Stochastic Flows and Jump-Diffusions [1 ed.] 978-981-13-3801-4

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Probability Theory and Stochastic Modelling  92

Hiroshi Kunita

Stochastic Flows and Jump-Diffusions

Probability Theory and Stochastic Modelling Volume 92

Editors-in-chief Peter W. Glynn, Stanford, CA, USA Andreas E. Kyprianou, Bath, UK Yves Le Jan, Orsay, France Advisory Board Søren Asmussen, Aarhus, Denmark Martin Hairer, Coventry, UK Peter Jagers, Gothenburg, Sweden Ioannis Karatzas, New York, NY, USA Frank P. Kelly, Cambridge, UK Bernt Øksendal, Oslo, Norway George Papanicolaou, Stanford, CA, USA Etienne Pardoux, Marseille, France Edwin Perkins, Vancouver, Canada Halil Mete Soner, Zürich, Switzerland

The Probability Theory and Stochastic Modelling series is a merger and continuation of Springer’s two well established series Stochastic Modelling and Applied Probability and Probability and Its Applications series. It publishes research monographs that make a significant contribution to probability theory or an applications domain in which advanced probability methods are fundamental. Books in this series are expected to follow rigorous mathematical standards, while also displaying the expository quality necessary to make them useful and accessible to advanced students as well as researchers. The series covers all aspects of modern probability theory including • • • • • •

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Hiroshi Kunita

Stochastic Flows and Jump-Diffusions

123

Hiroshi Kunita Kyushu University (emeritus) Fukuoka, Japan

ISSN 2199-3130 ISSN 2199-3149 (electronic) Probability Theory and Stochastic Modelling ISBN 978-981-13-3800-7 ISBN 978-981-13-3801-4 (eBook) https://doi.org/10.1007/978-981-13-3801-4 Library of Congress Control Number: 2019930037 Mathematics Subject Classification: 60H05, 60H07, 60H30, 35K08, 35K10, 58J05 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This book is dedicated to my family

Preface

The Wiener process and the Poisson random measure are fundamental to the study of stochastic processes; the former describes a continuous random evolution, and the latter describes a random phenomenon that occurs at a random time. It was shown in the 1940s that any Lévy process (process with independent increments) is represented by a Wiener process and a Poisson random measure, called the Lévy– Itô representation. Further, Itô defined a stochastic differential equation (SDE) based on a Wiener process. He defined also an equation based on a Wiener process and a Poisson random measure. In this monograph, we wish to present a modern treatment of SDE and diffusion and jump-diffusion processes. In the first part, we will show that solutions of SDE will define stochastic flows of diffeomorphisms. Then, we discuss the relation between stochastic flows and heat equations. Finally, we will investigate fundamental solutions of these heat equations (heat kernels), through the study of the Malliavin calculus. It seems to be traditional that diffusion processes and jump processes are discussed separately. For the study of the diffusion process, theory of partial differential equations is often used, and this fact has attracted a lot of attention. On the other hand, the study of jump processes has not developed rapidly. One reason might be that for the study of jump processes, we could not find effective tools in analysis such as the partial differential equation in diffusion processes. However, recently, the Malliavin calculus for Poisson random measure has been developed, and we can apply it to some interesting problems of jump processes. A purpose of this monograph is that we present these two theories simultaneously. In each chapter, we start from continuous processes and then proceed to processes with jumps. In the first half of the monograph, we present the theory of diffusion processes and jump-diffusion processes on Euclidean spaces based on SDEs. The basic tools are Itô’s stochastic calculus. In Chap. 3, we show that solutions of these SDEs define stochastic flows of diffeomorphisms. Then in Chap. 4, relations between a diffusion (or jump-diffusion) and a heat equation (or a heat equation associated with integro-differential equation) will be studied through properties of stochastic flow.

vii

viii

Preface

In the latter half of the monograph, we will study the Malliavin calculus on the Wiener space and that on the space of Poisson random measure. These two types of calculus are quite different in detail, but they have some interesting things in common. These will be discussed in Chap. 5. Then in Chap. 6, we will apply the Malliavin calculus to diffusions and jump-diffusions determined by SDEs. We will obtain smooth densities for transition functions of various types of diffusions and jump-diffusions. Further, we show that these density functions are fundamental solutions for various types of heat equations and backward heat equations; thus, we construct fundamental solutions for heat equations and backward heat equations, independent of the theory of partial differential equations. Finally, SDEs on subdomains of a Euclidean space and those on manifolds will be discussed at the end of Chaps. 6 and 7. Acknowledgements Most part of this book was written when the author was working on Malliavin calculus for jump processes jointly with Masaaki Tsuchiya and Yasushi Ishikawa in 2014–2017. Discussions with them helped me greatly to make and rectify some complicated norm estimations, which cannot be avoided for getting the smooth density. The contents of Sects. 5.5, 5.6, and 5.7 overlap with the joint work with them [46]. Further, Ishikawa read Chap. 7 carefully and gave me useful advice. I wish to express my thanks to them both. I would also like to thank the anonymous referees and a reviewer, who gave me valuable advices for improving the draft manuscript. Finally, it is my pleasure to thank Masayuki Nakamura, editor at Springer, who helped me greatly toward the publication of this book.

