval (a, b) is at most two. The proof of the lemma is finished.  Proof of Theorem 3.11 (Second part): Let λ ∈ Specd (H ) and let ψ(x) be the corresponding eigenfunction, i.e., H ψ(x) − λψ(x) = 0. We have Lψ(x) − λψ(x) =

N 

σi ψ(ai )δai (x)

i=1

or applying to this equation the resolvent operator (L − λ)−1 we get N 

ψ(x) =

σi ψ(ai )R(λ, x, ai ).

(3.17)

i=1

Taking consequently x = a1 , a2 , . . . , aN in Eq. (3.17), we obtain a homogeneous system of N linear equations with N variables ψ(aj ) =

N 

σi ψ(ai )R(λ, aj , ai )

(3.18)

i=1

or in the vector form → → " = R(λ, − a ,− a )",

(3.19)

where " = (ψ(ai ) : i = 1, . . . , N). The system (3.19) has a nontrivial solution if and only if → → a ,− a ) = 0. det( −1 − R(λ, −

(3.20)

Observe that the variable z := R(λ, ai , ai ) does not depend on ai , and its range is the whole interval ] − ∞, ∞[ when λ takes values in each of the open interval ]λk+1 , λk [. Equation (3.20) can be written as characteristic equation det(A − zI) = 0

(3.21)

70

A. Bendikov et al.

where A = (aij )N i,j =1 is symmetric N × N matrix with entries aij =

for i = j 1/σi . −R(λ, ai , aj ) for i = j

(3.22)

Let us compute R(λ, ai , aj ). For any two neighboring balls B ⊂ B , let us denote A(B) =

1 1 − . m(B) m(B )

Remember that we normalize m so that m(B) = diam(B) for any non-singleton ball B, whence for such B, A(B) =

1 1 − . diam(B) diam(B )

(3.23)

Let ai  aj be the minimal ball, which contains both ai and aj . Following the same line of reasons as in the proof of Eq. (3.9), we obtain 

R(λ, ai , ai ) =

B:ai ∈B

A(B) . λ(B) − λ

(3.24)

Similarly, for all i = j , we get R(λ, ai , aj ) = −

d(ai , aj )−1 + λ(ai  aj ) − λ

 B:ai aj ⊂B

A(B) . λ(B) − λ

(3.25)

Let λ > λ(ai aj ). Equations (3.23) and (3.25) and the fact S ⊂ T ⇒ λ(S) > λ(T ) imply that R(λ, ai , aj ) =

>

=

d(ai , aj )−1 − λ − λ(ai  aj )

 B:ai aj ⊂B

A(B) λ − λ(B)

d(ai , aj )−1 1 − λ − λ(ai  aj ) λ − λ(ai  aj ) 1 λ − λ(ai  aj )





A(B)

B:ai aj ⊂B

1 1 − d(ai , aj ) diam(ai  aj )

 > 0.

Hence, for λ > λ(ai  aj ), we obtain 0 < R(λ, ai , aj )
λk+1 . Let us choose δ > 1 such that if mini =j d(ai , aj ) ≥ δ, then λ(ai  aj ) < λk /2. Then, for all i = j , we get λ − λ(ai  aj ) > λk /2 and thus   R(λ, ai , aj ) < 2 := ε(δ) . δλk N

(3.27)

Let us increase if necessary δ so that the intervals {s : |1/σi − s| ≤ ε(δ)}, i = 1, 2, . . . , N, do not intersect. By Gershgorin circle theorem, the matrix A admits N different eigenvalues ai , each of which lies in the corresponding open interval {s : |1/σi − s| < ε(δ)}, i = 1, 2, . . . , N. The eigenvalues ai , i = 1, 2, . . . , N, are analytic functions of λ in each open interval (λs+1 , λs ), 1 ≤ s ≤ k; see [35, Theorem XII.1]. Whence, in each interval (λs+1 , λs ), the number of different solutions of the equations ai = R(λ, ai , ai ) is at least N. By Lemma 3.12, the number of different solutions is at most N. Thus, the number of different solutions is precisely N as claimed.  Theorem 3.13 The set Specd (H ) coincides with the set of solutions of Eq. (3.11). Each eigenfunction ψλ (x) corresponding to λ ∈ Specd (H ) can be represented as linear combination of functions R(λ, x, ai ), that is, ψλ (x) =

N 

ζi R(λ, x, ai ).

i=1

Thus, support of ψλ is the whole space X, whereas the eigenfunctions fB corresponding to the eigenvalues λ(B) ∈ Specess (H ) are compactly supported. Proof The proof is straightforward: we apply Eqs. (3.17) and (3.18) to get the result; see the first part of the proof of Theorem 3.11 (second statement).  Theorem 3.14 For λ ∈ / Spec(H ), the following identities hold true: → → → → RV (λ, x, y) = R(λ, x, y) + R(λ, x, − a )( −1 − R(λ, − a ,− a ))−1 R(λ, − a , y), 8 (3.28) → → → → a , y) = ( −1 − R(λ, − a ,− a ))−1 R(λ, − a , y) RV (λ, −

8 For

a matrix A and vectors ξ and η, we write ξ Aη :=

 i,j

aij ξi ηj .

(3.29)

72

A. Bendikov et al.

and → → → → → → RV (λ, − a ,− a ) = ( −1 − R(λ, − a ,− a ))−1 R(λ, − a ,− a ).

(3.30)

In particular, the operator T (λ) := (H − λI)−1 − (L − λI)−1 is of finite rank N. Its operator norm can be estimated as follows: 2 22 22 2 22 2 → → T (λ) ≤ 2( −1 − R(λ, − a ,− a ))−1 2 2(L − λI)−1 2 .

(3.31)

Proof Recall that Spec(H ) coincides with the union of two sets: Spec(L) and the → → a ,− a )) = 0. The proof of the theorem set of those λ ∈ R for which det( −1 −R(λ, − is similar to its one-dimensional version Theorem 3.8. Clearly, we can write the following equation: LRV (λ, x, y) − λRV (λ, x, y) = δy (x) +

N 

σj δaj (x)RV (λ, x, y)

i=1

= δy (x) +

N 

σj RV (λ, aj , y)δaj (x),

j =1

or equivalently RV (λ, x, y) = R(λ, x, y) +

N 

σj RV (λ, aj , y)R(λ, x, aj ).

(3.32)

j =1

Substituting consequently x = a1 , a2 , . . . , aN , we obtain system of N linear equations with N variables RV (λ, ai , y) = R(λ, ai , y) +

N 

σj R(λ, ai , aj )RV (λ, aj , y)

j =1

or in the vector form → → → → a , y) = R(λ, − a , y). (I − R(λ, − a ,− a ))RV (λ, −

(3.33)

→ → Assuming that λ ∈ / Spec(H ), in particular det(I − R(λ, − a ,− a )) = 0, we get → → → → a , y) = (I − R(λ, − a ,− a ))−1 R(λ, − a , y) RV (λ, −

(3.34)

Evidently Eqs. (3.32) and (3.34) imply Eqs. (3.28), (3.29) and (3.30). Equation T (λ) = (H − λI)−1 (L − H )(L − λI)−1 applies that T (λ) is of rank N. Finally, Eq. (3.31) follows from Eq. (3.28). Indeed, for f ∈ L2 (X, m),

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

→ we introduce (finite-dimensional) vectors R(λ, f, − a ) :=  − → − → R(λ, a , f ) := y f (x)R(λ, a , y) and then (T (λ)f, f ) =



 x

73

→ f (x)R(λ, x, − a ) and

→ → → → f (x)R(λ, x, − a )( −1 − R(λ, − a ,− a ))−1 R(λ, − a , y)f (y)

x,y

→ → → → = R(λ, f, − a )( −1 − R(λ, − a ,− a ))−1 R(λ, − a , f ). → → By symmetry, R(λ, − a , f ) = R(λ, f, − a ), whence 2 22 22 2 2 → → → |(T (λ)f, f )| ≤ 2( −1 − R(λ, − a , f )2 a ,− a ))−1 2 2R(λ, − 22 2 22 22 2 2 → → ≤ 2( −1 − R(λ, − a ,− a ))−1 2 2(L − λI)−1 f 2 2 22 22 2 22 2 → → ≤ 2( −1 − R(λ, − a ,− a ))−1 2 2(L − λI)−1 2 f 2 as desired. The proof of the theorem is finished.



3.4 Sparse Potentials We assume that the ultrametric measure space (X, d, m) is countably infinite and  homogeneous. Our analysis of finite rank potentials V = − N i=1 σi δai indicates that in the case of increasing distances between locations {ai } of the bumps Vi = −σi δai , their contributions to the spectrum of H = L + V are close to the union of the contributions of the individual bumps Vi (each bump contributes one eigenvalue in each gap (λm+1 , λm ) of the spectrum of the operator L). The development  of this idea leads to consideration of the class of sparse potentials V = − ∞ i=1 σi δai , where distances between locations {ai : i = 1, 2, . . .} form a fast increasing sequence. In the classical theory, this idea goes back to D. B. Pearson [33]; see also S. Molchanov [30] and A. Kiselev, J. Last, and B. Simon [20]. Throughout this section, we will assume that the sequence mini,j :≥n,i =j d(ai , aj ) tends to infinity with certain rate, which will be specified later.9 We will also assume that α < σi < β for all i and for some α, β > 0. For λ ∈ / Spec(L), we define the following infinite vectors and matrices: → • R(λ, x, − a ) := (R(λ, x, ai ) : i = 1, 2, . . .). − → → • R(λ, a , − a ) := (R(λ, a , a ) : i, j = 1, 2, . . .). i

j

•  := diag(σi : i = 1, 2, . . .),  −1 := diag(1/σi : i = 1, 2, . . .).

9 We

choose the ultrametric d(x, y) such that it coincides with the measure m(B) of the minimal ball B, which contains both x and y; see e.g. (3.23).

74

A. Bendikov et al.

Theorem 3.15 The following properties hold true: → (i) R(λ, x, − a ) ∈ l2. − → → (ii) R(λ, a , − a ),  and  −1 act in l 2 as bounded symmetric operators. → → (iii) If the operator B(λ) =  −1 − R(λ, − a ,− a ) has a bounded inverse, then → → a )B(λ)−1 R(λ, − a , y). RV (λ, x, y) = R(λ, x, y) + R(λ, x, −

(3.35)

Proof Let ξ = (ξi ) ∈ l 2 has finite number nonzero coordinates. Define function f = ξi δai . Evidently, f ∈ L2 = L2 (X, m) and f  = ξ . Let Rλ = (L − λI)−1 , λ ∈ / Spec(L), be the resolvent. Then, → R(λ, x, − a )ξ =

 R(λ, x, y)f (y)dm(y) = Rλ f (x)

whence   → R(λ, x, − a )ξ  ≤ Rλ  f  = Rλ  ξ  which clearly proves (i).To prove (ii), we write → → ξ R(λ, − a ,− a )ξ =

  f (x)R(λ, x, y)f (y)dm(y)dm(x)

= (f, Rλ f ) ≤ Rλ  f 2 = Rλ  ξ 2 → → which clearly proves boundedness of the symmetric operator R(λ, − a ,− a ) : l2 → 2 l . Since {σi } ∈ (α, β) for all i and some α, β > 0, boundedness of the operators  and  −1 follows. (iii) Assume that λ is such that the self-adjoint operator B(λ) has a bounded inverse; then, Eq. (3.35) follows from its finite dimensional version (3.28) by passage to limit.  Theorem 3.16 Spec(L) ⊂ Specess (H ). Proof Let V be the sum of all but finite number of bumps Vi and H = L + V . By Weyl’s essential spectrum theorem, Specess (H ) = Specess (H ). It follows that without loss of generality, we may assume that the sequence of distances n = mini,j :≥n,i =j d(ai , aj ) strictly increases to ∞. Having this in mind, we can choose for any given τ from the range of the distance function an infinite sequence {Bn } of disjoint balls of diameter τ such that Bn ∩ {ai } = ∅ for all n. Thanks to our choice, we obtain HfT = LfT = λ(T )fT for any ball T ⊂ Bn and for all n. In particular, each λ = λ(T ), such that T ⊆ Bn for some n, is an eigenvalue of the operator H having infinite multiplicity, whence it belongs to Specess (H ). 

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

75

Theorem 3.17 Let σ∗ be a limit point of the sequence {σi }. Fix m ∈ N and let λ∗m ∈ (λm+1 , λm ) be the unique solution of the equation 1 = R(λ, a, a).10 σ∗

(3.36)

Then, λ∗m belongs to the set Specess (H ). Before we embark on the proof of Theorem 3.17, let us state the Weyl’s characterization of the essential spectrum Specess (A) of a self-adjoint operator A; see [43] and [36, Ch. IX, Sect. 2(133)]. Lemma 3.18 A real number λ belongs to the set Specess (A) if and only if there exists a normed sequence {xi } ⊂ dom(A) such that xi → 0 weakly and Axi −λxi → 0 strongly. Proof of Theorem 3.17 To show that λ∗m ∈ Specess (H ), we construct a λ∗m sequence {fim } via Lemma 3.18. Let λim ∈ (λm+1 , λm ) be the unique solution of the equation 1/σi = R(λ, ai , ai ). Let ψim (x) = R(λim , x, ai )/ R(λim , ·, ai )2 be the normed solution of the equation Hi ψ = λim ψ, where Hi := L − σi δai is a one-bump perturbation of L. Clearly, λim → λ∗m . Passing if necessary to a subsequence of {σi }, we can assume that d(ai , 0) → ∞ monotonically. Let us put fim := ψim · 1Bi , where Bi is the maximal ball centered at ai , which does not contain ai−1 and ai+1 . Thanks to our choice fim → 0 weakly and  2 fim 2 = |ψim |2 dm → 1. Bi

Thus, what is left is to show that Hfim − λ∗m fim → 0 strongly. We have Hfim − λ∗m fim 2 ≤ Hfim − λim fim 2 + fim 2 |λim − λ∗m | ≤ Hfim − λim fim 2 + |λim − λ∗m | = Hfim − λim fim 2 + o(1), Hfim − λim fim 2 ≤ H ψim − λim ψim 2 + (H − λim I)(fim − ψim )2 ≤ H ψim − λim ψim 2 + (H − λim I)||||(fim − ψim )2 = H ψim − λim ψim 2 + o(1),

2 2 2 2 2 2 2 H ψim − λim ψim 2 ≤ Hi ψim − λim ψim 2 + 2 σ δ ψ j aj im 2 2 2 j =i 2

2

10 Recall

that the function λ → R(λ, a, a) does not depend on a.

76

A. Bendikov et al.

and 2 2 3 2 2 3 2 2 2 2 2 2 σj δaj ψim 2 σj |ψim (aj )| ≤ sup{σj } |ψim |2 dm. 2 2 = X\B 2 j =i 2 i j =i 2

The right-hand side of this inequality tends to zero as i → ∞, and we finally conclude that {fim } is the desired λ∗m -sequence in the sense of Lemma 3.18. The proof is finished.  Let us introduce the following notation: • ∗ is the set of limit points of the sequence {σi }. • 1/∗ := {1/σ∗ : σ∗ ∈ ∗ }. • R−1 (1/∗ ) := {λ : R(λ, a, a) ∈ 1/∗ }. Theorem 3.19 Assume that the following condition holds: lim sup

N→∞ i≥N

 j ≥N:j =i

1 = 0, d(ai , aj )

(3.37)

and then Specess (H ) = Spec(L) ∪ R−1 (1/∗ ).

(3.38)

Proof That Spec(L) and R−1 (1/∗ ) are subsets of Specess (H ) follows from Theorem 3.16 and Theorem 3.17. We are left to prove that Specess (H ) ⊂ Spec(L) ∪ R−1 (1/∗ ). Let us fix m ∈ N and choose a closed interval I from the spectral gap (λm+1 , λm ). We claim that I ∩ Specess (H ) = ∅. Indeed, since R(λ) := R(λ, a, a) is strictly increasing and continuous in the interval (λm+1 , λm ), closed sets R(I) and 1/∗ do not intersect. Hence, there exists only a finite number of σi such that 1/σi ∈ R(I). Let us choose N big enough so that the sets {1/σi : i > N} and R(I) do not intersect. Let us write H = H + V where V

is a finite number of bumps −σi δai , i ≤ N. By Weyl’s essential spectrum theorem, Specess (H ) = Specess (H ). Notice however that the sets Specd (H ) and Specd (H ), discrete parts of Spec(H ) and Spec(H ), may well be quite different. Observe that for the operators H and H , the sets of limit points, the function R, the set of gaps, etc. are the same. Thus, in all our further considerations, we may assume that {1/σi } ∩ R(I) = ∅.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

77

→ → Making this assumption, consider now the operator B(λ) =  −1 −R(λ, − a ,− a ), λ ∈ I. According to Identity (3.35), if B(λ) has a bounded inverse, then λ ∈ / Spec(H ). Let us write # % → → B(λ) =  −1 − R(λ, − a ,− a ) :=  −1 − R(λ)I − 4 R(λ). Since we assume that the closed bounded sets {1/σi } and R(I) do not intersect, the −1 −1 operator A(λ)  2 := 2 − R(λ)I has a bounded inverse A(λ) for all λ ∈ I. Clearly, −1 2 2 the norm A(λ) can be estimated by the reciprocal of the distance between sets {1/σi } and R(I); denote it by C1 . Thus, writing for λ ∈ I the identity B(λ) = A(λ)(I − A(λ)−14 R(λ)),

(3.39)

2 2 2 2 2 2 R(λ))2 ≤ C1 24 R(λ))2 . 2A(λ)−14

(3.40)

we get

Writing again H as H + V where V consists of a finite number, say N, of bumps and applying Inequality (3.26) for the operator H :   R(λ, ai , aj )
0, which depends only on I, and for N chosen big enough. Clearly, Inequalities (3.40) and (3.41) imply the fact that the operator I − A(λ)−14 R(λ) has bounded inverse for all λ ∈ I:  −1   k A(λ)−14 I − A(λ)−14 R(λ) R(λ) . = k≥0

This fact, in turn, implies that the operator B(λ) given by Eq. (3.39) has bounded inverse for all λ ∈ I; therefore, I ∩ Spec(H ) = ∅. In particular, since Specess (H ) = Specess (H ) by Weyl’s essential spectrum theorem, we finally get I ∩ Specess (H ) = ∅ as desired. The proof is finished.



78

A. Bendikov et al.

Remark 3.20 Theorem 3.19 does not contain any information about sets Specac (H ) and Specsc (H ), the absolutely continuous and singular continuous parts of Spec(H ). In the next section, we will show that under more restrictive assumption Specac (H ) and Specsc (H ) are indeed empty sets, that is, Spec(H ) is pure point. Moreover, the eigenfunctions of H decay exponentially in certain metric at infinity. This is the so-called spectral localization property.

3.5 Spectral Localization As in the previous section, the ultrametric measure space (X, d, m) is countably infinite and homogeneous. We consider the operator H = L + V where L, the deterministic part of H , is a hierarchical Laplacian and V =−



σ (a, ω)δa , ω ∈ (, F, P ),

a∈I

is a random potential defined by a family of locations I = {ai } and a family σ (ai , ω) of i.i.d. random variables. Henceforth, we assume that the probability distribution of σ (ai , ω) is absolutely continuous with respect to the Lebesgues measure and has a bounded density supported by a finite interval [α, β]. In the case when X is the Dyson lattice and L =Dα , the Dyson Laplacian (see Example 3.10), the perturbed operator H = Dα −



σ (a, ω)δa

a∈X

has a pure point spectrum for P −a.s. ω. This statement (the localization theorem) appeared first in the paper of Molchanov [29] (σ (a, ω) is the Cauchy random variable) and later in a more general form in the papers of Kritchevski [27] and [26]. The proof of this statement is based on the self-similarity property of the operator H. The localization Theorem 3.23 concerns the case where the family of locations I does not coincide with the whole space X, whence the operator H is not selfsimilar. The technique developed in [27, 29] and [26] does not apply here to prove Theorem 3.23. Our approach is based on a different technique: the abstract form of the Aizenman-Molchanov criterion for pure point spectrum, the Krein-type identity from the previous section, the technique of fractional moments, the decoupling lemma of Molchanov, and Borel-Cantelli type arguments; see papers [1, 28]. The Aizenman-Molchanov Criterion Let H = H0 + V be a self-adjoint operator in  l 2 () ( is a countable set of sites) with H0 a bounded operator and V = − a∈ σ (a, ω)δa . Assume that the collection of random variables {σ (a, ω) : a ∈

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

79

} has the property that for each site a the conditional probability distribution of σ (a, ω) (conditioned on the values of the potential at all other sites) is absolutely continuous with respect to the Lebesgues measure (in particular, this assumption holds if {σ (a, ω) : a ∈ } are mutually independent random variables having absolutely continuous w.r.t. the Lebesgues measure l probability distributions). $ Let H = λdEλ be the spectral resolution of the symmetric operator H . Let G(λ, x, y) be the integral kernel of the operator (H − λI)−1 . Then, for any fixed x, τ and  = 0, 

2 22  2 2 |G(τ + i, x, y)|2 = 2(H − (τ + i)I)−1 δx 2 =

y∈

d(Eλ δx , δx ) (λ − τ )2 +  2

(3.42)

As the left-hand side of Eq. (3.42) (as a function of ) decreases on the interval ]0, +∞[, the limit (finite or infinite) in Eq. (3.42) exists and equals lim ↓0



 |G(τ + i, x, y)| = 2

y∈

d(Eλ δx , δx ) . (λ − τ )2

Theorem 3.21 If for any x ∈  and Lebesgues a.a. τ ∈ [a, b]: lim ↓0



|G(τ + i, x, y)|2 < ∞,

(3.43)

y∈

for a.e. realizations of {σ (x, ·)}, then almost surely the operator H has only pure point spectrum in the interval [a, b]. Furthermore, if under Condition (3.43) the integral kernel G(τ + i0, x, y) := lim G(τ + i, x, y) ↓0

(which exists a.e. τ ) decays exponentially at infinity (in some metric ρ(x, y) on ), then do the eigenfunctions ϕτ (y), for τ ∈ [a, b].11 Proof The first part of the statement follows from Simon-Wolff Theorem 3.7. For completeness of exposition, we comment on the proof. To prove the second Part, one needs an ad hoc argument and we refer to the cited above paper [1, Theorems 3.1 and 3.3 in Sec. 3]). α α  Note that in the case of the Dyson-Vladimirov Laplacian D and H = D − σi (ai , ω)δa , one can use the metric ρ(x, y) = ln(1 + d(x, y)), where d(x, y) is the ultrametric generated by p-adic intervals as in Example 3.10. In this case, the exponential decay of eigenfunctions in ρ−metric follows directly from two facts: (1) each eigenfunction ϕτ (y) of H can be represented as a linear combination of

11 An

even more versatile version can be found in [1, Theorems 3.1 and 3.3 in Sec. 3].

80

A. Bendikov et al.

functions R(τ, ai , y), where R(λ, x, y) is the resolvent kernel of Dα (see Theorem 3.13) and (2) R(λ, x, y) has an exponential decay because the heat kernel p(t, x, y) does (see Eq. (1.2)). By the spectral theory, one can represent l 2 () as the direct sum of three H invariant subspaces: l 2 () = Hac ⊕ Hsc ⊕ Hpp , where Hac (resp. Hsc , Hpp ) is the set of all functions f ∈ l 2 () such that the spectral measure  σ f (A) = 1A (λ)d(Eλ f, f ) is absolutely continuous (resp. singular continuous, pure point) with respect to the Lebesgues measure. By Theorem 3.7, Condition (3.43) implies that for any x ∈  the probability measure  σ x (A) = 1A (λ)d(Eλ δx , δx ) is pure point, that is, σ x (A) = σ x (A ∩ Sx ) for any open set A and some at most countable set Sx . Set S := ∪x∈ Sx ; then, for any f ∈ l 2 () and measurable set A,  f σ (A) = 1A (λ)d(Eλ f, f ) = 1A (H )f 2 =



|f (x)|2 |(1A (H )f, δx )|2

x∈

and, if A lies in the complement of S, |(1A (H )f, δx )|2 ≤ |(1A (H )f, f )| |(1A (H )δx , δx )| = 1A (H )f 2 σ x (A) = 0. Thus, for any f ∈ l 2 () the spectral measure σ f is pure point, that is, f ∈ Hpp . That means that the operator H has a pure point spectrum.  Remark 3.22 The function z → G(z, x, y), analytic in the domain C+ , is represented by the Borel-Stieltjes transform of a signed measure of finite variation  G(z, x, y) =

d(Eλ δx , δy ) . λ−z

It follows that the limit G(τ + i0, x, y) exists and takes finite values for Lebesgues a.e. τ ; see e.g. [37, Theorem 1.4]. Moreover, the limit G(τ + i0, x, y) exists even in a more restrictive sense, as the non-tangential limit; see [34, Ch. III, Sec. 2.2, 3.1, and 3.2]. We will apply this fact in the proof of Theorem 3.23.

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

81

The Localization Theorem Coming back to our setting, let H = L+V where L is a hierarchical Laplacian and V a random potential of the form V = − i σi (ω)δai . Here, σi (ω) := σ (ai , ω) are i.i.d. random variables corresponding to the set of locations I = {ai }. Let d(x, y) be the ultrametric which is chosen such that it coincides with the measure m(B) of the minimal ball B containing both x and y. Let R(λ, x, y) be the integral kernel of the operator (L − λI)−1 , i.e., the solution of the equation Lu − λu = δy . The function λ → R(λ, x, x) does not depend on x; we denote its value R(λ). This is strictly increasing continuous in each spectral gap function; we denote by R−1 (ν) its inverse function. Theorem 3.23 The operator H has a pure point spectrum for P −a.s. ω provided for some (whence for all) y ∈ X the sequence d(ai , y) eventually increases and for some small r (say, 0 < r < 1/3): lim sup

M→∞ i≥M

 j ≥M:j =i

1 = 0.12 d(ai , aj )r

(3.44)

Proof The set of limit points of the sequence {σi (ω)} coincides (for P −a.s. ω) with the whole interval [α, β]. Hence, by Theorem 3.19, the closed set Specess (H ) consists (for P -a.a. ω) of two parts: (1) the set Spec(L) and (2) the collection of countably many disjoint closed intervals Ik = R−1 ([1/β, 1/α])∩ ]λk+1 , λk , [ and the interval I− = R−1 ([1/β, 1/α])∩ ] − ∞, 0[, i.e., Specess (H ) = Spec(L) ∪ I− ∪ I1 ∪ I2 . . . . Let RV (λ, x, y) be the integral kernel of the operator (H −λI)−1 , i.e., solution of the equation H u − λu = δy . Due to the Aizenman-Molchanov criterion, the operator H has only pure point spectrum (for P −a.s. ω) provided for each y ∈ X, for each interval Ik , and for Lebesgues a.e. τ ∈ Ik : lim

→+0



|RV (τ + i, x, y)|2 < ∞,

(3.45)

x

for a.e. realization of {σ (y, ω)}.We split the proof of Eq. (3.45) in seven steps. Step I When V has a finite rank, Theorem 3.11(i) and Theorem 3.14 imply that for each fixed ω, the function RV (τ +i0, x, y) = (H −τ I)−1 δy (x) belongs to L2 (X, m) for each y ∈ X and for all but finitely many τ ∈ Ik (which are eigenvalues of H ). In general, when the rank of V is infinite, we split V in two parts: V = −σ1 δa1

and V = − i>1 σi δai . Writing the set of locations as {a} = {a1 } ∪ {ai : i > 1},

12 Clearly,

this condition implies Condition (3.37).

82

A. Bendikov et al.

we get similarly to Eq. (3.35): for λ in the domain C+ , RV (λ, x, y) = RV

(λ, x, y) + RV

(λ, x, a1 )B(λ)−1 RV

(λ, a1 , y), where B(λ) = 1/σ1 − RV

(λ, a1 , a1 ) is a nonconstant analytic in the domain C+ function. It follows that RV (λ, ·, y)2 ≤ RV

(λ, ·, y)2 + |B(λ)|−1 RV

(λ, ·, a1 )2 |RV

(λ, a1 , y)|. Hence, the function RV (λ, x, y) satisfies Condition (3.45), i.e., RV (τ + i0, ·, y)2 is finite for all y and a.e. τ , provided the function RV

(λ, x, y) satisfies Condition (3.45), i.e., RV

(τ + i0, ·, a)2 is finite for all a and a.e. τ , and also one more restriction on τ, it does not belong to the exceptional set ϒ := {s : B(s + i0) = 0}. The function B(λ), analytic in the domain C+ , admits non-tangential boundary values B(s + i0) for a.e. s. By the Lusin-Privalov uniqueness theorem on boundaryvalues of analytic functions [34, Ch. IV, Sec. 2.5] (see also [37, Theorem 1.5]), the Lebesgues measure of the exceptional set ϒ equals to zero. Thus, we come to the conclusion that Condition (3.45) for the potential V can be reduced to the case of truncated potential V

. Repeating this argument finitely many times, we come to the final conclusion: in order to prove that (3.45) holds for V , we can consider, if necessary, any finitely truncated potential V

(the potential corresponding to the finitely truncated system of locations {ai : i > k}) and to prove that (3.45) holds for V

instead of V . Step II Writing for λ ∈ C+ equation H u−λu = δy in the form Lu−λu = δy −V u, we obtain RV (λ, x, y) = R(λ, x, y) +

∞ 

σj R(λ, x, aj )RV (λ, aj , y).

(3.46)

j =1

Equation (3.46) shows that to estimate the function y → RV (λ, ·, y)2 , it is enough to estimate the quantity |RV (λ, aj , y)| for j = 1, 2, . . . etc. Indeed, since R(λ, ·, y)2 does not depend on y, we get ⎛ RV (λ, ·, y)2 ≤ R(λ, ·, y)2 ⎝1 + β

∞  j =1

⎞ |RV (λ, aj , y)|⎠ .

(3.47)

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

83

Choosing x = ai , i = 1, 2, . . . , in Eq. (3.46) and setting R(λ, ai , ai ) = R(λ), we obtain RV (λ, ai , y) =

 σj R(λ, aj , ai )RV (λ, aj , y) R(λ, ai , y) + . 1 − σi R(λ) 1 − σi R(λ)

(3.48)

j :j =i

Step III Applying in Eq. (3.48) the inequality  s   ∞  ∞   s   Zj  , Zj ∈ C, 0 < s ≤ 1, Z ≤ j  j =1  j =1 we will get       σj R(λ, aj , ai ) s   R(λ, ai , y) s     RV (λ, aj , y)s  |RV (λ, ai , y)| ≤  +    1 − σi R(λ) 1 − σi R(λ) s

j :j =i

    R(λ, aj , ai ) s   R(λ, ai , y) s  s   RV (λ, aj , y)s .   ≤ +β    1 − σi R(λ) 1 − σi R(λ) j :j =i

Taking the expectation over {σi }, we obtain the following Inequality: s      1  R(λ, aj , y)s  E |RV (λ, ai , y)| ≤ E   1 − σi R(λ)   RV (λ, aj , y) s   s  R(λ, aj , ai )s . +β E   1 − σi R(λ) s

(3.49)

j :j =i

Step IV Due to Eq. (3.4), the random variable RV (λ, aj , y) can be represented in the form RV (λ, aj , y) = RV (λ, aj , y) +

σi RV (λ, aj , ai )RV (λ, ai , y) aσi + b := 1 − σi RV (λ, ai , ai ) cσi + d

 where the random variables V = − k:k =i σk (ω)δak , a, b, c , and d do not depend on σi (but they of course depend on the truncated sequence {σk : k = i}). From this observation and the following two general inequalities from Molchanov’s lectures [28, Chapter II, Lemma 2.2]): There exist constants c0 , c1 > 0 such that for all complex numbers a, b, c, d, σ

 0

1

dσ c0 , forall0 < s < 1, ≤ |σ − σ |s 1−s

84

A. Bendikov et al.

and    1  aσ + b s + b s dσ  dσ, forall0 < s < 1/2,     cσ + d  |σ − σ |s ≤ c1  0 cσ + d

1  aσ

 0

yield the following lemma, which is the fundamental point of our reasons. Lemma 3.24 (Decoupling Lemma) There exist constants C0 , C0 > 0 which depend on s, α, β, and k such that the inequalities   E 

s  1  ≤ C0 1 − σi R(λ) 

and    RV (λ, aj , y) s     ≤ C E RV (λ, aj , y)s E 0  1 − σi R(λ) hold for all 0 < s < 1/2 and all λ ∈ C+ such that Re λ ∈ Ik . Step V For any fixed y ∈ X and λ as above, let us denote ψi := E |RV (λ, ai , y)|s . Applying decoupling lemma to Inequality (3.49) and setting C1 := β s C0 , we get an infinite system of inequalities ψi ≤ C0 |R(λ, ai , y)|s + C1

   R(λ, aj , ai )s ψj . j :j =i

In the vector form, this system reads as follows: ψi ≤ gi + (Aψ)i ,i = 1, 2, . . . , s where ψ = (ψi ), g = (gi ) has entries gi = C0 |R(λ, s where A is an  ai , y)| and infinite matrix with nonnegative entries aij = C1 R(λ, aj , ai ) if i = j and 0 otherwise. Iterating formally this infinite system of inequalities, we get

      ψi ≤ gi + (Ag)i + A2 g + A3 g + . . . ≤ (I − A)−1 g . i

i

i

In particular, this would yield the following inequality (one of the fundamental points in the proof of (3.45)): ψ ≤ 2 g

(3.50)

Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials

85

given A : L → L is a bounded linear operator acting in some Banach space L of sequences such that A ≤ 1/2. For instance, choosing L = {ψ : ψ = 

 i

μi E |RV (λ, ai , y)|s ≤ 2C0

i

(3.51)

μi |ψi | < ∞}, we obtain



μi |R(λ, ai , y)|s

(3.52)

i

given A = sup



ψ=1 i

  μi (Aψ)i  ≤ 1/2.

(3.53)

For ψ such that ψ = 1, we have 

      μi (Aψ)i  ≤ μi aij ψj 

i

i

=

j



        μj ψj  μi aij /μj ≤ sup μi aij /μj .

j

i

j

i

In particular, Inequality (3.53) holds whenever ⎛ sup ⎝ j



i:i =j

⎞ s  1 μi R(λ, aj , ai ) ⎠ /μj ≤ . 2C1

(3.54)

Finally, (3.26) together with (3.54) allows us to conclude that (3.51) holds, provided ⎛ sup ⎝ j



i:i =j

⎞ 1 1 ⎠ /μj ≤ μi . s d(aj , ai ) 2C1 C2

(3.55)

Step VI For λ as above and εj > 0 which we will choose later, consider events   Aj = {RV (λ, aj , y) > εj }. Applying Chebyshev inequality, we will get, for each j = 1, 2, . . ., the following inequality: s  E RV (λ, aj , y) . P (Aj ) ≤ εjs

(3.56)

86

A. Bendikov et al.

Equations (3.56), (3.52), and (3.55) yield  j

P (Aj ) ≤

   E RV (λ, aj , y)s j

≤ 2C0

εjs    R(λ, aj , y)s εjs

j

≤ 2C0 C2

 j

1 d(aj , y)s εjs

provided εj are chosen such that ⎛

⎞  1 1 1 ⎠≤ sup εjs ⎝ . s s εi d(aj , ai ) 2C1 C2 j

(3.57)

i:i =j

Let us choose s = 1/2 − δ and εj = 1/d(aj , y)r . Then, truncating, if necessary, the potential V , i.e., passing to the potential V

= V − V with V of finite rank as explained in Step I, we can assume that the sequence εj is a strictly decreasing sequence. By the ultrametric inequality, we have d(ai , aj ) = d(ai , y). Hence, ⎛

⎞ ⎛ ⎞ ⎛ ⎞  1   d(ai , y)rs 1 1 ⎠ ≤ sup ⎝ ⎠ + sup εjs ⎝ ⎠ sup εjs ⎝ εis d(aj , ai )s d(aj , ai )s d(aj , ai )s j j j i:i =j

i:i 0, (7)  (γI + γA + k + 3μ) (k + μ) (γI + γA + 2μ) + (γI + μ) (γA + μ)   − Nk βI (1 − q) + βA q − (k + μ) (γI + μ) (γA + μ)   + Nk βI (1 − q) (γA + μ) + βA q (γI + μ) > 0.

(8)

Inequality (6) always holds. Inequality (7) can be equivalently written as R0 =

βA Nkq βI Nk (1 − q) + < 1. (k + μ) (γI + μ) (k + μ) (γA + μ)

(9)

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

97

By using Mathematica [10], we can confirm that inequality (8) holds for R0 < 1; thus, the disease-free steady state is stable.   Since we incorporate the demographic terms, we are interested in exploring the longer-term persistence and the endemic dynamics of the disease. Setting equal to zero the right-hand side of system (1), we find a unique endemic steady state. Then, we are interested in determining the conditions necessary for endemic steady-state stability. Theorem 2 If R0 > 1, the endemic steady state, (S ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of system (1) with S∗ = E∗ = I∗ = A∗ =

R∗ =

N (γI + μ) (γA + μ) (k + μ) = , k (βI (γA + μ) + q (βA (γI + μ) − βI (γA + μ))) R0   Nμ 1 1− , R0 (k + μ)   1 Nk (1 − q) μ 1− , R0 (γI + μ) (k + μ)   1 Nkqμ 1− , R0 (γA + μ) (k + μ)    Nk qγA μ + γI (γA + μ − qμ)  1 1− , R0 (γI + μ) (γA + μ) (k + μ)

(10)

is locally stable. Proof The characteristic equation of the Jacobian matrix (5) at the endemic steady state is λ4 + α3 λ3 + α2 λ2 + α1 λ + α0 = 0, with α3 = γI + γA + k + 3μ + μR0 > 0, α2 = μR0 (γI + γA + k + 3μ) + (γI + γA + 2μ) (k + μ) + (γI + μ) (γA + μ) −

N k (βI (1 − q) + βA q) , R0

α1 = μR0 (γI + γA + 2μ) (k + μ) + (γI + μ) (γA + μ) (μR0 + k + μ) −

N k (βI (1 − q) (γA + μ + 1) + βA q (γI + μ + 1)) , R0

α0 = μR0 (k + μ) (γI + μ) (γA + μ) − μ

N k (βI (1 − q) (γA + μ) + βA q (γI + μ)) . R0

98

V. Bitsouni et al.

From the Routh-Hurwitz criterion, the endemic steady (10) is locally stable if and only if α0 > 0, α1 > 0, α3 > 0 and α1 α2 α3 − α12 − α0 α32 > 0. We always have that α3 > 0, whereas a0 > 0 is equivalent to R0 > 1. By using Mathematica [10], we can confirm that the rest of the above relations hold for R0 > 1; thus, the endemic steady state is stable.  

2.4 Global Stability Analysis of the SEIAR Model Theorem 3 If R0 ≤ 1, then the disease-free steady state, (N, 0, 0, 0, 0), of system (1) is globally asymptotically stable. Proof We prove the global stability of the disease-free steady state (N, 0, 0, 0, 0) by constructing a Lyapunov function. We consider the function V1 : R+ × R3 → R with   NβA S NβI I+ A. V1 (S (t) , E (t) , I (t) , A (t)) = S − N − N ln +E + N γI + μ γA + μ We take the derivative of V1 with respect to t: V1

  NβI

NβA

N =S 1− + E + I + A S γI + μ γA + μ   μN 2 βI Nk (1 − q) βA Nkq = 2μN − μS − + E (k + μ) + −1 S (k + μ) (γI + μ) (k + μ) (γA + μ)   S N = −μN + − 2 + E (k + μ) (R0 − 1) . N S

From the arithmetic-geometric mean inequality, we have 1 2



S N + N S

;

 ≥

2

S N S N =1⇒ + − 2 ≥ 0. N S N S

Thus, if R0 ≤ 1, then V1 ≤ 0 for all t ≥ 0 and (S, E, I, A) ∈ R+ × R3 sufficiently close to (N, 0, 0, 0), and V1 (t) = 0 holds only for (S, E, I, A) = (N, 0, 0, 0). Hence, the singleton {(N, 0, 0, 0)} is the largest invariant set for which V1 = 0. Then, from LaSalle’s invariance principle [11], it follows that the disease-free steady state is globally asymptotically stable.   Theorem 4 If R0 > 1, then the endemic steady state, (S ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of system (1) is globally asymptotically stable.

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

99

4 Proof We consider the function V2 : R+ → R with     S E V2 (S (t) , E (t) , I (t) , A (t)) = S − S ∗ − S ∗ ln ∗ + E − E ∗ − E ∗ ln ∗ S E     βI S ∗ βA S ∗ I A + I − I ∗ − I ∗ ln ∗ + A − A∗ − A∗ ln ∗ . γI + μ I γA + μ A We take the derivative of V2 with respect to t:         βI S ∗

βA S ∗

S∗ E∗ I∗ A∗

+E 1− + I 1− + A 1− 1− S E γI + μ I γA + μ A     ∗ S E∗ = (μN − βI SI − βA SA − μS) 1 − + (βI SI + βA SA − (k + μ) E) 1 − S E   ∗ ∗ β S I + I (k (1 − q) E − (γI + μ) I ) 1 − γI + μ I   β S∗ A∗ . + A (kqE − (γA + μ) A) 1 − γA + μ A

V2 = S

After using the relations μN = βI S ∗ I ∗ + βA S ∗ A∗ + μS ∗ and βI S ∗ I ∗ +βA S ∗ A∗ = (k + μ) E ∗ , k (1 − q) E ∗ = (γI + μ) I ∗ , kqE ∗ = (γA + μ) A∗ , βI S ∗ I ∗ E βA S ∗ A ∗ E and , we have I E∗ AE ∗    ∗  S I E∗ I∗ E S S∗

∗ ∗ ∗ S − 2 − βI S I + ∗ ∗ + V2 = − μS + −3 S∗ S S S I E I E∗   ∗ S A E∗ A∗ E S − βA S ∗ A ∗ − 3 . + ∗ ∗ + S S A E A E∗ 2

2

and by adding and subtracting the terms

From the arithmetic-geometric mean inequality, we have that 1 3



S∗ S I E∗ I∗ E + ∗ ∗ + S S I E I E∗

;

 ≥

3

S∗ S I E∗ I ∗ E =1 S S∗ I ∗ E I E∗

100

V. Bitsouni et al.

and 1 3



S A E∗ A∗ E S∗ + ∗ ∗ + S S A E A E∗

;

 ≥

3

S ∗ S A E ∗ A∗ E = 1. S S ∗ A∗ E A E ∗

4 Hence, V2 ≤ 0 for all (S, E, I, A) ∈ R+ , and the equality holds only for the endemic steady state (S ∗ , E ∗ , I ∗ , A∗ ). We conclude again from LaSalle’s invariance principle that the endemic steady state is globally asymptotically stable.  

2.5 Numerical Simulations for the SEIAR Model We proceed to the estimation of the already known (discrete) epidemic curve of the dR (see, e.g., disease in Italy, as obtained from the data set [12], by (the continuous) dt [13] and [14]). We plot together the two functions in Fig. 2. The total population of Italy is 60,456,999. Once the restriction of movement (quarantine) during the manifestation of COVID-19 was applied, it limited the spread of the disease. To this end, we follow the approach in [15], and we consider as the total population N = 60,456,999/250.

Fig. 2 The number of confirmed cases per day in Italy until July 2020. The blue dots represent dR the data obtained from [12] and the red curve the graph of , as obtained by (1). The parameters dt used here are as follows: βI = 2.55/N, βA = 1.275/N, k = 0.07, μ = 0.001, γI = 0.0625, γA = 0.083 and q = 0.425, and the initial conditions (ICs) are S0 = N − 200, I0 = 100, A0 = 100 and E0 = R0 = 0

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

101

Fig. 3 The dynamics of the proportion of the values of model (1). (a) For all subgroups. (b) For the groups of both symptomatic and asymptomatic infectious individuals, I +A, towards the steady state I ∗ +A∗ . The parameters used are the same as in Fig. 2 (see also Table 1). The initial conditions are S0 = 1–0.0008, I0 = 0.0004, A0 = 0.0004 and E0 = R0 = 0

In Fig. 3 we show the dynamics of the proportion of the values of model (1) for the set of parameters used in Fig. 2. We see in Fig. 3a that, for this set of parameters, the solution of the system has an oscillatory behaviour towards the endemic steady state. This can be more clear in Fig. 3b, where the proportion of the infectious population (both symptomatic and asymptomatic) oscillates towards the proportion of the steady state I ∗ + A∗ .

3 Modelling Transmission Dynamics of COVID-19 in a Vaccinated Population In this section, we consider the subclass of the vaccinated-with-a-prophylacticvaccine (V ) individuals. We set 0 ≤ p ≤ 1 for the vaccine coverage as well as 0 ≤  < 1 for the vaccine efficacy [16]. Then, the model becomes dS dt dV dt dE dt dI dt dA dt

= (1 − p) μN − βI SI − βA SA − μS,

(11a)

= pμN − (1 − ) (βI V I + βA V A) − μV ,

(11b)

= (S + (1 − ) V ) (βI I + βA A) − (k + μ) E,

(11c)

= k (1 − q) E − (γI + μ) I,

(11d)

= kqE − (γA + μ) A,

(11e)

102

V. Bitsouni et al.

dR = γI I + γA A − μR, dt

(11f)

along with the initial conditions:

6 . (S (0) , V (0) , E (0) , I (0) , A (0) , R (0)) = (S0 , V0 , E0 , I0 , A0 , R0 ) ∈ R+ 0 (12) A flow diagram of the vaccination model is illustrated in Fig. 4. Following the same steps as before and using the disease-free steady state of the model, ((1 − p) N, pN, 0, 0, 0, 0), we have that the basic reproductive ratio for the model where vaccination is applied is 

RV0

βI Nk (1 − q) βA Nkq + = (1 − p) (k + μ) (γI + μ) (k + μ) (γA + μ)

Fig. 4 Flow diagram of the SVEIAR model (11)

 = (1 − p) R0 .

(13)

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

103

The endemic steady state, (S ∗ , V ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of model (11) is   (1 − p) Nk (1 − q) μ 1 1− V , 1− V , (γI + μ) (k + μ) R0 R0     (1 − p) Nk qγ μ + γ (γ + μ − qμ)   A I A (1 − p) Nkqμ 1 1 1− V , 1− V . (γA + μ) (k + μ) (γI + μ) (γA + μ) (k + μ) R0 R0



(1 − p) N pN Nμ , V, (k + μ) RV R 0 0



1



We prove the global stability of the model following the same steps as before. Theorem 5 If RV0 ≤ 1, then the disease-free steady state, ((1−p) N,pN, 0, 0, 0, 0), of system (11) is globally asymptotically stable. Proof We prove that the disease-free steady state of system (11) is globally asymptotically stable by applying again the LaSalle’s invariance principle for the

2 Lyapunov function V1V : R+ × R3 → R, with  V1V (S (t) , V (t) , E (t) , I (t) , A (t)) = S − (1 − p) N − (1 − p) N ln

S (1 − p) N



  V (1 − p) NβI (1 − p) NβA + V − pN − pN ln +E+ I+ A, pN γI + μ γA + μ

and following the steps corresponding to the proof of Theorem 3.

 

Theorem 6 If RV0 > 1, then the endemic steady state, (S ∗ , V ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of system (11) is globally asymptotically stable. Proof We prove that the endemic steady state of system (11) is globally asymptotically stable by applying again LaSalle’s invariance principle for the Lyapunov

5 function V2V : R+ → R, with     S V V2V (S (t) , V (t) , E (t) , I (t) , A (t)) = S − S ∗ − S ∗ ln ∗ + V − V ∗ − V ∗ ln ∗ S V     β S ∗ + (1 − ) βI V ∗ E I I − I ∗ − I ∗ ln ∗ + E − E ∗ − E ∗ ln ∗ + I E γI + μ I   A βA S ∗ + (1 − ) βA V ∗ A − A∗ − A∗ ln ∗ , + γA + μ A

and following the steps corresponding to the proof of Theorem 4.

 

104

V. Bitsouni et al.

3.1 Numerical Simulations for the SVEIAR Model To assess the vaccine effectiveness, we focus on three important epidemiological measures [17]: (1) the risk of infection spread, represented by RV0 , (2) the peak prevalence of infection and (3) the time at which the peak prevalence occurs. Relation (13) shows that the vaccine coverage, p, and vaccine efficacy, , act multiplicatively on R0 . As the proportion of asymptomatic cases is still unknown, in Fig. 5a, we present a contour plot of the dependence of RV0 on the vaccine coverage and vaccine efficacy, for different proportion of asymptomatic cases. The coloured curves represent the threshold RV0 = 1 in (13), . . . , between scenarios in which the infection is expected to spread (represented by the area below the threshold; RV0 > 1) or not spread (represented by the area above the threshold; RV0 < 1). The plot indicates that the vaccine efficacy and coverage need to be greater for small proportion of asymptomatic cases. As the number of symptomatic cases increases, a more effective vaccine is needed. We see that even for a severe COVID-19 epidemic, as in our case, the vaccine can prevent the infection spread if both the vaccine efficacy and the vaccine coverage are high. Considering, however, that the data reflect a period where the severity of COVID-19 was not yet known and the average number of close contacts between individuals was very high due to occasions and events, the transmission rate, βI , as obtained by the data is not the most appropriate index to predict the vaccine effectiveness, as the situation has changed dramatically and close contacts have been significantly reduced. Hence, in Fig. 5b, we present a corresponding contour plot for a lower βI . We see that, in the case of a reduced

(a)

(b)

Fig. 5 Assessing the vaccine effectiveness: A contour plot showing the dependence of RV0 on the vaccine efficacy, vaccine coverage and proportion of asymptomatic cases for (a) βI = 2.55 and (b) βI = 0.2. The rest of the model parameters are given in Table 1

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

105

Fig. 6 Assessing the vaccine effectiveness: The effect of vaccination on the prevalence of infection, with ICs, S0 = V0 = 0.4996 and I0 = A0 = 0.0004, and for (a) βI = 2.55 and (b) βI = 0.2. The rest of the model parameters are given in Table 1

transmission rate, the vaccine can prevent the infection spread, even for imperfect vaccines and small vaccine coverage. In Fig. 6 we see the effect of vaccine efficacy on the proportion of the infection dynamics for high (Fig. 6a) and low (Fig. 6b) transmission rates. Higher vaccine efficacy leads to milder but prolonged epidemics due to the slower rate of infection transmission. Moreover, it causes later occurrence of the first infection incidence and peak prevalence and a slower rate of postpeak prevalence decline.

4 Conclusions We presented an ad hoc SEIAR model with horizontal transmission and demographic terms for the epidemic spread of COVID-19, and we extended the model to include vaccination. The stability of both models is proved by implementing suitable Lyapunov functions; the model is fitted to real data from the epidemic in Italy. We studied the condition under which a vaccine can prevent disease spread. We accessed the vaccine effectiveness, focusing on the risk of infection spread, the peak prevalence of infection and the time at which the peak prevalence occurs. Future work includes further investigation of the vaccine model, by incorporating different vaccination strategies, and if possible the comparison with biological data. An extension of the model will also include additional important factors of COVID19 spread, such as the population age, the geographical spread of the epidemics (see, e.g. Refs [28–30] and other references therein) and the waning immunity gained by the infected individuals, as well as vertical transmission and migration terms for the infected individuals.

106

V. Bitsouni et al.

Table 1 Model parameters, values, units and relevant references Param. Description μ Birth/death rate βI Transmission rate of symptomatic infectious individuals βA

k

q

γI

γA

p 

Transmission rate of asymptomatic infectious individuals Incubation rate (rate of latent individuals becoming infectious) Proportion of the asymptomatic infectious individuals Recovery rate of the symptomatic infectious individuals Recovery rate of the asymptomatic infectious individuals Proportion of vaccinated individuals Vaccine efficacy

Value Unit Reference 0.001(2 · 10−5 –0.001) days−1 [18, 19] 2.55(0.14–2.55) individuals−1 ·days−1 [5, 15, 20–22] βI 2

individuals−1 ·days−1 [23]

0.071(0.071–0.2)

days−1

[24, 25]

0.425(0–1)

–

[3]

0.0625(0.02–0.0625)

days−1

[4, 26, 27]

0.083(0.083–0.33)

days−1

[4, 27]

0.5(0–1)

–

Estimated

0.5(0–1)

–

Estimated

References 1. Gorbalenya, A.E., Baker, S.C., Baric, R.S., de Groot, R.J., Drosten, C., Gulyaeva, A.A., et al.: The species Severe acute respiratory syndrome-related coronavirus: classifying 2019-nCoV and naming it SARS-CoV-2. Nat. Microbiol. 5(4), 53–544 (2020) 2. Worldometer. www.worldometers.info 3. Lavezzo, E., et al.: Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo’. Nature 584, 425–429 (2020) 4. Yang, R., Gui, X., Xiong, Y.: Comparison of clinical characteristics of patients with asymptomatic vs symptomatic coronavirus disease 2019 in Wuhan, China. JAMA Netw. Open 3(5), e2010182–e2010182 (2020) 5. Li, R., Pei, S., Chen, B., Song, Y., Zhang, T., Yang, W., Shaman, J.: Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science 368(6490), 489–493 (2020) 6. Heneghan, C., Brassey, J., Jefferson, T.: COVID-19: What proportion are asymptomatic? CEBM (2020) 7. Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28(4), 365–382 (1990) 8. Diekmann, O., Heesterbeek, J.A.P., Roberts, M.G.: The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface 7(47), 873–885 (2010) 9. Edelstein-Keshet, L.: Mathematical Models in Biology. SIAM (2005) 10. Wolfram Research, Inc., Mathematica, Version 12.1. Champaign (2020) 11. La Salle, J.P.: The Stability of Dynamical Systems. SIAM (1976)

A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy

107

12. European Centre for Disease Prevention and Control (ECDC) (2020). https://www.ecdc. europa.eu/en/publications-data/download-todays-data-geographic-distribution-covid-19cases-worldwide 13. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Math. Phys. Sci. 115(772), 700–721 (1927) 14. Braun, M.: Differential Equations and Their Applications, 4th edn. Springer, New York (1993) 15. Ndairou, F., Area, I., Nieto, J.J., Torres, D.F.: Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos, Solitons Fractals 135, 109846 (2020) 16. McLean, A.R.: Vaccination, evolution and changes in the efficacy of vaccines: a theoretical framework. Proc. Biol. Sci. 261(1362), 389–393 (1995) 17. Feng, Z., Towers, S., Yang, Y.: Modeling the effects of vaccination and treatment on pandemic influenza. AAPS J. 13(3), 427–437 (2011) 18. UNdata: Crude birth/death rate (per 1,000 population). United Nations (2020) 19. Keeling, M.J., Rohani, P.: Modeling Infectious Diseases in Humans And Animals. Princeton University Press, Princeton (2011) 20. Pribylova, L., Hajnova, V.: SEIAR model with asymptomatic cohort and consequences to efficiency of quarantine government measures in COVID-19 epidemic (2020). arXiv:2004.02601 21. Castilho, C., Gondim, J.A., Marchesin, M., Sabeti, M.: Assessing the efficiency of different control strategies for the COVID-19 epidemic. EJDE 2020(64), 1–17 (2020) 22. Calafiore, G.C., Novara, C., Possieri, C.: A modified SIR model for the COVID-19 contagion in Italy (2020). arXiv:2003.14391 23. Sypsa, V., Roussos, S., Paraskevis, D., Lytras, T., Tsiodras, S., Hatzakis, A.: Modelling the SARS-CoV-2 first epidemic wave in Greece: social contact patterns for impact assessment and an exit strategy from social distancing measures (2020). medRxiv 24. World Health Organization: Coronavirus disease 2019 (COVID-19): situation report, 72 (2020) 25. Lauer, S.A., et al.: The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application. Ann. Intern. Med. 172(9), 577– 582 (2020) 26. Zhou, B., She, J., Wang, Y., Ma, X.: The duration of viral shedding of discharged patients with severe COVID-19. Clin. Infect. Dis. 71(16), 2240-2242 (2020) 27. Zhou, R., Li, F., Chen, F., Liu, H., Zheng, J., Lei, C., Wu, X.: Viral dynamics in asymptomatic patients with COVID-19. Int. J. Infect. Dis. 96, 288–290 (2020) 28. Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6(2), 109 (1978) 29. Khachatryan, K.A., Narimanyan, A.Z., Khachatryan, A.K.: On mathematical modelling of temporal spatial spread of epidemics. Math. Model. Nat. Phenom. 15(6), 1–14 (2020) 30. Sergeev, A., Khachatryan, K.: On the solvability of a class of nonlinear integral equations in the problem of a spread of an epidemic. Trans. Mosc. Math. Soc. 80, 95–111 (2019)

Rate of Convergence to the Poisson Law of the Numbers of Cycles in the Generalized Random Graphs Sergey G. Bobkov, Maria A. Danshina, and Vladimir V. Ulyanov

√ Abstract Convergence of order O(1/ n) is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is based on the Chen–Stein approach and on the derived properties of the ratio of the sum of squares of random variables and the sum of these variables. These properties can be applied to other asymptotic problems related to generalized random graphs. Keywords Generalized random graphs · Poisson law · Rate of convergence · Cycles

S. G. Bobkov University of Minnesota, Minneapolis, MN, USA National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected] M. A. Danshina Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] V. V. Ulyanov () National Research University Higher School of Economics, Moscow, Russia Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_5

109

110

S. G. Bobkov et al.

1 Introduction Complex networks attract increasing attention of researchers in various fields of science. In the last years, numerous network models have been proposed. With the uncertainty and the lack of regularity in real-world networks, these models are usually random graphs. Random graphs were first defined by Paul Erd˝os and Alfred Rényi in their 1959 paper “On Random Graphs” (see [10]) and independently by Gilbert in [12]. The suggested models are closely related: there are n isolated vertices, and every possible edge occurs independently with probability p : 0 < p < 1. It is assumed that there are no self-loops. Later, the models were generalized. A natural generalization of the Erdös–Rényi random graph is that the equal edge probabilities are replaced by probabilities depending on the vertex weights. Vertices with high weights are more likely to have more neighbors than vertices with small weights. Vertices with extremely high weights could act as the hubs observed in many real-world networks. The following generalized random graph (GRG) model was first introduced by Britton et al.; see [5]. Let V = {1, 2, .., n} be the set of vertices and Wi > 0 be the weight of vertex i, 1 ≤ i ≤ n. The edge probability of the edge between any two vertices i and j , for i = j , is equal to pij =

Wi Wj Ln + Wi Wj

(1)

 and pii = 0 for all i ≤ n. Here, Ln = ni=1 Wi denotes the total weight of all vertices. The weights Wi , i = 1, 2, ..., n can be taken to be deterministic or random. If we take all Wi − s as the same constant Wi ≡ nλ/(n − λ) for some 0 ≤ λ < n, it is easy to see that pij = λ/n for all 1 ≤ i < j ≤ n. That is, the ErdHos–Rényi random graph with p = λ/n is a special case of the GRG. There are many versions of the GRG, such as Poissonian random graph (introduced by Norros and Reittu in [19] and studied by Bhamidi et al. [3]), rank1 inhomogeneous random graph (see [4]), random graph with given prescribed degrees (see [8]), and Chung–Lu model of heterogeneous random graph (see [7]). The Chung–Lu model is the closest to the model of generalized random graph. Two vertices i and j are connected with probability pij = Wi Wj /Ln and independently of other pairs of vertices, where W = (W1 , W2 , ..., Wn ) is a given sequence. It is necessary to assume that Wi2 ≤ Ln , for all i. Under some common conditions (see [15]), all of the abovementioned versions of the GRG are asymptotically equivalent, meaning that all events have asymptotically equal probabilities. The updated review on the results about these inhomogeneous random graphs can be seen in Chapter 6 in [21]. One of the problems that arise in real networks of various nature is the spread of the virus. In [6], the authors proposed an approach called nonlinear dynamic system (NLDS) for modeling such processes. Consider a network of n vertices represented by an undirected graph G. Assume an infection rate β > 0 for each connected edge

Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG

111

that is connected to an infected vertex and a recovery rate of δ > 0 for each infected individual. Define the epidemic threshold τ as a value such that β/δ < τ ⇒ infection dies out over time β/δ > τ ⇒ infection survives and becomes an epidemic. τ is related to the adjacency matrix A of the graph. The matrix A = [aij ] is an n × n symmetric matrix defined as aij = 1 if vertices i and j are connected by an edge and aij = 0 otherwise. Define a walk of length k in G from vertex v0 to vk to be an ordered sequence of vertices (v0 , v1 , ..., vk ), with vi ∈ V , such that vi and vi+1 are connected for i = 0, 1, ..., k − 1. If v0 = vk , then the walk is closed. A closed walk with no repeated vertices (with the exception of the first and last vertices) is called a cycle. For example, triangles, quadrangles, and pentagons are cycles of length three, four, and five, respectively. In the following, the cycle will be denoted by the first k vertices, without specifying the vertex vk , which is the same as v0 : (v0 , v1 , ..., vk−1 ). In Theorem 1 in [6], it has been stated that τ is equal to 1/λ1 , where λ1 is the largest eigenvalue of the adjacency matrix A. The following lower bound for λ1 (A) was shown in [20] λ1 (A) ≥

6) +



36)2 + 32e3 /n , 4e

where n, e, and ) are the number of vertices, edges, and triangles in G, resp. Moreover, using information about the cycle numbers of higher orders, one can get more precise upper bounds for τ . In [13], the central limit theorems were proved for the total number of edges in GRG. There are also many results on asymptotic properties of the number of triangles in homogeneous cases. For example, for the ErdHos–Rényi random graph, the upper tails for the distribution of the triangle number had been studied in [2, 9, 14, 16]. Recently, in [18], it was shown for GRG model that asymptotic distribution of the triangle number converges to a Poisson distribution under strong assumption that the vertex weights are bounded random variables. A lot of real-world networks such as social or computer networks in the Internet (see, e.g. [11]) follow a so-called scale-free graph model; see Chapter 1 in [21]. In Chapter 6, in [21], it was shown that when the vertex weights have approximately a power-law distribution, the GRG model leads to scale-free random graph. In the present paper, we prove √ not only the convergence, but we get the convergence rate of order O(1/ n) for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in GRG with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is based on the Chen–Stein approach and on the derived properties of the ratio of the sum of squares of random variables and the sum of these variables. These properties can be applied to other asymptotic problems related to GRG.

112

S. G. Bobkov et al.

The main results are formulated in Sect. 2. For their proofs, see Sect. 4. Section 3 contains auxiliary lemmas, some of which are of independent interest.

2 Main Results Let {1, 2, ..., n} be the set of vertices and Wi be a weight of vertex i : 1 ≤ i ≤ n. The probability of the edge between vertices i and j is defined in (1). Let Wi , i = 1, 2, ..., n, be independent identically distributed random variables distributed as a random variable W . For k ≥ 3, denote by I (k) the set of potential cycles of length k. We have that the number of elements in I (k) is equal to (n)k /(2k), where (n)k = n(n − 1)...(n − k + 1) is the number of ways to select k distinct vertices in order, and the factor 1/(2k) appears since, for k > 2, a permutation of k vertices corresponds to a choice of a cycle in I (k) together with a choice of any of two orientations and k starting points. For example, all six cycles {1, 3, 4}, {3, 4, 1}, {4, 1, 3}, {4, 3, 1}, {1, 4, 3}, and {3, 1, 4} are, in fact, one cycle of length 3. For α ∈ I (k), let Yα be the indicator that α occurs as a cycle in GRG. For example, P(Y{1,3,4} = 1) = p13 p34 p41 . For any integer-valued nonnegative random variables Y and Z, denote the total variation distance between their distributions L(Y ) and L(Z) by  L(Y ) − L(Z) ≡ suph=1 |Eh(Y ) − Eh(Z)|,

(2)

where h is any real function defined on {0, 1, 2, ...} and  h ≡ supm≥0 |h(m)|.  For k ≥ 3, put Sn (k) = α∈I (k) Yα , that is, Sn (k) is the number of cycles of length k. Let Zk be a random variable having Poisson distribution with parameter λ(k) = (EW 2 /EW )k /(2k). Theorem 1 For any k ≥ 3, one has  L(Sn (k)) − L(Zk ) = O(n−1/2 ),

(3)

P(W > x) = o(x −2k−1), as x → +∞.

(4)

provided that

Remark 1 Relation (3) holds under condition that W has power-law distribution. The condition on the tail behavior of the distribution of W can be replaced by stronger moment condition: the finiteness of expectation EW 2k+1 . Remark 2 Recently in [18], the convergence in distribution of the number of triangles Sn (3) in a generalized random graphs to the Poisson random variable Z3 was proved by method of moments under assumption that the vertex weights Wi -s are bounded random variables. In Theorem 1, we have used the Chen– Stein approach; see, e.g., [1] and [2]. This allows us not only to extend the

Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG

113

Triangles number Poisson (338.72)

100

80

60

40

20

0 280

300

320

340

360

380

400

Fig. 1 Histogram of the number of triangles in GRG with 2000 vertices. The distribution of vertex weights Wi ∼ U ni(10, 15), forall i ≤ 2000. The number of realizations is 500 400 380

Ordered Values

360 340 320 300 280

280

300

320

340

360

380

400

Theoretical quantiles Fig. 2 Q–Q plot for the number of triangles in GRG with 2000 vertices and the Poisson variable P ois(338.72). Wi ∼ U ni(10, 15), forall i ≤ 2000. The number of realizations is 500

convergence result to cycles of any fixed length k but also to get the rate of convergence. Moreover, we replace the assumption about the boundness of Wi -s with the condition that Wi has a power-law distribution. As we noted in Introduction, this condition better matches real-world networks. Figures 1 and 2 illustrate the results of Theorem 1, with the example of the number of triangles distribution.

114

S. G. Bobkov et al.

The next results are not directly connected with number of cycles in GRG. They are an important part of the proof of Theorem 1. At the same time, the results are of independent interest. They describe the asymptotic properties of ratio of a sum of the squares of n i.i.d. random variables and a sum of these random variables. These properties can be applied to other asymptotic problems related to GRG. Given i.i.d. positive random variables X, X1 , . . . , Xn , define the statistics Tn =

X12 + · · · + Xn2 . X1 + · · · + Xn

Assume that X has a finite second moment, so that, by the law of large numbers, with probability one p

lim Tn =

n→∞

 EX2 p EX

for any p ≥ 1. Here, we describe the tail-type and moment-type conditions which ensure that this convergence also holds on average. Theorem 2 Given an integer p ≥ 2, the convergence p

lim ETn = (EX2 /EX)p

n→∞

(5)

is equivalent to the tail condition P{X ≥ x} = o(x −p−1 ) as x → ∞.

(6)

Moreover, if P{X ≥ x} = O(x −p−3/2) as x → ∞, then ETn − (EX2 /EX)p = O(n−1/2) p

(7)

The finiteness of the moment EXp+1 is sufficient for (5) to hold, while the finiteness of the moments EXq is necessary for any real value 1 ≤ q < p + 1. Let Mn = max1≤i≤n Xi . For p ≥ 2, define (p)

Rn

p

= Tn Mn2 /(X1 + X2 + ... + Xn ).

(8)

(p)

By the law of large numbers, Rn → 0 as n → ∞ a.s., under mild moment (p) assumptions. The next theorem gives the order of convergence of ERn to zero under tail-type and moment-type conditions. Theorem 3 Given an integer p ≥ 2, if P(X ≥ x) = O(x −p−7/2 ) as x → +∞, then (p)

ERn

= O(n−1/2).

(9)

Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG

115

When p > 8 and EXp+4 is finite, the rate can be improved to (p)

ERn

= O(n−(p−2)/(p+4) ).

(10)

Moreover, if E eεX < ∞ for some ε > 0, then (p)

ERn

=O

 (log n)2  n

(11)

.

3 Auxiliary Lemmas Lemma 1 Let Sn = η1 + · · · + ηn be the sum of independent random variables ηk ≥ 0 with finite second moment, such that ESn = n and Var(Sn ) = σ 2 n. Then, for any 0 < λ < 1, one has  (1 − λ)2   n . (12) P{Sn ≤ λn} ≤ exp − 2 σ 2 + maxk (Eηk )2 Proof We use here the standard arguments. Fix a parameter t > 0. We have E e−t Sn ≥ e−λt n P{Sn ≤ λn}. Every function uk (t) = E e−t ξk is positive and convex and admits Taylor’s expansion near zero up to the quadratic form, which implies that uk (t) ≤ 1 − t Eξk +

 t2 t2 Eξk2 ≤ exp − t Eξk + Eξk2 . 2 2

Multiplying these inequalities, we get

bt 2  E e−t Sn ≤ exp − tn + , 2

b=

n 

Eξk2 .

k=1

The two bounds yield

 P{Sn ≤ λn} ≤ exp − (1 − λ)nt + bt 2 /2 , and after optimization over t (in fact, t =

1−λ b

n), we arrive at the exponential bound

(1 − λ)2  n2 . P{Sn ≤ λn} ≤ exp − 2b

116

S. G. Bobkov et al.

Note that b = Var(Sn ) +

n 

  (Eξk )2 ≤ σ 2 + max (Eξk )2 n, k

k=1

 

and (12) follows.

For further lemmas, we need additional notation. Denote by F (x) = P{X ≤ x} (x ∈ R) the distribution function of the random variable X, and put εq (x) = x q (1 − F (x)),

x ≥ 0, q > 0.

Raising the sum Un = X12 + · · · + Xn2 to the power p with n ≥ 2p, we have p

Un =



Xi21 . . . Xi2p ,

(13)

where the summation is performed over all collections of numbers i1 , . . . , ip ∈ {1, . . . , n}. For r = 1, . . . , p, we denoted by C(p, r) the collection of all tuples γ = (γ1 , . . . , γr ) of positive integers such that γ1 + · · · + γr = p. For any γ ∈ C(p, r), there are n(n − 1) . . . (n − r + 1) sequences Xi1 , . . . , Xip with r distinct terms that are repeated γ1 , . . . , γr times, resp. Therefore, by the i.i.d. assumption, p

ETn =

p  n(n − 1) . . . (n − r + 1) r=1

np



Eξn (γ ),

(14)

γ ∈C(p,r)

where 1 1 2γ 2γ ξn (γ ) = X1 1 . . . Xr r /( Sr + Sn,r )p n n and Sr = X1 + · · · + Xr ,

Sn,r = Xr+1 + · · · + Xn .

In the following lemmas, without loss of generality, let EX = 1. p

Lemma 2 For the boundedness of the sequence ETn , it is necessary that the moment EXp be finite. Moreover, for the particular collection γ = (p) with r = 1, we have Eξn (γ ) ≥ 2−p np EXp 1{X≥n} .

(15)

Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG 2p

Proof Since ξn (γ ) = X1 /( n1 X1 +

1 n

117

Sn,1 )p , applying Jensen’s inequality, we get 2p

Eξn (γ ) ≥ EX1 =E

X1 ( n1 X1 +

1 n

ESn,1 Sn,1 )p

X2p ( n1 X +

n−1 p n )

≥ 2−p np EXp 1{X≥n} .  

In the sequel, we use the events

n−r An,r = Sn,r ≤ 2

n−r and Bn,r = Sn,r > . 2

(16)

By Lemma 1, whenever n ≥ 2p, for some constant c > 0 independent of n, P(An,r ) ≤ e−c(n−r) ≤ e−cn/2 .

(17)

Lemma 3 If EXp is finite, then Eξn → (EX2 )p as n → ∞, where 1 1 ξn = X12 . . . Xp2 /( Sp + Sn,p )p . n n 2p

p

(18) p

Proof Using X1 . . . Xp ≤ Sp , we have ξn ≤ Sp /( n1 Sn )p ≤ np Sp . Hence, E ξn 1An,p ≤ np E Sp P(An,p ) = o(e−cn ) p

(19)

for some constant c > 0 independent of n. Here, we applied (17) with r = p and p Lemma 2 which ensures that E Sp < ∞. Further, ξn 1Bn,p ≤ 2p X12 . . . Xp2 .Hence, the random variables ξn 1Bn,p have an integrable majorant. Since also ξn → X12 . . . Xp2 (the law of large numbers) and 1Bn,p → 1 a.s. (implied by (17)), one may apply the Lebesgue dominated convergence theorem, which gives Eξn 1Bn → (EX2 )p . Together with (19), we get Eξn → (EX2 )p .   Lemma 4 If the moment EXp is finite, then for any γ = (γ1 , . . . , γr ) ∈ C(p, r), 2γ

E ξn (γ ) = 4p E

2γr

X1 1 . . . Xr

( n1 Sr + 1)p

+ o(1).

118

S. G. Bobkov et al.

Proof Using an elementary bound X1 1 . . . Xr r ≤ (X1 + · · · + Xr )2γ1 +···+2γr = 2p p p Sr and applying Jensen’s inequality, we see that ξn (γ ) ≤ np Sr ≤ np r p−1 (X1 + p · · · + Xr ). Hence, 2γ

Eξn (γ ) 1An,r ≤ n r

p p−1

r 

2γ



EXk 1An,r = np r p EXp P(An,r ) = o(e−c n ). p

k=1

(20) On the other hand, on the set Bn,r , there is a point-wise bound 2γ

2γr

X1 1 . . . Xr

ξn (γ ) 1Bn,r ≤

( n1 Sr +

n−r p 2n )

2γ

≤ 4p

2γr

X1 1 . . . Xr

( n1 Sr + 1)p

(21)

.

  Our task is reduced to the estimation of the expectation on the right-hand side of (21). Let us first consider the shortest collection γ = (p) of length r = 1. Lemma 5 Under the condition (6), 2p

E

X1

( n1 X1 + 1)p

= o(np−1 ).

(22)

In addition, if P{X ≥ x} = O(x −q ) for some real value q in the interval p < q < 2p, then 2p

E

X1

( n1 X1 + 1)p

= O(n2p−q ).

(23)

Proof We have 2p

E

X1

( n1 X1 + 1)p

2p

=E

X1

( n1 X1 + 1)p

2p

1{X1 ≥n} + E

X1

( n1 X1 + 1)p

1{X1 0 ∀n ≥ 1 |rn | ≤

cp (1 + n)p

 .

Random Tempered Distributions on Separable LCAG

151

2. The space of slowly increasing complex number sequences is given by the following:

 si = (sn )n≥1 ∈ CN0 : ∃q ≥ 1 ∃cq > 0 ∀n ≥ 1 |sn | ≤ cq (1 + n)q . We now have the important isomorphism theorem between the spaces si and rd (see [26, p. 59]). Theorem 3 The space rd with the family of seminorms (ρk )k∈N – with ρk (r) =  +∞ k n=1 (1+n) |rn | for r = (rn )n≥1 ∈ rd—is a locally convex Hausdorff space having a dual isomorphic to the space si. The duality may be written for r = (rn )n≥1 ∈ rd and s = (sn )n≥1 ∈ si, by < r, s >= +∞ n=1 rn sn . We may now define on G an analogue of Schwartz space of tempered functions on the real line. Definition 2 The space S of rapidly decreasing functions on the group G is given by the following:  2

S = φ ∈ L (G) : ∃(an )n≥1 ∈ rd, φ =L2

+∞ 

 a n en

n=1

+∞  For such a φ = +∞ n=1 an en ∈ S, we write φ = n=1 rn (φ)en to emphasize the growth properties of the coefficients. Remark 1 For φ ∈ S and any g ∈ G, we have that +∞  +∞      |φ(g)| =  |rn (φ)| < +∞ , rn (φ)en (g) ≤   n=1

n=1

by reason of |en | = 1 and (rn (φ))n≥1 ∈ rd. Moreover, we have unicity of the representation; in fact, ifwe suppose that φ ∈ S admits two representations φ = +∞ a b r (φ)e and φ = +∞ n n=1 n n=1 rn (φ)en , due to the speed of convergence of the series of the representations, we have that for every character ek , 0 =< φ, ek > − < ψ, ek > =

+∞ 

ran (φ)< en , ek >

n=1

−

+∞ 

rbn (φ)< en , ek > = ran (φ) − rbn (φ) ,

n=1

due to, both the Parseval duality formula (2), and, by the orthogonality relations, < en , ek >= δnk , with δnk the Kronecker’s delta.

152

M. L. Esquível and N. P. Krasii

By defining an obvious isomorphism between S and the sequence space rd, we now have the following result. Theorem 4 With family of seminorms (σk )k∈N —given for any φ ∈ the increasing k |r (φ)|—the space S is a Hausdorff locally convex S by σk (φ) = +∞ (1 + n) n n=1 topological vector space. Let us now consider a space of Schwartz generalized functions—continuous linear forms over S (see [27, p. 51])—over the group G which is isomorphic to the sequence space si. Definition 3 Let S denote the dual space of S, that is, the space of continuous linear maps from S to C, more precisely S = {T ∈ L(S, C) : ∃k > 0 ∃c > 0 ∀φ ∈ S |< T , φ >| ≤ cρk (φ)}

(3)

with L(S, C) the space of linear maps from S to C. Theorem 5 S is a locally convex topological vector space and admits a description given by

S =

+∞ 

 bn en : (bn )n≥1 ∈ si

n=1

 emphasize the slow meaning that, if for T ∈ S we write T = +∞ n=1 sn (T )en —to  growth properties of the coefficients—the image of T on φ = +∞ n=1 rn (φ)en ∈ S +∞ is given by the duality formula < T , φ >= n=1 sn (T )rn (φ) that encompasses Formula (2). Proof This is a consequence of Theorem 3.

 

2.2 Random Tempered Distributions We first introduce a space of random Schwartz distributions over G by means of series of L2 (G) basis functions having as coefficients random variables satisfying a certain growth condition. Our approach of a random distribution is akin to the definition of a generalized random process in [13, p. 242]. The results presented here generalize those presented in [9]—where the particular case of periodic distributions was studied—and in [10] for the case of the real numbers. Let (Ω, A, P) be a complete probability space. Let M denote the space of random variables taking complex values and MN the space of sequences of elements of M indexed by the integers N. For A, an element of M, let E[|A|] represent the expectation of A. Let Cm be the space of sequences (cn )n∈N0 ∈ MN such that for

Random Tempered Distributions on Separable LCAG

153

A ∈ M with A ≥ 0 and E[A] < +∞, we have the following: ∀n ∈ N0 |cn | ≤ A(1 + n)k a. s. on Ω .

(4)

Condition given in Formula (4) is the analogue—for random variables—to the defining condition of slowly increasing sequences in Definition 1; the constant in the right-hand side is, in the present case, a random variable. This space can be described by the equivalent conditions given in the next theorem that we recall from [8]. Theorem 6 For (cn )n∈N0 ∈ MN , the following are equivalent: 1. (cn )n∈N0 ∈ Cm . 2. There exists an integrable random variable A and a real bounded random variable K, defined on Ω, such that ∀n ∈ N0 |cn | ≤ A (1 + n)K a. s. on Ω 3. The sequence (E[|cn |])n∈N0 is a sequence of slow growth or, in an equivalent rephrasing, (∃a > 0) (∃k ∈ N0 ) (∀n ∈ N0 ) E[|cn |] ≤ a (1 + n)k Remark 2 (Coefficients of a Random Tempered Distribution) Associated with every sequence of random variables (cn )n∈N0 in Cm , there is a random Schwartz distribution T ∈ S on G, defined for a φ ∈ S, represented by (rn (φ))n∈N0 , by < T , φ >=

+∞ 

cn rn (φ) .