Fukuoka, Japan

Hiroshi Kunita

Introduction

Stochastic differential equations (SDEs) based on Wiener processes have been studied extensively, after Itô’s work in the 1940s. One purpose was to construct a diffusion process satisfying the Kolmogorov equation. Results may be found in monographs of Stroock–Varadhan [109], Ikeda–Watanabe [41], Oksendal [90], and Karatzas–Shreve [55]. Later, the geometric property of solutions was studied. It was shown that solutions of an SDE based on a Wiener process define a stochastic flow of diffeomorphisms [59]. In 1978, Malliavin [77] introduced an infinite-dimensional differential calculus on a Wiener space. The theory had an interesting application to solutions of SDEs based on the Wiener process. He applied the theory to the regularity of the heat kernel for hypo-elliptic differential operators. Then, the Malliavin calculus developed rapidly. Contributions were made by Bismut [9], Kusuoka–Stroock [69, 71], Ikeda–Watanabe [40, 41], Watanabe [116, 117], and many others. At the same period, the Malliavin calculus for jump processes was studied in parallel (see Bismut [10], Bichteler–Gravereau–Jacod [7], and Leandre [74]). Later, Picard [92] proposed another approach to the Malliavin calculus for jump processes. Instead of the Wiener space, he developed the theory on the space of Poisson random measure and got a smooth density for the law of a “nondegenerate” jump Markov process. Then, Ishikawa–Kunita [45] combined these two theories and got a smooth density for the law of a nondegenerate jump-diffusion. Thus, the Malliavin calculus can be applied to a large class of SDEs. In this monograph, we will study two types of SDEs defined on Euclidean space and manifolds. One is a continuous SDE based on a Wiener process and smooth coefficients. We will define the SDE by means of Fisk–Stratonovich symmetric integrals, since its solution has nice geometric properties. The other is an SDE with jumps based on the Wiener process and the Poisson random measure, where coefficients for the continuous part are smooth vector fields and coefficients for jump parts are diffeomorphic maps. These two SDEs are our basic objects of study. We want to show that both of these SDEs generate stochastic flows of diffeomorphisms and these stochastic flows define diffusion processes and jump-diffusion processes. In the course of the argument, we will often consider backward processes, i.e., ix

x

Introduction

stochastic processes describing the backward time evolution. It will be shown that inverse maps of a stochastic flow (called the inverse flow) define a backward Markov process. Further, we show that the dual process is a backward Markov process and it can be defined directly by the inverse flow through an exponential transformation. In each chapter, we will start with topics related to Wiener processes and then proceed to those related to Poisson random measures. Investigating these two subjects together, we can understand both the Wiener processes and Poisson random measures more strongly. Chapters 1 and 2 are preliminaries. In Chap. 1, we propose a method of studying the smooth density of a given distribution, through its characteristic function (Fourier transform); it will be applied to the density problem of infinitely divisible distributions. Then, we introduce some basic stochastic processes and backward stochastic processes. These include Wiener processes, Poisson random measures, martingales, and Markov processes. In Chap. 2, we discuss stochastic integrals. Itô integrals and Fisk–Stratonovich symmetric integrals based on continuous martingales and Wiener processes are defined, and Itô’s formulas are presented. Then, we define stochastic integrals based on (compensated) Poisson random measures. Further, we will give Lp -estimates of these stochastic integrals; these estimates will be used in Chaps. 3 and 6 for checking that stochastic flows have some nice properties. The backward stochastic integrals will also be discussed. In Chap. 3, we will revisit SDEs and stochastic flows, which were discussed by the author [59, 60]. A continuous SDE on d-dimensional Euclidean space Rd based  on a d  -dimensional Wiener process (Wt1 , . . . , Wtd ) is given by 

dXt =

d 

Vk (Xt , t) ◦ dWtk + V0 (Xt , t) dt,

(1)