(5)

n=0

The distribution T is well defined because, by the second condition of the Theorem 6, the sequence (cn )n∈N0 is almost surely a sequence of slow growth. Besides, the sequence (rn (φ))n∈N0 , representing φ, is a rapidly decreasing sequence, and so, the series, in the left-hand side of Formula (5), converges almost surely. Let us mention some properties of the random Schwartz distributions just defined. Theorem 7 (On the Uniqueness of the Series Representation in Formula (5)) Let T be given, as in Formula (5), by a sequence (cn )n∈N0 . Then, the following are equivalent: 1. T = 0 a. s. on Ω . 2. ∀n ∈ N0 cm = 0 a. s. on Ω . The proof is similar to the one in [8] and we omit it. As a consequence, for every random Schwartz distribution, defined this way, there is, up to equality almost surely, a unique sequence of random variables such that Formula (5) is verified

154

M. L. Esquível and N. P. Krasii

almost surely. Such a sequence will be denoted by the following: (sn (T ))n∈N0 . An obvious corollary of this theorem gives a necessary and sufficient condition for equality of two random Schwartz distributions obtained by the way just explained. Theorem 8 (On the Uniqueness of the Series Representation in Formula (5)) Let T and U be two random Schwartz distributions obtained by Formula (5) from the sequences (sn (T ))n∈N0 and (sn (U ))n∈N0 , respectively. Then, the following are equivalent: 1. T = U a. s. on Ω . 2. ∀n ∈ N0 sn (T ) = sn (U ) a. s. on Ω . In this case also, the proof is simple and similar to the one in [8], and so, we omit it. Due to the hypotheses made on a sequence, in Cm , a random Schwartz distribution given, by way of Formula (5), has a mean in the sense of the next definition (see also [13, p. 246]). Definition 4 Let T be a random Schwartz distribution associated to a complex sequence (sn (T ))m∈N , by Formula (5). Then, T admits T¯ as a mean if and only if: 1. ∀φ ∈ S 2. ∀φ ∈ S

< T , φ >∈ M ∩ L1 (Ω) . E[< T , φ >] =< T¯ , φ > .

A random Schwartz distribution, built with a sequence (cm )m∈N0 ∈ Cm by Formula (5), does admit a mean. This mean has a representation as series in the complete orthonormal set of L2 (G) denoted by (en )n≥1 having, as coefficients, the sequence of expectations (E[cm ])m∈N0 . This simple result follows from an application of Lebesgue dominated convergence theorem as we will show next.  Theorem 9 If T = +∞ n=0 sn (T ) en is a random Schwartz distribution such that the sequence (sn (T ))n∈N0 is inthe class Cm , then T admits the nonrandom Schwartz distribution given by T¯ = +∞ n=0 E[sn (T )] en as a mean. Proof For every φ ∈ S, the sequence rn (φ))n∈N0 is a rapidly decreasing sequence. By Theorem 6, the sequence (E[|sn (T )|])n∈N0 is a slow growth sequence, and so, the series +∞  n=0

E[|sn (T )|] |rn (φ)| ,

Random Tempered Distributions on Separable LCAG

155

converges almost surely. Now, by the Lebesgue dominated convergence theorem, < T¯ , φ >=

+∞ 

E [|sn (T )|] rn (φ) = E

n=0

+ +∞ 

, sn (T ) rn (φ) = E [< T , φ >] ,

m=0

where the last equality results from the duality formula (5).

 

We next prove a converse of Theorem 9 that holds, in a sense, that we now proceed to explain. Let T be a measurable map from (Ω, A) into S , which we consider endowed with the Kolmogoroff σ -algebra associated to the dual countably Hilbertian (or Schwartz) topology on S (for the case of the real line, see [20, p. 6–16]). Then, for almost every ω ∈ Ω, the Hermite coefficients of T (ω), which we denote, as usual, by sn (T )(ω) =< T (ω), en > , for n ∈ N0 , are all well defined, and furthermore, we can consider (sn (T ))n∈N0 as a well-defined sequence of random variables. For a general T , no growth condition on the sequence (sn (T ))n∈N0 is verified so as to ensure that this sequence is in the class Cm . For instance, consider a sequence in which only a finite number of terms are nonzero and having one of these as a non-integrable random variable. Another example, in the real line, is given by a sequence of random variables taking small values with a big probability and big values with small probability such as (sn (T ))n∈N0 verifying P[sn (T ) = nen ] =

1 1 , P[sn (T ) = 0] = 1 − 2 . n2 n

 n As the series +∞ n=1 P[sn (T ) = ne ] converges, then, by Borel-Cantelli lemma, T is almost surely given by a finite sum of terms. That is, for almost all ω ∈ Ω, there exists a N ∈ N0 , N = N(ω) such that T (ω) =

N 

sn (T )(ω) en .

n=1

Now, as we have ∀n ∈ N0 E[sn (T )] =

en , n

which does not define a sequence of slow growth, the sequence sn (T ))n∈N0 cannot be in the class Cm , as a result of Theorem 6. The next theorem shows that the first condition in Definition 4 is a sufficient condition on T for the sequence (sn (T ))n∈N0 to be in the class Cm .

156

M. L. Esquível and N. P. Krasii

Theorem 10 Let T be defined (almost surely) in S, by a sequence of random variables (sn (T ))n∈N0 , such that ∀φ ∈ S

+∞ 

< T , φ >=

sn (T ) rn (φ) a. s. on Ω .

n=1

Then, if ∀φ ∈ S

< T , φ >∈ L1 (Ω) ,

the sequence (sn (T ))n∈N0 is in the class Cm . Proof The proof is an adaptation of the proof given to a similar result for Schwartz distributions in the real numbers (see [10]). Let us show first, using the closed graph theorem, that the map ΛT defined for every test function φ ∈ S by ΛT (φ) =< T , φ > is continuous from S into L1 (Ω). As T is, almost surely, a random tempered distribution, if (φl )l∈N0 is a sequence of test functions converging to zero inS, then the sequence of random variables (Ul )l∈N0 , defined for l ∈ N, by Ul =< T , φl > , converges a.s. to zero. So, this sequence (Ul )l∈N0 converges also in probability to zero. Now, suppose that (Ul )l∈N0 converges to U in L1 (Ω); then, the sequence converges also in probability to U , and so, U = 0. By the closed graph theorem (see [38, p. 51]), the map ΛT is continuous. Taking in account the topologies of S and L1 (Ω), the continuity of ΛT can be expressed in the following way: ∃k ∈ N0 , ∃ck > 0 ∀φ ∈ S || < T , φ > ||L1 (Ω) ≤ ck sup (1 + |n|)k |rn (φ)| . n∈N0

(6) As a second step, we use the sequence of Rademacher functions [41, p. 212] defined in [0, 1] by ∀t ∈ [0, 1] ∀n ∈ N radn (t)(t) = σ (sin(2n+1 πt)) , where σ (t) denotes the sign of t defined by the following:  σ (t) =

|t | t

ift = 0

0

otherwise .

(7)

And, we also consider rd, the space of rapidly decreasing complex sequences in Definition 1 with the topology induced by the family of seminorms (ρk )k∈N in

Random Tempered Distributions on Separable LCAG

157

Theorem 3 which, we recall, are defined for r = (rn )n≥1 ∈ rd by the following: ρk (r) =

+∞ 

(1 + n)k |rn | .

n=1

Then, for every t ∈ [0, 1], the map from rd to rd which associates to each sequence r = (rn )n≥1 ∈ rd, the sequence w = (wn )n∈N0 defined by ∀n ∈ N0 wn = radn−1 (t) rn , is a homeomorphism such that ∀k ∈ N ρk (w) = ρk (r) . As a consequence of this observation and of the expression of the continuity of ΛT , given by (6) and of Parseval formula, we have that (∃k ∈ N, ck > 0) (∀φ ∈ S), (∀t ∈ [0, 1]) , ++∞     E  radn−1 (t)(t)sn (T )rn (φ) ≤ ck sup (1 + n)k |rn (φ)| .   n∈N0

(8)

n=1

In the third step of the proof, we will show that the left-hand side of the inequality in (8) can be replaced by the following expression: ⎡  12 ⎤ +∞  |sn (T )|2 |rn (φ)|2 ⎦ . E⎣ n=1

In order to do as stated, we observe that, as the sequence (rn (φ))n∈N is rapidly decreasing and, almost surely, (sn (T ))n∈N is a sequence of slow growth, we have almost surely +∞ 

|sn (T )|2 |rn (φ)|2 < +∞ .

n=1

Now, by the standard inequality for Rademacher functions, namely, KhintchineKahane inequality (see [41, p. 213]), we have almost surely for some constant c +∞  n=1

 12 |sn (T )| |rn (φ)| 2

2



   

+∞ 1 

≤ c 0

n=1

   radn−1 (t) sn (T )rn (φ) dt . 

158

M. L. Esquível and N. P. Krasii

To conclude as desired, it is enough to apply Fubini theorem, to get for k, ck , and φ as in (8) ⎡  12 ⎤ +∞  |sn (T )|2 |rn (φ)|2 ⎦ ≤ c ck sup (1 + n)k |rn (φ)| . E⎣

(9)

n∈N0

n=1

In the fourth step, we remark that the left-hand side of the inequality, in (9), can be replaced by ++∞ , √ E αn |sn (T )| |rn (φ)| , n=1

where (αn )n∈N0 is an arbitrary sequence of strictly positive numbers such that  n=+∞ αn = 1. This statement results from the fact that, for almost every ω ∈ Ω, n=1 the expression +∞ 

|sn (T )|2 |rn (φ)|2 =

n=1

+∞ 

αn |sn (T )|2

n=1

|rn (φ)|2 , αn

can be considered as an integral over N0 , of the function defined by |sn (T )|2

∀n ∈ N0

|rn (φ)|2 , αn

with respect to the measure over (N0 , P(N0 )) that puts a√mass αn on each integer n. Applying Jensen inequality, with the convex function − x on the interval [0, +∞[, we have that a.s. +∞ 

 12 |sn (T )|2 |rn (φ)|2

≥

n=1

+∞  n=1

|rn (φ)| αn |sn (T )| √ . αn

As a consequence, for k, ck , and φ as in (8), E

++∞ √

, αn |sn (T )| |rn (φ)|

≤ c ck sup (1 + n)k |rn (φ)| .

(10)

n∈N0

n=1

√ This expression shows that the sequence (E[|sn (T )| αn ])n∈N0 is of slow growth at infinity. In order to conclude now, it is enough to consider, for instance, the sequence (αn )n∈N0 defined by αn = 1/n2 A with A = π 2 /6. This sequence satisfies the made in the fourth step, and it is clear that if with this

hypothesis  √  sequence, E |sn (T )| αn n∈N , is a sequence of slowly increasing at infinity, then 0

Random Tempered Distributions on Separable LCAG

159

(E [|sn (T )|])n∈N0 is also slowly increasing, thus showing that the random Schwartz distribution T is in the class Cm .   As a consequence of this theorem, we can now formulate the result which gives a characterization of random Schwartz distributions having a mean or a first moment. Theorem 11 Let T be a measurable random Schwartz distribution. Then, T has a first moment if and only if the sequence (sn (T ))n∈N0 of its coefficients defined in Remark 2 is in the class Cm . Proof That the last condition is sufficient was already shown in Theorem 9. The condition is necessary as a consequence of Theorem 10.   Remark 3 This last result can also be read as a characterization of the stochastic processes with a first moment which have as trajectories tempered distributions. In fact, let X be a generalized stochastic process (see [15, p. 115]). This will mean for us, according to the reference quoted, that X = (Xφ )φ∈S is a family of random variables on a probability space (Ω, A, P) such that, for all ω ∈ Ω, the map from S into C, given by X. (ω), is a tempered distribution. We can then consider that X has, as trajectories, tempered distributions. Obviously, such an object defines a random distribution, in the sense, we have been using, and moreover, the sequence (sn (X))m∈N is a sequence of random variables. As a consequence, Theorem 11 can be applied giving the characterization mentioned.

3 Derivatives of Tempered Generalized Functions on Some LCA Groups The main purpose of this section is to show how to recover previous results on random Schwartz tempered distributions of the real line and the torus with the formalism here presented. Moreover, we will show that we can define Schwartz tempered distributions—and consequently random Schwartz tempered distributions also—over the Cantor group, and so, we may extend the theory to a nonclassical setting. Let (en )n∈N be a complete orthonormal sequence in L2 (G). Usually, distribution theory is associated to the possibility of extending the usual notion of derivative to wider spaces. We next define a notion of derivative that covers the classical derivation operators and allow the definition of new ones. Definition 5 An operator D : D ⊂ L2 (G) ,→ L2 (G) defined on a vectorial space domain S ⊆ D is a derivative operator on L2 (G) if the following conditions are verified: 1. The elements of (en )n∈N are eigenvectors of D with real eigenvalues—that is, D(en ) = λn en and λn ∈ R—and the operator D acts on the basis elements with controlled growth, that is, ∃α ∈]0, +∞[ , ∀n ∈ N |D(en )| ≤ (1 + n)α |en | .

(11)

160

M. L. Esquível and N. P. Krasii

2. The operator is linear over the domain D: ∀φ1 , φ2 ∈ D ∀λ1 , λ2 ∈ C D(λ1 φ1 + λ2 φ2 ) = λ1 D(φ1 ) + λ2 D(φ2 ) . Remark 4 The restriction α ∈]0, +∞[ in Expression (11) allows for derivatives of fractional order which are important in applications. Theorem 12 A derivative operator D on D ⊆ L2 (G) admits an extension to S

that verifies, for all T ∈ S , ∀φ ∈ S , < D(T ), φ >=< T , D(φ) > . Proof If T is represented by (sn (T ))n∈N and φ is represented by (rn (φ))n∈N , as D(T ) is represented by (λn sn (T ))n∈N , we have, by Theorem 5, that < D(T ), φ >=

+∞ 

λn sn (T )rn (φ) =< T , D(φ) > .

n=1

as D(φ) is represented by (λn rn (φ))n∈N which is a rapidly decreasing sequence.   We consider next three instances of LCA separable groups and of derivative operators in these groups. For simplicity, we presented only the unidimensional torus and the real line. The presentation of the multivariate case is straightforward at the price of a more cumbersome notation.

3.1 The Real Line We consider the additive LCA group (R, λ) with λ the Lebesgue measure, that is, our chosen Haar measure. We follow [39, p. 260–263] for a brief summary of some needed results. For L2 (R), there is an orthonormal complete sequence of Schwartz functions and the Hermite functions. The Hermite polynomials on R are the polynomials Hm (x), for m ∈ N, defined by the following equations: √ 1 m d m −2πx 2 2 m e = (−1) m! 2m− 4 π 2 Hm (x) e−2πx . dx m Associated with these polynomials are the Hermite functions given, for m ∈ N, by Hm (x) = Hm (x)e−πx . 2

We may immediately observe that the family (Hm )m∈N is an orthonormal set of functions in L2 (R) and that it is complete by recovering the representation for the

Random Tempered Distributions on Separable LCAG

161

elements of L2 (R), as a series of Hermite functions by means of the special form of the Fourier transform of the Hermite functions. Theorem 13 The Fourier transform of a Hermite function is given by F(Hm )(ξ ) = (−1)m Hm (ξ ). For every φ ∈ L2 (R), the following equality in the L2 sense holds: φ=

+∞ 

am (φ) Hm ,

m=0

where the coefficients (am (φ))m∈N are given by  am (φ) =

φ(x)Hm (x) dλ(x) , R

and the following Plancherel type result holds: +∞ 

 |am (φ)| =

|φ(x)|2 dλ(x) .

2

m=0

R

The Hermite functions are products of polynomials by an exponential function. By using the usual derivation, we can define natural operators given by (τ+ f )(x) = (

d f )(x) + 2πxf (x) dx

(τ− f )(x) = −(

d f )(x) + 2πxf (x) . dx (12)

for sufficiently regular functions f . The recurrence relations that hold among the Hermite functions imply the next result about the action of τ+ , τ− , and the composition τ+ ◦ τ− operating on the Hermite functions. Theorem 14 For every m ≥ 1, √ τ+ (Hm ) = 2 πm Hm−1 , τ− (Hm ) = 2 π(m + 1) Hm+1 , (τ+ ◦ τ− ) (Hm ) = 4π(1 + m)Hm .

(13)

It is now clear that D ≡ τ+ ◦ τ− satisfies the conditions of Definition 5 and so is a derivative operator over S by Theorem 12. Let us now quote a classical result of Laurent Schwartz (see [39, p. 262]). Theorem 15 A necessary and sufficient condition for T being a classical tempered distribution, that is, a continuous form over the space of rapidly decreasing functions over the line, is that the sequence (am (T ))m∈N is a slowly increasing sequence.

162

M. L. Esquível and N. P. Krasii

Theorem 15 shows that the classical tempered distributions on the line are exactly the elements of S .

3.2 The Torus We follow [22, p. 1–30] for the notations for the model of the torus that we adopt (an alternative would be to follow [39, p. 224–231]). We consider the torus to be the quotient group T = R/2πZ where 2πZ is the group of integral multiples of 2π. The Haar measure on T is the normalized Lebesgue measure, and so, we have that   1 |f (t)|2 dt < +∞ , L2 (T) = f : T ,→ C : 2π T as functions over T can be identified with periodic function over R of period 2π. The family (eint )n∈Z is a complete orthonormal system in L2 (T) which is denumerable. Following the usual notations, we have with 0 = 1 ∀n ∈ Z , φ(n) 2π

 φ(t)eint dt T

 0(n)eint in L2 (T) and that given any sequence of complex numbers that φ = n∈Z φ  (an )n∈Z such that n∈Z |an |2 < +∞, there exists φ ∈ L2 (T) such that 0 φ (n) = an for all n ∈ Z. Now, the exponential functions eint , for n ∈ Z, are also characters, and so, by defining the operator D : S ,→ S by its operation on the characters of the L2 (T) basis by D(eint ) = −i

d int e = neint . dt

we have that the operator D satisfies Definition 5. Again, Theorem I in [39, p. 225] shows—for the multivariate case—that there is a topological isomorphism between the Schwartz distributions over the torus and the space of the slowly increasing sequences, and so we, again, recover the classical case.

3.3 The Cantor Group This last example shows that the approach here presented is more general than the classical examples of the torus and the real line. A brief introduction to the Cantor group can be read in [22]. The connection with the Cantor set can be seen in [19, pp. 13,70]. A study of some other properties of this group can be found in [7]. Let us briefly summarize some basic facts. By definition, we have that C := {0, 1}N0 , and

Random Tempered Distributions on Separable LCAG

163

so, as {0, 1}, with the given that the discrete topology is a compact group, we have that C is compact by Tychonoff’s theorem and it is also a group with coordinate-wise addition modulo 2. The map Φ that to c = (cn )n∈N0 ∈ C associates Φ(c) = 2

+∞  cn , 3n n=1

is a homeomorphism onto the middle-thirds Cantor set on the line. The group C is metrizable as for c = (cn )n∈N0 , c = (cn )n∈N0 ∈ C, and the distance d(c, c ) =

2− inf{n∈N0 :cn =cn } that defines the topology of C. A canonical Haar measure on C is the Φ pulled back measure of the Lebesgue-Stieltjes measure on the Cantor set defined by the devil’s staircase Cantor function (see [6]), and it is also the product of the normalized Haar measures over each of the factors {0, 1} of C. That is, if we consider over each factor {0, 1} of C := {0, 1}N0 the probability μn such that μn ({0}) = 1/2 = μn ({1}), then a normalized Haar measure over C is given by ⊗n∈N0 μn . As a consequence, we have that each c = (cn )n∈N0 ∈ C is a sequence of independent random variables with Bernoulli distribution of parameter 1/2. To describe the dual group, we consider, for each k ∈ N, the map ωk : C ,→ {z ∈ C : |z| = 1} such that ωk (c) = 2ck+1 − 1 ∈ {−1, 1} for c = (cn )n∈N0 ∈ C. Now, for each finite choice of distinct integers 0 ≤ j1 < j2 < · · · < jk , we define, with n = 2j1 + 2j2 + · · · + 2jk , Γn = Γ2j1 +2j2 +···+2jk := ωj1 ◦ ωj1 ◦ · · · ◦ ωjk . It is clear that for c = (cn )n∈N0 ∈ C, Γn (c) = (2cj1 +1 − 1)(2cj2 +1 − 1) · · · (2cjk +1 − 1) = (−1)



cjm +1 =0,m=1,...,k

1

= (−1)

+∞

n=1 cn dn

,

and so, (Γn )n∈N0 is a sequence of characters of the group C. All characters are of this form, and (Γn )n∈N0 is an orthonormal family (see [11, 376–380]). We observe that the dual group 0 C of C is the set of sequences d = (dn )n∈N ∈ {0, 1}N such that only a finite number of the terms are equal to one. As C is compact, the set of characters 0 C are a complete orthonormal family in L2 (C). Now, we can apply the proposed construction of tempered distributions (in Sect. 2) over C. Let us define the operator D : S ,→ S by the way it evaluates on the characters, that is, ∀n ≥ 1 , D(Γn ) = nΓn .

164

M. L. Esquível and N. P. Krasii

It is clear that this operator satisfies the conditions of Definition 5 as we have that  2

L (C) = φ =

+∞ 

a n Γn :

n=1

 = T =

+∞ 

+∞ 

 |an | < +∞ ⊂ S

2

n=1



sn Γn : (sn )n≥1 ∈ si

n=1

and so, as (nsn )n≥1 ∈ si, we have that D(T ) =

+∞ 

nsn Γn ∈ S .

n=1

We have thus defined Schwartz distributions over C—a set homeomorphic to the Cantor set—despite the fact that a usual theory of differentiation over this group encounters natural difficulties (see [24] for a discretization approach). The proposed theory of random Schwartz distributions over C is now straightforward, and in future work, we will develop some aspects of this theory that can be applied to modeling on fractals (see [12] for a problem in optics) but on a random environment. Acknowledgments This work was partially supported, for the first author, through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) and, for the second author, through Russian Foundation for Basic Research (RFBR) (grant no. 19-01-00451).

References 1. Abdik, A.: Dérivation sur un groupe localement compact et abélien. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25, 269–290 (1971) 2. Akbarov, S.S.: Smooth structure and differential operators on a locally compact group. Izv. Ross. Akad. Nauk Ser. Mat. 59(1), 3–48 (1995) 3. Bohnenblust, F.: An axiomatic characterization of lp -spaces. Duke Math. J. 6(3), 627–640 (1940) 4. Bruhat, F.: Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes ℘-adiques. Bull. Soc. Math. France 89, 43–75 (1961) 5. de Nápoles, Silva distributions for certain locally compact groups. Portugal Math. 51(3), 351– 362 (1994) 6. Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24(1), 1–37 (2006) 7. Edwards, R.E.: Fourier Series. A Modern Introduction, vol. 2. Graduate Texts in Mathematics, vol. 85, 2nd edn. Springer, New York, Berlin (1982) 8. Esquível, M.L.: Sur une classe de distributions aléatoires périodiques. Ann. Sci. Math. Québec 17(2), 169–186 (1993)

Random Tempered Distributions on Separable LCAG

165

9. Esquível, M.L.: Applications of Fourier methods to the analysis of some stochastic processes. Universidade Nova de Lisboa Departamento de Matemática, Lisbon, 1996. Dissertation, Universidade Nova de Lisboa, Lisbon, 1996, Text in English and Portuguese 10. Esquível, M.L.: On the space of random tempered distributions having a mean. In: Classical Analysis (Kazimierz Dolny, 1995), pp. 13–47. Wars Agriculture University Press, Warsaw (1996) 11. Fine, N.J.: On the Walsh functions. Trans. Am. Math. Soc. 65, 372–414 (1949) 12. Golmankhaneh, A.K., Baleanu, D.: Diffraction from fractal grating cantor sets. J. Mod. Opt. 63(14), 1364–1369 (2016) 13. Gel’fand, I.M., Vilenkin, N.Ya.: Generalized Functions, vol. 4. AMS Chelsea Publishing, Providence, RI (2016). Applications of harmonic analysis. Translated from the 1961 Russian original [ MR0146653] by Amiel Feinstein, Reprint of the 1964 English translation [ MR0173945] 14. Hida, T.: Stationary Stochastic Processes. Mathematical Notes. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1970) 15. Hida, T.: Brownian Motion. Applications of Mathematics, vol. 11 . Springer, New York, Berlin (1980). Translated from the Japanese by the author and T. P. Speed 16. Havin, V.P., Nikolski, N.K., Gamkrelidze, R.V. (eds.) Commutative Harmonic Analysis II. Group Methods in Commutative Harmonic Analysis. Transl. from the Russian., vol. 25. Springer, Berlin (1998) 17. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Die Grundlehren der mathematischen Wissenschaften, Band 152. Springer, New York, Berlin (1970) 18. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, 2nd edn. Springer, Berlin, New York (1979) 19. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable. Springer, New York (1965) 20. Ito, K.: Foundations of Stochastic Differential Equations in Infinite-Dimensional Spaces. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 47. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1984) 21. Kac, G.I.: Generalized functions on a locally compact group and decompositions of unitary representations. Trudy Moskov. Mat. Obšˇc. 10, 3–40 (1961) 22. Katznelson, Y.: An Introduction to Harmonic Analysis, corrected edition. Dover Publications, Inc., New York (1976) 23. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Vol. 2: Measure. The Lebesgue Integral. Hilbert Space. Translated from the first (1960) Russian ed. by Hyman Kamel and Horace Komm. Graylock Press, Albany, NY (1961) 24. Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001) 25. Khavin, V.P., Nikol’skij, N.K., Gamkrelidze, R.V. (eds.) Commutative Harmonic Analysis I. General Survey. Classical Aspects. Transl. from the Russian, vol. 15. Springer, Berlin (1991) 26. Koan, V.-K.: Distributions, Analyse de Fourier, Opérateurs aux Dérivées Partielles, vol. II. Librairie Vuibert, Paris (1972) 27. Koan, V.-K.: Distributions, Analyse de Fourier, Opérateurs aux Dérivées Partielles, vol. I. Librairie Vuibert, Paris (1972) 28. Lacey, H.E.: The Isometric Theory of Classical Banach Spaces. Springer, New York, Heidelberg (1974). Die Grundlehren der mathematischen Wissenschaften, Band 208 29. Loomis, L.H.: An Introduction to Abstract Harmonic Analysis. D. Van Nostrand Company, Inc., Toronto, New York, London (1953) 30. Malliavin, P., Kay, L., Airault, H., Letac, G.: Integration and Probability. Graduate Texts in Mathematics. Springer, New York (2012)

166

M. L. Esquível and N. P. Krasii

31. Marcus, M.B., Pisier, G.: Random Fourier Series with Applications to Harmonic Analysis. Annals of Mathematics Studies, vol. 101. Princeton University Press, Princeton, NJ/University of Tokyo Press, Tokyo (1981) 32. Reid, G.A.: A theory of distributions for locally compact Abelian groups and compact groups. Proc. Lond. Math. Soc. (3) 16, 415–455 (1966) 33. Reiter, H.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (1968) 34. Riss, J.: Éléments de calcul différentiel et théorie des distributions sur les groupes abéliens localement compacts. Acta Math. 89, 45–105 (1953) 35. Robert, A.: Introduction to the Representation Theory of Compact and Locally Compact Groups, vol. 80. Cambridge University Press/London Mathematical Society, Cambridge/London (1983) 36. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987) 37. Rudin, W.: Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York (1990). Reprint of the 1962 original, A Wiley-Interscience Publication 38. Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, Inc., New York (1991) 39. Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966). Nouv. éd., entièrement corr 40. Schwartz, L.: Analyse Hilbertienne. Collection Méthodes. Hermann, Paris (1979) 41. Zygmund, A.: Trigonometric Series. Vol. I, II. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1979 edition

On Solutions of Stochastic Equations with Current and Osmotic Velocities Yuri E. Gliklikh

Abstract The main aim of this paper is to collect the results connected with existence of solutions of equations with current and osmotic velocities, published in several articles, and supply all constructions and results with complete proofs. Some new properties of both types of equations and their interrelation are described. Keywords Nelson’s mean derivatives · Current velocities · Osmotic velocities · Equations with current and osmotic velocities · Existence of solutions

1 Introduction Current and osmotic velocities are symmetric and antisymmetric, respectively, Nelson’s mean derivatives (see [1–3] and the details below). Note that classical Nelson’s mean derivatives give some information about the drift of a diffusion process but no information about the diffusion coefficient. In order to make it possible to recover a process from its mean derivatives, in [4] (see also [5]), we introduced an additional mean derivative called quadratic. Note that current velocities are natural analogs of ordinary physical velocities of deterministic processes and osmotic velocities in some sense measure how fast “the randomness” of a process growths up. Thus, the investigation of equations given in terms of current and quadratic velocities as well as osmotic and quadratic velocities is important for applications to mathematical physics and other branches of science. The main aim of this paper is to collect the results connected with existence of solutions of equations with current and osmotic velocities, published in several articles (mainly in [6–8] and some others) and supply all constructions and results with complete proofs. Some new properties of both types of equations and their interrelation are described.

Y. E. Gliklikh () Voronezh State University, Voronezh, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_8

167

168

Y. E. Gliklikh

We deal with equations in Euclidean space Rn . By L(Rn , Rn ), we denote the space of n × n matrices. By S+ , we denote the set of symmetric positive definite matrices. Its closure, the set of symmetric positive semi-definite matrices, we denote by S¯+ . The symbol ∂x∂ i denotes both the partial derivative (i.e., the differential operator) and the ith vector of basis in Rn . For a vector field v and a function f , the symbol vf denotes the derivative of f in the direction of v. Everywhere below, we use Einstein’s summation convention, i.e., the existence of repeated upper and lower indices means the sum with respect to such index from 1 to n.

2 Preliminaries on Mean Derivatives Here, we describe the notions and facts from the theory of mean derivatives and close to it that is necessary for our purpose. More details can be found in [3, 5]. Consider a stochastic process ξ(t) in Rn , t ∈ [0, T ] given on a certain probability space (Ω, F , P) such that ξ(t) is an L1 random element for all t. ξ Denote by symbol Nt the σ -sub-algebra in F generated by pre-images of Borel n sets in R under the mapping ξ(t) : Ω → Rn . ξ ξ The conditional expectation with respect to Nt is denoted by Et (·). Ordinary mathematical expectation is denoted by E. Recall that, strictly speaking, almost surely (a.s.) the sample paths of ξ(t) are not differentiable for almost all t. Hence, the “classical” derivatives in time of ξ(t) exist only in the sense of distributions. In order to avoid using the distributions, following Nelson (see., e.g., [1–3]), we give the next: Definition 1 (i) The forward mean derivative Dξ(t) of ξ(t) at time instant t is an L1 random element of the form   ξ ξ(t + Δt) − ξ(t) Dξ(t) = lim Et (1) Δt →+0 Δt where the limit is supposed to exist in L1 (Ω, F , P) and Δt → +0 means that Δt tends to 0 and Δt > 0. (ii) The backward mean derivative D∗ ξ(t) of ξ(t) at time instant t is an L1 random element   ξ ξ(t) − ξ(t − Δt) (2) D∗ ξ(t) = lim Et Δt →+0 Δt where the conditions and notation are the same as in (i). Note that usually, Dξ(t) = D∗ ξ(t), but if, say, ξ(t) a.s. has smooth sample paths, those derivatives evidently coincide.

On Solutions of Stochastic Equations with Current and Osmotic Velocities

169

From the properties of conditional expectation (see [9] ), it follows that Dξ(t) and D∗ ξ(t) can be represented via superpositions of ξ(t) and Borel measurable vector fields (regressions)  Y 0 (t, x) = lim E Δt →+0

 Y∗0 (t, x) = lim E Δt →+0

ξ(t + Δt) − ξ(t) | ξ(t) = x Δt ξ(t) − ξ(t − Δt) | ξ(t) = x Δt

  (3)

on Rn . This means that Dξ(t) = Y 0 (t, ξ(t)) and D∗ ξ(t) = Y∗0 (t, ξ(t)). Definition 2 The derivative DS = 12 (D + D∗ ) is called symmetric mean derivative. The derivative DA = 12 (D − D∗ ) is called the antisymmetric mean derivative. Consider the vector fields v ξ (t, x) =

 1 0 Y (t, x) + Y∗0 (t, x) 2

and uξ (t, x) =

1 0 Y (t, x) − Y∗0 (t, x)v). 2

Definition 3 v ξ (t) = v ξ (t, ξ(t)) = DS ξ(t) is called the current velocity of ξ(t); uξ (t) = uξ (t, ξ(t)) = DA ξ(t) is called the osmotic velocity of ξ(t). For stochastic processes, the current velocity is a direct analog of ordinary (“physical”) velocity of deterministic processes (see, e.g., [1–3, 5]). The osmotic velocity measures how fast “the randomness” is growing up. Following [4] (see also [5]), we introduce the differential characteristic D2 of ξ(t) by the formula  D2 ξ(t) = lim

)t →+0

ξ Et

 (ξ(t + )t) − ξ(t)) ⊗ (ξ(t + )t) − ξ(t)) , )t

(4)

where the limit is supposed to exist in L1 (Ω, F , P). On the basis of properties of Itô integrals, it is shown that after taking the limit, D2 ξ(t) becomes a symmetric positive semi-definite matrix function on [0, T ] × Rn . Definition 4 D2 is called the quadratic mean derivative. Below, it is convenient to use the following definition of diffusion process.

170

Y. E. Gliklikh

Definition 5 By diffusion process, we call a strong solution of a stochastic differential equation in Itô form 

t

ξ(t) = ξ0 +



t

a(s, ξ(s))ds +

0

A(s, ξ(s))dw(s)

(5)

0

where a(t, x) and A(t, x) are C ∞ -smooth mappings from [0, T ] × Rn , to Rn , to L(Rn , Rn ), respectively. Theorem 1 ([4, 5]) For diffusion process (5), the forward mean derivative Dξ(t) exists and takes the form Dξ(t) = a(t, ξ(t)). Theorem 2 ([4, 5]) For diffusion process (5), the quadratic mean derivative D2 ξ(t) exists and takes the form D2 ξ(t) = α(t, ξ(t)) where α = AA∗ is the diffusion coefficient.

3 The Formulae for Current and Osmotic Velocities Let a C ∞ -smooth nonautonomous (2, 0)-tensor field with values in S¯+ (n) be given on Rn . In other words, it is a C ∞ -smooth nonautonomous field of symmetric positive definite matrices α(t, x) = (α ij (t, x)). Since all the matrices are nondegenerate and the field is C ∞ -smooth, there exists unique C ∞ -smooth field of symmetric positive definite inverse matrices (αij (t, x)) ((0, 2)-tensor field). The latter field generates a new time-dependent Riemannian metric αt (·, ·) = αij (t, x)dx i ⊗ dx j on Rn . Note that αt (·, ·) can be also considered as the restriction on the level surface t = const of the Riemannian metric α(·, ¯ ·) on R×Rn with the matrix of metric tensor at the point (t, x) ∈ R×Rn in the form   1 0 . (α¯ kl (t, x)) = 0 (αij (t, x)) Note that the determinants of the matrices (α¯ kl (t, x)) and (αij (t, x)) coincide. It is well-known that the volume form of metric αt (·, ·) on the level surface t = const is described as = Λαt = det(αij (t, x))dx 1 ∧ dx 2 ∧ · · · ∧ dx n , and so the volume form of α(·, ¯ ·) is described as Λα¯ =

= det(αij (t, x))dt ∧ dx 1 ∧ dx 2 ∧ · · · ∧ dx n = dt ∧ Λαt .

For simplicity for the latter form, we shall use the notation dt ∧ Λαt .

On Solutions of Stochastic Equations with Current and Osmotic Velocities

171

Below, we deal with C ∞ -smooth fields of nondegenerate linear operators A(t, x) : R × Rn → Rn and t ∈ R, x ∈ Rn (i.e., with C ∞ -smooth nonautonomous (1, 1)-tensor fields on Rn ). For example, let ξ(t) be a diffusion process (5) where A(t, x) is C ∞ -smooth and nondegenerate. Then, its diffusion coefficient A(t, x)A∗(t, x) is a C ∞ -smooth field of symmetric positive definite matrices α(t, x) = (α ij (t, x)) ((2, 0)-tensor field on Rn ), i.e., on its basis, we can construct the Riemannian metric αt (·, ·) as above. Note that the transitions from the field (α ij (t, x)) as above to the field A(t, x) is also possible. Lemma 1 ([4, 5]) Let α(t, x) be jointly continuous (measurable, smooth) mapping from [0, T ] × Rn to S+ (n). Then, there exists jointly continuous (measurable and smooth, respectively) mapping A(t, x) from [0, T ] × Rn to L(Rn , Rn ) such that for all t ∈ R and x ∈ Rn the equality A(t, x)A∗ (t, x) = α(t, x) holds. Denote by ρ(t, ˆ x) the probabilistic density of the random element ξ(t) with respect to the Lebesgue measure on R × Rn , i.e., with respect to Euclidean volume form ΛE = dt ∧ dx 1 ∧ · · · ∧ dx n . This means that for every [0, T ] ⊂ R and every bounded continuoua function f : [0, T ] × Rn → R, the following relation holds: 

T

 E(f (t, ξ(t))) dt =

0

T

0

Ω

 =

 f (t, ξ(t))dP dt



[0,T ]×Rn

f (t, x)ρ(t, ˆ x)dt ∧ ΛE .

Theorem 3 ([10, Theorem 1.1] or [11, Formula (2.8)]) Let ξ(t) be a diffusion ˆ Then, under the assumpprocess with the diffusion coefficient (a ij ) and density ρ. tion that ρˆ is smooth and nowhere equals to zero, u(t, x) =

1 2

∂ (α ij (x)ρ(t, ˆ x)) ∂x j

ρ(t, ˆ x)

∂ . ∂x i

(6)

Proof Here, we suggest the proof, alternative to [10, Theorem 1.1] and [11, formula (2.8)]. Take an arbitrary smooth function f (t, x) with compact support on Rn . Note that ξ f (t, ξ(t)) is Nt -measurable. Hence,   ξ E f (ξ(t))Et



t t −Δt

A(t, ξ(t))dw(t)

Since f (ξ(t − Δt)) and

$t

t −Δt

 E



= E f (ξ(t))



t t −Δt

 A(t, ξ(t))dw(t) .

A(t, ξ(t))dw(t) are independent and t t −Δt

A(t, ξ(t))dw(t) = 0,

172

Y. E. Gliklikh

we obtain  E(f (ξ(t))



t t −Δt

A(t, ξ(t))dw(t)

  = E (f (ξ(t)) − f (ξ(t − Δt)))



t t −Δt

A(t, ξ(t))dw(t)

.

$t By the Ito formula, f (ξ(t)) − f (ξ(t − Δt)) = t −Δt (df · a(s, ξ(s)))ds + $ $ t 1 t

2 t −Δt tr f (ξ(s))ds + t −Δt (df · A(s, ξ(s)))dw(s) (by ·, we denote the action of 1-forms on vectors). Thus,   E f (ξ(t)) =E ˜



+

t t −Δt

1 2 

+



t −Δt

t −Δt

t −Δt

A(t, ξ(t))dw(t)

(df · a(s, ξ(s)))A(s, ξ(s))dsdw(s)

t

t



t

tr f

(ξ(s))(A(s, ξ(s)), A(s, ξ(s)))dsdw(s)

 (df · A(s, ξ(s)))A(s, ξ(s))ds .

The first two integrals in the right-hand side are equal to zero. The calculations in coordinates yield the equality (df · A)A = df · (AA∗ ). On the other hand, 

T

  E f (t, ξ(t))uξ (t, ξ(t)) dt

0

1 =− 2 =− = = =

1 2 1 2 1 2

1 2  





T

E f (t, ξ(t)) lim

Δt →+0

0



T 0

T



T



0

T 0

Rn

Rn

0



E(df · AA∗ )dt = −

$t ξ t −Δt Et ( 1 2



T 0



 f · d(AA∗ · ρˆ ξ )Λαt dt f

A(s, ξ(s))dw(s)

Rn

Δt

 ) dt

df · AA∗ · ρˆ ξ Λαt

 dt

  d(AA∗ · ρˆ ξ ) ξ 1 T  d(AA∗ · ρˆ ξ )  dt ρ ˆ Λ E f dt = αt ρˆ ξ 2 0 ρˆ ξ

∂ ij ˆ ξ )  ∂  j (α ρ E f (t, ξ(t)) ∂x ξ dt. ρˆ ∂x i

On Solutions of Stochastic Equations with Current and Osmotic Velocities

173

Since it is valid for an arbitrary function f as above, this means that uξ = 1 d(AA∗ ·ρˆ ξ ) 2 ρˆ ξ

=

∂ ∂x j

1 2

(α ij ρˆ ξ ) ∂ . ρˆ ξ ∂x i

  ij

∂ Remark 1 Denote by Ξ (t, x) the vector field with coordinate presentation ∂α . ∂x j ∂x i 1 1 ξ ξ One can easily derive from (6) that u (t, x) = 2 Gradt log ρ (t, x) + 2 Ξ (t, x) where Gradt denotes the gradient with respect to the metric αt (·, ·). Indeed, ∂ ∂x j

(α ij ρ ξ (t,x))

ρ ξ (t,x) ∂α ij ∂ = ∂x j ∂x i

∂ ∂x i

= 12 α ij

∂ρ ξ ∂x j ρξ

∂ ∂x i

+

1 ∂α ij ∂ 2 ∂x j ∂x i ,

where α ij

∂ρ ξ ∂x j ρξ

∂ ∂x i

= Gradt log ρ ξ and

Ξ . Take into account that here, the gradient is found only with respect to the space variables, i.e., here, t is a parameter. Denote by symbol ρ ξ (t, x) the probability distribution density of the random element ξ(t) with respect to the volume form dt ∧ Λαt on [0, T ] × Rn , i.e., for every continuous bounded, function f : [0, T ] × Rn → R the relation 

T



T

E(f (t, ξ(t)))dt =

0

 =

 f (t, ξ(t))dP dt



0

Ω



 ξ

[0,T ]

Rn

f (t, x)ρ (t, x)Λαt dt

holds. √ Note that by the construction, ρ(t, ˆ x) = ρ(t, x) det (at (x)). Lemma 2 For v ξ (t, x) and ρ ξ (t, x) of process (5), the following relation of continuity equation type in the form ∂ρ ξ (t, x) = − Divt (ρ ξ (t, x)v ξ (t, x)), ∂t

(7)

holds where Divt denotes the divergence with respect to Riemannian metric αt (·, ·) on the level surface t = const. Proof Denote by symbol ΛE the Euclidean volume form dx 1 ∧ · · · ∧ dx n on Rn . Thus, Λαt = det(αij )ΛE . Recall that Divt (ρ ξ v ξ ) = d((ρ ξ v ξ )  Λαt ), where  is the interior product of vector (ρ ξ v ξ ) and n-form Λαt and the exterior differential d is considered on the level surface t = const, differentiation only in space variables x i . i.e., it contains n ξ ξ Then, (ρ v )  Λαt = det(αij ) i=1 (ρ ξ v ξ )i dx 1 ∧· · ·∧dx i−1 ∧dx i+1 ∧· · ·∧dx n , and so, (ρ ξ v ξ )i ∂ det(αij ) ∂(ρ ξ v ξ )i + . Divt (ρ v ) = ∂x i ∂x i det(αij ) ξ ξ

(8)

174

Y. E. Gliklikh

Specify a smooth real function f (t, x) with compact support and two numbers 0 ≤ s < t ≤ T . By symbol df , we denote the differential with respect to space ∂f i coordinates x i : df = ∂x i dx . By calculations in coordinates, we obtain  

[s,t ]×Rn

[s,t ]×Rn

  df · (ρ ξ v ξ (τ, ξ(τ ))) dτ ∧ Λαt = =   df · (ρ ξ v ξ (τ, ξ(τ ))) det(αij ) dτ ∧ ΛE = 

, ∂(ρ ξ v ξ )i ξ ξ i ∂ det(αij ) det(αij ) + (ρ v ) f (τ, x) dτ ∧ ΛE = − ∂x i ∂x i [s,t ]×Rn  + ,   ∂(ρ ξ v ξ )i (ρ ξ v ξ )i ∂ det(αij ) = f (τ, x) − + det(αij ) · ∂x i ∂x i det(αij ) [s,t ]×Rn 

+ =

· dτ ∧ ΛE =  + ,  ∂(ρ ξ v ξ )i (ρ ξ v ξ )i ∂ det(αij ) f (t, x) dτ ∧ Λαt = + − ∂x i ∂x i det(αij ) [s,t ]×Rn    f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt . − [s,t ]×Rn

By the Itô formula,   E f (t, ξ(t)) − f (s, ξ(s)) = 

t

E s

∂f dτ + ∂τ



t

df · Y 0 (τ, ξ(τ ))dτ +

s

1 2



t

tr f

(A, A)dτ



s

and by the backward Itô formula,   E f (t, ξ(t)) − f (s, ξ(s)) = 

t

E s

∂f dτ + ∂τ



t s

df · Y∗0 (τ, ξ(τ ))dτ −

1 2



t

 tr f

(A, A)dτ .

s

Hence, 



E f (t, ξ(t)) − f (s, ξ(s)) = E

 s

t

∂f dτ + ∂τ



t s

 df · v ξ (τ, ξ(τ ))dτ .

On Solutions of Stochastic Equations with Current and Osmotic Velocities

175

But 

 t  ∂f E dτ + df · v ξ (τ, ξ(τ ))dτ = s ∂τ s   ∂f  ρ ξ + [df · (ρ ξ v ξ (τ, ξ(τ )))] dτ ∧ Λαt = [s,t ]×Rn ∂τ     ∂ρ ξ  ∂  f (τ, x) (f (τ, x)ρ ξ dτ ∧ Λαt − dτ ∧ Λαt ∂τ [s,t ]×Rn ∂τ [s,t ]×Rn    − f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt = t

[s,t ]×Rn

   E f (t, ξ(t)) − f (s, ξ(s)) −  − Thus,

[s,t ]×Rn



$ [s,t ]×Rn

 [s,t ]×Rn

f (τ, x)

∂ρ ξ  dτ ∧ Λαt ∂τ

  f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt .

 

$ ξ ξ ξ f (τ, x) ∂ρ ∂τ dτ ∧ Λαt + [s,t ]×Rn f (τ, x) Div(ρ v ) dτ ∧ Λαt =

0. Since this is valid for every function f (t, x) as above, we obtain that − Divt (ρ ξ v ξ ).

∂ρ ξ ∂τ

=  

Lemma 3 The following formula takes place: Divt (ρ ξ v ξ ) = αt (v ξ , Gradt ρ ξ ) + ρ Divt v ξ

(9)

where Gradt denotes the gradient with respect to Riemannian metric αt (·, ·) on the level surface t = const. Proof In complete analogy with the derivation of formula (8), we obtain (v ξ )i ∂ det(αij ) ∂(v ξ )i + . Divt (v ) = ∂x i ∂x i det(αij ) ξ

Recall that

∂(ρ ξ v ξ )i ∂x i

ξ i

(10)

ξ

) = ρ ξ ∂(v + (v ξ )i ∂ρ . Then, from formula (8), taking into ∂x i ∂x i ξ

account also formula (10), we derive that Divt (ρ ξ v ξ ) = ρ ξ Divt v ξ + (v ξ )i ∂ρ , ∂x i ξ

where (v ξ )i ∂ρ is the derivative of density ρ ξ in the direction of vector field v ξ ∂x i on the level surface. But by the definition of gradient Gradt , the latter derivative is equal to αt (v ξ , Gradt ρ ξ ).   In the particular case of autonomous α(x), it generates the unique Riemannian metric α(·, ·) on Rn with the corresponding Div and Grad.

176

Y. E. Gliklikh

Corollary 1 If α(x) is autonomous, relation (7) takes the form ∂ρ ξ (t, x) = − Div(ρ ξ (t, x)v ξ (t, x)) ∂t

(11)

and formula (9) takes the form Div(ρ ξ v ξ ) = α(v ξ , Grad ρ ξ ) + ρ Div v ξ

(12)

4 Equations with Current Velocities As it is said above, the current velocity is a direct analog of ordinary velocity of deterministic physical processes. Thus, the case of equations with current velocities is apparently the most natural from the physical point of view. Let v : R × Rn → Rn and α : R × Rn → S¯+ (n) be Borel measurable mappings. The system of the form

DS ξ(t) = v(t, ξ(t)) D2 ξ(t) = α(t, ξ(t))

(13)

is called the first-order equation with current velocities. Definition 6 We say that (13) has a solution on the interval [0, T ], if there exists a probability space (Ω, F , P) and a stochastic process ξ(t) given on it for t ∈ [0, T ], such that it satisfies (13). Theorem 4 Let v : [0, T ] × Rn → Rn and α : [0, T ] × Rn → S+ (n) be jointly C ∞ -smooth in all variables (in particular, this means that for every t, the field α determines the time-dependent Riemannian metric αt (·, ·) on Rn , introduced in Sect. 3). Let also the following estimates take place for a certain K > 0: v(t, x) < K(1 + x),

(14)

tr α(t, x) < K(1 + x2 )

(15)

|Ξ (t, x)| < K(1 + x).

(16)

Let ξ0 be a random element with values in Rn whose probability distribution density ρ0 with respect to the volume form Λα0 of metric α0 (·, ·) on Rn (see Sect. 3) is C ∞ -smooth and nowhere equal to zero. Then, for the initial condition ξ(0) = ξ0 , Eq. (13) has a solution well posed on the entire interval t ∈ [0, T ] that is unique as a diffusion process.

On Solutions of Stochastic Equations with Current and Osmotic Velocities

177

Proof Since v(t, x) is C ∞ -smooth and estimate (14) is valid for it, its flow gt is well-defined on the entire interval t ∈ [0, T ]. By symbol gt (x), we denote the orbit of this flow (i.e., the solution of equation x (t) = v(t, x)) with initial condition g0 (x) = x. Since v(t, x) is C ∞ -smooth, its flow is C ∞ -smooth as well. By the use of Lemma 3, we transform equation of continuity (7) to the form ∂ρ(t, x) = −αt (v, Gradt ρ) − ρ Divt v. ∂t

(17)

Suppose that ρ(t, x) is nowhere equal to zero in [0, T ] × Rn . Then, we can divide (17) by ρ and so transform it to the form ∂p = −αt (v, Gradt p) − Divt v ∂t

(18)

where p = log ρ. Introduce p0 = log ρ0 . Show that the solution of Eq. (18) with initial condition p(0) = p0 is described by the formula  p(t, x) = p0 (g−t (x)) −

0

t

(Divτ v)(s, gs (g−t (x)) ds.

(19)

Introduce the product [0, T ] × Rn , and consider function p0 as given on the level surface t = 0. Consider the vector field (1, v(t, x)) on [0, T ] × Rn . The orbits of its flow gˆ t , starting at the points of surface t = 0, take the form gˆt (0, x) = (t, gt (x)), and the flow gˆt is C ∞ -smooth as well as gt . Consider on [0, T ]×Rn the Riemannian metric α(·, ¯ ·) (see Sect. 3). Note that for every (t, x), the point gˆ−t (t, x) belongs to the level surface t = 0, where the function p0 is given. Thus, on the one hand, (1, v)p(t, x) (the derivative of function p(t, x) in the direction of vector field (1, v)) equals − Divt v(t, x) by definition. But on the other hand, one can easily calculate that (1, v)p(t, x) = ∂t∂ p(t, x) + αt (v(t, x), Gradt p(t, x)). So, (18) is satisfied. We emphasize that ρ = ep is indeed nowhere equal to zero, and so, our arguments are correct. Recall that the process having density ρ and diffusion coefficient α has the osmotic velocity u, uniquely and independently of v calculated by formula (6). Hence, the process having current velocity v and diffusion coefficient α (and consequently density ρ) has the drift (forward mean derivative) a(t, x) = v(t, x) + √ u(t, x) = v(t, x)+ 12 Gradt p+ 12 Ξ = v(t, x)+Gradt log ρ + 12 Ξ (see Remark 1), uniquely determined by this formula. Thus, the process having current velocity v and diffusion coefficient α is a solution of the following stochastic differential equation: 

t

ξ(t) = ξ0 + 0

 a(s, ξ(s))ds +

t

A(s, ξ(s))dw(s), 0

(20)

178

Y. E. Gliklikh

where C ∞ -smooth field A(t, x) such that A(t, x)A∗ (t, x) = α(t, x) exists by Lemma 1. Since the coefficients of this equation are C ∞ -smooth and estimates (14), (15), and (16) are fulfilled, from results of [12], it follows that this equation has a unique strong solution that is well defined on the entire interval [0, T ].   Lemma 4 Let v(t, x), α(x), and Λα be the same as in Theorem 4 but α(x) be autonomous. Let ρ(t, x) be the smooth nowhere equal to zero probability density whose existence is proved in Theorem 4. Then, the flow of vector field (1, v(t, x)) on R×Rn preserves the volume form ρ(t, x)dt ∧Λα , i.e., L(1,v(t,x))ρ(t, x)dt ∧Λα = 0, where L(1,v(t,x))ρ(t, x)dt ∧ Λα is the Lie derivative of the form ρ(t, x)dt ∧ Λα in the direction of vector field (1, v(t, x)). p(t,x) . Recall that ρ(t, x)dt∧ Proof Introduce p(t, x) = log ρ(t, x), i.e., ρ(t, x) = e Λα = ρ(t, x) det(aij )(x)dt ∧ ΛE . Since ρ(t, x) det(aij )(x)dt ∧ ΛE is a volume form,

= =   L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = d (1, v)  ρ(t, x) det(aij )(x)dt ∧ ΛE , where  denotes the inner product of the vector (1, v) and the form ρ(t, x) det(aij )(x)dt ∧ ΛE (see [13]). From the latter formula, one can easily derive that = L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 

=  =  ∂ρ + vρ + ρ div v det(aij )(x)dt ∧ ΛE + v det(aij )(x) ρdt ∧ ΛE = ∂t (21) 

 = = ∂p + vp + div v + v log det(aij )(x) ρ det(aij )(x)dt ∧ ΛE . ∂t

But div v+v log det(aij )(x) = Div v, and vp is the derivative of p in the direction of v that by definition equals α(Grad p, v), where Grad is the gradient with respect to the metric α(·, ·). Thus, = L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE =

∂p ∂t

=  + Div v + α(Grad p, v) ρ(t, x) det(aij )(x)dt ∧ ΛE .

(22)

On the other hand, it is easy to calculate that Div(ρv) = ρ v + det(aij ) + v ρ + ρ div v. The continuity equation (11) is transformed to the form ∂ρ ∂t =

On Solutions of Stochastic Equations with Current and Osmotic Velocities

179

−α(Grad ρ, v) − ρ Div v, i.e., after dividing by ρ to the form ∂p = −α(Grad p, v) − Div v. ∂t On substituting this relation for

∂p ∂t

to (22), we obtain that

= L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 0.   Corollary 2 Let α(x) and Λα be the same as in Lemma 4 and ρ(t, x) be a certain smooth nowhere equal to zero probability density. Let a smooth vector field v(t, x) on Rn be such that L(1,v) dt ∧ Λα = 0 (i.e., (1, v) preserves the form dt ∧ Λα on R × Rn ). Then, ρ(t, x) and v(t, x) satisfy continuity-type equation (11). Proof Introduce p = log ρ. Use the arguments as at the end of the proof of Lemma 4, applying them in opposite direction. This allows us to derive from L(1,v)dt ∧ Λα = L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 0 that ∂p ∂t = −α(Grad p, v) − Div v. Replace in the latter relation p by log ρ. Recall that by definition, α(Grad log ρ, v) = v log ρ (the derivative of log ρ in the direction of v). ∂p So, α(Grad log ρ, v) = vρ ρ , and since vρ = α(Grad ρ, v), ∂t = −α(Grad p, v) − ∂ρ

Div v transforms into ∂tρ = − ρ1 α(Grad ρ, v) − Div v. After multiplying by ρ and taking into account (12), we obtain (11).  

5 Equations with Osmotic Velocities Consider C ∞ -vector field u(t, x) and C ∞ -smooth (2, 0)-tensor field of symmetric positive definite matrices α(x) = (α ij )(x) ON Rn . Since the field (α ij )(x) is C ∞ smooth and positive definite, the field of inverse matrices (αij ) exists, and it can be considered as a Riemannian metric on Rn (see Sect. 3). To avoid some technical difficulties, we suppose that u(t, x), α(x) and Ξ (x) (see Remark 1) are uniformly bounded with respect to the corresponding norms. Definition 7 The system

DA ξ(t) = u(t, ξ(t)) D2 ξ(t) = α(ξ(t))

is called the first-order equation with osmotic velocities. By technical reasons, here, we consider autonomous field α(x).

(23)

180

Y. E. Gliklikh

Definition 8 System (23) has a solution on the interval [0, T ], if there exists a probability space (Ω, F , P) and a stochastic process ξ(t) given on it for t ∈ [0, T ] and having values in Rn , such that a.s. (23) holds. Taking into account formula (6) and Remark 1, one can easily see that (23) can have solutions not for all right-hand sides and initial values since u and α should ˆ x)) with probabilistic be connected by formula u(t, x) = 12 (Ξ (x) + Grad log ρ(t, density ρ(t, ˆ x) with respect to the Lebesgue measure on Rn for any specified t (see (6)). In particular, this relation should hold for the initial density ρ0 (smooth and nowhere equal to zero), initial value of the osmotic velocity u(0, x), and the quadratic velocity α(x). Lemma 5 For any smooth u(t, x) and smooth positive definite α(x) for any specified t, there exists a positive function ρ(t, ˆ x), given up to a multiplicative constant, such that (6) holds. Proof Introduce pˆ(t ) (x) = log ρ(t, ˆ x) (hence, ρ(t, ˆ x) = epˆ (t) (x) ). Now, we can 1 transform (6) into the form u(t, x) = 2 Grad pˆ (t )(x) + 12 Ξ (x), where Ξ (x) is known since its coordinates consist of the first partial derivatives of the known smooth field α(x). Recall that Grad is given only by space variables and t here is only a parameter. Note that u(t, x) is known from the hypothesis. So, Grad p(t, ˆ x) = 2u(t, x)−Ξ (x). Denote the ith coordinate of vector 2u(t, x)−Ξ (x) by (2u(t, x) − Ξ (x))i and the j th coordinate of 1-form d pˆ(t ) (differential of pˆ (t ) ; here, d is the exterior differential) by (d pˆ (t ))j . Then, (d pˆ (t ))j = αij (2u(t, x) − Ξ (x))i , and so, d pˆ (t ) is known. Emphasize that d pˆ(t ) is given for any specified t and that d pˆ (t ) is smooth in t by construction. By the standard method, it is possible to recover pˆ(t ) from d pˆ (t ) on Rn up to an additive constant. For this, specify a certain value of pˆ(t ) at a certain point x ∈ Rn for a certain specified t. For simplicity, let x be the origin in Rn , and let the value of pˆ (t ) at x be zero. Construct the value of pˆ(t ) (x) at every x ∈ Rn for any specified t by the following way. Let σ (s) be a curve connecting the origin with x. Determine $ pˆ (t )(x) = σ d pˆ (t ) . Take another curve σ1 (s) connecting the origin with x. Change the orientation on σ1 „ and consider the union of σ and σ1 as the boundary ∂Θ of an arbitrary specified two-dimensional sub-manifold Θ in Rn . Then, ? $$ ? by the Stokes theorem (see, e.g., [14]), ∂Θ d pˆ (t ) = Θ dd pˆ (t ). Since d 2 = 0, ∂Θ dp(t) = 0, and so, the constructed value of pˆ(t ) (x) does not depend on the choice of curve σ . In particular, it is possible to construct p(t, ˆ x) by using the abovementioned integrals along the straight rays starting from the origin. By the construction, ρˆ = epˆ is positive for all (t, x). Since pˆ is defined up to an additive constant, ρˆ is defined up to a multiplicative constant.   Note that among the functions ρ(t, ˆ x) constructed in Lemma 5 for any$specified t, ˆ E= there should exist a probabilistic density on Rn , i.e., a function such that Rn ρΛ 1. Since the functions ρ(t, ˆ x) are constructed up to multiplicative constant, if for any $ t the integral Rn ρΛ ˆ E has finite value for some function, it has finite values for all other functions. In this case, there exists a function, for which the integral equals to 1. But if the integral has infinite value (the integral does not exist) for a certain

On Solutions of Stochastic Equations with Current and Osmotic Velocities

181

function, it does not exist for all others. In this case, the problem of existence of solution to Eq. (23) is not well posed. We can formulate a condition, under which this problem is well posed, and moreover, it is possible to prove the existence of solution. Condition 1 Among the function ρ(t, ˆ x), whose existence is shown in Lemma 5 up to a multiplicative constant, there exists a function that is a probabilistic density with respect to ΛE on Rn for every t ∈ [0, T ], and this function is C ∞ -smooth jointly in all variables. If we know ρ(t, ˆ x) from Condition 1, we can find the corresponding density ρ(t, x) with respect to Λα by formula ρ(t, ˆ x) = ρ(t, x) det(αij (x)). In particular, with the same system of arguments as above, the fields u(0, x) and α(x) determine unique ρˆ0 (see formula (6)) and so unique ρ0 that is the initial density of the only initial value ξ0 under which the solution can exist. Remark 2 Note that density ρ does not determine the process uniquely; for unique determination of the process, we need the measure on the space of sample paths. For given ρ and (α ij ), the osmotic velocity u(t, x) is uniquely defined by formula (6). From Corollary 2, one can easily see that any vector field v(t, x) such that L(1,v(t,x))ρ(t, x)dt ∧ Λα = 0 is a current velocity of a certain process with density ρ and diffusion coefficient (α ij ), since for this v and given ρ equality (11) holds. Theorem 5 Let the vector field u(t, x) be smooth and uniformly bounded and α(x) be the smooth and uniformly bounded autonomous field of symmetric positive definite matrices. Let also Condition 1 be fulfilled. Then, for the initial condition ξ(0) = ξ0 , there exists a solution of Eq. (23), and this solution is not unique. Proof Since Condition 1 is fulfilled, the density ρ(t, ˆ x) with respect to ΛE is found, and so, the density ρ(t, x) with respect to dt ∧ Λα is found as well by formula ρ(t, ˆ x) = ρ(t, x) det(αij (x)). The function ρ(t, x) allows us to find at least one vector field v(t, x) such that it is a current velocity of a certain possible solutions that we are looking for. We will find it by using the fact that by Remark 2, this field should satisfy the relation L(1,v)ρ(t, x)dt ∧ Λα = 0. Recall that ρ(t, x)dt ∧ Λα = ρ(t, x) det(aij )(x)dt ∧ ΛE . By formula (21), = L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 

 = = ∂p + vp + div v + v log det(aij )(x) ρ det(aij )(x)dt ∧ ΛE ∂t

and if ∂p ∂t + vp + div v + v log det(aij )(x) = 0, by Remark 2, v(t, x) is the current velocity of a certain solution. We look for such a vector field among the vector fields that have only one nonzero coordinate (say, the first one) while all other coordinates are equal to zero. In this

182

Y. E. Gliklikh

case, the latter formula takes the form ∂ log det(aij )(x) ∂p ∂p ∂v 1 + v1 1 + 1 + v1 = 0. (24) ∂t ∂x ∂x ∂x 1 √ ∂ log det(aij )(x) ∂p Note that in (24), the functions ∂p , , and are known. So, (24) ∂t ∂x 1 ∂x 1 for every specified t can be considered as a first-order ordinary differential equation with respect to v 1 . Determine the initial condition v01 for this equation as a smooth real valued function on the subspace x 1 = 0 in Rn , and use this initial condition for all t. The obtained solution v 1 (t, x 1 ) depends on the point in the subspace x 1 = 0, from which it started, and the value of the initial condition at this point, as on the parameters. Set the other coordinates equal to zero. As a result, the vector field v(t, x) on Rn is constructed that is the current velocity of a certain solution of (23). Now, find the forward mean derivative by the formula a(t, x) = v(t, x)+u(t, x). It follows from Lemma 1 and from the hypothesis of Theorem that there exists a smooth and uniformly bounded A(x) such that A(x)A∗ (x) = α(x). Then, from the general theory of equation with forward mean derivatives, it follows that ξ(t), having the density ρ(t, x), as above, must satisfy the following stochastic differential equation in Itô form  ξ(t) = ξ0 + 0

t



t

a(s, ξ(s))ds +

A(s, ξ(s))dw(s).

(25)

0

From the hypothesis and the results of [12], it follows that (25) with the initial condition ξ0 has a unique strong solution ξ(t), well defined on t ∈ [0, T ]. The fact that DA ξ(t) = u(t, ξ(t)) and D2 ξ(t) = α(ξ(t)) follows from the construction. It is clear that the current velocity of the type constructed above can be found among the vector fields having another nonzero coordinate. That is why this solution is not unique.   Acknowledgement The research is supported by Russian Foundation for Basic Research (RFBR) Grant No. 18-01-00048

References 1. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 50, 1079–1085 (1966) 2. Nelson, E.: Dynamical theory of Brownian motion. Princeton University Press, Princeton (1967) 3. Nelson, E.: Quantum Fluctuations. Princeton University Press, Princeton (1985) 4. Azarina, S.V., Gliklikh, Yu.E.: Differential inclusions with mean derivatives. Dyn. Syst. Appl. 16(1), 49–71 (2007) 5. Gliklikh, Yu.E.: Global and Stochastic Analysis with Applications to Mathematical Physics. Springer, London (2011)

On Solutions of Stochastic Equations with Current and Osmotic Velocities

183

6. Azarina, S.V., Gliklikh, Yu.E.: On existence of solutions to stochastic differential equations with current velocities. Bulletin of the South Ural State University. Ser. Math. Model. Programm. Comput. Softw. 8 (4), 100–106 (2015) 7. Azarina, S.V., Gliklikh, Yu.E.: On the solvability of nonautonomous stochastic differential equations with current velocities. Math. Notes. 100(1), 3–10 (2016) 8. Gliklikh, Yu.E. On solvability of stochastic differential equations with osmotic velocities. Probabil. Theory Appl. 65, 806–817 (2020) 9. Parthasarathy, K.R.: Introduction to Probability and Measure. Springer, New York (1978) 10. Cresson, J., Darses, S.: Stochastic embedding of dynamical systems. J. Math. Phys. 48, 072703-1–072303-54 (2007) 11. Haussmann, U.G., Pardoux, E.: Time reversal diffusions. Ann. Probabil. 14(4), 1188–1206 (1986) 12. Gihman, I.I., Skorohod, A.V.: Theory of Stochalic Processes, vol. 3. Springer, New York (1979) 13. Schutz, B.F.: Geometrical methods of mathematical physics. Cambridge University Press, Cambridge (1982) 14. Sternberg, S.: Lectures on Differential Geometry. Prentice Hall, Englewood Cliffs (1964)

Stochastic Methods in Investigation of Modern Networks Vladimir A. Gorlov and Alla V. Makarova

Abstract There are lots of methods to analyze complex network systems; one of them is to use the stochastic analysis. This approach is more accurate than others, and with its help, we can get the best solution of optimization (Bander and White, Transport Sci 36(2):218–230, 2002; Bertsekas, Dynamic programming and optimal control. Athena Scientific, Belmont, 1995). In the current article, we use Kolmogorov equation to find out the optimal parameters of the system in various conditions. It also helps us to predict the productivity of the networks and to build the self-defined networks with the ability to rebuilt or restore their operation automatically. To solve this problem, it is necessary to describe all possible conditions for the system with automatic rebuilding or restoring and without these characteristics. The important part of the research is to implement the graph theory and on its base to build the mathematical model of the whole system. Then, the methods of stochastic analysis using Kolmogorov equation solve the problem of the system elements productivity. At the end of this approach, we obtain the accurate probabilities for each condition. The significance of the method in such problems lies in the fact that we can create on its base the automatic algorithm for complex networks with the vast amount of conditions. Keywords SDN · Kolmogorov equations · Markov process · Efficiency · Network · Differential equation · Probability mathematical modelling · Optimal control

1 SDN Network Without Reservation The research is supported by the RFBR (Russian Foundation for Basic Research) grant 18-01-00048. Let’s consider the SDN (Self-Defined Network) network based on principles and architecture used for building those kind of networks. Also, let the network V. A. Gorlov · A. V. Makarova () N.E. Zhukovsky and Y.A. Gagarin Air Force Academy, Voronezh, Russian Federation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_9

185

186

V. A. Gorlov and A. V. Makarova

consists of one server and two switchers. As well known, in SDN, the switchers have limited option list, and all control functions, routing and other operations, lie on SDN controller. SDN controller is a technically complex device for the network, so it’s important to keep it in online mode, and we need to minimize the errors and the faults. Reservation is one of the methods that can help us to solve the problem of the faults and unstable processing of SDN network. To estimate the efficiency of reservation, we need to build the mathematical models for both systems without reservation and with it [1, 2]. In previous study [6], we made some assumptions: If one route is offline, the system stays stable; the SDN network is unworkable if SDN controller or all switchers are offline; at the same time, only one server is fault; the restore time of controller is more than the restore time of switchers. We also defined the list of conditions for the network: the system is online; one switcher is fault; two switchers are fault; restoring switcher if the server is online; restoring two switchers if the server is online; server is offline; server and switcher are offline; server and two switchers are offline; restoring controller; controller from set of servers is unworkable.

1.1 Transfers Between Conditions in SDN To realize the processing of SDN network, it’s necessary to describe the system conditions and how it transfers from one condition to another and what it means. To answer this question, we built the spreadsheet with conditions and description (look at Fig. 1).

1.2 Modelling of SDN Without Reservation Furthermore, if we combine all these condition in one whole, we obtain the state graph for the system without reservation (Fig. 2). If we notice that we have the system with reservation, there is one new state where the system transfers to condition 10—SDN controller from the set of servers is unworkable. So, now, we can build a new state graph for the system with reservation (Fig. 3). Further using the state graph for the system without reservation, we can write the equations that will help us to define the probabilities for each state in considered Markov process [3, 4, 7–9].

Stochastic Methods in Investigation of Modern Networks

Transfers 1

1

λsw

λsw

187

Description 2 Switcher failure or two switchers failures, failure rate is equal 3

λsr 1

2

3

λsr

λsr

2

3

6

7

8 λ0

λ0

λ0 7

λsw 6 λsw

Transfer in restore condition

5 λ0

λ0

6 Server restore, switcher restore in background

λ0 9

μm 9

If we have server or switcher failure and happens server or switcher failure

6

4

9

8

Server failure

6 μ

1

5 Failures removal

μm 9

μ 1

4

Fig. 1 Transfers between conditions in SDN with description

188

V. A. Gorlov and A. V. Makarova

Fig. 2 State graph for SDN without reservation

Then, we obtain the following system of Kolmogorov differential equations: ⎧ dP1 (t ) ⎪ = −(2λsr )P1 + μP3 + μP4 + μP8 , ⎪ ⎪ dt ⎪ ⎪ dP (t ) 2 ⎪ ⎪ dt = λsw P1 − (λ0 + λsr )P2 , ⎪ ⎪ ⎪ ⎪ dP3 (t ) = λsw P1 + (λ0 + λsr )P3 , ⎪ dt ⎪ ⎪ ⎪ dP4 (t ) ⎪ ⎪ ⎨ dt = λ0 P2 − μP4 , dP5 (t ) (1) = λ0 P3 − μP5 , dt ⎪ ⎪ dP (t ) ⎪ 6 ⎪ dt = λsr P1 − (2λsw + λ0 )P6 , ⎪ ⎪ ⎪ ⎪ dP7 (t ) ⎪ = λsr P2 + λsw P6 − λ0 P7 , ⎪ dt ⎪ ⎪ ⎪ dP8 (t ) ⎪ = λsr P3 + λsw P6 − λ0 P8 , ⎪ ⎪ dt ⎪ ⎩ dP9 (t ) = λ0 P6 + λ0 P7 − λ0 P8 − μP9 . dt The solution of this system could be obtained by well-known algorithm for differential equations systems. However, we can simplify the calculation if we notice

Stochastic Methods in Investigation of Modern Networks

189

Fig. 3 State graph for SDN with reservation

that in this network, there is a stationary Markov process, so then, dPi = 0, i.e., probabilities aren’t changing through time. Now, we overwrite the left side of the equations to zero and put in the system normalization condition: n 

Pi = 1.

i=0

Considering all these remarks, we obtain the following: ⎧ ⎪ −(2λsr )P1 + μP3 + μP4 + μP8 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsw P1 − (λ0 + λsr )P2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsw P1 + (λ0 + λsr )P3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ0 P2 − μP4 = 0, λ0 P3 − μP5 = 0, ⎪ ⎪ ⎪ ⎪ λsr P1 − (2λsw + λ0 )P6 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsr P2 + λsw P6 − λ0 P7 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsr P3 + λsw P6 − λ0 P8 = 0, ⎪ ⎪ ⎪ ⎩ λ0 P6 + λ0 P7 − λ0 P8 − μP9 = 0.

(2)

190

V. A. Gorlov and A. V. Makarova

As we see from the system of differential equations written above, it can be simplified, because some of probabilities are equal. So, we have the following: ⎧ ⎪ λsw P1 − (λ0 + λsr )P2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λ0 P2 − μP4 = 0, ⎪ ⎪ ⎪ ⎨λ P − (2λ + λ )P = 0, sr 1 sw 0 6 ⎪λsr P2 + λsw P6 − λ0 P7 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ0 P6 + λ0 P7 − λ0 P8 − μP9 = 0, ⎪ ⎪ ⎩ P1 + 2P2 + 2P4 + P6 + 2P7 + P9 = 1.

(3)

To find out the solution, we involve Mathcad where we define matrix X and vector Y . Furthermore, if we have the intensity points, we obtain the accurate probabilities for each state, illustrated on Fig. 4.

2 Modelling of SDN Network with Reservation In previous section, we have considered the SDN network without reservation and built the mathematical model for this configuration. Using all links and elements of graph theory from Fig. 1, we can obtain the state graph for system without reservation [6] and state graph for the system with reservation (Fig. 3). Notice that the system with reservation has one more condition: SDN controller from the set of servers is unworkable. Further using the algorithm from [6] and the state graph for system with reservation from Sect. 1, we can write the equations that will help us to define the probabilities for each state in considered Markov process [3, 8, 9]. Therefore, we obtain the following system of Kolmogorov differential equations: ⎧ dP (t ) ⎪ ⎪ dt1 ⎪ ⎪ ⎪ ⎪ dP2 (t ) ⎪ ⎪ dt ⎪ ⎪ dP3 (t ) ⎪ ⎪ ⎪ dt ⎪ ⎪ dP4 (t ) ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎨ dP5 (t )

= −(2λsw + λsr )P1 + μP3 + μP4 + μP8 , = λsw P1 − (λ0 + λsr )P2 , = λsw P1 − (λ0 + λsr )P3 , = λ0 P2 − μP4 ,

= λ0 P3 − μP5 , dt dP6 (t ) ⎪ = λsr P1 − (2λsw + λ0 + λsr )P6 , ⎪ dt ⎪ ⎪ ⎪ dP7 (t ) ⎪ ⎪ = λsr P2 + λsw P6 − (λ0 + λsr )P7 , ⎪ dt ⎪ ⎪ dP8 (t ) ⎪ ⎪ = λsr P3 + λsw P6 − (λ0 + λsr )P8 , ⎪ dt ⎪ ⎪ ⎪ dP9 (t ) = λ P + λ P − λ P − μP + λ P 0, ⎪ 0 6 0 7 0 8 9 0 1 ⎪ dt ⎪ ⎪ ⎩ dP1 0(t ) = λ P + λ P − λ P − λ P 0. dt

sr 6

sr 7

sr 8

0 1

(4)

7.74281408640 1218*10-6

1.17642861530 2545*10-6

9.10916313551 40848*10-12

9.10916313551 40848*10-12

3.52934050088 6445*10-6

7.7428531712 94175*10-6

5.8821920816 87397*10-7

2.2773060177 35746*10-12

2.2773060177 35746*10-12

3.5293425766 84651*10-6 1

7.74281408640 1218*10-6

7.7428531712 94175*10-6

1

3.87140704320 0608*10-6

1.9357132928 23544*10-6

1

3.87140704320 0608*10-6

1.9357132928 23544*10-6

4

1

3.5293363492 97348*10-6

3.6436165059 83808*10-11

3.6436165059 83808*10-11

2.3528180272 01446*10-6

7.7427359176 77193*10-6

7.7427359176 77193*10-6

7.7427359176 77192*10-6

7.7427359176 77192*10-6

0.9999631468 2908

Fig. 4 Current probabilities for the states in SDN without reservation

9

8

7

6

5

4

3

2

1

2

0.99997206577 0406

1

0.9999765253 00732

0.5

0.9999787 55080811 9.6785908 92341742 *10-7 9.6785908 92341742 *10-7 7.7428727 13873395 *10-6 7.7428727 13873395 *10-6 2.9511082 92252383 *10-7 5.6932840 87269202 *10-13 5.6932840 87269202 *10-13 3.5293436 1458467* 10-6

5

1

3.5293342735 06462*10-6

5.6931127069 03878*10-11

5.6931127069 03878*10-11

2.9409980323 3458*10-6

7.7426968338 46117*10-6

7.7426968338 46117*10-6

9.6783710423 07647*10-6

9.6783710423 07647*10-6

0.9999586874 1808

10

1

3.5293238945 88615*10-6

2.2771689178 6334*10-10

2.2771689178 6334*10-10

5.8817510571 97453*10-6

7.7425014199 99912*10-6

7.7425014199 99912*10-6

1.9356253559 99978*10-5

1.9356253559 99978*10-5

0.9999363909 59674

15

1

3.5293135157 31758*10-6

5.1234587037 96127*10-10

5.1234587037 96127*10-10

8.8222590975 88638*10-6

7.7423060150 01899*10-6

7.7423060150 01899*10-6

2.9033647556 25712*10-5

2.9033647556 25712*10-5

0.9999140954 95552

20

1

3.52930331369 3589*10-6

9.10806640335 57052*10-10

9.10806640335 57052*10-10

1.17625221765 0559*10-5

7.74211061885 1476*10-6

7.74211061885 1476*10-6

3.87105530942 5739&10-5

3.87105530942 5739&10-5

0.99989180102 5647

24

1

3.52929483394 3107*10-6

1.31152647103 2283*10-9

1.31152647103 2283*10-9

1.41145562828 3036*10-5

7.74195430830 1015*10-6

7.74195430830 1015*10-6

4.64517258498 0608*10-5

4.64517258498 0608*10-5

0.99987396616 5514

Stochastic Methods in Investigation of Modern Networks 191

192

V. A. Gorlov and A. V. Makarova

The solution of this system could be obtained by well-known algorithm for differential equations systems. However, we can simplify the calculation if we notice that in this network, there is a stationary Markov process, so then, dPi = 0, i.e., probabilities aren’t changing through time. Now, we overwrite the left side of the equations to zero and put in the system normalization condition: n 

Pi = 1.

i=0

Thus, we obtain the following: ⎧ ⎪ −(2λsr )P1 + μP3 + μP4 + μP8 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsw P1 − (λ0 + λsr )P2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λsw P1 + (λ0 + λsr )P3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λ0 P2 − μP4 = 0, ⎪ ⎪ ⎪ ⎨λ P − μP = 0, 0 3 5 ⎪ λsr P1 − (2λsw + λ0 + λsr )P6 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λsr P2 + λsw P6 − (λ0 + λsr )P7 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λsr P3 + λsw P6 − (λ0 + λsr )P8 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪λsr P6 + λsr P7 − λ0 P8 − μP9 + λ0P1 0 = 0, ⎪ ⎩λ P + λ P + λ P − λ0P 0 = 0. sr 6 sr 7 sr 8 1

(5)

The last system of differential equations can be simplified, because some of probabilities are equal. Thereby, we have the following: ⎧ ⎪ λsw P1 − (λ0 + λsr )P2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λ0 P2 − μP4 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λsr P1 − (2λsw + λ0 + λsr )P6 = 0, λsr P2 + λsw P6 − (λ0 + λsr )P7 = 0, ⎪ ⎪ ⎪ ⎪ λ0 P6 + 2λ0 P7 − μP9 + λ0 P1 0 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ λ ⎪ sw P6 + 2λsr P7 − λ0 P1 0 = 0, ⎪ ⎪ ⎩ P1 + 2P2 + 2P4 + P6 + 2P7 + P9 + P1 0 = 1.

(6)

To find out the solution, we involve Mathcad where we define matrix X and vector Y . Moreover, if we have the intensity points, we obtain the accurate probabilities for each state, illustrated on Fig. 5.