k=1

where ◦dWtk denotes the symmetric integral based on the Wiener process Wtk . If coefficients Vk (x, t), k = 0, . . . , d are of Cb∞,1 -class, it is known that the family of solutions {Xtx,s , s < t} of the SDE, starting from x at time s, have a modification {Φs,t (x), s < t}, which is continuous in s, t, x and satisfies (a) Φs,t : Rd → Rd are C ∞ -diffeomorphisms, (b) Φs,u = Φt,u ◦ Φs,t for any s < t < u almost surely, and (c) Φs,t and Φt,u are independent. {Φs,t } is called a continuous stochastic flow of diffeomorphisms defined by the SDE. A similar problem was studied for SDE based on the Wiener process and the Poisson random measure. Let N(dt dz) be a Poisson random measure on the space  U = [0, T ] × (Rd \ {0}) with the intensity measure n(dt dz) = dtν(dz), where ν is a Lévy measure having a “weak drift.” We consider an SDE driven by a Wiener process and Poisson random measure: 



dXt =

d  k=1

Vk (Xt , t) ◦ dWtk

+ V0 (Xt , t) dt +

(φt,z (Xt− ) − Xt− )N (dt dz),

|z|>0+

(2)

Introduction

xi

where {φt,z } is a family of diffeomorphic maps on Rd with some regularity conditions (precise conditions will be stated in Sect. 3.2). It was shown that solutions define a stochastic flow of diffeomorphisms stated above (Fujiwara–Kunita [30]). In this monograph, we will prove these facts through discussions of the “master equation” and “backward SDE.” By using them, some complicated arguments in previous works [30, 59] are simplified. Further, we will define a backward SDE based on a Wiener process and another based on Wiener process and Poisson random measure. These backward SDEs define backward stochastic flows of diffeomorphisms. The solution of an SDE (or a backward SDE) based on a Wiener process defines a diffusion process (continuous strong Markov process) (or backward diffusion process). Further, the solution of an SDE (or backward SDE) based on a Wiener processes and a Poisson random measure defines a jump-diffusion (or a backward jump-diffusion). We will study these diffusion and jump-diffusion processes. Let {Ps,t } be the semigroup defined by Ps,t f (x) = E[f (Φs,t (x))]. In the case of a diffusion process on Rd , its generator is given by a second-order differential operator d

A(t)f (x) =

1 Vk (t)2 f (x) + V0 (t)f (x), 2

(3)

k=1

where Vk (t), k = 0, 1, . . . , m are first-order partial differential operators defined by Vk (t)f (x) = i Vki (x, t) ∂x∂ i f (x). In the case of a jump-diffusion process on Rd , the generator is given by an integro-differential operator of the form d

AJ (t)f =

1 Vk (t)2 f (x) + V0 (t)f (x) + 2 k=1

 |z|>0+

{f (φt,z (x)) − f (x)}ν(dz),

(4) where the last integral is an improper integral. In Chap. 4, we study the relation between stochastic flows and time-dependent heat equations and backward heat equations associated with the differential operator A(t) of (3) and integro-differential operator AJ (t) of (4), respectively. For a given time t1 and a bounded smooth function f1 (x), the function v(x, s) := Ps,t1 f1 (x) = E[f1 (Φs,t1 (x))] is a smooth function of x. Further, in the case of diffusions, v(x, s) is the unique solution of the final value problem of the time-dependent backward heat equation: ∂ v(x, s) = −A(s)v(x, s) for s < t1 , ∂s

v(t1 , x) = f1 (x).

(5)

This fact will be extended to a more general class of the operator A(t). Consider d

Ac (t)f =

1 (Vk (t) + ck (t))2 f + (V0 + c0 )f, 2 k=1

(6)

xii

Introduction

where c = (ck (x, t); k = 0, 1, . . . , m) are bounded smooth functions. We show that the semigroup with the generator Ac (t) is obtained by an exponential transformation c f (x) = E[f (Φ (x))G (x)], where based on ck (x, t); it is given by Ps,t s,t s,t Gs,t (x) = exp

 k≥1 s

t

 ck (Φs,r (x), r) ◦ dWrk

+

t

 c0 (Φs,r (x) dr .

s

c f (x) is the unique solution of the final value problem (5) Further, v(x, s) := Ps,t 1 1 associated with the operator Ac (t). For jump-diffusion processes, we will also extend the integro-differential operator AJ (t) to another one, which will be denoted by AJc,d (t). Then, we will study the final value problem of the backward heat equation associated with the operator AJc,d (t) (see Sect. 4.5). We are also interested in the initial value problem of the time-dependent heat equation associated with the operator Ac (t) given by (6):

∂ u(t, x) = Ac (t)u(t, x) for t > t0 , ∂t

u(t0 , x) = f0 (x).