7.74281408640 1216*10-6

7.74281408640 1216*10-6

1.17642723128 1225*10-6

9.10914706001 2178*10-12

9.10914706001 2178*10-12

3.52934050088 6445*10-6

1.38405347008 8641*10-12

7.7428531712 94174*10-6

7.7428531712 94174*10-6

5.8821886215 8984*10-7

2.2773040084 12807*10-12

2.2773040084 12807*10-12

3.5293425766 84652*10-6

3.4601377456 88239*10-13 1

3.87140704320 0608*10-6

1.9357132928 23544*10-6

1

3.87140704320 0608*10-6

1.9357132928 23544*10-6

1

0.99997206577 0406

0.9999765253 00732

0.9999787 55080811 9.6785908 92341742 *10-7 9.6785908 92341742 *10-7 7.7428727 13873395 *10-6 7.7428727 13873395 *10-6 2.9411074 27222461 *10-7 5.6932815 75550584 *10-13 5.6932815 75550584 *10-13 3.5293436 14584669 *10-6 8.6503494 52310621 *10-14 1

5.5362008548 95788*10-12

3.5293363492 297349*10-6

3.6436036463 00351*10-11

3.6436036463 00351*10-11

2.3528124912 57784*10-6

7.7427359176 77192*10-6

7.7427359176 77192*10-6

7.7427359176 77192*10-6

7.7427359176 77192*10-6

0.9999586874 1808

4

Fig. 5 Current probabilities for the states in SDN with reservation

10

9

8

7

6

5

4

3

2

1

2

1

0.5

1

8.6503036596 61937*10-11

3.5293238945 88615*10-6

5.6930875905 533343*10-11

5.6930875905 33343*10-11

2.9409893825 33247*10-6

7.7426968338 46119*10-6

7.7426968338 46119*10-6

9.6783710423 07647*10-6

9.6783710423 07647*10-6

0.9999586874 1808

5

1

3.4601011117 47004*10-11

3.5293238945 88615*10-6

2.2771488257 27233*10-10

2.2771488257 27233*10-10

5.8817164602 04763*10-6

7.7425014199 99912*10-6

7.7425014199 99912*10-6

1.9356253549 99978*10-5

1.9356253549 99978*10-5

0.9999363909 59674

10

1

7.7851817095 69678*10-10

3.5293135157 31758*10--6

5.1233908959 74683*10-10

5.1233908959 74683*10-10

8.8221812593 33107*10-6

7.7423060150 01899*10-6

7.7423060150 01899*10-6

2.9033647556 25712*10-5

2.9033647556 25712*10-5

0.9999140954 95552

15

1

1.38402416321 9997*10-10

3.52930313693 589*10-6

9.10790568139 968*10-10

9.10790568139 968*10-10

1.17623838062 337*10-5

7.74211061885 1476*10-6

7.74211061885 1476*10-6

3.87105530942 5739&10-5

3.87105530942 5739&10-5

0.99989180102 5647

20

1

1.99298541702 0808*10-10

3.52929483394 3107*10-6

1.31149869927 8956*10-9

1.31149869927 8956*10-9

1.41143570398 3217*10-5

7.74195430830 1014*10-6

7.74195430830 1014*10-6

4.64517258498 0608*10-5

4.64517258498 0608*10-5

0.99987396616 5514

24

Stochastic Methods in Investigation of Modern Networks 193

194

V. A. Gorlov and A. V. Makarova

Thus, every complex network can be simulated by mathematical modelling and applying the stochastic analysis to investigation of efficiency and reliability networks. As a result, we obtain accurate characteristics.

3 Reservation Effect Estimation To estimate the efficiency, let’s calculate the availability factors for the model with and without reservation applying the following formula: Kr (t) =

n 

Pi (t),

i=1

where n is the number of processing conditions and Pi (t) is the probability of processing condition. As we can see from Figs. 1, 2, and 3, there are five conditions for the system without reservation and eight conditions for the system with reservation. So, we obtain the availability factor for the system without reservation: Kr (t) =

5 

Pi (t)

i=1

and Kr (t) =

8 

Pi (t)

i=1

for the system with reservation. Let’s simplify the last equations because some probabilities are equal. Thus, for the system without reservation, we got the following: Kr (t) = P1 + 2P2 + 2P4 for the system with reservation: Kr (t) = P1 + 2P2 + 2P4 + P6 + 2P7 . Using Mathcad, we obtain the availability factors for all failure detection rates. Analyzing the results, we can say that, firstly, the system with reservation is more stable at the same numerical values, because the availability factor is more, and, secondly, the reservation process is less dependent on failure detection rates.

Stochastic Methods in Investigation of Modern Networks

195

References 1. Bander, J.L., White, C.C.: A heuristic search approach for a nonstationary stochastic shortest path problem with terminal cost. Transport. Sci. 36(2), 218–230 (2002) 2. Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont (1995) 3. Gihman, I.I., Skorohod, A.V.: Theory of Stochastic Processes, vol. 3. Springer, New York, NY (1979) 4. Gliklikh, Yu.E.: Global and Stochastic Analysis with Applications to Mathematical Physics. Springer, London (2011) 5. Gorlov, V.A.: Anisotropic diffusion in anisotropic Stepanov spaces. In: Bulletin of the South Ural State University. Mathematical Modelling, Programming and Computer Software (Bulletin SUSU MMCS), vol. 12(3), pp. 153–160 (2019) 6. Gorlov, V.A., Makarova, A.V.: Stochastic analysis in modelling and efficiency estimation of modern networks. Global and Stochastic Analysis Vol. 7(2) (July–December, 2020), pp.131– 137 7. Makarova, A.V.: Stochastic inclusions with forward mean derivatives having decomposable right-hand sides. Bull. South Ural State Univ. Ser. Math. Modell. Programm. Comput. Softw. (Bulletin SUSU MMCS) 12(2), 143–149 (2019) 8. Parthasarathy, K.R.: Introduction to Probability and Measure, Springer-Verlag, New York, NY (1978) 9. Yosida, K.: Functional Analysis. Springer, Berlin (1965)

Double-Barrier Option Pricing Under the Hyper-Exponential Jump Diffusion Model S. M. Grudsky and O. A. Mendez-Lara

Abstract This work is devoted to the study of the valuation problem for a two-barrier option where the stock price is modeled by a Lévy process whose characteristic function is rational (hyper-exponential jump diffusion (HEJD) case). In a standard way, this problem is reduced to a convolution equation on a finite interval, which is solved explicitly using the modified Wiener–Hopf method. On the basis of the obtained explicit formulas, an algorithm for calculating the price of the option under consideration is developed and numerically implemented. On the other hand, with the help of the theory of generalized Toeplitz operators, theorems are proved about the existence and uniqueness of the solution of the original convolution equation and whether it belongs to a certain natural class of functions associated with the Sobolev spaces. Keywords Lévy processes · Barrier options · Toeplitz operators · Wiener–Hopf method Mathematical Subject Classification (2010) Primary 47G20; Secondary 60G40, 60G51

S. M. Grudsky () Department of Mathematics, CINVESTAV–IPN, Mexico City, Mexico Regional Mathematical Center of the Southern Federal University, Rostov-on-Don, Russia e-mail: [email protected] O. A. Mendez-Lara Department of Mathematics, CINVESTAV–IPN, Mexico City, Mexico e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_10

197

198

S. M. Grudsky and O. A. Mendez-Lara

1 Introduction The pricing problem of a double-barrier option when the log stock price is modeled by the Brownian motion (classical hypothesis) is considered in [1–7]. The articles [4–7] are devoted to an approach connected with a solution of the Black–Scholes (partial) differential equation on a strip of finite width. But it should be noted that for many cases, the Brownian motion is not an adequate model for the log stock price. Therefore, in recent years, many authors have used Lévy processes as models for the log stock price [8–24]. In the papers [15–24], the problems of pricing one-barrier and two-barrier options in the case of Lévy processes are considered. The most natural way for solving the one-barrier option problem is the Wiener–Hopf method, various modifications of which are proposed in [15–21]. This method allows one to obtain explicit formulas for the price of options in the form of multiple integrals which are numerically implemented using different approximations. The situation changes significantly in the case of a two-barrier option; the matrix Wiener–Hopf equation that arises, generally speaking, does not have an explicit solution. Therefore, special cases when the explicit solution can be obtained are of great importance. The hyper-exponential jump diffusion (HEJD) model, which has a rational symbol (see [19]), is attractive. On one hand, it is well known in the theory of options [19–24], and on the other hand, it makes possible to solve explicitly the associated matrix Wiener–Hopf equation, also known as the modified Wiener–Hopf equation. Note also that the HJED can also be used as an approximation for other stochastic processes used in the modern option theory (see [24]). In the present article, we consider the problem of pricing a double-barrier option in the case where the stock price is modeled by the HEJD process. Following the monograph [8], we use the generalized Black–Scholes equation approach. That is, we reduce the original option problem to a partial pseudodifferential equation. This equation is then reduced to the Wiener–Hopf equation, which is solved explicitly taking into account the zeros and poles of the characteristic function. Based on the obtained explicit formulas, an algorithm for calculating the option price is developed and implemented numerically. In the second part of the work, a theoretical substantiation of the obtained solution is carried out. This justification is based on the method developed in [21]. Using the method of sectorial operators in the theory of generalized Toeplitz operators, the existence and uniqueness of the solution of the partial pseudodifferential equation in Sobolev spaces are proved. In Sect. 2, the necessary definitions are introduced, the partial pseudo-differential equation is formulated, and it is reduced to the convolution equation on a finite interval and then to the generalized Wiener–Hopf equation. In Sect. 3, these problems are solved for the case of the HEJD process, and the formula for the price of a double-barrier option is constructed in an explicit form. Section 4 is devoted to the numerical implementation of the obtained formulas. We analyze here the accuracy and speed of the obtained algorithm. One reduces in Sect. 5 the problem under consideration to the so-called generalized Toeplitz operator, and using the

Double-Barrier Option Pricing under the HEJD Model

199

theory of sectorial operators, one proves that this Toeplitz operator is invertible in the space L2 (R, 2s ). On this basis, in Sect. 6, the existence and uniqueness of a solution of the generalized Wiener–Hopf equation are proved, and the formula for the option price is written in terms of this solution.

2 Preliminaries The Fourier transform of a function f is defined by  eixξ f (x) dx, (F f ) (ξ ) =

∀ ξ ∈ R;

R

and the Laplace transform of a function f as  ∞ e−τ ω f (τ ) dτ, f4(ω) = (Lf ) (ω) =

∀ ω ∈ iR.

0

Let X be a Lévy process under the probability P; according to the LévyKhintchine representation # %  P eiξ x dμXt (x) = e−t ψ (ξ ) , ξ ∈ R, t ≥ 0, E P eiξ Xt = R

the function ψ P is known as the characteristic exponent of X and has the form ψ P (ξ ) =

σ2 2 ξ − iμξ + ϕ (ξ ) , 2

with  ϕ (ξ ) = −

∞ −∞

#

% eiuξ − 1 − iuξ χ(−1,1) (u)  (du) ,

 $ where  is a measure on R satisfying R max x 2 , 1  (dx) < ∞. ϕ characterizes the pure non-Gaussian component of the process. Taking the positive sign in the exponent in the Fourier transform as defined above matches the Fourier-Stieltjes transform in the definition of the characteristic exponent. Definition 1 (See [8, p. 59]) Let X be a Lévy process and q > 0; the resolvent operator U q is defined by 

∞

U q f (x) = E

e−qt f (x + Xt ) dt



0

for every measurable nonnegative function f . The Lévy process X satisfies the ACP condition if the resolvent operators have the strong Feller property, that is, for every q > 0 and f ∈ L∞ (R), U q f is continuous.

200

S. M. Grudsky and O. A. Mendez-Lara

Consider r > 0 the risk-free interest and an arbitrage-free market and it is true that the existence of the solution of the valuation problem on such market depends on the existence of a locally equivalent martingale measure (LEMM), let’s see how the representations of a Lévy process between the historic measure P and the LEMM Q are related. Definition 2 Let P and Q be measures on the space (Ω, F , {Ft }) with right continuous filtration {Ft }. Denote the restriction measures Pt = P|Ft , Qt = Q|Ft , dPt and Zt = dQ as the density of Qt with respect to Pt . If 0 < Zt < ∞ a.e, then the t measures P and Q are called locally equivalent. Definition 3 A measure Q that is locally equivalent to the measure P such that the discounted process St∗ = e−rt St is a Q-martingale is called a locally equivalent martingale measure. Since for a non-Gaussian process it is typical for infinitely many LEMMs to exist, we assume that such measure has been chosen, so we will work the Lévy process X under the LEMM Q, with characteristic exponent ψ Q . The formula of the pseudo-differential operator given the characteristic exponent ψ of the Lévy process is   (Lf ) (x) = F −1 [−ψ (−·) (F f )] (x) . Lemma 1 (See [9]) Let X be a Lévy process under the probability P and ψ P its characteristic exponent; then, /ψ P (ξ ) ≥ 0 for arbitrary ξ ∈ R.

2.1 Boundary Value Problem Let us consider a general-type barrier option with deterministic expiration date T > 0, if X is a Lévy process that satisfies the ACP-condition, the stock prices {St }t ≥0 are given by the relation St = S0 eXt where S0 is the spot price of the stock at the time of the valuation. Consider the sets D ⊂ R × (−∞, T ], D T = {x ∈ R | (x, T ) ∈ / D} , c B = {(x, t) ∈ / D | t ∈ (−∞, T )} . and the functions g D ∈ L∞ (D) ,

  g T ∈ L∞ D T ,

 g c ∈ L∞ B c .

(1)

D stands as the barrier region where the option gets terminated by reaching it before the expiration date; g D serves as a rebate paid out as a fraction of the premium. D T are the values at the expiration time where the payoff g T is paid. B c is the

Double-Barrier Option Pricing under the HEJD Model

201

region where the option remains valid before the expiration date, and g c is an early exercise compensation. Let’s use the hitting time TD of D by the process Yt % := (x + Xt % , t + t % ), and for (x, t) ∈ R × [0, T ], define the stochastic integral under a chosen measure % #$ T ∧T V (x, t) := E 0 D e−rs g c (Ys ) ds | Y0 = (x, t) # %  

(2) +E e−rTD g YTD χ{T T } D as the valuation of the option with conditions defined as above. For the set of functions with support on D, define C0 (D) as the space of continuous functions vanishing at infinity, S (D) as the space of infinitely differentiable functions vanishing at infinity faster than any negative power of |x| together with all the derivatives, and S (D) as the set of all continuous linear functionals, or distributions, on S (D). Theorem 1 ([8, p. 65]) Let X be a Lévy process satisfying the ACP condition. Then, the stochastic integral V is a bounded solution to the following boundary value problem ⎧ ⎨ [r − ∂t − L] V (x, t) = g c (x, t) , ∀ (x, t) ∈ B c ; V (x, T ) = g T (x) , ∀ x ∈ D T ; ⎩ V (x, t) = g D (x, t) , ∀ (x, t) ∈ D.

(3)

The first equation is understood in the sense of generalized functions $V , [r − ∂t − L] u% = 0, for all u ∈ S (R × R) such that supp u ⊆ B c . In the remaining of this work, we will specifically study a double-barrier knockout option with expiration date T (see [19]). Perform the change of variable τ = T −t; thus, with x = ln Sτ , the observed state (x, τ ) becomes the initial spot of the stochastic process. Let’s set two predefined values H1 < H2 as the barriers of the option; let h1 := ln H1 , h2 := ln H2 , and h = h2 −h1 . Then, for this particular option D = {0, h}×(0, ∞), B c = (0, h) ×(0, ∞), D T = (0, h), g c = g D = 0, and g T = g is a defined payoff. So, we can rewrite (3) as the pseudo-differential problem ⎧ [r + ∂τ − L] V (x, τ ) ⎪ ⎪ ⎨ V (x, 0) ⎪ V (0, τ ) ⎪ ⎩ V (h, τ )

= 0, (x, τ ) ∈ (0, h) × (0, ∞) ; = g (x) , x ∈ (0, h) ; = 0, = 0.

(4)

202

S. M. Grudsky and O. A. Mendez-Lara

2.2 The Convolution Equation Applying the Laplace transform, the problem becomes ⎧ 4 (x, ω) = g (x) , ⎨ [r + ω − L] V 4 (0, ω) = 0, V ⎩ 4 V (h, ω) = 0. Let’s define, for s ∈ R, H s (D), the subset of distributions on S (D) with norm

s $ ·H s f 2H S = D |(F f ) (ξ )|2 1 + |ξ |2 dξ < ∞. Since C∞ = H s (D), the 0 (D) problem can be written as the convolution equation  

 4 P(0,h) F −1 [r + ω + ψ (−·)] F V (x, ω) = g (x) , 4 (·, ω) ∈ H s1 (0, h), where P(0,h) is the restriction operator to the interval (0, h), V s 2 and g ∈ H (0, h), for some s1,2 ∈ R. Let’s notice that the convolution equation acts on the finite interval (0, h) with the symbol a (ξ, ω) := r + ω + ψ (−ξ ) . It is desirable to find functions with supports (−∞, 0] and [h, ∞), such that 4 (x, ω) will be continuous on the whole real line and their values on {0, h} match. LV 4 (0, ω) = V 4 (h, ω) = 0. Consider This will include the boundary conditions V 4[h,∞) such functions and denote V 4[0,h] := V 4 and 4 4(−∞,0] and V g := χ[0,h] g. Then, V the problem can be stated (∀ x ∈ R) (∀ ω ∈ C) as the convolution equation 4(−∞,0] (x, ω) + [r + ω − L] V 4[0,h] (x, ω) + V 4[h,∞) (x, ω) = 4 V g (x) .

2.3 The Modified Wiener–Hopf Equation

s Definition 4 For s ∈ R, consider the weight function 2 (ξ ) := ξ 2 + 1 (see [25]). Let 1 < p < ∞ and define Lp (R, s) as the 2-weighted Banach space with norm  φp,s :=

∞ −∞

1/p |φ (ξ )|p 2 (ξ ) (ξ ) dξ

< ∞.

Applying the Fourier transform, the convolution equation can be stated as the modified Wiener–Hopf equation (mWH equation) φ− (ξ, ω) + a (ξ, ω) φh (ξ, ω) + eihξ φ+ (ξ, ω) = G (ξ ) ,

(5)

Double-Barrier Option Pricing under the HEJD Model

203

where φh (·, ω) ∈ L2 (R, s1 ) ,

φ+ (·, ω) , φ− (·, ω) , G ∈ L2 (R, s2 ) .

(6)

The real numbers s1 ands2 will be chosen later. This problem, known as the modified Wiener–Hopf equation with symbol a, has for solution a triple (φ− , φh , φ+ ) of unknown functions such that the supports before the Fourier transform is applied are (−∞, 0], [0, h] and [h, ∞), respectively.

3 HEJD Symbols In terms of the model for the stock price, let’s focus on a specific set of the rational symbols, those of HEJD type. Let’s first introduce a wider variety of processes. Definition 5 Let − < 0 < + and ν ∈ (0, 2]. A Lévy process is called a regular Lévy process of exponential type [− , + ] and order ν > 0 if the following two conditions are satisfied: (i) The characteristic exponent admits a representation ψ(ξ ) = −iμξ + φ(ξ ), where φ is holomorphic in the strip 0ξ ∈ (Λ− , Λ+ ), is continuous up to the boundary of the strip, and admits a representation φ(ξ ) = c|ξ |ν + O(|ξ |ν1 ) O(|ξ |ν1 ) as ξ → ∞ in the strip (ii) There exist ν2 < ν and C such that the derivative of ψ in (i) admits a bound |ψ (ξ )| ≤ C(1 + |ξ |)ν2 , 0ξ ∈ [Λ− , Λ+ ]. It is well known that an RLPE satisfies the ACP condition (see [8, p. 84–85]), so the problem in the previous section is well defined for RLPEs. Grudsky [21] is devoted to RLPEs with σ = 0 and ν ∈ (0, 2). The aim of this work is the HJED symbols; this processes are RLPEs of order 2, if σ > 0; they have nontrivial Gaussian component ϕ with asymptotic behavior ϕ (ξ ) ∼ ξ . The characteristic exponent of a HEJD process is given by (see [19]) , + M 2  λˇ m σ , ξ 2 − i μξ ˇ +ξ ψ P (ξ ) = 2 ξ − i ηˇ m m=1

where σ > 0, μ ∈ R, λˇ m , and ηˇ m represent the intensity and the tail-decay behavior of the m-th jump, all λˇ m are positives, and ηˇ m is positive or negative if it is an positive or negative jump, respectively. P is the historic measure, but the stochastic integral in (2) will be under the measure Q, the LEMM associated to the Esscher transform (see [8, p. 98]), so the new characteristic exponent is given by ψ Q (ξ ) = ψ P (ξ − iθ ) − ψ P (−iθ ) ,

204

S. M. Grudsky and O. A. Mendez-Lara

where θ is the solution of ψ P (−i [1 + θ ]) − ψ P (−iθ ) + r = 0 (see [8, p. 100]). With the adjusted parameters μ = μˇ + σ 2 θ,

λm = λˇ m

ηˇ m , θ + ηˇ m

4 ηm = ηˇ m + θ,

ψ Q also has the form , + M  λm σ2 2 ψ (ξ ) = . ξ − iμξ + ξ 2 ξ − i4 ηm Q

(7)

m=1

The sign of the 4 ηm remains for positive and negative jumps. Let’s use the Wiener–Hopf factorization a (ξ, ω) = a + (ξ, ω) a − (ξ, ω), where a ± and a1± are analytic on the half-planes ± . Given that the symbol is expressed ηm . Now, we can order the parameters in terms of ψ Q (−ξ ), let’s denote ηm := −4 M {ηm }M 1 (coupled with {λm }1 ) in a manner such that from 1 to J ≤ M, the parameters are positive, and thus, corresponding to the poles on the upper half-plane and the remaining K = M − J , negative parameters correspond to the poles on the lower half-plane. Notice that a has for numerator a polynomial with two degrees more than the denominator, and by Lemma 1, /a (ξ, ω) ≥ 0; hence, both a + and a − have one − − K more zero than poles, that is, a + has poles {iηn− }K 1 ⊂  and zeros {ξk (ω)}0 ⊂ + J + − + + − K  and a has poles {iηl }1 ⊂  and zeros {ξj (ω)}0 ⊂  . So, we have the following representations for a ± and also the partial fraction decompositions a + (ξ, ω) = a − (ξ, ω) =

  − K σ 2 k=0 ξ−ξk (ω) 2 K ξ −iηn− , #n=1 % Jj=0 ξ −ξj+ (ω)   , Jl=1 ξ −iηl+

1 a + (ξ,ω)

=

1 a − (ξ,ω)

=

a + (ξ, ω) =

2 σ2

K

J

bk (ω) k=0 ξ −ξ − (ω) , k

cj (ω) j =0 ξ −ξ + (ω) , j

σ2 2

#

ρ (ξ, ω) +

K

4 bn (ω) n=1 ξ −iηn−

% ,

with ρ (ξ, ω) = ξ − d (ω), where d (ω) is determined in the operation of partial fraction decomoposition of a + and {cj (ω)}, {bk (ω)}, {4 bn (ω)} ⊂ C. Remark Without losing perspective of the dependence on ω ∈ C, we will drop its notation when possible. Divide the mWH equation by a − , the problem can now be separated according to the support of the inverse Fourier transform of the summands of the equation, and the φ− , φ+ , and φh can be expressed in terms of the unknown values  functions 

 −  + and φh iηn . In particular, the terms corresponding to the segment φ+ ξj

Double-Barrier Option Pricing under the HEJD Model

205

[0, h] yield the formula

φh (ξ ) =

 J

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

1 + a + (ξ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩−



cj j =0 ξ −ξ + G (ξ ) j ⎡

−  η− h 1 ⎤ σ 2 K 4 n n=1 bn φh iηn e 2 ξ −iηn− ⎦ ihξ   ⎣ J e + 1 − j =0 cj φ+ ξj ξ −ξj+ #     % J ihξj+ + + 1 c G ξ − φ ξ e j + j =0 j j ξ −ξ +

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

(8)

j

Once the unknown values are solved, the functions φ− , φ+ , and φh can be explicitly determined. To do so, the domains where φh and φ + are analytic can be used to get the linear system ⎡

A00 ⎢A ⎢ 10 ⎢ . ⎢ .. ⎢ ⎢ ⎢ AK0 ⎢ ⎢ A10 ⎢ ⎢ A20 ⎢ ⎢ .. ⎣ . AJ 0

A01 A11 .. . AK1 A11 A21 .. .

· · · A0J · · · A1J . .. . .. · · · AKJ · · · A1J · · · A2J . .. . ..

B01 B02 B11 B12 .. .. . . BK1 BK2 B11 B12 B21 B22 .. .. . .

AJ 1 · · · AJ J BJ 1 BJ 2

⎤⎡ ⎤

⎤ ⎡ · · · B0K φ+ ξ0+  E0 + ⎥ ⎥ ⎢ ⎢ · · · B1K ⎥ ⎥ ⎢ φ+ ξ1 ⎥ ⎢ E1 ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ . .. .. ⎥ ⎢ ... ⎥ ⎢ . .. ⎥ . ⎥⎢ ⎥ ⎥ ⎢

⎥ ⎢ ⎥⎢ ⎥ · · · BKK ⎥ ⎢ φ+ ξJ+  ⎥ ⎢ EK ⎥ ⎥⎢ ⎥, − ⎥=⎢ · · · B1K ⎥ ⎢ φh iη1  ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ ⎥ ⎢ − ⎥ ⎥ ⎢ ⎢ · · · B2K ⎥ ⎥ ⎢ φh iη2 ⎥ ⎢ 0 ⎥ . . . ⎥ ⎥ ⎥ ⎢ ⎢ .. .. ⎦ ⎣ .. ⎦ . .. ⎦ ⎣

− · · · BJ K φh iηK 0

where Ek = Ak,j =

+

J

 ,

 G ξk− −G ξj+

j =0 cj ξk− −ξj+   ihξ + ihξ − cj e k −e j

,

ξk− −ξj+

− ihξ − b n e ηn h e k σ2 4 − 2 ξk −iηn− cj iηl+ −ξj+ − b n e ηn h  σ 2 4 2 i η+ −ηn− l

Bk,n = − Al,j =

Bl,n = −

,

,

,

, k = 0, . . . , K; k = 0, . . . , K, j = 0, . . . , J ; k = 0, . . . , K, n = 1, . . . , K; l = 1, . . . , J,

j = 0, . . . , J ;

l = 1, . . . , J.

n = 1, . . . , K.

Solving the system (for each ω), φh is known, and the valuation can be obtained by applying the inverse Fourier transform followed by the Inverse Laplace Transform. Once a payoff is chosen, numerical results can be obtained by a direct computation of the line integral. Let’s consider as an example the super-share option.

206

S. M. Grudsky and O. A. Mendez-Lara

3.1 Super-Share Option Let K1 < S < K2 be fixed parameters; the payoff is given by 0 g (S) =

S/K1 S ∈ [K1 , K2 ] 0 c.c.

Let S = ex and k1,2 := ln K1,2 and then g (x) = e−k1 ex χ[k1 ,k2 ] (x). Remark k1 and k2 are interpreted as barriers of the payoff and are not necessary equals to the ones in the option contract (h1 , h2 ). It is then useful to set b2 := g (x) = e−k1 ex χ[b1 ,b2 ] (x) . min {h2 , k2 } > 0 and b1 := max {h1 , k1 } < 0, so 4 To make the notation easier and making use of our assumptions for the option barriers,  hletihξh1 = k1 and h2 = k2 . Applying the Fourier transform, G (ξ ) = i e e − 1 . For this particular option, the exact valuation is given by − ξ −i V[0,h] (x, τ ) =

K J  1  bk (ω) cj (ω) fk,j (x, ω) eτ ω dω, iπσ 2 iR

(9)

k=0 j =0

fk,j

⎢ ⎢ ⎢ ⎢ (x, ω) = ⎢ ⎢ ⎢ ⎢ ⎣

+



  ⎤ ⎤ ⎤ + − iφ ξ (ω) + j ⎢ ⎥⎥ ihξ + (ω) ⎢ ⎢ e j ⎣ # % ⎦⎥⎥ ⎢ ⎥⎥ −ixξj+ (ω) 1 −ixξk− (ω) × e − e ⎢ ⎥⎥ − + ξk (ω)−ξj (ω) ⎢ ⎥⎥   ⎣ ⎦⎥ ⎥, 1 1 −ixξk− (ω) ⎥ − +e − + ξk (ω)−i ξj (ω)−i ⎥ ⎦ x e ⎡

⎡

⎡

eh ξj+ (ω)−i

% #  − ξk (ω)−i ξj+ (ω)−i

where K and J represent the number of downward and upward jumps in the model as stated in Sect. 3.

4 Numerical Example There are several problems with formula (9). There are no closed formulas for ξj+ (ω) and ξk− (ω), and numerical values are obtained for each ω by a root-finding algorithm for polynomials. The values bk (ω) and cj (ω) are obtained numerically from linear systems along   the discretized contour. The same situation is true for the values φ+ ξj+ (ω) ; they are solutions of the linear system in Sect. 3, which is dependent on ω. A discretization of a finite segment is used as the integration contour, and discretized integrals are valuated. The result is an approximation, but it will serve us as a benchmark for accuracy and computation time of other approximation methods.

Double-Barrier Option Pricing under the HEJD Model

207

As the value of the roots is given numerically as a modulus-sorted array, it is necessary to classify them to form continuous paths. The asymptotic behavior of the roots is helpful. Two roots, one in each half-plane, behave as the roots in the Wiener–Hopf factorization of the Black–Scholes model’s symbol; the remaining roots tend to the poles iηm as |0ω| → ∞. Thus, the finite segment needs to contain at least [−iW, iW ] where W = max {|ηm |}. This segment is not enough, as the roots that behave as in the Black–Scholes model do not change as rapidly as the ones that tend to the mentioned poles; this led us to use the segment [−20W i, 20W i] over the imaginary axis as the contour of integration to perform the numerical valuations. Given the oscillatory nature of the integrand, we decided to use a nonuniform grid on the finite contour. As all the factors decay exponentially, the discretization consists of 2X points on [−2i, 2i], Y points in [−10W i, −2i) and (2i, 10W i], and Z points on [−20W i, −10W i) and (10W i, 20W i]. Each subinterval was discretized with equidistant points, and the values of X, Y, andZ were determined for a particular numerical example. We consider the following parameters: H1 = K1 = 80, H2 = K2 = 140, τ = 2.0, σ = 0.16, μˇ = 0.12, r = 0.05,  {λˇ m } = {0.03, 0.04, 0.02, 0.025, 0.02}, ˇ {ηˇ m } = {−10, 15, −15, 20, −20}, and λ = M 1 λm . A grid of initial stock values is formed for S ∈ [90, 130] with increments of 0.25, i.e., 160 values. As the number of points X, Y, and Z in the discretization affects the accuracy and the calculation time of the valuation, we calculated them for several values presented in Table 1, Figs. 1 and 2. Notice that all the linear systems are independent of the pairs (S, τ ) and the time that took to solve the systems to get the values of bk , cj , ξj+ , andξk− is on the 2nd column. The next two columns reflect the time to calculate the valuation once the previous values were computed. The last column shows the percent error against the finest discretization (FD) at S = 110. The present algorithm of valuation takes a considerable amount of time, but it will serve as a benchmark to measure precision for other approximation methods. If needed, a coarser discretization can be used to save time; for example, if X = 2000, Y = 1000, and Z = 500, the valuations are approximately within 5% of the finest discretization for S ∈ [90, 130], and the computation time is 4–5 times faster. A Table 1 Computation times for several discretizations of the finite segment Discretization X Y 10,000 5000 8000 4000 5000 2500 2000 1000 1000 400 750 300 500 200

Z 1000 800 500 500 200 150 100

Solving Lin-Sys (s) 46.4897 37.7685 23.2303 10.0892 6.4721 3.5796 2.3412

Integration (s) S = 110 S ∈ [90, 130] 6.0143 968.3023 4.7670 767.4870 2.7245 438.6445 1.2691 204.3251 0.8309 133.7749 0.4194 67.5234 0.2872 46.2392

Percent error vs FD at S = 110 – 0.1070 0.4175 1.4484 3.1461 5.3019 6.7165

208

S. M. Grudsky and O. A. Mendez-Lara

Fig. 1 HEJD valuations for several discretizations

Fig. 2 HEJD: Comparison of valuation errors (%) for several discretizations against the finest

coarser discretization can also be viewed as benchmark for computation time against other methods once a percent error is fixed. When the jump intensities are multiplied by 1, 0.5, 0.25, and 0.1, so, as the global intensity of jumps λ decreases, the HEJD valuations using the algorithm tend to the Black–Scholes valuation; at S = 110, the difference with the Black–Scholes valuation is of 22.68%, 7.94%, 4.10%, and 1.78%, respectively.

Double-Barrier Option Pricing under the HEJD Model

209

Note The computations were performed on an Arch Linux® machine with an Intel® Core™ i5-4570T Processor (4M Cache, up to 3.60 GHz). The algorithm was developed with Python 3.8.

5 General Toeplitz Operators: Sectoriality Approach In the previous sections, we presented an explicit solution and implemented it numerically with an algorithm for solving (4) in the case of a HEJD process. In the subsequent sections, we justify this solution in the sense of proving the existence and uniqueness of the solution of (4) in the Sobolev spaces that arise naturally. The theory of generalized Toeplitz operators and the concept of sectoriality are presented in this section. Let H be a Hilbert space and L (H ) the algebra of bounded linear operators in H. Definition 6 A ∈ L (H ) is called normally solvable if im A = im A. Lemma 2 If A is normally solvable, H = im A ⊕ ker A∗ .

 Definition 7 A is called left-invertible (right-invertible) if A−L A−R ∈ L (X)

 exists such that A−L A = I AA−R = I . If there exists A−1 ∈ L (H ) such that A−1 A = AA−1 = I , A is called an invertible operator. Remark Since the kernel of a left-invertible operator is {0} and for a right-invertible operator its image is H , a one-side invertible operator is also normally solvable. Definition 8 The analytic projectors are given by P ± = the Cauchy-Lebesgue singular integral operator given by 1 (SR f ) (ξ ) = iπ

 R

f (x) dx, x−ξ

1 2

[I ± SR ], where SR is

ξ ∈ R,

and such integral is understood in the sense of the principal value. The operators SR andP ± are well defined on Lp (R) for 1 < p < ∞. The operator SR is bounded in the space Lp (R, s) for 1 < p < ∞ and −1/2 < s < (p − 1)/2 (see [26]). Given (6), in the remainder of the work, let’s work with p = 2, so the boundedness is given for |s| < 1/2.

2 On L2 (R, s) for |s| < 1/2, the analytic projectors are bounded, satisfy P ± = P ± , P + P − = P − P + = 0, P + + P − = I , and generate the spaces L± 2 (R, s) := P ± (L2 (R, s)). Since χ(0,∞) is the Heaviside step function, then it is easy to see that P + := F χ(0,∞) F −1 ,

P − := F χ(−∞,0) F −1 .

210

S. M. Grudsky and O. A. Mendez-Lara

If |s| ≥ 1/2, the analytic projectors are not bounded; nevertheless, one can define s L± 2 (R, s) := F (H (R± )), where R+ = (0, ∞) , R− = (−∞, 0). Definition 9 For a ∈ L∞ (R) , introduce the operator Ma given by Ma f = af for all f ∈ L2 (R, s). The operator + Ta := T (a) := P + Ma : L+ 2 (R, s) → L2 (R, s) ,

is called a Toeplitz operator with symbol a. + ∗ Its adjoint operator Ta∗ : L+ 2 (R, s) → L2 (R, s) is given by Ta = Ta . Note that Ta , Ta∗ = Ta are bounded operators on L+ 2 (R, s) for |s| < 1/2. Proposition 1 (See [27]) If a ∈ L∞ (R) satisfies a ≡ 0, for Ta to be normally solvable in L+ 2 (R, s) for |s| < 1/2, it is necessary that ess inf ξ ∈R |a (ξ )| > 0.

5.1 The Douglas Algebra Let’s define the set of all continuous functions f such that limx→−∞ f (x) = z = limx→∞ f (x) as C• (R) and H ∞ (R) the class of functions of bounded norm with f H ∞ = supz∈R |f (z)|. Consider the set H ∞ (R) + C• (R) ⊂ L∞ (R) of all functions f such that f = h + c where h ∈ H ∞ (R) and c ∈ C• (R); this set is called the Douglas algebra. Introduce the functions γ − (ξ ) := ξ − i, γ + (ξ ) := ξ + i, u := γ + /γ − , and eh (ξ ) := eihξ , so uh := eh u is element of the Douglas algebra. The following result assures the normal solvability of Toeplitz operators with symbols on the Douglas algebra, in particular for uh . Lemma 3 (See, e.g., [27]) If the symbol a is in H ∞ (R) + C• (R)   H ∞ (R) + C• (R) and satisfies ess inf ξ ∈R |a (ξ )| > 0 and a1 is not in   H ∞ (R) + C• (R) H ∞ (R) + C• (R) , thenTa is left-invertible (right-invertible) in L+ 2 (R, s) for |s| < 1/2.

5.2 Sectoriality Definition 10 Let A ∈ L (H ). A is called a ε-sectorial operator for ε > 0 if infxH =1 $Ax, x%H = ε. Clearly, if a ∈ L∞ (R), the multiplication operator Ma is ε-sectorial if and only if ess inf ξ ∈R /a (ξ ) = ε > 0.

Double-Barrier Option Pricing under the HEJD Model

211

Remark Lemma 1 and the last fact enable us to use the sectoriality approach on the valuation problem. Definition 11 A function a ∈ L∞ (R) is called ε-sectorial if there is θ ∈ (−π, π) such that Maθ is a ε-sectorial operator where aθ (ξ ) := eiθ a (ξ ). Theorem 22 (See [21]) Let A be a ε-sectorial operator on H ; then, A is invertible, 2 and 2A−1 2L(H ) < 2ε−1 . Let |s| < 1/2, given G is a subspace of the Hilbert space L+ 2 (R, s) and its as the orthoprojectors onto each orthogonal complement G⊥ ; consider PG and P⊥ G subspace, that is, for f ∈ L+ ∈ G, f2 ∈ G⊥ s), there are unique functions f (R, 1 2 + ⊥ such that f = f1 + f2 . Let’s note that P = PG + PG . + For a ∈ L∞ (R), introduce the operator D : L+ 2 (R, s) → L2 (R, s) given by + D := P⊥ G + P aPG .

Theorem 3 (See [21], Theorem 4.2) If a is a ε-sectorial function, D is invertible, and the solution g of f ∈ L+ 2 (R, s) , |s| < 1/2,

Dg = f, satisfies the estimate

g1 L+ (R,s) ≤ 2ε−1 f1 L+ (R,s) , 2

2

where g1 := PG g and f1 := PG f . ⊥ Moreover, if g2 := P⊥ G g and f2 := PG f and we consider the operator D1 := PG aPG : G → G, D1 is also invertible, and the unique solution of the equation has the form % # −1 ⊥ f. g = D1−1 PG + P⊥ − P aD P G G G 1

6 Unique Solvability of Boundary Value Problems for HEJD Symbols In this section, we use properties of the symbol to construct an equivalent problem to our pricing problem and whose solvability is given by Theorem 3; then, some analytic properties of the solution are explored. Introduce the function φ (ξ ) 4 c (ξ ) := = 1 + ξ2

σ2 2 2 ξ

+ ϕ (−ξ ) . 1 + ξ2

It is clear that 4 c ∈ L∞ (R) and ∃ M > 0, ε1 > 0 | inf|ξ |≥M /4 c (ξ ) = ε1 .

212

S. M. Grudsky and O. A. Mendez-Lara

The assumption ∀ ω ∈ C ∃ ε2 (ω) > 0 | r + /ω ≥ ε2 (ω) .

(10)

is trivially checked at the contour for the inverse Laplace transform: Γ = iR. Let’s add the assumption that ε2 is independent of ω. Define the function c (ξ, ω) :=

a (ξ, ω) r + ω + ψ (−ξ ) = . 1 + ξ2 1 + ξ2

(11)

Condition (10) and definition (11) seem arbitrary and of technical nature but follow directly from Lemma 1, which is a standard tool in the theory of Lévy process, and make it possible to implement the concept sectoriality. The following statements are proved in the same way as Lemma 5.1 and Lemma 5.2 in [21]. Lemma 4 If ψ has the form (7) and ω ∈ iR, the function (11) is ε-sectorial for some ε > 0 nondependent of ω. Let’s use an alternative to condition (10). Lemma 5 If ψ has the form (7) and exists, θ0 > 0 such that, for ω = 0,

 π Kθ0 := ω ∈ C | |arg ω| ≤ + θ0 . 2 There exists ε > 0 such that c (·, ω) is ε-sectorial for all ω ∈ Kθ0 .

6.1 Solvability of Equivalent Problems Since we introduced the function γ (ξ ) := γ + (ξ ) γ − (ξ ) = ξ 2 + 1 with c(·, ω), let’s consider s1 = s + 1/2 and s2 = s − 1/2, where |s| < 1/2; then, (6) turns into φh (·, ω) ∈ L2 (R, s + 1/2) , φ+ (·, ω) , φ− (·, ω) , G ∈ L2 (R, s − 1/2) .

(12)

Equation (5) can be rewritten as % # φ− (ξ, ω) + 1 + ξ 2 c (ξ, ω) φh (ξ, ω) + eihξ φ+ (ξ, ω) = G (ξ ) . Notice that

1 γ ± (ξ )

(13)

are analytic on ± , so dividing Eq. (12) by γ − (ξ ) and denoting ∈ L± Ψ ± (ξ, ω) := φγ±±(ξ,ω) 2 (R, s) , (ξ ) + + Ψh (ξ, ω) := γ (ξ ) φh (ξ, ω) ∈ L+ 2 (R, s) ;

(14)

Double-Barrier Option Pricing under the HEJD Model

Ψ − (ξ, ω) + c (ξ, ω) Ψh+ (ξ, ω) + eihξ u (ξ ) Ψ + (ξ, ω) =

213 G(ξ ) . γ − (ξ )

(15)

The following result will help us with the spaces where the solutions must belong. Lemma 6 (See [25]) 1. The set of functions where the unknown function φh (ξ, ω) must coincide with the set of "+"-components of solutions of the boundary value problem e−ihξ φh+ (ξ, ω) = φh− (ξ, ω) , φh± (·, ω) ∈ L± 2 (R, s + 1/2)

(16)

2. The set of functions where the unknown function Ψh+ (ξ, ω) must coincide with the set of "+"-components of solutions of the boundary value problem e−ihξ u (ξ ) Ψh+ (ξ, ω) = Ψh− (ξ, ω) , Ψh± (·, ω) ∈ L± 2 (R, s)

(17)

That is, the set where the function Ψh+ lies coincides with the subspace ker Tuh L+ (R,s) . 2

In particular, Problems (16) and (17) are called Riemann boundary value problems with coefficients e−h and uh = e−h u, respectively. Now, the classes of functions where the unknown functions can be found have been narrowed by the images of the analytic projectors, for Eq. (15), ⎧ ⎪ ⎨ ⎪ ⎩

= Ψ − (ξ, ω) + c (ξ, ω) Ψh (ξ, ω) + eihξ u (ξ ) Ψ + (ξ, ω) ,  Ψh+ (·, ω) ∈ ker Tuh L+ (R,s) , G(ξ ) γ − (ξ )

2

Ψ ± (·, ω) ∈ L± 2 (R, s) .

) If we apply P + to Eq. (15) and define g + (ξ ) := P + γG(ξ − (ξ ) ,

P + (c (ξ, ω)) Ψh+ (ξ, ω) + P + uh (ξ ) Ψ + (ξ, ω) = g + (ξ ) . Noting that P + uh (ξ ) = Tuh , the problem becomes P + (c (ξ, ω)) Ψh+ (ξ, ω) + Tuh Ψ + (ξ, ω) = g + (ξ ) .

(18)

Given the construction of Problem (18), Problem (13) has a solution if and only if Problem (18) has it; furthermore, one can obtain the solutions of one problem according to the other by using the definitions of Ψ ± and Ψh+ . By Lemma 3, Tuh is left-invertible in L+ 2 (R, s). Hence, Tuh is normally solvable, and according to Lemma 2, L+ s) = im Tuh ⊕ ker Tuh . (R, 2

214

S. M. Grudsky and O. A. Mendez-Lara

Taking the corresponding orthogonal projectors Puh : L+ 2 (R, s) → ker Tuh ,

+ P⊥ uh : L2 (R, s) → im Tuh ,

and the operator + Duh : L+ 2 (R, s) → L2 (R, s) + Duh := P c (·, ω) Puh + P⊥ uh ,

(19)

we can formulate the equation 

Duh Y + (ξ ) = g + (ξ ) ,

(20)

and state the following equivalence of problems. Lemma 7 (See [21], Lemma 5.4) Equation (18) has solution if and only if Problem a left-inverse of Tuh and Y + a solution (20) has a solution. Furthermore, given Tu−L h of Problem (20), the pair 

Ψh+ (ξ ) = Puh Y + (ξ ) ,

  ⊥ + P Y Ψ + (ξ ) = Tu−L (ξ ) , uh h

is a solution of Problem (18).

6.2 Uniqueness of the Solution The existence of the solutions and the equivalency of problems are granted; the uniqueness and the relationship between the solutions are given by the next theorem. Theorem 4 If the function c has the form (11) with ψ as in (7) and ω ∈ Kθ0 : 1. The operator Duh on L+ 2 (R, s) defined as above is invertible, and the solution Y + of Problem (20) satisfies the estimate 2 2 2 22 2 Pu Y + 2 + ≤ 2Puh g + 2L+ (R,s) , h L2 (R,s) 2 ε where ε does not depend on ω. 2. Problem (18) has for unique solution the pair   + Ψh+ (ξ, ω) = Puh Du−1 g (ξ ) , h

  ⊥ −1 + P D g Ψ + (ξ, ω) = Tu−R (ξ ) . uh uh h

Double-Barrier Option Pricing under the HEJD Model

215

3. Problem (13) has for unique solution the triple φh+ (ξ, ω) = γ +1(ξ ) Ψh+ (ξ, ω) , φ + (ξ, ω) = γ + (ξ ) Ψ + (ξ, ω) , φ − (ξ, ω) = γ − (ξ ) Ψ − (ξ, ω) , where Ψ − (ξ, ω) can be obtained from Eq. (15). Proof Statement 1 is a consequence of Theorem 3 and Lemma 5. Statement 2 follows from the previous one, and Lemma 7 gives Y + = Du−1 g + , which is h the unique solution of Problem (20). Finally, for Statement 3, the equivalency of Problems (13)–(18) is straightforward, and the relation between its solutions

is given

by (14). In particular, Ψ − , Ψh+ , Ψ + satisfies Eq. (15) if and only if Ψh+ , Ψ + satisfies the equation in Problem (18) and −

Ψ (ξ, ω) = P

−



G (ξ ) γ − (ξ )



 



− P − uh (ξ ) Ψ + (ξ, ω) − P − c (ξ, ω) Ψh+ (ξ, ω) .  

φh+ (ξ, ω)

Once is obtained, the inverse Fourier transform should be applied, and given that ω must be in Kθ0 for small enough θ0 in order to use the Theorem 4, the contour in which the inverse Laplace transform is performed is Γθ0 = ∂Kθ0 . So, the solution of the Problem (4) has the form   1 eτ ω e−ixξ φh+ (ξ, ω) dξ dω. V (x, τ ) = (21) (2π)2 i Γθ0 R Let’s take a moment to study the functional space where V as obtained with the sectoriality approach lies. For the first variable, fix ω in Kθ0 , since φh+ (ξ, ω) is the solution of Problem (16), φh+ (·, ω) ∈ L± 2 (R, s + 1/2), and its norm is bounded uniformly; therefore, 4 (·, ω) ∈ H s1 (0, h) = H s+1/2 (0, h). This remains true for F −1 φh+ (·, ω) = V 4 (·, τ ) V (·, τ ) = L−1 V For the second variable, fix x, V H s1 (0,h) is continuous on [0, ∞) as a function of τ ; furthermore, it vanishes as τ → ∞; hence, V (x, ·) ∈ C0 [0, ∞). We will resume the behavior of V on both variables by using the notation   V ∈ H s+1/2 (0, h) , C0 [0, ∞) ,

|s| < 1/2,

which is consistent with our initial definition of the problem. Once the payoff function is assumed in the proper space, the existence, uniqueness, form, and space where the solution is can be stated in the following result.

216

S. M. Grudsky and O. A. Mendez-Lara

Theorem 5 Let g (x) ∈ H s−1/2 (0, h) for |s| < 1/2, and let ψ Q be the characteristic exponent (under a chosen LEMM) of

a HEJD process, i.e., it has the form (7). Problem (4) has a unique solution in C 0 [0, ∞), H s+1/2 (0, h) , and it has the form (21). This theorem is a direct consequence of Theorem 4 and given that /ω → −∞ ⇒ eτ ω → 0, /ω < 0 ∀ ω ∈ Γθ0 , ω ∈ Γθ0 , |ω| → ∞ ⇒ /ω → −∞.

We have obtained and exact formula for the pricing problem specified, the computational implementation of (21) is easier when expressed as (9), and the numerical result is in reality an approximation; nevertheless, this will serve as a benchmark for accuracy against other approximation methods. As shown in Sect. 4, the discretization and the method to find the roots of the symbol can be tweaked to consume less time at the cost of accuracy, but this approximations will also provide a reference to compare accuracy to time with alternative methods.1

References 1. Kunitomo, N., Ikeda, M.: Pricing options with curved boundaries. Math. Financ. 2(4), 275–298 (1992). https://doi.org/10.1111/j.1467-9965.1992.tb00033.x 2. Geman, H., Yor, M.: Pricing and hedging double-barrier options: a probabilistic approach. Math. Financ. 6(4), 365–378 (1996). https://doi.org/10.1111/j.1467-9965.1996.tb00122.x 3. Sidenius, J.: Double barrier options: valuation by path counting. J. Comput. Financ. 1(3), 63– 79 (1998). https://doi.org/10.21314/jcf.1998.012 4. Pelsser, A.: Pricing double barrier options using Laplace transforms. Financ. Stoch. 4(1), 95– 104 (2000). https://doi.org/10.1007/s007800050005 5. Baldi, P., Caramellino, L., Iovino, M.G.: Pricing general barrier options: a numerical approach using sharp large deviations. Math. Financ. 9(4), 293–321 (1999). https://doi.org/10.1111/ 1467-9965.t01-1-00071 6. Hui, C.H.: One-touch double barrier binary option values. Appl. Financ. Econ. 6(4), 343–346 (1996). https://doi.org/10.1080/096031096334141 7. Hui, C.H.: Time-dependent barrier option values. J. Futur. Mark. 17(6), 667–688 (1997). https://doi.org/10.1002/(sici)1096-9934(199709)17:63.0.co;2-c 8. Boyarchenko, S.I., Levendorskii, S.Z.: Non-Gaussian Merton-Black–Scholes Theory. World Scientific, Singapore (2002). https://doi.org/10.1142/4955 9. Satõ, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2013) 10. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Press, Boca Raton (2015). https://doi.org/10.1201/9780203485217 11. Kou, S.G.: A jump diffusion model for option pricing. SSRN Electron. J. (2000). https://doi. org/10.2139/ssrn.242367 12. Boyarchenko, S.I., Levendorskii, S.Z.: Option pricing for truncated Lévy processes. Int. J. Theor. Appl. Financ. 3(3), 549–552 (2000). https://doi.org/10.1142/s0219024900000541

1 We are deeply grateful to the referees for the useful comments that helped improve the presentation of the article.

Double-Barrier Option Pricing under the HEJD Model

217

13. Kou, S.G., Wang, H.N.: Option pricing under a double exponential jump diffusion model. SSRN Electron. J. (2001). https://doi.org/10.2139/ssrn.284202 14. Cont, R., Voltchkova, E.: Integro-differential equations for option prices in exponential Lévy models. Financ. Stoch. 9(3), 299–325 (2005). https://doi.org/10.1007/s00780-005-0153 15. Kudryavtsev, O.E.: Advantages of the Laplace transform approach in pricing first touch digital options in Lévy-driven models. SSRN Electron. J. (2015). https://doi.org/10.2139/ssrn. 2713193 16. Boyarchenko, S.I., Levendorskii, S.Z.: Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab. 12(4), 1261–1298 (2002). https://doi. org/10.1214/aoap/1037125863 17. Kudryavtsev, O.E., Levendorskii, S.Z.: Fast and accurate pricing of barrier options under Levy processes. SSRN Electron. J. (2007). https://doi.org/10.2139/ssrn.1040061 18. Kudryavtsev, O.E.: Finite difference methods for option pricing under Lévy processes: Wiener– Hopf factorization approach. Sci. World J. 2013, 1–12 (2013). https://doi.org/10.1155/2013/ 963625 19. Boyarchenko, M., Boyarchenko, S.I.: User’s guide to pricing double barrier options. Part I: Kou’s model and generalizations. SSRN Electron. J. (2008). https://doi.org/10.2139/ssrn. 1272081 20. Boyarchenko, S.I.: Two-point boundary problems and perpetual american strangles in jumpdiffusion models. SSRN Electron. J. (2006). https://doi.org/10.2139/ssrn.896260 21. Grudsky, S.M.: Double barrier options under Lévy processes. In: Erusalimsky, Y.M., Gohberg, I., Grudsky, S.M., Rabinovich, V., Vasilevski, N. (eds.) Modern Operator Theory and Applications: The Igor Borisovich Simonenko Anniversary, vol. 2007. Birkhäuser, Basel, pp 107–135 (2006). https://doi.org/10.1007/978-3-7643-7737-3_8 22. Sepp, A.: Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform. Int. J. Theor. Appl. Financ. 7(2), 151– 175 (2004). https://doi.org/10.1142/s0219024904002402 23. Kirkby, J.L.: Robust barrier option pricing by frame projection under exponential Levy dynamics. SSRN Electron. J. (2014). https://doi.org/10.2139/ssrn.2541980 24. Crosby, J., Saux, N.L., Mijatovic, A.: Approximating Levy processes with a view to option pricing. SSRN Electron. J. (2009). https://doi.org/10.2139/ssrn.1403919 25. Dybin, V., Grudsky, S.M.: Introduction to the theory of Toeplitz operators with infinite index. Birkhäuser, Basel (2002). https://doi.org/10.1007/978-3-0348-8213-2 26. Gohberg, I., Krupnik, N.I.: One-dimensional linear singular integral equations. Birkhäuser, Basel (1992). https://doi.org/10.1007/978-3-0348-8647-5 27. Böttcher, A., Silbermann, B., Karlovich, A.: Analysis of Toeplitz operators. Springer, Berlin (2006). https://doi.org/10.1007/3-540-32436-4

Single Jump Filtrations: Preservation of the Local Martingale Property with Respect to the Filtration Generated by the Local Martingale Alexander A. Gushchin and Assylliya K. Zhunussova

Abstract Let M be a local martingale with respect to a so-called single jump filtration F = F(γ , F ) generated by a random time γ on a probability space (Ω, F , P). It was recently mentioned by Herdegen and Herrmann (2016) that M is also a local martingale with respect to the filtration H = FM that it generates if F is the smallest σ -field with respect to which γ is measurable. We provide an example of a local martingale with respect to a general single jump filtration which is not a local martingale with respect to H. Then, we find necessary and sufficient condition for preserving the local martingale property with respect to H. The main idea of our constructions and the proofs is that H is also a single jump filtration generated, in general, by other random time and σ -field. Finally, we prove that every σ -martingale in considered models is still a σ -martingale with respect to the filtration that it generates. Keywords Filtration · Local martingale · σ -martingale · Single jump filtration

1 Introduction Let γ be a random variable with values in [0, +∞] on a probability space (Ω, F , P). We tacitly assume that P(γ > 0) > 0. According to [3], we call a filtration F = (Ft )t ∈R+ a single jump filtration generated by γ and F if A ∈ Ft , t ∈ R+ , if and only if A ∈ F and A ∩ {t < γ } is either ∅ or coincides with {t < γ }. This means that all randomness occurs at time γ . A special case where

A. A. Gushchin () Steklov Mathematical Institute, Moscow, Russia Lomonosov Moscow State University, Moscow, Russia e-mail: [email protected] A. K. Zhunussova Faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University, Moscow, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_11

219

220

A. A. Gushchin and A. K. Zhunussova

F = σ {γ }, i.e., F is the smallest σ -field with respect to which γ is measurable, will be referred to as the Dellacherie model. In fact, Dellacherie [2] assumes that Ω = R+ and γ (ω) = ω, but this specification is not essential. In the Dellacherie model, the single jump filtration is the smallest one with respect to which γ is a stopping time. Additional randomness admitted in a general single jump filtration makes the considered model more flexible. As it is shown in [3], every σ -martingale is a local martingale in the Dellacherie model, while there are σ -martingales which are not local martingales with respect to a general single jump filtration. In this paper, we are interested in a similar phenomenon. It is well known that, if M is a local martingale on an arbitrary stochastic basis (Ω, F , F, P), it is not necessarily a local martingale relative to the filtration FM that it generates. However, Herdegen and Herrmann [4] show that every local martingale in the Dellacherie model is also a local martingale with respect to FM . We aim to show that it is not the case for a general model with a single jump filtration F, and our main goal is to characterize all F-local martingales which are not local martingales relative to FM . This may happen if a local martingale with a positive probability equals a constant on a time interval starting before γ and including γ . Then, the filtration FM has no information about γ at all, and it may happen that there are not enough FM -stopping times to provide a localizing sequence for M. Our example resembles Ruf’s counterexample [5] where a concrete single jump filtration is considered. It has a property that, for every finite stopping time σ , σ ∧ γ is bounded. In our example, the filtration FM is also a single jump filtration where there are not enough stopping times. A σ -martingale on an arbitrary stochastic basis also needs not be a σ -martingale with respect to the filtration that it generates; see [1]. Our final result states that a σ -martingale with respect to a general single jump filtration preserves this property relative to the filtration that it generates. We use two facts that simplify proofs and constructions. The first is that, for a local martingale M relative to a single jump filtration F, FM is also a single jump filtration. The second fact is a simple characterization of all local martingales with respect to single jump filtrations.

2 Main Results We assume unless otherwise stated that there is given a probability space (Ω, F , P) with a random variable γ with values in [0, +∞]. Let G(t) = P(γ ≤ t) be a distribution function of γ , G(t) = 1 − G(t), tG = sup {t ∈ R+ : G(t) < 1}, and T = {t ∈ R+ : P(γ ≥ t) > 0}. By default, (Ω, F , P) is equipped with the single jump filtration F = (Ft )t ∈R+ generated by γ and F . We will also write F(γ , F ) to distinguish it among other filtrations. Recall that A ∈ Ft if and only if A ∈ F and A ∩ {t < γ } is either ∅ or {t < γ }. A convenient way to characterize an Ft -measurable random variable is that it is a constant on the set {γ > t} (and F -measurable). No

Single Jump Filtrations: Preservation of the Local Martingale Property

221

completeness assumption is made. Every σ -martingale is assumed to have all paths right continuous everywhere. For ease of references, we reformulate some statements from [3] in a convenient form. Theorem 1 (i) Every F-σ -martingale M has a representation Mt = F (t)1{t t) = 0 for t ≥ tG , replacing F by another right-continuous function on [tG , ∞) changes X only on an evanescent set. However, random variables γ˜ may differ significantly for different versions of F on [tG , ∞). This is the price we pay for exact coincidence of filtrations and not up to null sets. Remark 2 Without loss of generality, we may assume that L = 0 on the set {γ = ∞}. Then, L is F X -measurable.

Single Jump Filtrations: Preservation of the Local Martingale Property

223

Proof Since >

{γ˜ < t} =

{Xr = F (r)} ∈ FtX

rrational r t. This means that the traces of the σ -fields F X and FtX on {γ˜ ≤ t} are equal. Now, let A ∈ F X . We have just proved that A ∩ {γ˜ ≤ t} ∈ FtX . If A belongs to Ht , then either A ∩ {γ˜ > t} = {γ˜ > t} or A ∩ {γ˜ > t} = ∅. In both cases, A ∩ {γ˜ > t} ∈ FtX and A ∈ FtX . The converse inclusion FtX ⊆ Ht is obvious because we can rewrite (7) as Xt = F (t)1{t 0} is equal to R+ . Since the filtration generated by M is the single jump filtration generated by γ˜ and F M by Theorem 2, it is necessary for M to be a local martingale with respect to its own filtration that E|Mt | < ∞ for all t ∈ R+ by Theorem 1 (iii). However, for t ≥ 1, 

 2

E|Mt | = E |ε|S(γ ) = E S(γ ) = ∞. 3 We conclude that M is not a local martingale with respect to the filtration FM that it generates.

2.3 Examples of Appropriate Localizing Sequences Let us assume that tG < ∞, P(γ = tG ) = 0, and M has form (1) and is a martingale on [0, tG ). Take an arbitrary sequence tn  tG . In [3], see the proof of Theorem 1; the following sequence of F-stopping times is constructed: Tn =

tn , if γ > tn ; +∞, otherwise.

It is proved that Tn → ∞ a.s. and M Tn = M tn , which implies that M is an Flocal martingale with a localizing sequence {Tn }. We will show how this idea can be modified to prove that M is an FM -local martingale; see also [4, Proof of Theorem 3.5 (a)]. These constructions aim to explain preliminarily how some conditions appear in our main result in the next subsection. They are not used in its proof. Assume that tG < ∞, P(γ = tG ) = 0, and F is not constant on any interval [t ∗ , tG ). Then, there is an increasing sequence {tn } converging to tG and such that F (tn ) = F (tn−1 ) for all n. Put Sn =

tn , if Mtn = Mtn−1 ; +∞, otherwise,

where M is as above. Then, obviously, Sn is an FM -stopping time and Sn ≤ Tn for all n. Therefore, M Sn is an F(γ , F )-martingale and hence an FM -martingale. Moreover, since P(γ < tG ) = 1, we have P(tn > γ ) → 1 as n → ∞. We conclude that Sn ↑ ∞ a.s. and M is an FM -local martingale. Now, assume that F (t) ≡ c on [t ∗ , tG ) and P(L = c) = 0. Take an increasing sequence t ∗ < t1 < · · · < tn < · · · < tG , tn → tG , and define Rn by Rn =

tn , if Mtn = c; +∞, otherwise.

Single Jump Filtrations: Preservation of the Local Martingale Property

225

Then, Rn = Tn a.s. and Rn is an FM -stopping time for all n. We arrive at the conclusion that again, M is an FM -local martingale.

2.4 Necessary and Sufficient Conditions In this subsection, we give necessary and sufficient conditions for a local martingale with respect to a single jump filtration to be a local martingale with respect to the filtration that it generates. Theorem 3 Let M be an F(γ , F )-local martingale of the form (1). Denote by FM the filtration generated by M; see (8). Then, M is not an FM -local martingale if and ∗ only if tG < ∞, P(γ = tG ) = 0, there is t ∗ < tG such that F (t)  ≡ c on [t , tG ) for ∗   some constant c, P(L = c, γ ≥ t ) > 0, and E L − E(L|γ ) = ∞. Corollary 1 Let M be an F(γ , F )-local martingale and G = (Gt )t ≥0 be any filtration such that FtM ⊆ Gt ⊆ Ft for every t ≥ 0. Suppose that the assumptions of Theorem 3 are not satisfied. Then, M is a G-local martingale. Corollary 2 (Herdegen and Herrmann [4]) In the Dellacherie model, every F(γ , F )-local martingale is an FM -local martingale. Proof of Theorem 3 According to Theorem 1 (i), M has a representation (1). Obviously, every F(γ , F )-martingale is an FM -martingale. Thus, we may and will assume that M is not an F(γ , F )-martingale. By Theorem 1 (iv), this means that tG < ∞, P(γ = tG ) = 0, and either (5) or (6) holds. Put γ˜ = inf{t ∈ R+ : Mt = F (t)}

(inf ∅ = +∞).

By Theorem 2, the filtration FM coincides with the single jump filtration F(γ˜ , F M ). Rewriting (1) in the form Mt = F (t)1{t 0},

t ∈ T M.

226

A. A. Gushchin and A. K. Zhunussova

Obviously, γ˜ ≥ γ ; hence, T ⊆ T M . Therefore, if M is not an FM -local martingale, it is necessary that T  T M , i.e., tG ∈ T M . In other words, P(γ˜ ≥ tG ) > 0. But {γ˜ ≥ tG } ⊆ {ω : L(ω) = F (t) for all t ∈ [γ (ω), tG )} =: B. Put B ∗ := B ∩ {γ < tG }. Since P(γ < tG ) = 1, we have P(B ∗ ) > 0. We obtain that F (t) is constant on any interval [γ (ω), tG ) if ω ∈ B ∗ . Thus, putting t ∗ = inf∗ γ (ω), ω∈B

we obtain that F (t) ≡ c := F (t ∗ ) on [t ∗ , tG ), and the set {L = c, γ ≥ t ∗ } contains B ∗ and, hence, has a positive probability. 

 To complete the proof of necessity, it remains to show that E L − E(L|γ ) = ∞. Note that F is bounded in the left neighborhood of tG and G(t) → 0 as t  tG . Hence, the second condition in (6) is not satisfied. Thus, we have (5). On the other hand, denote H (t) = E[L|γ = t]; see details in [3]. Then,  



 E E(L|γ ) = E H (γ ) =

 [0,tG )

|H (s)| dG(s)

and  E(Mt ) = F (t)G(t) +

[0,t ]

H (s) dG(s),

t < tG ,

(10)

$ in particular, [0,t ] |H (s)| dG(s) < ∞. Differentiating (10) in t and taking ∗ (4) $ into account, we obtain that H (t) = c dG(t)-a.s. on (t , tG ). Thus, [t ∗ ,tG ) |H (s)| dG(s) < ∞. The claim follows. Conversely, let us show that the conditions of the theorem are sufficient. If ∗ F (t) ≡ c on [t ∗ , tG ), then γ˜ ≥  tG on the set  = c, t ≤ γ ≤  tG }.Finally,

 {L     it follows from E L − E(L|γ ) = ∞ that E L = ∞. Hence, E MtG  = ∞, and M is not an F(γ˜ , F M )-local martingale by Theorem 1 (iii).  

2.5 Preservation of the σ -Martingale Property In this subsection, we prove that every F-σ -martingale is also a σ -martingale with respect to the filtration that it generates. Theorem 4 Let M be a σ -martingale with respect to a single jump filtration F. Then, it is also a σ -martingale with respect to the filtration FM that it generates.

Single Jump Filtrations: Preservation of the Local Martingale Property

227

Corollary 3 Let M be an F(γ , F )-σ -martingale and G = (Gt )t ≥0 be any filtration such that FtM ⊆ Gt ⊆ Ft for every t ≥ 0. Then, M is a G-σ -martingale. Corollary 4 Every F(γ , F )-local martingale is an FM -σ -martingale. Proof of Theorem 4 Every F(γ , F )-σ -martingale has a representation (1). Without loss of generality, we may assume that F (t) ≡ y, t ≥ tG , where y is a number such that P(L = y) = 0. Put γ˜ = inf{t ∈ R+ : Mt = F (t)}

(inf ∅ = +∞).

The above assumption guarantees that P(γ˜ > tG ) = 0. By Theorem 2, the filtration FM coincides with the single jump filtration F(γ˜ , F M ). Rewrite (1) in the form Mt = F (t)1{t 0, then for any ε ∈ (0, a) there is s ∈ (a −ε, a) with F (s) = c; (iii) if b < tG then for any ε ∈ (0, tG −b) there is s ∈ (b, b + ε) with F (s) = c. Obviously, there are at most countable numbers of such intervals, and they do not intersect. Enumerate those of them that satisfy P(γ ∈ J, L = F (a)) > 0 as J1 , . . . , Jk , . . . . Let ak < bk be the end points of Jk and ck = F (ak ). If γ˜ (ω) = γ (ω), then γ˜ (ω) > γ (ω) and F (t) = L(ω),

for all t such that γ (ω) ≤ t < γ˜ (ω).

Excluding P-null sets such as {γ > tG } and {γ = tG , L = F (tG )}, we obtain {γ < γ˜ } =

>

Dk

a.s.,

k

where Dk = {ω : γ (ω) ∈ Jk , L(ω) = ck } = {ω : γ (ω) ∈ Jk , γ˜ (ω) = bk , L(ω) = ck }. Our purpose is to check that the conditions of Theorem 1 (ii) imply the corresponding conditions for representation (11) and γ˜ .

228

A. A. Gushchin and A. K. Zhunussova

Assume that there exists a strictly positive function ψ : (0, +∞) → (0, +∞) such that  |L|ψ(γ ) dP < ∞, {0 0, ⎩ 2 c + 4γ

(51)

262

A. Kh. Khachatryan

H0 (s) =

⎧ æ ⎪ eμ1 s , if s ≤ 0, √ ⎪ ⎪ 2 ⎨ c + 4æ ⎪ ⎪ æ ⎪ ⎩√ eμ2 s , if s > 0. c2 + 4æ

(52)

ν2 < 0 < ν1 and μ2 < 0 < μ1 are the roots of the following characteristic equations, respectively, ν 2 − cν − γ = 0,

(53)

μ2 − cμ − æ = 0.

(54)

It is easy to check that ∞ H (s)ds = 1,

(55)

H0 (s)ds = 1.

(56)

−∞

∞ −∞

Note that Eq. (49) for a given ϕ coincides with Eq. (1). Our main goal is to study and solve the system of nonlinear Eqs. (49) and (50) with boundary conditions (47) and (48). Observe that the corresponding nonlinear integral operator of the systems (49) and (50) does not pass the monotonicity property. The latter makes it very difficult to construct a fixed point for the above system. Below, we suggest the approximate (iterative) way to solve it.

3.3 Auxiliary Nonlinear Integral Equation In this section, we will consider the so-called SIS model, which is simple comparing with SIRS model discussed in the previous paragraph. In the framework of the SIS model of epidemics, the corresponding equations can be written as follows: ∂ 2 I (t, x) ∂I (t, x) = −βS(t, x)I (t, x) + − γ I (t, x), ∂t ∂x 2

(57)

S(t, x) = 1 − I (t, x).

(58)

In fact, we divide the population into two classes, S and I , and for simplicity, we assume (inexorable generality) that the removed rate from class I to class S is the

Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics

263

same as the removed rate from class I to class R, considered in the previous SIRS model. From the solution of Eq. (57), we search in the form of travelling wave: I (t, x) = I (x + ct) = y(z).

(59)

Equation (57) is easily reduced to the following boundary value problem:   y

− cy − γ y = −β y − y 2 ,

(60)

y(−∞) = 0, y(+∞) = η,

(61)

which is in turn can be transformed into following integral equation: ∞ H (z − s)g(y(s))ds

y(z) =

(62)

−∞

with nonlinearity g(u) =

β (u − u2 ) = α(u − u2 ). γ

(63)

Observe that Eq. (62) coincides with (49) if we take ϕ = 0 (first approximation). Let η be the fixed point of the function g, i.e., g(η) = η.

(64)

From (63), it follows that η= Since g(u) =

α−1 β −γ = , (β > γ ). β α

(65)

β (u − u2 ) monotonically increasing on u ∈ (0, 12 ), we have γ β ∈ (γ , 2γ ).

(66)

Notice that y = 0 and y = η satisfy Eq. (62) (trivial solutions). Our main goal is to construct a nontrivial solution for Eq. (62), satisfying boundary conditions (61). It should be noted that Eq. (62) with symmetric kernel in the class of bounded functions has only trivial solutions y = 0 and y = η. Observe that kernel H is asymmetrical (c > 0). The existence of a nontrivial solution for Eq. (62) immediately follows from Diekman’s Theorem 6.1 (see [5]). Hereinafter, we’ll use this theorem for our consideration.

264

A. Kh. Khachatryan

We introduce the Diekman’s function [5]: ∞



L(σ ) = g (0)

e−σ s H (s)ds, σ ≥ 0

(67)

−∞

and consider the characteristic equation L(σ ) = 1.

(68)

Using representations (51) and (63), integrating (67), from (68) we get σ1,2 =

c∓



c2 − 4(β − γ ) . 2

(69)

Observe that ν1 > max(σ1 , σ2 ) = σ2 , which provides convergence of integral (67). The relation (69) contains the so-called threshold condition. If c > cm = 2(β − γ ) (minimum speed), then there is an epidemic and an epidemic wave passes through the population; otherwise, (0 < c < cm ) the epidemic does not occur. Let us consider the following iteration for Eq. (62): ∞ yn+1 (z) =

H (z − s)g(yn−1 (s))ds, z ∈ R,

(70)

−∞

as a zero approximation was chosen  y0 (z) =

η,

for z ≥ 0,

ηeσ1 z ,

for z < 0,

(71)

where σ1 ∈ (0, σ0 ) is the root of Eq. (68) and σ0 is the positive number for which L(σ0 ) < 1.

(72)

The sequence of functions {yn (z)}∞ n=0 is monotonically decreasing. The sequence of these functions is bounded from below by function (z): yn (z) ≥ (z) =

 ηeσ1 z − Me(σ1 +δ)z , for z ≤ 0,

for z >

1 δ 1 δ

η ln M , η ln M ,

(73)

Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics

265

where 0 < δ < min{σ1 , σ0 − σ1 }, M ≥ max

(74)

 σ1 η η2 ; . σ1 + δ 1 − L(σ1 + δ)

(75)

Since yn (z) is monotonically decreasing and bounded from below, there exists a pointwise convergence: lim yn (z) = y(z),

n→∞

where y(z) is the monotonically increasing solution of Eq. (62) and satisfies boundary condition (61). Several numerical simulations concerning with construction of function y(z) are reported in Sect. 4.

3.4 Scheme of Solving a System of Eqs. (49) and (50) Now we are ready to describe the approach for solving of system of nonlinear integral Eqs. (49) and (50), satisfying the boundary conditions (47) and (48). In Eq. (50), we replace the function (z) with y(z) (first approximation for ϕ(z)). We have ∞ ϕ(z) =

H0 (z − s)y(s)ds, z ∈ R,

(76)

−∞

where y(z) is the solution of Eq. (62). From (76), it is obvious that function ϕ(z) will “inherit” all the properties of function y(z). 1. Since y(z) ↑ in z on R, the function will also be monotonically increasing on R. 2. Taking into consideration (4), (28), (56), and (61) from (76), we get ∞ ϕ(+∞) = y(+∞)

H0 (s)ds = η

−∞

γ = ε, æ

(77)

∞ ϕ(−∞) = y(−∞) −∞

Note that ϕ(z) satisfies conditions (4) and (5).

H0 (s)ds = 0.

(78)

266

A. Kh. Khachatryan

Now we substitute the already constructed solution ϕ(z) in (49). All conditions of Theorem 1 are satisfied. So Eq. (49) has bounded nonincreasing solution and satisfying double estimate η − ε ≤ (z) ≤ η, z ∈ R

(79)

(−∞) = η, (+∞) = η − ε.

(80)

and limit relations

4 Numerical Results In this section, we present several numerical results, which are based on real data. For influenza epidemic, a blood test was performed among students during 1 year at Yale University. It leads to a contact number estimate of α = γβ = 1, 5 (see [19, 20]).

4.1 SIS Model We start with Eq. (70). It should be noted that all approximations can be calculated analytically. For the first iteration, we get ⎧ # % η 1−η 1 ⎪ y1+ (z) = η + √ βη , for z ≥ 0, eν2 z σ1 −ν − + ⎪ ⎪ 2σ −ν ν 2 2 1 2 2 ⎪ c +4γ ⎪ ⎪   ⎨ − βη η 1 ν1 z 1−η + y1 (z) = y1 (z) = √c2 +4γ e ν1 σ1 −ν1 − 2σ1 −ν1 + ⎪ ⎪     ⎪ ⎪ ⎪ 1 1 1 1 σ z 2σ z 1 1 ⎪ , for z < 0. ⎩ +e σ1 −ν2 − σ1 −ν1 + ηe 2σ1 −ν1 − 2σ1 −ν2 As for the initial parameters, we took β = 9, γ = 6, and c = 4. Then, from relations (53), (63), (65), (69), and (51), we obtain ν1,2 = 2 ±

√ 3 10, g (0) = , η = 2 ⎧ ⎨ √3 eν1 s , 10 H (s) = ⎩ √3 eν2 s , 10

1 , σ1 = 1, σ2 = 3, 3 if s ≤ 0, if s > 0.

Minimization of function L(σ ) leads to σ0 = 2. So δ = 1, L(δ + σ1 ) =

9 10 .

Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics



η2 η , M = max 2 1 − L(2)  (z) =

zmax =



10 , 9

=

1 z 3e

−

10 2z 9 e ,

z0 =

267

η 1 ln = −1, 204, δ M

for z ≤ −1, 204, for z > −1, 204,

0,

ησ1 1 ln = −1, 897, max = 0, 025, δ M(σ1 + δ)  y0 (z) =

1 3, 1 z 3e ,

for z ≥ 0,

(81)

for z < 0.

Recently in work [10], it was proved the following uniform estimate for iterations (70):     n = e−(δ+σ1 )z |yn+1 (z) − yn (z)| ≤ α0 Ln (σ1 + δ), (82) where 

α := sup e−(δ+σ1 )z |y1 (z) − y0 (z)| .

(83)

From (70) and (71), one can easily calculate the first iteration:  y1 (z) =

√ √ 1 10+3 (2− 10)z , 3 − 60 e √ √ 1 z e2z 10−3 (2+ 10)z , 3 e − 10 − 60 e

if z ≥ 0, if z < 0.

(84)

Taking into account (81), (84), and (82), we get  α0 = 0.102, 4 ≤

9 10

4 · α0 ≈ 0.066.

Observe that after the fourth iteration, the error is less than 6.6%. In Fig. 1 is plotted the solution of Eq. (62) y(z) for the given parameters. It can be easily proved that this solution is unique for the given cone segment (z) ≤ y(z) ≤ y0 (z).

Fig. 1 Graph of the solution of Eq. (62) for the parameters β = 9 and γ = 6

268 A. Kh. Khachatryan

Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics

269

4.2 SIRS Model Now, we are ready to construct the function ϕ(z) using the solution y(z). We rewrite Eq. (76) in the form of

ϕ (x) = √

 0

γ

+

e

y (t)dt +

c2

+ 4æ

∞

 eμ1 (x−t )y + (t)dt ,

+

x

μ2 (x−t ) −

−∞

eμ2 (x−t )y + (t)dt

0

(85)

x

ϕ (x) = √

 x

γ

−

e

y (t)dt +

c2

+ 4æ

∞

 eμ1 (x−t )y + (t)dt .

+

0

μ2 (x−t ) −

−∞

eμ1 (x−t )y − (t)dt

x

(86)

0

From (4), (77) follows that æ > 3γ η. As for æ, we choose æ = 7. Therefore, from (54) and (52) we have √ ηγ 11, ε = = 0, 29, æ ⎧ ⎨ √3 eμ2 s , if s ≥ 0,

μ1,2 = 2 ±

H0 (s) =

⎩

11 √3 e μ1 s , 11

if s < 0.

Notice that the intersection of sets (6) and (66) will be  β∈

 2γ γ , . 1−ε 1+ε

(87)

So one can be seen that all parameters are self-consistent. Figure 2 shows the graphs of functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η which are determined via relations (86) and (49), respectively.

270

A. Kh. Khachatryan

Fig. 2 Graphs of the functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η for the parameters β = 9, γ = 6, and æ = 7

Fig. 3 Graphs of the functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η for the parameters β = 9, γ = 6, and æ = 14

In Fig. 3 are plotted graphs of the functions ϕ(z) and (z) for the parameters β = 9, γ = 6, and æ = 14. Observe that solution (z) in this cone segment [η − ε, η] is unique (see Theorem 2).

Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics

271

Acknowledgments The author is grateful to Prof. Kh.A. Khachatryan for the useful discussions and comments in preparing this paper. The author thanks the GKN MON RA for the support in the framework of the project №SCS 18T-1A004.