(7)

For this problem, we solve SDE (1) to the backward direction and obtain a backward c f (x) := stochastic flow {Φˇ s,t }; then, we define a backward semigroup by Pˇs,t   ˇ s,t (x) , where G ˇ s,t (x) is the exponential functional associated with E f (Φˇ s,t (x))G the backward flow {Φˇ s,t }. Then, if f0 (x) is a bounded smooth function, the solution of the forward equation (7) exists uniquely, and it is represented by u(x, t) = Pˇtc0 ,t f0 (x). We stress that the final function f1 (or initial function f0 ) is smooth in these c f (x), etc. with respect studies. Indeed, the smoothness of functions v(x, s) = Ps,t 1 to x follows from the smoothness of f1 (x) and the stochastic flow Φs,t (x) with respect to x, a.s. If the function f1 is not smooth, we need additional arguments for the solution of equations (5) and (7), which will be discussed in Chap. 6 using the Malliavin calculus. Another subject of Chap. 4 is the investigation of the dual of a given diffusion and a jump-diffusion with respect to the Lebesgue measure. It will be shown that the dual of these processes can be constructed through the change-of-variable formula concerning stochastic flows {Φs,t }; the stochastic process defined by the inverse −1 maps Xˇ s = Ψˇ s,t (x) = Φs,t (x) should be a dual process of Xt = Φs,t (x), and it is a backward diffusion or a backward jump-diffusion with respect to s, where t is the initial time of the process. The dual semigroup of the semigroup {Ps,t } is then defined by using the inverse flow {Ψˇ s,t } as ∗ g(x) = E[g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x)], Ps,t

where ∇ Ψˇ s,t is the Jacobian matrix of the diffeomorphism Ψˇ s,t ; Rd → Rd .

Introduction

xiii

In the latter half of this monograph, we will study the Malliavin calculus on the Wiener space and the space of Poisson random measure, called Poisson space; we will apply it for proving the existence of fundamental solutions for heat equations discussed in Chap. 4. In Chap. 5, we will discuss the Malliavin calculus on the Wiener space and that on the Poisson space (space of Poisson random measure) separately. Then, we will combine these two. For the Malliavin calculus on the Wiener space, we will restrict our attention to the problem of finding the smooth density for laws of Wiener functionals. Our discussion is motivated by Malliavin and Watanabe (Watanabe [116, 117]), but we will take a simple and direct approach; we will not consider the Ornstein–Uhlenbeck semigroup on the Wiener space. Instead, we study the derivative operator Dt and its adjoint δ (Skorohod integral) directly. Then, we give an estimate of Skorohod integrals using Lp -Sobolev norms; a new proof is given for Theorem 5.2.1. A criterion for the smooth density of laws of Wiener functional will be given in terms of the celebrated “Malliavin covariance” in Sect. 5.3. Another reason why we do not follow Ornstein–Uhlenbeck semigroup argument is that a similar fact is not known for Poisson space; in fact, we want to bring together the Malliavin calculus on the Wiener space and that on the Poisson space in a unified method. In Sects. 5.4, 5.5, 5.6, 5.7, and 5.8, we will discuss the Malliavin calculus on the Poisson space, which is characterized by a Lévy measure ν. A basic assumption for the Poisson random measure is that the characterizing Lévy measure is nondegenerate and satisfies the order condition at the center. Here, the origin 0 is  regarded as the center of the Lévy measure defined on Rd \ {0}. We will see that the difference operator D˜ u and its adjoint operator δ˜ defined by Picard [92] work well as Dt and δ do on the Wiener space. Criteria of the smooth density for Poisson functionals are more complicated. We need a family of Lp -Sobolev norms conditioned to a family of neighborhoods of the center of the Lévy measure. It will be given in Sects. 5.5, 5.6, 5.7, and 5.8. Further, in Sects. 5.9, 5.10, and 5.11, we will study the Malliavin calculus on the product of the Wiener space and the Poisson space. A criterion for the smooth density of the law of a Wiener–Poisson functional will be given after introducing the “Malliavin covariance at the center.” In the application of the Malliavin calculus to solutions of an SDE, properties of stochastic flows defined by the SDE are needed. In Chap. 6, we study the existence of smooth densities of laws of a nondegenerate diffusion and a nondegenerate jump-diffusion defined on a Euclidean space. The class of nondegenerate diffusions includes elliptic diffusions and hypo-elliptic diffusions. Further, the class of nonc (or degenerate jump-diffusions includes pseudo-elliptic jump-diffusions. Let Ps,t c,d c,d Ps,t ) be the semigroup associated with the generator Ac (t) (or AJ (t)). It will be c (x, ·) (or P c,d (x, ·)) have densities p c (x, y) shown that its transition functions Ps,t s,t s,t c,d (or ps,t (x, y)), which are smooth with respect to both variables x and y, and further, the family of the densities is the fundamental solution of the backward heat equation associated with the operator Ac (t) (or AJc,d (t)); the fundamental solution of the heat equation will be obtained as a family of density functions of a backward transition