References 1. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. J. Proc. R. Soc. Lond. A 115, 700–721 (1927). https://doi.org/10.1098/rspa.1927.0118 2. Anderson, R.M., May, R.M. (eds.): Population Biology of Infectious Diseases. Springer, Berlin (1982) 3. Atkinson, C., Reuter, G.: Deterministic epidemic waves. J. Math. Proc. Camb. Philos. Soc. 80(2), 315–330 (1976). https://doi.org/10.1017/S0305004100052944 4. Badshah, V.H., Porwal, P., Tiwazi, V.: Mathematical modeling and role of dynamics in epidemiology. Int. J. Compat. Sci. Math. 5(1), 1–10 (2013) 5. Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978). https://doi.org/10.1007/BF02450783 6. Khachatryan, A.Kh., Khachatryan, Kh.A.: On the solvability of some nonlinear integral equations in problems of epidemic spread. Proc. Steklov Inst. Math. 306, 271–287 (2019). https://doi.org/10.1134/S0081543819050225 7. Sergeev, A.G., Khachatryan, Kh.A.: On the solvability of a class of nonlinear integral equations in the problem of a spread of an epidemic. Trans. J. Mosc. Math. Soc. 80, 95–111 (2019). https://doi.org/10.1090/mosc/286 8. Khachatryan, Kh.A., Narimanyan, A.Zh., Khachatryan, A.Kh.: On mathematical modelling of temporal spatial spread of epidemics. Math. Model. Nat. Phenom. 15(6), 1–13 (2020). https:// doi.org/10.1051/mmnp/2019056 9. Webb, G.: A reaction-diffusion model for a deterministic diffusive epidemic. J. Math. Anal. Appl. 84, 150–161 (1981) 10. Khachatryan, Kh.A., Petrosyan, H.S.: On the solvability of a class of nonlinear Hammerstein– Stieltjes integral equations on the whole line. Proc. Steklov Inst. Math. 308, 238–249 (2020) 11. Capasso, V.: Global solution for a diffusive nonlinear deterministic epidemic model. SIAM J. Appl. Math. 35(2), 274–284 (1978). https://doi.org/10.1137/0135022 12. Gomez, C., Prado, H., Trofimchuk, S.: Separation dichotomy and wavefronts for a nonlinear convolution equation. J. Math. Anal. Appl. 420(1), 1–19 (2014). https://doi.org/10.1016/j.jmaa. 2014.05.064 13. Williams, S.A., Chow, P.L.: Nonlinear reaction-diffusion models for interacting populations. J. Math. Anal. Appl. 62, 157–169 (1978). https://doi.org/10.1016/0022-247X(78)90227-5 14. Volkov, D., Lui R.: Spreading speed and travelling wave solutions of a partially sedentary population. JMA, J. Appl. Math. 72(6), 801–816 (2007). https://doi.org/10.1093/imamat/ hxm025 15. Schumacher, K.: Travelling-front solutions for integro-differential equations. I., J. für die reine und angewandte Math. 316, 54–70 (1980). https://doi.org/10.1515/crll.1980.316.54 16. Rudin, W.: Functional Analysis. McGraw-Hill Science/Engineering/Math (1991) 17. Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow (1981); Dover Publ., Mineola (1999) 18. Engibaryan, N.B.: Conservative systems of integral convolution equations on the haline and the entire line. Sb. Math. 193(6), 847–867 (2002). https://doi.org/10.1070/ SM2002v193n06ABEH000660

272

A. Kh. Khachatryan

19. Evans, A.S.: Viral infections of Humans. Epidemiology and Control. Plenum Medical Book Company, New York (1982); Springer (1989) 20. Cauchemez, S., Horby, P., Fox, A., Mai, L.Q., Thanh, L.T., Thai, P.Q., Hoa, L.N., Hien, N.T., Ferguson, N.M.: Influenza infection rates, measurement errors and the interpretation of paired serology. PLoS Pathog. 8(12). 1–14 (2012); e1003061. https://doi.org/10.1371/journal.ppat. 1003061

A Simple Wiener-Hopf Factorization Approach for Pricing Double-Barrier Options Oleg Kudryavtsev

Abstract This paper suggests a new approach to pricing double-barrier options under pure non-Gaussian Lévy processes with jumps of finite variation. The key idea behind the method is to represent the process under consideration as a difference between two subordinators. We use such a splitting rule to the Lévy process at exponentially distributed randomized time points. Then, we obtain the doublebarrier option price by solving recurrent simple Wiener-Hopf equations on the interval defined by the barriers. Keywords Wiener-Hopf factorization · Barrier options · Levy processes · Numerical methods · Laplace transform

1 Introduction A standard option pricing problem in computational finance deals with the numerical methods to compute an expectation of a specific-type function G, which depends on a stochastic process St modeling a stock price dynamic. If the payoff function G depends not only on the stock price at the terminal date but also on its observed trajectory, we obtain the typical setting for pricing path-dependent options in finance. The most popular path-dependent options are barrier options, which include double-barrier options. Recall that a double-barrier option is a contract which pays the specified amount G(ST ) at the terminal date T , provided during the lifetime of the contract; the price of the stock does not cross specified constant barriers D from

O. Kudryavtsev () Russian Customs Academy, Rostov Branch, Rostov-on-Don, Russia Southern Federal University, Rostov-on-Don, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_15

273

274

O. Kudryavtsev

above and U from below. When at least one of the barriers is crossed, the option expires worthless, or the option owner is entitled to some rebate. In recent years, researchers pay much attention to stochastic models of financial markets that depart from the classical Black-Scholes model. At this moment, a broad range of models is available. We concentrate on the class of one-factor non-Gaussian exponential Lévy processes that admit jumps in asset prices and can reproduce the well-known volatility smile phenomena. For an introduction to Lévy models in application to finance, we refer to [1, 2]. From a probabilistic viewpoint, double-barrier option prices can be expressed in terms of conditional expectation on a payoff function that depends on the underlying stochastic process and its supremum and infimum. In analytical terms, a doublebarrier derivative’s value is the solution to the Kolmogorov backward equation of a particular-type subject to appropriate initial and boundary conditions. In the case of Lévy models, one needs to solve complex partial integrodifferential equations where a spatial variable belongs to the interval specified by the lower and upper barriers. By now, several large groups of relatively universal numerical methods exist for pricing single-barrier options under Lévy processes. However, the known results on pricing double-barrier options are somewhat limited. Existing numerical methods in literature can be categorized into three groups: Monte Carlo simulations (see, e.g., [3–6]), backward induction methods (see, e.g., [7–13]), and numerical methods for solving integrodifferential equations (see, e.g., [14–20]). Standard Monte Carlo methods for pricing path-dependent options under Lévy models are rather slow. Incorporating the Wiener-Hopf factorization technique into the Monte Carlo algorithms leads to efficient pricing single-barrier and lookback options. In [6], the joint distribution of a Lévy process and its supremum and infimum under the Laplace transform was computed for a particular case of a Lévy measure whose restriction on a positive half line has a rational Laplace transform. In the general case, such results are unavailable. The backward induction methods are based on the fact that the risk-neutral valuation formula for the European option can be seen as a convolution of the payoff function with the transition density. The key idea is to set up a time lattice and view the option as of the European type between two adjacent dates. In the case of doublebarrier options, we need to consider boundary conditions at each time step. As a theoretical background behind the procedure, the third group uses the method of horizontal lines [21], which includes a time discretization while a space variable remains continuous. Carr suggested a meaningful probabilistic interpretation of the method in [22], which we call a time randomization technique or Carr’s randomization. After the time discretization, a sequence of specific stationary boundary problems for integrodifferential equations on an interval arises. In the case of continuously monitored options, one can also reduce the initial Kolmogorov backward equation to a convolution one on an interval applying the Laplace transform in time variable (see, e.g., [15, 17, 19]). To solve them, one may apply either finite difference methods like in [14] or Wiener-Hopf factorization method to solve two coupled integrodifferential equations (see, e.g., [15–17]). In both cases, the numerical

A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options

275

methods are rather more involved than in the single-barrier case and require approximate formulas for the Wiener-Hopf factors (see, e.g., [23, 24]). Analytical solutions can be found in a few Lévy models only (see, e.g., [19]). The doublebarrier problem is related to a matrix Wiener-Hopf factorization (see details in [18, 25]), which is not analytically available yet. Therefore, pricing double-barrier options in exponential Lévy models remains a computational challenge. The goal of the current paper is to suggest a new, easy, and effective method to price double-barrier options under pure non-Gaussian Lévy processes with jumps of finite variation. The main advantage of the approach is applying explicit WienerHopf factorization formulas similar to the ones suggested in [26] for options with a single barrier. The key idea behind our method is to represent the process under consideration as a sequence of positive and negative jumps. We apply such splitting rule to the process at consecutive exponentially distributed randomized time points and reduce the problem to iterative solving of simple Wiener-Hopf equations on the interval.

2 Theoretical Background 2.1 Lévy Processes and Wiener-Hopf Factorization: Basic Facts A Lévy process is a stochastically continuous process with stationary independent increments (for general definitions, see, e.g., [27]). A Lévy model may have a Gaussian component and pure jump component. The latter is characterized by the density of jumps, which is called the Lévy density. A Lévy process Xt can be completely specified by its characteristic exponent, ψ, definable from the equality E[eiξ X(t )] = e−t ψ(ξ ) (we confine ourselves to the one-dimensional case). The Lévy-Khintchine formula gives the characteristic exponent ψ(ξ ) =

σ2 2 ξ − iμξ + 2



+∞ −∞

(1 − eiξy + iξy1[−1;1](y))F (dy),

(1)

where σ 2 ≥ 0 is the variance of the Gaussian component, 1A is the indicator function of the set A, and the Lévy measure F (dy) satisfies  min{1, y 2 }F (dy) < +∞. R\{0}

If the jump component is a process of finite variation, equivalently, if  min{1, |y|}F (dy) < +∞, R\{0}

(2)

276

O. Kudryavtsev

then (1) can be simplified σ2 2 ξ − iμξ + ψ(ξ ) = 2



+∞

−∞

 1 − eiξy F (dy),

(3)

with a different μ, and the new μ is the drift of the Gaussian component. Assume that the riskless rate r is constant and, under a risk-neutral measure chosen by the market, the underlying process evolves as St = S0 eXt , where Xt is a Lévy process. Then, we must have E[eXt ] < +∞, and therefore, ψ must admit the analytical continuation into the strip 0ξ ∈ (−1, 0) and continuous continuation into the closed strip 0ξ ∈ [−1, 0]. Further, if d ≥ 0 is the constant dividend yield on the underlying asset, then the following condition (the EMM-requirement) must hold: E[eXt ] = e(r−d)t . Equivalently, r − d + ψ(−i) = 0,

(4)

which can be used to express the drift μ via the other parameters of the Lévy process: σ2 + μ=r −d − 2



+∞ −∞

(1 − ey + y1[−1;1](y))F (dy),

(5)

where 1A (x) = 1 if x ∈ A, and otherwise, it is equal to 0. In the examples below, we list some popular classes of Lévy processes in empirical studies of financial markets. Example 1 (Tempered Stable Lévy Processes) The characteristic exponent of a pure jump KoBoL process of order ν ∈ (0, 2), ν = 1 is given by ψ(ξ ) = −iμξ + cΓ (−ν)[λν+ − (λ+ + iξ )ν ]

(6)

+cΓ (−ν)[(−λ− )ν − (−λ− − iξ )ν ], where c > 0, μ ∈ R, and λ− < −1 < 0 < λ+ . The characteristic exponent (6) is derived in [2] from the Lévy-Khintchine formula with the Lévy densities of negative and positive jumps, F∓ (dy), given by F∓ (dy) = ceλ± y |y|−ν−1 dy;

(7)

in the first two papers, the name extended Koponen family was used. Later, the same class of processes was used in [28] under the name CGMY model. The following relations between parameters of KoBoL model and C, G, M, Y parameters of CGMY model are valid: C = c, Y = ν, G = λ+ , M = −λ− .

A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options

277

More general version with c± instead of c and the different exponents ν± is known as a tempered stable Lévy model [1]. In this case, we have for ν+ , ν− ∈ (0, 2), ν+ , ν− = 1   ν ψ(ξ ) = −iμξ + c+ Γ (−ν+ ) λ++ − (λ+ + iξ )ν+ +c− Γ (−ν− )[(−λ− )

ν−

(8)

− (−λ− − iξ ) ], ν−

where c+ , c− > 0, μ ∈ R, and λ− < −1 < 0 < λ+ . Example 2 (Variance Gamma Processes) The Lévy density of a variance gamma process is of the form (7) with ν = 0, and the characteristic exponent is given by (see [29]) ψ(ξ ) = −iμξ + c[ln(λ+ + iξ ) − ln λ+ + ln(−λ− − iξ ) − ln(−λ− )],

(9)

where c > 0, μ ∈ R, and λ− < −1 < 0 < λ+ . Example 3 (Kou Model) If F∓ (dy) are given by exponential functions on negative and positive axis, respectively, F∓ (dy) = c± (±λ± )eλ± y , where c± ≥ 0 and λ− < 0 < λ+ , then we obtain Kou model. The characteristic exponent of the process is of the form ψ(ξ ) =

ic− ξ ic+ ξ σ2 2 ξ − iμξ + + . 2 λ+ + iξ λ− + iξ

(10)

The two-sided version was introduced in [30]. Example 4 (β-Class) The β-class of Lévy processes, introduced in [31], is a tenparameter Lévy process which has a characteristic exponent 1 ψ(ξ ) = aξ + σ 2 ξ 2 2  c1 + B(α1 , 1 − λ1 ) − B α1 − β1  c2 + B(α2 , 1 − λ2 ) − B α2 + β2

iξ , 1 − λ1 β1 iξ , 1 − λ2 β2

 

278

O. Kudryavtsev

with parameter range a, σ ∈ R, c1 , c2 , α1 , α2 , β1 , β2 > 0 and λ1 , λ2 ∈ (0, 3) \ {1, 2}. Here, B(x, y) = Γ (x)Γ (y)/Γ (x + y) is the beta function. The density of the Lévy measure is given by π(x) = c1

e−α1 β1 x eα2 β2 x 1 + c 1{x0} 2 (1 − e−β1 x )λ1 (1 − eβ2 x )λ2

2.2 Wiener-Hopf Factorization There are several forms of the Wiener-Hopf factorization. Let X t = sup0≤s≤t Xs and X t = inf0≤s≤t Xs be the supremum and infimum of Lévy process Xt . The Wiener-Hopf factorization formula used in probability reads     iξ X   E eiξ XTq = E eiξ XTq E e Tq ,

∀ ξ ∈ R,

(11)

where Tq ∼ Exp q is an exponentially distributed random variable and independent of X. Notice that the following useful relations hold (see details in [27]): XTq ∼ XTq − X Tq

(12)

XTq ∼ XTq − XTq .

(13)

One can easily show that the r.h.s. of (11) can be represented in terms of the characteristic exponent ψ(ξ ) of Xt as follows: 

E e

iξ XTq 



∞

= qE

e

−qt iξ Xt

e

0





∞

dt = q

e−qt E[eiξ Xt ]dt =

0

q . q + ψ(ξ )

Introducing the notation φq+ (ξ ) = qE φq− (ξ ) = qE

 

∞

 % # e−qt eiξ Xt dt = E eiξ XTq ,

∞

 # % iξ X e−qt eiξ Xt dt = E e Tq ,

0

0

we can rewrite (11) as q = φq+ (ξ )φq− (ξ ). q + ψ(ξ )

(14)

A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options

279

If XTq = XTq , then X Tq = 0, and we obtain a trivial factorization identity: φq− (ξ ) =

q , q + ψ(ξ )

φq+ (ξ ) = 1.

(15)

In another particular case, when X Tq = XTq , we have X Tq = 0, and the factorization factors have a simple form: φq+ (ξ ) =

q , q + ψ(ξ )

φq− (ξ ) = 1.

(16)

In order to write the factorization identity (11) in operator form, we define the normalized resolvent of X for a function u(x) as Eq u(x) = E[u(x + XTq )].

(17)

In [32], Eq is referred as the expected present value operator (EPV operator) due to the observation that, for a stream u(Xt ), 

+∞

Eq g(x) = E

qe

−qt

 u(x + Xt )dt .

(18)

0

Replacing in (18) process X with the supremum X and infimum X of process X, we obtain the EPV operators E± q under X and X, respectively. Equivalently,   E+ q u(x) = E u(x + X Tq ) ,   E− q u(x) = E u(x + X Tq ) .

(19) (20)

It follows from the definition that + E+ q 1(−∞,h) (·) = 1(−∞,h) (·)Eq 1(−∞,h) (·);

(21)

− E− q 1(0,+∞) (·) = 1(0,+∞) (·)Eq 1(0,+∞) (·).

(22)

The operator form of the Wiener-Hopf factorization is written as follows (see details in [32]): − − + Eq = E+ q Eq = Eq Eq .

(23)

Note that (23) are understood as equalities for operators in appropriate function spaces, for instance, in the space of semi-bounded Borel functions. Under appropriate conditions on the characteristic exponent, the EPV operators are defined as operators in spaces of functions of exponential growth at infinity, and (23) holds in these spaces.

280

O. Kudryavtsev

Let Z be a continuous random variable with probability density. Then, for a wide class of functions u(x), the following useful relation holds E[u(x + Z)] = (2π)−1



+∞ −∞

# % eixξ E eiξ Z u(ξ ˆ )dξ,

(24)

where uˆ is the Fourier transform of a function u:  +∞ u(ξ ˆ )= e−ixξ u(x)dx. −∞

− Hence, we can interpret Eq , E+ q , andEq as pseudo-differential operators (PDO) q + − with the symbols q+ψ(ξ ) , φq (ξ ), andφq (ξ ), respectively. Denote by Dx the difd . Recall that a PDO a(Dx ) with the symbol a(ξ ) acts as ferential operator −i dx follows:  +∞ −1 a(Dx )u(x) = (2π) eixξ a(ξ )u(ξ ˆ )dξ, (25) −∞

−1 is the Notice that Dx = Fξ−1 →x ξ Fx→ξ , where F is the Fourier transform and F inverse Fourier transform:  +∞ −1 −1 ˆ = (2π) eixξ u(ξ ˆ )dξ = u(x). (F u)(x) −∞

Hence, one can represent (25) shortly as Fξ−1 →x a(ξ )Fx→ξ u(x). ± Taking into account that Eq = q(q + ψ(Dx ))−1 and E± q = φq (Dx ), we may write (17) and (19)–(20) as −1 Eq u(x) = Fξ−1 →x q(q + ψ(ξ )) Fx→ξ u(x);

(26)

−1 ± E± q u(x) = Fξ →x φq (ξ )Fx→ξ u(x).

(27)

If φq± (ξ ) are known explicitly, then each operator on the rightmost part of (26) and (27) can be numerically implemented by means of the fast Fourier transform (FFT).

3 Splitting Rule and Wiener-Hopf Factorization Let T , K, D, and U be the maturity, strike, lower barrier, and upper barrier, and the stock price St = DeXt be an exponential Lévy process under a chosen riskneutral measure (see (4)) which has no diffusion component (σ = 0) and only jumps of finite variation (see (2)). Without loss of generality, we confine ourselves

A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options

281

to a double-barrier put option. Set the riskless rate and the dividend rate equal to r and d, respectively. As a basic example to illustrate our method, we consider an approach to pricing continuously monitored double-barrier put options without rebate under a Lévy process with the characteristic exponent (3) that satisfies (2).

3.1 The Problem Setup and General Pricing Formulas Let us introduce h = ln U/D. Then, the payoff at maturity is 1(0,h)(XT )G(XT ), where G(x) = (K − Dex )+ , and the no-arbitrage price of the double-barrier put option at the beginning of a period under consideration (t = 0) and Xt = x with x ∈ (0, h) given by # % V (T , x) = E x e−rT 1XT >0 1XT 0 1Xt 0 1Xt 0 1Xt 0 1XT

q+r 0 1Xt 0 1XT(n,q+r) 0 and the rate parameter q > 0. Taking into account that T(n, q + r) ∼ T(n − 1, q + r) + Tq+r , we can represent XT(n,q+r) as a sum n , XT 1 + . . . + XTq+r q+r

1 ,. . . ,T n are consecutive time increments being independent exponenwhere Tq+r q+r tially distributed random variables with the parameter q + r. Using the relations

1XT(n,q+r) >0 = 1XT(n−1,q+r) >0 1XT(n−1,q+r) +XT n

q+r

>0 ,

and 1XT(n,q+r) 0 ] =

1 E− 1(0,+∞) E+ q+r vn−1 (q, x). (1 + r/q) q+r

(37)

Unfortunately, in the case of double-barrier options (with h < +∞), the formulas similar to (37) are not available for general Lévy models. Instead, one needs to factorize the following matrix: 

exp(iξ h) 0 (q + r + ψ(ξ ))/(q + r) exp(−iξ h),



284

O. Kudryavtsev

(see, e.g., [25])) or to find an iterative solution to the pair of certain coupled integral equations (see details in [13, 15]). The new approach to calculating (35) requires the following steps. First, we represent Xt as a difference between two subordinators Xt+ and −Xt− : Xt = Xt+ − (−Xt− ). Recall that a subordinator is a Lévy process with sample paths being almost surely nondecreasing. According to [1, Proposition 3.10], a subordinator has no diffusion component, only positive jumps of finite variation and nonnegative drift. Denote by ψ+ (ξ ) and ψ− (ξ ) the characteristic exponents of Xt+ and Xt− , respectively. If μ ≥ 0 in (3), we define ψ+ (ξ ) and ψ− (ξ ) as follows:  ψ+ (ξ ) = −iμξ +  ψ− (ξ ) =

0 −∞

+∞

(1 − eiξy )F (dy),

0

(1 − eiξy )F (dy),

otherwise,  ψ+ (ξ ) =

+∞

 1 − eiξy F (dy),

0



ψ− (ξ ) = −iμξ +

0 −∞

 1 − eiξy F (dy).

Notice that Xt− has almost surely nonincreasing sample paths. Hence, we have that +

Xt = Xt+ ,

(38)

− X− t = Xt .

(39)

Let Xt+,1 and Xt+,2 be Lévy processes with the same characteristic exponent ψ+ (ξ ), i.e., Xt+,1 ∼ Xt+ and Xt+,2 ∼ Xt+ . Due to the property of increments of a − Lévy process being stationary independent, we conclude that Xt and Xt+,1 /2 + Xt +

Xt+,2 /2 are identically distributed. It means that for a fixed t > 0, the current position of Xt with starting point x has the same distribution as the final position of the discrete-time process Yjt with the following dynamics: • • • •

Y0t Y1t Y2t Y3t

= x. = Y0t + Xt+,1 /2 —an upward movement. − t = Y1 + Xt —a downward movement. = Y2t + Xt+,2 /2 —an upward movement.

A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options

285

For a short time period [0, t], we may approximate the value of X and X at a given time t by the correspondent maximum and minimum of Y t at the time j = 3. Let us denote by Y t = min{Y0t , Y1t , Y2t , Y3t } and Y t = max{Y0t , Y1t , Y2t , Y3t }. Notice that extrema of Y can be easily defined as follows:   − Y t = min x, x + Xt+,1 /2 + Xt ,  +,1 +,2  − Y t = max x, x + Xt+,1 /2 , x + Xt /2 + Xt + Xt /2 .

(40)

Let a natural number N be sufficiently large and q = N/T . As explained in Sect. 3.1, vN (N/T , x) defined iteratively by (35) gives an approximate price of the double-barrier option V (T , x) (see (28)). For computing (35), we approximate XTq+r in (35) with Y Tq+r , since the randomized time Tq+r converges in quadratic mean to 0 as N → +∞. Notice that in this case, the following relations hold: Y3 q+r = x + XT+,1 + XT−q+r + XT+,2 ; q+r /2 q+r /2 T

(41)

1Y Tq+r >0 = 1(0,+∞)(x) × 1(0,+∞) (x + XT+,1 + XT−q+r ); q+r /2

(42)

1Y Tq+r 0 1Y Tq+r 0

=

Tq+r /2 0}. Most known example of such family is the normal and the logistic distribution families. Another class of the distribution family proposed in [12] is R = {R((x/β)α ), α > 0, β > 0, x ∈ X ⊂ [0, ∞)}. The limit distribution of the Cramér–von Mises statistic for this class does not depend on the unknown parameters α and β. Most known examples of such family are the Weibull and the lognormal distribution families. Both families are generated

Cramér–von Mises Test for Parametric Families

297

Table 1 Correspondence between the distribution families G and R Y = e−X (Y = eX )

X = −ln Y (X = ln Y ) 1 FX (x) = G( x−θ θ2 ), θ2 G(x) = 1 − R(e−x )

FY (x) = R(( βx )α ), α, β > 0

>0

R(x) = 1 − G(−ln x) α = 1/θ2 , β = e−θ1

θ1 = −ln β, θ2 = 1/α Normal distribution family Negative exponential distribution F (x) = ex/θ2 , −∞ < x < 0 Extreme value distribution

Lognormal distribution family Pareto distribution F (x) = 1 − x α , x > 1 Weibull distribution α F (x) = 1 − e−(x/β) ,

x−θ − θ 1

F (x) = e−e 2 , x > θ1 Exponential distribution F (x) = 1 − e−x/θ2 , x > 0 Logistic distribution  1  , −∞ ≤ x ≤ ∞ F (x) = π x−θ1 1+exp − √

3

θ2

x>0 Power distribution F (x) = x α , x ∈ [0, 1] Heavy-tailed distribution 1 √ F (x) = 1 − , x≥0 (π/ 3)α 1+(x/β)

by the root distribution functions G(x) and R(x). They are related by the forms G(x) = 1 − R(e−x ) and vice versa R(x) = 1 − G(−ln x). The corresponding random variables X and Y are transformed by the formulas X = −ln Y and Y = e−X or X = ln Y and Y = eX . The parameters of the two families are converted by the formulas θ1 = −ln βand θ2 = 1/α and vice versa α = 1/θ2 and β = e−θ1 . The results are shown in Table 1. When testing a hypothesis, a sample can be converted from one family to another. Moreover, the distributions of the statistics are the same in both cases. It can be seen that although the Table 1 presents many important distributions, it does not contain the gamma distribution. The distribution of the Cramér–von Mises statistic for gamma distribution is studied in Sect. 2. It turns out that the limiting distribution in this case depends on unknown parameters. What to do in this case? Section 2 proposes some method to overcome this difficulty. Section 1.4 contains the Khmaladze method for the same purpose. Both methods have their own advantages and disadvantages.

1.3 Formula for Calculation of the Cramér–von Mises Statistic The simple formulas for computing the value of the Cramér–von Mises statistic exist (see, e.g., [2, 3]): For traditional Cramér–von Mises type tests, there is a very simple formula for calculating the values of their statistic. This is the formula: ωn2

 n   1 i − 1/2 2 + = t(i) − 12n n i=1

(3)

298

G. Martynov

where t(i) = G(X(i) ), X(1) ≤ X(2) ≤ · · · ≤ X(n) , designed for the simple hypothesis, and t(i) = G(X(i) , θˆn ), X(1) ≤ X(2) ≤ · · · ≤ X(n) , designed for the parametric hypothesis. Here, X(1) ≤ X(2) ≤ · · · ≤ X(n) is an ordered sample X.

1.4 Khmaladze Transformation of the Parametric Empirical Process Khmaladze has proposed in [8] √ (see also [9]) the following transformation of the parametric empirical process n(Fn (x) − G(x, θˆn )) to the form wn (x) =

√ n{Fn (x) − Kn (x, θˆn )},

where compensator Kn is Kn (x, θˆn ) =



x

#

q 5 (y, θˆn )

∞



∞

×

q(s, θˆn )q (s, θˆn )dG(s, θˆn ) 5

−1 

y

∞

, q(z, θˆn )dFn (z)

y

 × dG(y, θˆn ) . The analog of the Cramér–von Mises statistic is  ωn2 =

0

1

2 wn (x)|t =G(x,θ) ˆ

dt.

Here, the empirical process weakly converges to the Wiener process, that is, in the limit, it does not depend on the unknown parameters of sufficiently regular families and from the families themselves. The advantage is that in practice, only a few percentage point values need to be remembered. The disadvantage is that each parametric family requires a separate algorithm to be developed, which can be very complex.

Cramér–von Mises Test for Parametric Families

299

1.4.1 Application to the Exponentiality Test We will now consider the problem of testing hypothesis H0 : F (x) ∈ {G(x, λ) = 1 − e−λx }, x > 0, λ > 0. Then, the compensator Kn (x, θn ) can be represented as (see [6]) λˆ  ˆ Xi (2 − (λ/2)X i )) n

Kn (x, θn ) =

i:Xi≤t

1 ˆ ˆ +λx(2 + (λ/2)x)(1 − Fn (x)) − λˆ 2 x n



Xi

i:Xi≤t

where λˆ is the sample mean.

2 Cramér–von Mises Test for the Gamma Distribution Family In this section, a short derivation of the formula for the traditional empirical process covariance function for the parametric family of the gamma distribution will be given. Exact distribution tables of Cramer–Mises statistics for the family of gamma distributions will be presented here. The Tables 2, 3 and 4 are obtained by exact calculations. Table 2 Exact critical values of the significance levels 0.1 for the Cramér-von Mises statistic for gamma distribution with κ >= 1.0 κ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

t0.1,κ 0.1112 0.1105 0.1099 0.1093 0.1089 0.1085 0.1082 0.1079 0.1077 0.1074

κ 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

t0.1,κ 0.1072 0.1069 0.1066 0.1063 0.1061 0.1059 0.1058 0.1056 0.1055 0.1054

κ 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

t0.1,κ 0.1053 0.1051 0.1049 0.1048 0.1047 0.1046 0.1045 0.1045 0.1044 0.1043

κ 9.0 10 12 14 18 20 25 40 50 90

t0.1,κ 0.1043 0.1042 0.1041 0.1040 0.1039 0.1039 0.1038 0.1037 0.1037 0.1036

300

G. Martynov

Table 3 Exact critical values of the significance levels 0.05 for the Cramér-von Mises statistic for gamma distribution with κ >= 1.0 κ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

t0.05,κ 0.1362 0.1353 0.1344 0.1338 0.1332 0.1327 0.1322 0.1318 0.1315 0.1312

κ 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

t0.05,κ 0.1309 0.1304 0.1300 0.1297 0.1294 0.1292 0.1290 0.1288 0.1286 0.1285

Table 4 Exact critical values of the significance levels 0.1 for the Cramér-von Mises statistic for gamma distribution with κ < 1.0

κ 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 κ 0.01 0.02 0.03 0.04 0.05 0.06

t0.05,κ 0.1284 0.1281 0.1279 0.1277 0.1275 0.1274 0.1273 0.1272 0.1271 0.1271 t0.1,κ 0.1711 0.1679 0.1649 0.1621 0.1595 0.1571

κ 0.07 0.08 0.09 0.10 0.20 0.30

κ 9.0 10 12 14 18 20 25 40 50 90 t0.1,κ 0.1548 0.1526 0.1506 0.1488 0.1351 0.1271

t0.05,κ 0.1270 0.1269 0.1268 0.1266 0.1265 0.1264 0.1263 0.1262 0.1262 0.1261 κ 0.40 0.50 0.60 0.70 0.80 0.90

t0.1,κ 0.1221 0.1187 0.1163 0.1145 0.1131 0.1120

Let’s define the density function and the gamma distribution function as follows: g(x; θ, κ) =

1 (κ, x/θ ) . x κ−1 e−(x/θ), G(x; θ, κ) = κ (κ)θ (κ)

We will derive the covariance function in accordance with the formula (2). First, we find, using formulas from [1], the derivatives of the distribution function with respect to κ and θ :

Gκ (x; θ, κ) = −

1 W (x, θ, κ), (κ)

where W (z, θ, κ) =

  z  1  z κ 2 F2 {κ, κ}, {1 + κ, 1 + κ}; − 2 κ θ θ  z    z − (κ) ln − ψ(κ) , +  κ, θ θ

2 F2 (·) is a generalized hypergeometric function, (κ, z) is the incomplete gamma function,

(4)

Cramér–von Mises Test for Parametric Families

301

ψ(z) is the digamma function:

Gθ (x; θ, κ) = −e− θ

x

 x κ θ

/(θ (κ)).

The vector q(x; θ, κ) as a result will be



q(x; θ, κ) = (Gκ (x; θ, κ), Gθ (x; θ, κ)), The information matrix will be as follows:  I (θ, κ) =

ψ (κ) 1 θ

1 θ κ θ2

 .

Covariance function of the standard empirical process is K(t, τ ; θ, κ) = min(t, τ ) − t τ

  − q(x; θ, κ)T I (θ, κ)q(y; θ, κ)

t =G(x;θ,κ), τ =G(y;θ,κ)

.

Theorem 1 The covariance function of the considered limiting empirical process can be represented as follows: K(t, τ, θ, κ) = min(t, τ ) − tτ ⎛  xy κ − x + y  k − x 1 2 x ⎝ θ κ − 4 2 e e θ W (y, θ, κ) +θ θ  (κ) θ2 θ ⎞   y κ − y  e θ W (x, θ, κ) + θ 4 ψ (κ)W (x, θ, κ)W (y, θ, κ)⎠ +θ 2 θ  x = G−1 (t; θ, κ)

y = G−1 (τ ; θ, κ)

where W (x, θ, κ) is determined by the formula (4). From the formulas obtained, it is easy to conclude that the variation function depends only on the κ parameter. This fact was noted for the first time in [17] and then by many authors. In work [17] and more recent works, tables obtained by modeling are given. Here for the first time, four-digit tables are obtained by precise calculations. The change in the critical value when changing the value θ at a given significance level is relatively small. Since the distribution of the statistic depends on an unknown parameter, the hypothesis will be tested according to the following general procedure. Procedure 1 If there is a table of critical value corresponding to the exact unknown values of the vector of parameters under the null hypothesis, then we select the level

302

G. Martynov

in it corresponding to the estimate of this vector. If there is no such table, then we carry out a sufficiently exact modeling, from which we find an approximate critical value. Statistics are calculated using the formula (3). The advantage of this criterion is the ease of calculating the statistic after finding the parameter estimates. The disadvantage is that it is necessary to find the critical value in one way or another each time. Both the proposed procedure and Khmaladze’s method [8] face the problem of finding parameter estimates, which, with a large dimension of the parameter vector, can be a difficult task. The disadvantage of Khmaladze’s method, on the contrary, lies in the difficulty of calculating the statistics itself, expressed in the form of a multiple integral.

3 Hypernormal Distribution Family In this section, we will consider a three-parameter hypernormal distribution. An increase in the number of parameters in the family of distributions leads to the fact that, using goodness-of-fit tests, we approximate the unknown distribution of the sample by some distribution from this family. The probability of rejecting the hypothesis with such a family of distributions will be negligible. The family under consideration here includes several simpler families.

3.1 Test with the Maximum Likelihood Estimation There, we will consider the three parametric distribution functions with the density h(x) =

1 2θ1(1 +

1 θ2 )

e

   x−θ θ2 − θ 0  1

(see [10, 14, 16]). In Fig. 1, there are several representatives of the family of hypernormal distributions. Let X = (X1 , X2 , . . . , Xn ) be a sample from a random variable having a hypernormal distribution. To estimate the parameters, we will maximize the logarithm of the likelihood function Ln ,     i=n   Xi − θ0 θ2 1   , − log Ln = −n log 2 + log θ1 + log (1 +  θ  θ2 1 i=1

Cramér–von Mises Test for Parametric Families

303

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Fig. 1 Three parametric hypernormal distributions

Derivatives with respect to parameters are   i=n  Xi − θ0 θ2 −1 ∂ θ2    log Ln = sign(Xi − θ0 )  , ∂θ0 θ1 θ1  i=1

 i=n  ∂ n θ2   Xi − θ0 θ2 −1 log Ln = − + ,  θ  ∂θ1 θ1 θ1 1 i=1

ψ(1 + ∂ log Ln = n ∂θ2 θ22

1 θ2 )

−

   i=n    Xi − θ0 θ2     log  Xi − θ0  .  θ    θ i=1

1

1

We obtain equations for estimates ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

   Xi −θ0 θ2 −1 sign(X − θ ) = 0,   i 0 i=1 θ 1 θ 2    Xi −θ0 −1 + θn2 i=n i=1  θ1  = 0,       Xi −θ0 θ2  Xi −θ0  − n1 i=n log    i=1 θ1 θ1  = 0.

i=n

⎪ ⎪ 1 ⎪ ⎪ ⎩ ψ(1+2 θ2 ) θ2

Figure 2 presents the empirical distribution functions for the Cramér–von Mises statistic at n = 5000 for each function. The simulation was carried out at θ0 = 0, θ1 = 1, and at various values of θ2 . The empirical distribution functions are condensed at θ2 greater than five. Estimates for other θ2 were found to be strongly biased. This can be explained by both the irregularity of the family and insufficient sample sizes.

304

G. Martynov

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

Fig. 2 Empirical distribution of the Cramér–von Mises statistic for different values of θ2 with θ0 = 0 and θ1 = 1. Sample size is equal to 5000

3.2 Test with Moments Estimation Classic variants of the parametric Cramér–von Mises tests use maximum likelihood estimates. These estimates are convenient for the traditional parametric families discussed in the first section. However, their use for the family of hypernormal distributions considered in the previous subsection leads, when calculating distribution tables for each specific vector of parameters, to the need for a large number of calculations. In this section, the maximum likelihood estimate is replaced by the method of moments. Let us first express the first four moments of the hypernormal distribution in terms of its parameters. The kth central moment is obtained as follows:  Eh (x − θ0 ) =

∞

k

=

−∞

(x − θ0 )k 2θ1 (1 +

⎧ ⎨$∞ ⎩

0

e

   x−θ θ2 − θ 0 

1 θ2 )  θ 2 − θx

xk e θ1 (1+ θ1 )

1

1

dx,

dx, k = 0, 2, 4, . . . ,

2

0, otherwise,

Cramér–von Mises Test for Parametric Families

305

  ⎧ k  k+1 ⎪ θ ⎪ ⎨ 1  θ2  , k = 0, 2, 4, . . . , = θ  1+ 1 ⎪ θ2 ⎪ 2 ⎩ 0, otherwise.

(5)

It used the formula 

∞

k −x θ2

x e 0

  1 k+1 dx =  . θ2 θ2

Finally, we get the following formula Eh (x − θ0 ) = k

θ1k qk (θ2 ), k = 0, 2, 4, . . . , 0, otherwise,

where  qk (θ2 ) =



k+1 θ2

 θ2  1 +

 1 θ2

.

(6)

It follows from the identities x ≡ θ0 + (x − θ0 ), x 2 ≡ θ02 + 2θ0(x − θ0 ) + (x − θ0 )2 , x 3 ≡ θ03 + 3θ02 (x − θ0 ) + 3θ0(x − θ0 )2 + (x − θ0 )3 , x 4 ≡ θ04 + 4θ03 (x − θ0 ) + 6θ02 (x − θ0 )2 + 4θ0 (x − θ0 )3 + (x − θ0 )4 and formula (5) the following equalities for the sample moments μ1 = θ 0 , μ2 = θ02 + θ12 q2 (θ2 ), μ3 = θ03 + 3θ0 θ12 q2 (θ2 ), μ4 = θ04 + 6θ02 θ12 q2 (θ2 ) + θ14 q4 (θ2 ). Replacing μ1 , μ2 , μ3 , and μ4 with the sample moments about the origin m1 , m2 , m3 , and m4 , correspondingly, we obtain the following equations for the estimates θ˜0 , θ˜1 , and θ˜2 of the parameters θ0 , θ1 , and θ2 : m1 = θ˜0 , m2 = θ˜02 + θ˜12 q2 (θ˜2 ),

306

G. Martynov

Table 5 Critical values of 0.1 for the Cramér–von Mises statistic with parameters estimated by the method of moments 5000 T0,1,2.5

0.090

5000 5000 T0.5,1,4 = 0.185 T−0.5,1,4 = 0.167 2500 2500 2500 2500 2500 = T0,1,3 = 0.084 T0,1,4 = 0.082 T0,1,5 = 0.082 T0,1,7 = 0.082 T0,1,9 = 0.093 5000 = 0.081 T 5000 = 0.090 T0,1.1,4 0,1.5,4

m3 = θ˜03 + 3θ˜0 θ˜12 q2 (θ˜2 ), m4 = θ˜04 + 6θ˜02 θ˜12 q2 (θ˜2 ) + θ˜14 q4 (θ˜2 ). In this system of equations, the second and third equations are inconsistent. We put θ˜12 q2 (θ˜2 ) = (m2 − m21 ) in the fourth equation. We get the equation m4 = m41 + 6m1 (m2 − m21 ) + (m2 − m21 )2

q4 (θ˜2 ) q 2 (θ˜2 ) 2

to find θ˜02 . Finally, we get with using formula (6) the following equation   S(θ˜2 ) ≡ θ˜2



5 θ˜2

  1+  

2

3 θ˜2

1 θ˜2

 −

m4 − m41 (m2 − m21 )2

+

6m21 m2 − m21

= 0.

(7)

Having calculated the estimate θ˜2 from this equation and having the estimate θ˜0 = m1 , we obtain the estimate θ1 as θ˜1 =

=

(m2 − m21 )/q2 (θ˜2 ).

We denote by Tθn0 ,θ1 ,θ2 critical values of the Cramér–von Mises statistic, where n is the sample size. Some critical values of the significance level 0.1 are presented in Table 5. Figure 3 represents the graphs of the left-hand sides of Eq. (7) to find the estimate θ˜2 of the parameter θ2 . From the results of this and the previous subsection, it can be concluded that the verification of the described hypotheses with the applied estimation methods can be carried out only in the region of the family, remote from its irregular part.

Cramér–von Mises Test for Parametric Families

307

1.5

1

0.5

0

-0.5

-1

-1.5

1

2

3

4

5

6

7

8

9

10

Fig. 3 Graph of the function S(θ˜2 ) depending from sample moments m1 , m2 , and m4 under conditions that the parameter vector (θ0 , θ1 , θ2 ) takes values (0, 1, 2), (0, 1, 3), (0, 1, 4), (0, 1, 5), and (0, 1, 7)

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Integrals, 1046 pp. Dover, New York (1965) 2. Anderson, T.W., Darling, D.A.: Asymptotic theory of certain “Goodness of Fit” criteria based on stochastic processes. Ann. Math. Stat. 23(2), 193–212 (1952) 3. Durbin, J.: Weak Convergence of the sample distribution function when parameters are estimated. Ann. Stat. 1(2), 279–290 (1973) 4. Durbin, J., Knott, M., Taylor, C.C.: Components of Cramer–von Mises statistics. II. J. R. Stat. Soc. Ser. B 37(2), 216–237 (1975) 5. Gikhman, I.I.: One conception from the theory of ω2 -test [in Ukrainian]. Nauk. Zap. Kiiv Univ. 13, 51–60 (1954) 6. Haywood, J., Khmaladze, E.: On distribution-free goodness-of-fit testing of exponentiality. J. Econ. 143, 5–18 (2005) 7. Kac, M., Kiefer, J., Wolfowitz, J.: On tests of normality and other tests of goodness-of-fit based on distance methods. Ann. Math. Stat. 30, 420–447 (1955) 8. Khmaladze, E.V.: A martingale approach in the theory of parametric goodness-of-fit tests. Theor. Prob. Appl. 26(2), 240–257 (1981) 9. Koul, H.L., Swordson, E.: Khmaladze transformation. In: International Encyclopedia of Statistical Science, pp. 715–718. Springer, Berlin 10. Lemeshko, B.Yu., Lemeshko, S.B., Postovalov, S.N.: Statistic distribution models for some nonparametric goodness-of-fit tests in testing composite hypotheses. Commun. Stat. Theory Methods 39(3), 460–471 (2010) 11. Martynov, G.V.: The Omega Square Tests, 80pp. Nauka, Moscow (1979) 12. Martynov, G.: Note on the Cramer–von Mises test with estimated parameters. Publ. Math Debrezen. 76(3), 341–346 (2010)

308

G. Martynov

13. Martynov, G.: Weighted Cramer–von Mises test with estimated parameters. Commun. Stat. Theory Methods 40(19–20), 3569–3586 (2011) 14. Nadarajah, S.: A generalized normal distribution. J. Appl. Stat. 32(7), 685–694 (2005) 15. Neuhaus, G.: Asymptotic properties of the Cramér–von Mises statistic when parameters are estimated. In: Proc. Prague Symp. Asymptotic Stat., vol. 2, pp. 257–297. Charles University, Prague (1973/1974) 16. Roenko, A.A., Lukin, V.V., Djurovic, ´ I., Simeunovic, ´ M.: Estimation of parameters for generalized Gaussian distribution. In: 6th International Symposium on Communications, Control and Signal Processing (ISCCSP), Athens, pp. 376–379 (2014) 17. Stephens, M.A.: Tests based on EDF statistics. In: D’Agostino, R.B., Stephens, M.A. (eds.) Goodness-of-Fit Techniques, pp. 97–193. Marcel Dekker, New York (1986) 18. Van der Vaart, A.W., Wellner, J.A.: Weak Converge and Empirical Processes, 508 pp. Springer, Berlin (1996)

CVaR Hedging in Defaultable Jump-Diffusion Markets Alexander Melnikov and Hongxi Wan

Abstract This paper deals with the problem of conditional value-at-risk (CVaR)based optimal partial hedging in a defaultable jump-diffusion market. The minimal superhedging costs of claims with a zero recovery rate are derived. Meanwhile, we convert the CVaR minimization problem into a static optimization problem in the corresponding default-free market, and the solution of such a problem is given with the help of the Neyman-Pearson lemma. Furthermore, our method is implemented to derive minimal values of CVaR and optimal hedging strategies of defaultable equity-linked life insurance contracts, whose payoffs are equal to the maximum of two risky assets conditioned by the occurrence of default events. Keywords Partial hedging · Conditional value-at-risk · Jump-diffusion model · Equity-linked life insurance contracts · Default

1 Introduction Hedging and risk management are crucial topics in financial mathematics. In a complete market, any contingent claim can be replicated with sufficient initial wealth. However, if a market is incomplete, the initial costs of superhedging are often too high. In this case, a hedger usually allocates initial capitals less than the superhedging costs while accepts the possibility of shortfall. Such a hedging strategy is called partial hedging. Föllmer and Leukert are pioneers in this filed. They studied quantile hedging and efficient hedging [3, 4] in semimartingale financial market models. In their papers, explicit solutions in complete markets were provided with the help of the classical Neyman-Pearson lemma, while solutions in incomplete markets were given according to the convex duality approach. Another reason that

A. Melnikov · H. Wan () Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics 358, https://doi.org/10.1007/978-3-030-76829-4_17

309

310

A. Melnikov and H. Wan

makes partial hedging interesting is that although it has some downside risk, it brings opportunities to gain benefits. Some financial institutions such as insurance companies indeed exploit risk to make profits. The recent book of Melnikov and Nosrati [8] discussed several partial hedging methods and their applications in pricing and hedging insurance contracts. In this paper, we consider the partial hedging problem in defaultable markets, which may be incomplete. There is a list of references in this area. For instance, Nakano [13] solved problems of optimal quantile hedging and efficient hedging with a linear loss function for claims with a single default time. Later, Melnikov and Nosrati [7] focused on the efficient hedging problem with more general loss functions for claims with several independent default times and provided a closed form solution in the special case that recovery rates were zeros. In our paper, we would discuss partial hedging by employing a coherent risk measure called conditional value-at-risk (CVaR) which provides information about the average loss that exceeds the value-at-risk (VaR) level. Melnikov Smirnov [11] studied partial hedging with such a measure and provided a semi-explicit solution in complete markets. Aforementioned papers dealt with option pricing and hedging in Brownian market models. However, a growing number of evidences show that pure diffusion models are not accurate enough to represent real-life assets’ dynamics. It is not rare to observe jumps in stock prices when some significant financial or political announcements are published. In order to address this drawback, a jump-diffusion model was proposed by Merton [12]. Melnikov and Skornyakova [10] and Kirch and Melnikov [5] discussed quantile hedging and efficient hedging problems, respectively, in a two-factor jump-diffusion model. However, to our knowledge, a jump-diffusion market model with default has not been well studied, and CVaR hedging problems have not been discussed in this literature. Our main objective in this paper is to derive a hedging strategy that minimizes CVaR of the hedging loss subject to a constraint on the initial wealth in a jumpdiffusion defaultable market. The paper is organized as follows. In Sect. 2, we introduce our financial model. Several useful properties regarding the default time are listed. Most importantly, similar to the discussion about martingales in a defaultable Brownian market in Bielecki and Rutkowski [1], we derive densities of martingale measures in our model. Furthermore, utilizing properties of equivalent martingale measures, we show that the minimal superhedging costs of a defaultable claim with a zero recovery rate coincide with the perfect hedging costs of the corresponding non-defaultable claim. In Sect. 3, the explicit form of the optimal CVaR hedging strategy in our incomplete market is derived. We prove that the CVaR minimization problem of a zero-recovery rate defaultable claim can be converted to a problem of finding the optimal randomized test in the default-free complete market and the optimal strategy is then given by the option decomposition of a modified claim. In Sect. 4, a numerical example is provided to illustrate the implementation of our method in life insurance contracts that have stochastic guarantees. Section 5 gives a conclusion for the paper.

CVaR Hedging in Defaultable Jump-Diffusion Markets

311

2 Model Setup and Preliminaries Let (, G, P ) be a standard probability space. Consider a financial market with terminal time T ∈ (0, ∞) consisting one riskless asset (St0 )t ∈[0,T ] and two risky assets, (St1 )t ∈[0,T ] and (St2 )t ∈[0,T ] , described by a two-factor jump-diffusion model: dSt0 = rSt0 dt, S00 = 1, dSti = Sti− (μi dt + σi dWt − υi dNt ), S0i > 0,

i = 1, 2,

(1)

where r ≥ 0 is the risk-free interest rate. Constants μi ∈ R, σi > 0, and υi < 1 are drifts, volatilities, and jump parameters, respectively. W and N are independent Wiener process and Poisson process with a filtration F = (Ft )t ≥0 , Ft = σ (Ns , Ws ; s ≤ t). Here, we assume σ1 > σ2 , and the intensity of the Poisson process is a nonnegative constant λ. In addition, let a positive random variable τ denote the default time such that P (τ > t) > 0, ∀t ≥ 0. The default indicator process is defined as Ht = I{τ ≤t } , t ≥ 0, and the corresponding filtration generated by it is H = (Ht )t ≥0, Ht = σ (Hs ; s ≤ t). Let us specify that τ is independent of W and N. Moreover, let G = (Gt )t ≥0 be a joined filtration, i.e., G = H ∨ F. For simplicity, we assume G = GT . It is worth to mention that since H and F are independent, any F-martingale is also a G-martingale (see Bielecki and Rutkowski [1] Section 6.1.1). In this paper, credit risk is modeled with the help of a hazard process, that is, the survival probability is defined by P (τ > t) = e−t ,

(2)

$t where (t )t ≥0 is the hazard process such that t = 0 βu du and βu is a nonnegative deterministic function called the hazard rate of the random time τ . Let us summarize some crucial results regarding the default time and the default indicator process from Bielecki and Rutkowski [1] for the reader convenience: 1. The process 

t ∧τ

Mt = Ht − 0



t

βu ds = Ht −

βu (1 − Hu )du,

(3)

0

is both a H-martingale and a G-martingale (Proposition 5.1.3). 2. The process Qt = (1 − Ht )et , t ≥ 0 follows a G-martingale with the dynamic dQt = −Qt − dMt . Moreover, for any bounded F-martingale m, the product Qm and the quadratic covariation [Q, m] are G-martingales (Lemma 5.1.7).

312

A. Melnikov and H. Wan

3. For any Gt -measurable random variable Y , there is a Ft -measurable random variable E(I{τ >t } Y |Ft ) , Y˜ = P (τ > t)

(4)

such that Y I{τ >t } = Y˜ I{τ >t } , P − a.s. (Lemma 5.1.2). Note that the market (1) with default may be incomplete due to the additional random source τ , so that there is a set of martingale measures. Let us start with the structure of martingales in our setting. Theorem 1 For a G-measurable integrable random variable XT , we define a Gmeasurable martingale Xt = E(XT |Gt ), t ∈ [0, T ]. Then, it admits the following representation:  Xt = X0 +

t

0

 ξuW dWu

t

+ 0

ξuN d Nˆ u



t

+ 0

 ξuM dMu

t

+ 0

ˆ u, ξuMN d[M, N]

(5)

where Nˆ t = Nt − λt is the compensated Poisson process and ξ W , ξ N , ξ MN , ξ M are G-predictable processes. Proof Since GT = HT ∨ FT , according to Bielecki and Rutkowski [1], it is sufficient to consider a random variable XT = (1 − Hs )Y for some fixed s ≤ T and some FT -measurable random variable Y : XT = (1 − Hs )Y = (1 − Hs )es Y¯ = Qs Y¯ , where Y¯ = e−s Y . Let us define a F-martingale m: mt = E(Y¯ |Ft ) = E(Y¯ ) +



t 0



t

ξu dWu +

ζu d Nˆ u ,

0

where ξ and ζ are F-predictable processes. The second equality is because of the martingale representation property in the filtration F. Since Q is a process with finite variation, we have [Q, m]t =

 u≤t

Qu mu =

 u≤t

−Qu− ζu Mu Nˆ u = −



t 0

ˆ u. Qu− ζu d[M, N]

CVaR Hedging in Defaultable Jump-Diffusion Markets

313

Obviously, mT = Y¯ and according to the integration by parts formula, we get 

T

XT = Qs mT = Q0 m0 + = E(Y¯ ) +



0 T

T

− 0

T 0



T

Qu− ξu dWu +

0



 Qu− dmu +

mu− I[0,s] (u)dQu + [Q, m]s

Qu− ζu d Nˆ u −

0



T

mu− I[0,s] (u)Qu− dMu

0

ˆ u. ζu Qu− I[0,s] (u)d[M, N]

With the choice ξuW = Qu− ξu , ξuN = Qu− ζu , ξtM = −mu− I[0,s] (u)Qu− , ξ MN = −ζu Qu− I[0,s] (u), (5) is proved.   A simple modification of Theorem 1 implies that the Radon-Nikodym density of a martingale measure P˜ which is equivalent to P on (, G) has the form  t d P˜ ˆ ˆ u ), |G = 1 + Zt = Zu− (θu dWu + ψu d Nˆ u + ku dMu + γu d[M, N] dP t 0

(6)

where θ , ψ, k, γ are G-predictable processes. And the solution of (6) is Zˆ t = Et (



·



·

θu dWu +

0

0

ψu d Nˆu +



·

 ku dMu +

0

·

ˆ u ), γu d[M, N]

(7)

0

where Et (·) is the Doleans exponential. Note that, in order for Zˆ to be a positive martingale, we have to impose the following restrictions: (a) kt > −1, ψt > −1, kt + ψt + γt > −1, ∀t ∈ [0, T ]; (b) E(Zˆ t ) = 1, ∀t ∈ [0, T ]. With the help of Girsanov theorem, we conclude that processes Wt∗ = Wt −



t 0

θu du, N˜ t = Nt −



t

(1 + ψu )λdu,

0

are martingales under the measure P˜ defined by (6). To derive processes θ , ψ, k, and γ , let us consider discounted value processes S˜ti = e−rt Sti , (i = 1, 2), which can be rewritten as   d S˜ti = S˜ti− (μi − r)dt + σi dWt − vi dNt   = S˜ti− (μi − r + σi θt − vi (1 + ψt )λ)dt + σi dWt∗ − vi d N˜ t , (i = 1, 2). (8)

314

A. Melnikov and H. Wan

They are martingales under the measure P˜ if drift terms vanish, i.e.,

μ1 − r + σ1 θt − v1 (1 + ψt )λ = 0, μ2 − r + σ2 θt − v2 (1 + ψt )λ = 0,

(9)

which implies θ=

(μ1 − r)v2 − (μ2 − r)v1 (μ1 − r)σ2 − (μ2 − r)σ1 − 1. , ψ= σ2 v1 − σ1 v2 (σ2 v1 − σ1 v2 )λ

(10)

However, processes k and γ cannot be determined uniquely, and hence the market (1) with default is incomplete. We denote M as the set containing all martingale measures and L is the set of Radon-Nikodym densities of martingale measures. In particular, since Nˆ and M arepurely discontinuous martingales, their  ˆ t = quadratic covariation is [M, N] Mu Nˆ u = Hu Nu , and hence u≤t

ˆ t I{τ >T } is 0, t ∈ [0, T ]. Therefore, we arrive to [M, N] Zˆ T I{τ >T } = ET (θ W + ψ Nˆ +



= ET (θ W )ET (ψ Nˆ +

u≤t

· 0

ku dMu )I{τ >T }



· 0

= ET (θ W )ET (ψ Nˆ )ET (

ku dMu )I{τ >T }



· 0

ku dMu )I{τ >T }

  θ2 = exp θ WT − T + (λ − λ∗ )T + (lnλ∗ − lnλ)NT 2  T ∧τ  T  exp ln(1 + ku )dHu − ku βu du I{τ >T } = ZT∗ ZTk I{τ >T } , 0

0

(11) where λ∗ = (1 + ψ)λ = ZT∗ = exp(θ WT −

(μ1 − r)σ2 − (μ2 − r)σ1 , (σ2 v1 − σ1 v2 )

θ2 T + (λ − λ∗ )T + (lnλ∗ − lnλ)NT ), 2 

ZTk = exp(−

T

ku βu du). 0

CVaR Hedging in Defaultable Jump-Diffusion Markets

315

The second and third equalities are both due to the multiplication rule of Doleans ˆ M]t I{τ >T } = 0, ∀t ≤ T . exponential and the fact [W, M]t = 0, [W, Nˆ ]t = 0, [N, ∗

∗ Remark 1 The probability measure P ∗ defined by dP dP = ZT that satisfies ψ > −1, and σ2 v1 −σ1 v2 = 0 is the unique martingale measure in the non-defaultable market (1) on the probability space (, FT , P ). Moreover, W ∗ and N are independent Wiener process and Poisson process (with intensity λ∗ ) under this measure. We denote E ∗ (·) as the expectation under the measure P ∗ .

With notations introduced above, Sti , i = 1, 2, can be rewritten as

 1 Sti = S0i exp σi Wt + (μi − σi 2 )t + Nt ln(1 − υi ) 2

 1 = S0i exp σi Wt∗ + (r + υi λ∗ − σi 2 )t + Nt ln(1 − υi ) . 2

(12)

A G-strategy is a G-predictable process π := (πt0 , πt1 , πt2 )t ∈[0,T ] such that 

T 0

 |πt0 |dt

T

< ∞, 0

(πti Sti )2 dt < ∞, P − a.s (i = 1, 2),

and the value process corresponding to the strategy π at time t ∈ [0, T ] is Vt = πt0 St0 + πt1 St1 + πt2 St2 .

(13)

In addition, for a given initial value v ≥ 0, a trading strategy is called self-financing admissible if its value process satisfies 

t

Vt = v + 0

 πu0 dSu0 +

0

t

 πu1 dSu1 +

t 0

πu2 dSu2 , and Vt ≥ 0, ∀t ∈ [0, T ].

(14)

We denote the set of all admissible self-financing strategies with initial value v as A(v). Remark 1 indicates that the default-free market (1) is complete, and hence the fair price of any contingent claim with payoff C which is nonnegative FT -measurable is defined as E ∗ (e−rT C), and there is a self-financing strategy π that duplicates it (see [5] for the form of the duplication strategy). However, we would like to investigate the hedging problem of a defaultable claim C 0 = CI{τ >T } in the enlarged filtration G described before. As we know from the option pricing theory in incomplete markets, the minimal initial superhedging costs of such a claim are defined as ˜

U0 = sup E P (e−rT C 0 ). P˜ ∈M

(15)

316

A. Melnikov and H. Wan

Let us denote ˜ U˜ t = Ut e−rt = ess supE P (e−rT C 0 |Gt ), t ∈ [0, T ],

(16)

P˜ ∈M

which is a supermartingale with respect to any P˜ ∈ M and represents the discounted value process of the minimal superhedging strategy of the claim C 0 . According to the optional decomposition theorem (see Kramkov [6]), there is an admissible strategy (U0 , π) and a discounted optional consumption process D with D0 = 0 such that U˜ t = U0 +



t 0



πu1 d S˜u1

t

+ 0

πu2 d S˜u2 − Dt .

(17)

In our setting, the minimal superhedging costs U0 can be derived explicitly. Lemma 1 Assume E ∗ (C) < +∞. In the market (1) with default, the discounted superhedging value process of C 0 satisfies 

 ˜

U˜ t = ess supE P e−rT C 0 |Gt = E ∗ e−rT C|Ft I{τ >t } , ∀t ∈ [0, T ].

(18)

P˜ ∈M

In particular, U0 = U˜ 0 = E ∗ (e−rT C), i.e., the minimal superhedging costs equal to the fair price of the non-defaultable contingent claim C. Proof For a given martingale measure P˜ with a density Zˆ T , we know that E[e ˜ U˜ t = ess sup E P [e−rT CI{τ >T } |Gt ] = ess sup P˜ ∈M

−rT Z ˆ

T CI{τ >T } |Gt ]

Zˆ t

ˆ Z∈L

.

Moreover, by (11), we have E[Zˆ T CI{τ >T } |Gt ] = e−

$t

0 ku βu du

E[ZT∗ e−

$T t

ku βu du

CI{τ >T } |Gt ].

By (4), the Ft -measurable random variable % $t # $T E E[ZT∗ e− t ku βu du CI{τ >T } |Gt ]I{τ >t } |Ft e 0 βu du   $t $T = E ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du satisfies $T $T



 $t E ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du I{τ >t} = E ZT∗ e− t

ku βu du

 CI{τ >T } |Gt I{τ >t} .

CVaR Hedging in Defaultable Jump-Diffusion Markets

317

Thus, U˜ t can be rewritten as U˜ t = ess sup

e−

$t

0 ku βu du

ˆ Z∈L

= ess sup

$T  $t

E e−rT ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du I{τ >t } Zˆ t

E(e−rT ZT∗ e−

$T t

ˆ Z∈L

ku βu du

Zt∗

CI{τ >T } |Ft )

e

$t 0

βu du

I{τ >t } .

In particular, choosing k constant and k 7 −1, we have U˜ t ≥ lim

e−

$T t

kβu du

k7−1

E(e−rT ZT∗ CI{τ >T } |Ft ) $ t βu du e0 I{τ >t} = E ∗ (e−rT C|Ft )I{τ >t} , Zt∗ (19)

where the second equality is due to the independence of τ and FT . On the other hand, since kt > −1, we get U˜ t ≤

E(e−rT ZT∗ e

$T t

βu du

Zt∗

CI{τ >T } |Ft )

e

$t 0

βu du

 I{τ >t } = E ∗ e−rT C|Ft I{τ >t } . (20)

Combing (19) and (20), we arrive to

 U˜ t = E ∗ e−rT C|Ft I{τ >t } .

(21)  

3 CVaR Partial Hedging If a hedger allocates less capitals than the minimum superhedging costs U0 , there is a possibility of shortfall characterized by L = C 0 − VT , where VT is the value of a hedging portfolio at T . Conditional value-at-risk (CVaR) is a famous criterion to measure the hedging loss, and it is widely used in financial institutions. Definition 1 The CVaR of the loss L at a confidence level α ∈ (0, 1) is defined as the following: CV aRα (L) =

1 1−α



1

V aRs (L)ds, α

where V aRα (L) = inf{s ∈ R : P (L ≤ s) > α}.

(22)

318

A. Melnikov and H. Wan

CV aRα (L) measures the expected loss for a hedging strategy given that the loss exceeds its α quantile. Rockafellar and Uryasev [14] have indicated that it can also be represented as 

 1 E (L − z)+ : z ∈ R . (23) 1−α



1 Moreover, they showed that V aRα (L) = min y|y ∈ argmin z + 1−α E (L − z∈R  + z) . Our goal is to find a self-financing admissible strategy with an initial budget constraint v0 < U0 that minimizes hedging losses under the measure CVaR, i.e.,

CV aRα (L) = inf z +

min CV aRα (L),

(v,π)∈A0

(24)

where A0 = {(v, π)|(v, π) ∈ A(v), v ≤ v0 } is the set of self-financing admissible strategies with the initial hedging capital no more than v0 . Melnikov and Smirnov [11] have indicated that we can interchange the order of two minimization problems: min CV aRα (L) =

(v,π)∈A0

# min min z +

(v,π)∈A0 z∈R

# = min z + z∈R

% 1 E (C 0 − VT − z)+ 1−α %

min E (C 0 − VT − z)+ .

1 1 − α (v,π)∈A0

(25)

If the inner minimization problem in (25) for each z is solved, then the initial problem (24) is reduced to a one-dimensional optimization problem over z. Thus, for a fixed z, let us consider the problem

 min E (C 0 − VT − z)+ .

(v,π)∈A0

(26)

We focus on z ≥ 0 because z is corresponding to the V aRα of the hedging loss, and it is nonnegative when α is close to 1. Consequently, we have (C 0 − VT − z)+ = ((C 0 − z)+ − VT )+ = (C 0 (z) − VT )+ , where C 0 (z) = (C 0 − z)+ = (C − z)+ I{τ >T } = C(z)I{τ >T } with C(z) = (C − z)+ . Obviously, C 0 (z) is a GT -measurable nonnegative random variable, so it can be treated as a contingent claim, and thus the problem (26) is equivalent to an optimal efficient hedging problem of the contingent claim C 0 (z). Föllmer and Leukert [3] studied this kind of problem and proved that if a random variable ϕ solves

 maxE ϕC 0 (z) , ϕ∈R

(27)

CVaR Hedging in Defaultable Jump-Diffusion Markets

319

 ˜

where R = {ϕ :  → [0, 1]| GT − measurable, sup E P e−rT C 0 (z)ϕ ≤ v0 }, P˜ ∈M

then the optimal hedging strategy π is obtained from the optional decomposition (17) for the modified claim ϕ C(z). However, usually the optimal randomized test does not admit an explicit form in incomplete markets. To address such a difficulty, we transfer the optimization problem (27), which is in the enlarged filed GT into a problem in the filed FT . Lemma 2 If a FT -measurable random variable ϕ˜ ∈ R˜ solves the problem

 maxE ϕC(z) , ϕ∈R˜

(28)

 where R˜ = {ϕ :  → [0, 1]| FT − measurable, E ∗ e−rT C(z)ϕ ≤ v0 }, then ϕ = ϕI ˜ {τ >T } is the solution of the problem (27). Proof Suppose ϕ˜ is the solution for (28). Let us define ϕ = ϕI ˜ {τ >T } . Obviously ϕ ∈ [0, 1] and is GT -measurable. Also, by Lemma 1, we know that  

 ˜

˜

sup E P e−rT C 0 (z)ϕ = sup E P e−rT C(z)ϕI ˜ {τ >T } = E ∗ e−rT C(z)ϕ˜ ≤ v0 ,

P˜ ∈M

P˜ ∈M

and hence we have ϕ ∈ R. $T On the other hand, for any other ϕ ∈ R, define ϕ0 = E(I{τ >T } ϕ|FT )e 0 βu du and by (4) we conclude that ϕI{τ >T } = ϕ0 I{τ >T } . Thus, E ∗ (e−rT C(z)ϕ0 ) = ˜ ˜ Furthermore, sup E P (e−rT C 0 (z)ϕ) ≤ v0 , which indicates that ϕ0 ∈ R. P˜ ∈M



 



E ϕC 0 (z) = E ϕC(z)I{τ >T } = E ϕ0 C(z)I{τ >T }







 = E ϕ0 C(z) E(I{τ >T } ) ≤ E ϕC(z) ˜ E I{τ >T } = E ϕ C 0 (z) , (29) where the last line is due to the independence of FT and τ and the optimal property of ϕ. ˜ The Inequality (29) indicates that ϕ is the solution of (27). Lemma 2 is proved.   The solution of the problem (28) is given with the help of the classical NeymanPearson lemma (see Föllmer and Leukert [3]) such that the optimal randomized test has the form ϕ ∗ (z) = I{a(z) T ) be the survival probability for the next T years of an insured. There are two sources of risk: market risk associated with underline assets prices and the default time and insurance risk reflected by the insured mortality. Since usually the insurance risk and the financial market risk have no effect on each other, we would take a natural assumption that (, G, P ) and (, G2 , P2 ) are independent. Instead of simple call or put options, we consider a pure endowment life insurance contract with payoff C 0 = max(ST1 , ST2 )I{τ >T } provided that an insured is alive at T . This kind of contract has a flexible guarantee S 2 and a potential for future gains associated with S 1 . It is popularly traded in insurance companies. According to Brennan and Schwartz [2], the premium for such a contract is defined as T Ux

˜

= sup E P (e−rT C 0 )E P2 (I{T (x)>T } ) = T px U0 ,

(33)

P˜ ∈M

where x is the insured’s age, T is the maturity time of the contract, and U0 is the minimal superhedging costs of the defaultable claim C 0 . Notice that T Ux < U0 , which means the premium that the insurance company collects would be less than the minimal superhedging costs and therefore only a partial hedging strategy can be constructed. In our setting, we assume that the insurance company constructs the optimal hedging strategy described in Sect. 3 with the premium v0 = T px U0 . Theorem 3 (A) The minimal superhedging costs U0 of the defaultable claim C 0 max(ST1 , ST2 )I{τ >T } are U0 =

∞ 

# √ √ % ∗ 1 2 s0,n p0,n (1 (n) + σ1 T ) + s0,n (−1 (n) − σ2 T ) .

=

(34)

n=0

(B) The optimal CVaR hedging strategy for C 0 = max(ST1 , ST2 )I{τ >T } is given by the perfect hedging of the modified contingent claim (max(ST1 , ST2 ) − zˆ )+ I{Z ∗−1 >a(ˆ ˜ z)} during [0, τ ] while holding zero positions in risky assets after T

default, where a(z) ˜ is the unique solution of e−rT E ∗ (C(z)I{aT } ], i.e., f (z) = e−

$T 0

βu du

∞ 

√ √  p0,n S01 (1 − v1 )n eμ1 T 2 (5,6 (n) + σ1 T , 7 (n) + σ1 sign(θ) T , 3 )

n=0

√ √ T , 7 (n) + σ2 sign(θ) T , 8 (n) + σ2 T , 4 )

 − z 2 (5,6 (n), 7 (n), 3 ) + 3 (−6 (n), 7 (n), 8 (n), 4 ) . (37)

+ S02 (1 − v2 )n eμ2 T 3 (−6 (n) − σ2

√

Meanwhile, d(ˆz) is the corresponding value of minimal CVaR. The parameter z∗ is determined from E ∗ (e−rt C(z)) = v0 , that is, ∞ #  √ ∗ 1 2 s0,n p0,n (1,3 (z, n) + σ1 T ) + s0,n 2 (−1 (n) n=0

√ √ − σ2 T , 4 (z, n) + σ2 T , 5 )

% − ze−rT (1,3 (z, n)) + 2 (−1 (n), 4 (z, n), 5 ) = v0 .

(38)

Here,(·) is the distribution function of a standard normal random variable, and J (·, ) denotes the distribution function of J jointly normally distributed random variables with zero means, unit variances, and the correlation matrix . ∗ (T −t)

i st,n = Sti evi λ

ln( 1 (n) =

1 s0,n 2 s0,n

)+

σ22 −σ12 2 T

√ (σ1 − σ2 ) T

(1 − vi )n , 2

, 2 (a, n) =

− ln(aZt∗ ) − ( θ2 + λ − λ∗ )T − (ln λ∗ − ln λ)n √ , |θ| T

CVaR Hedging in Defaultable Jump-Diffusion Markets

ln

3 (z, n) =

5 (n) =

7 (n) =

ln

1 s0,n z

+ (r − √ σ1 T

S01 (1−v1 )n z

+ (μ1 − √ σ1 T

ln a(z) ˜ −

θ2 2 T

σ12 2 )T

σ12 2 )T

323

, 4 (z, n) =

ln , 6 (n) =

ln

2 s0,n z

S01 (1−v1 )n S02 (1−v2 )n

+ (r − √ σ2 T

σ22 2 )T

+ (μ1 − μ2 + √ (σ1 − σ2 ) T

ln + (λ − λ∗ )T + (ln λ∗ − ln λ)n , 8 (n) = √ |θ| T

, σ22 −σ12 2 )T

S02 (1−v2 )n z

+ (μ2 − √ σ2 T

,

σ22 2 )T

,

1,3 (z, n) = min{1 (n), 3 (z, n)}, 5,6 (n) = min{5 (n), 6 (n)}, pt,n = exp(−λ(T − t))  1 =

1

−sign(θ)

−sign(θ)

1

 3 =

(λ(T − t))n (λ∗ (T − t))n ∗ , pt,n , = exp(−λ∗ (T − t)) n! n!

1 sign(θ) sign(θ) 1



⎛

⎞ 1 sign(θ) −1 ⎜ ⎟ , 2 = ⎝ sign(θ) 1 −sign(θ) ⎠ , −1



−sign(θ)

1

⎞ 1 −sign(θ) −1 ⎟ ⎜ , 4 = ⎝ −sign(θ) 1 sign(θ) ⎠ , −1 sign(θ) 1 ⎛

 5 =

1 −1 −1 1

 .

 

Proof See Appendix 2. Remark 4 Note that distribution functions be expressed in terms of (·):

2 (x, y, 2± )

and

2 (x, y, z, 3± )

 2 (x, y, 2+ ) =  min(x, y) ;

(x) − (−y), if x > −y, 0, otherwise;



 min(x, y) − (−z), if min(x, y) > −z, 3 (x, y, z, 3+ ) = 0, otherwise; 2 (x, y, 2− ) =

3 (x, y, z, 3− ) =

(x) − (max(−y, −z)), if x > max(−y, −z), 0, otherwise;

can

(39) (40) (41) (42)

324

A. Melnikov and H. Wan

where 2±

 =

⎛ ⎞  1 ±1 −1 1 ±1 ± , 3 = ⎝ ±1 1 ∓1 ⎠ . ±1 1 −1 ∓1 1

Example 1 We consider the Russell 2000 (RUT-I) and the Dow Jones Industrial Average (DJIA) as S 1 and S 2 . Melnikov and Skornyakova [10] estimated (μi , σi ) (i = 1, 2) for those two risky assets using daily observations and assume υ1 = υ2 = 0, i.e., the Black–Scholes model. In our paper, we assume underlying risky assets with same drifts and volatilities as Melnikov and Skornyakova [10] but with nonzero jump components. Values of parameters are given as the following: μ1 = 0.0481, σ1 = 0.2232,

μ2 = 0.0417, σ2 = 0.2089, S01 = S02 = 100,

υ1 = −0.05, υ2 = −0.1, λ = 0.1, r = 0. We assume the hazard rate β is constant. The insured’s age, contract maturities, and levels of β would differ in this example. According to the most recently published United States 2015 Life Table (National Vital Statistics Reports volume 67, Number 7), the survival probability T px of a given insured can be found, and then for a fixed confidence level α = 0.95, utilizing results in Theorem 3, we can derive the minimal CVaR that can be achieved with the hedging capital T px U0 . Results are displayed in Table 1. It is observed that the minimal CVaR increases as the insured’s age or the contract maturity increases. This is because for an older client and a long-term contract, the survival probability of the insured would decrease so as the premium which leads to an increase in hedging losses. Moreover, for a given insured and a contract maturity time, the minimal CVaR decreases as the hazard rate increases. In addition, the decrease trend is more significant for long-term contracts. This can be explained by the fact that there is a higher chance that contracts would mature with a zero payoff if the hazard rate increases and for long-term contracts, there is an even higher possibility that a default event would occur during contract periods, which reduces the shortfall risk further. Table 1 Minimal CV aR0.95 for contracts with different maturities, insured’s age, and levels of β Age = 20 T = 10 T = 15 0.9902 0.9839 T px β=0 0.996 1.6431 β = 0.01 0.9958 1.6378 β = 0.015 0.9956 1.6337 β = 0.02 0.9954 1.6283

T = 20 0.9764 2.4136 2.3884 2.3684 2.3411

Age = 30 T = 10 T = 15 0.9860 0.9758 1.4318 2.4745 1.4316 2.4698 1.4314 2.4662 1.4312 2.4613

T = 20 0.9609 4.017 3.9978 3.9818 3.9596

Age = 40 T = 10 T = 15 0.97462 0.9509 2.5966 5.0477 2.5964 5.045 2.5963 5.0427 2.5961 5.0395

T = 20 0.9166 8.6448 8.6394 8.6335 8.624

CVaR Hedging in Defaultable Jump-Diffusion Markets

325

5 Conclusion This paper focuses on the problem of CVaR-based partial hedging in a defaultable jump-diffusion model. We first provide the set of martingale measures in this incomplete market which admit special forms Z ∗ Z k conditioned on {τ > T }, and then by their properties, the minimal superhedging costs of a defaultable claim with a zero recovery rate are given that coincide with the initial wealth required for perfectly hedging the non-defaultable claim. Most importantly, we prove that the optimal CVaR hedging problem in the defaultable market can be converted to a problem of finding an optimal randomized test in the corresponding default-free market. The hedging strategy can be explained as constructing the perfect hedging of a modified non-defaultable claim during [0, τ ] while investing nothing in risky assets and depositing all cash into the saving account after the default. Furthermore, our method is applied to the area of life insurance. Numerical results show that for a given insured and a contract maturity time, the minimal CVaR decreases as the hazard rate increases. Acknowledgments The authors are grateful to anonymous reviewers and the editor for fruitful suggestions to improve the paper. This research was supported by the Natural Sciences and Engineering Council of Canada under Discovery Grant NSERC RES0043487.

Appendix 1: Proof of Theorem 2 For a fixed z, from Lemma 2 and Eqs. (30)–(32), we know that the optimal hedging strategy of the efficient hedging problem (27) is a superhedging strategy of the modified claim ϕ ∗ (z)C(z)I{τ >T } , and hence the shortfall is % #

 (C 0 (z) − VT )+ = C 0 (z) − VT C 0 (z) = C 0 (z) − ϕ ∗ (z)C(z)I{τ >T } = C(z) 1 − ϕ ∗ (z) I{τ >T } .

Thereby, # CV aR(L) = min z + z∈R

% 

 1 E C(z) 1 − ϕ ∗ (z) I{τ >T } . 1−α

 

 1 With the notation d(z) = z + 1−α E C(z) 1 − ϕ ∗ (z) I{τ >T } , it is clear that CV aR(L) = d(ˆz), where zˆ is the point of minimum of the function d(z).

326

A. Melnikov and H. Wan

Furthermore, applying Lemma 1 to the modified claim ϕ ∗ (ˆz)C(ˆz)I{τ >T } , the discounted value process of the superhedging strategy of such a claim is

 U˜ t = E ∗ e−rT ϕ ∗ (ˆz)C(ˆz)|Ft I{τ >t }



 = E ∗ e−rT ϕ ∗ (ˆz)C(ˆz)|Ft − E ∗ e−rT ϕ ∗ (ˆz)C(ˆz)|Ft I{τ ≤t } .

(43)

Let us define Vt e−rt = E ∗ (e−rT ϕ ∗ (ˆz)C(ˆz)|Ft ). It is the discounted value process for ϕ ∗ (ˆz)C(ˆz) in the default-free market.$This non-defaultable claim has a replica$t t tion strategy π such that Vt e−rt = v0 + 0 πu 1 d S˜u1 + 0 πu 2 d S˜u2 . Applying the integration by parts formula for E ∗ (e−rT ϕ ∗ (ˆz)C(ˆz)|Ft )I{τ ≤t } = Vt e−rt Ht and because Ht has finite variation, we get Vt e−rt Ht =



t 0

Hu− πu 1 d S˜u1 +



t 0

Hu− πu 2 d S˜u2 +



t 0

Vu e−ru dHu .

Notice that Hu− = lim I{τ ≤t } = I{τ