xiv

Introduction

c (x, E) associated with the semigroup {Pˇ c }, etc. Thus, initial–final function Pˇs,t s,t value problems (5) and (7) are solved for any bounded continuous functions f0 and f1 , respectively. In Sects. 6.7 and 6.8, for elliptic diffusions and pseudo-elliptic jump-diffusions, we will discuss the short-time asymptotics of the transition density functions as t ↓ s, making use of the Malliavin calculus. Our Malliavin calculus cannot be applied to jump processes or jump-diffusion processes which admit big jumps. Indeed, in order to apply the Malliavin calculus, solution Xt of the SDE should be at least an element of L∞− = p>1 Lp , and the fact excludes solutions of SDEs with big jumps. In Sect. 6.9, we consider such processes: we first truncate big jumps and get the smooth density. Then, we add big jumps and show that this action should preserve the smooth density, where the short-time asymptotics of the fundamental solution will be utilized. It is hard to apply the Malliavin calculus directly to (jump) diffusions on a bounded domain of a Euclidean space or those defined on a manifold. In Sect. 6.10, we consider a process killed outside of a bounded domain of a Euclidean space. In order to get a smooth density for the killed process, we need two facts. One is a short-time estimate of the density of a non-killed process. The other is a potential theoretic argument of a strong Markov process using hitting times. We show that the c (x, y) of the killed transition function is smooth with respect density function qs,t c (x, y) is the fundamental solution for the to x and y; further, we show that qs,t backward heat equation (5) on an arbitrary bounded domain of a Euclidean space with the Dirichlet boundary condition. Finally, in Chap. 7, we study SDEs on a manifold. Stochastic flow generated by an SDE on a manifold will be discussed in Sect. 7.1. Diffusions, jump-diffusions, and their duals will be treated in Sects. 7.2 and 7.3. Then, the smooth density for a (jump) diffusion on a manifold will be obtained by piecing together killed densities on local charts. It will be discussed in Sects. 7.4 and 7.5.

A Guide for Readers Discussions of the Malliavin calculus for Poisson random measures contain some complicated and technical arguments. For the beginner or the reader who is mainly interested in Wiener processes and diffusion processes, we suggest to skip these arguments at the first reading. After Chap. 4, we could proceed in the following way: Chapter 5, Sects. 5.1, 5.2, 5.3 −→ Chap. 6, Sects. 6.1, 6.2, 6.3, 6.7, 6.8, 6.10 −→ Chap. 7, Sects. 7.1, 7.2, 7.3, 7.4. The author hopes that this course should be readable.

Contents

1

2

3

Probability Distributions and Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . 1.1 Probability Distributions and Characteristic Functions . . . . . . . . . . . . . . 1.2 Gaussian, Poisson and Infinitely Divisible Distributions . . . . . . . . . . . . 1.3 Random Fields and Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Wiener Processes, Poisson Random Measures and Lévy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Martingales and Backward Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Quadratic Variations of Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Markov Processes and Backward Markov Processes . . . . . . . . . . . . . . . . 1.8 Kolmogorov’s Criterion for the Continuity of Random Field . . . . . . .

1 1 8 14 15 25 31 37 41

Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Itô’s Stochastic Integrals by Continuous Martingale and Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Itô’s Formula and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Regularity of Stochastic Integrals Relative to Parameters . . . . . . . . . . . 2.4 Fisk–Stratonovitch Symmetric Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Stochastic Integrals with Respect to Poisson Random Measure . . . . 2.6 Jump Processes and Related Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Backward Integrals and Backward Calculus . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 49 55 59 64 67 73

Stochastic Differential Equations and Stochastic Flows . . . . . . . . . . . . . . . . . 3.1 Geometric Property of Solutions I; Case of Continuous SDE . . . . . . . 3.2 Geometric Property of Solutions II; Case of SDE with Jumps . . . . . . 3.3 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Lp -Estimates and Regularity of Solutions; C ∞ -Flows . . . . . . . . . . . . . . 3.5 Backward SDE, Backward Stochastic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Forward–Backward Calculus for Continuous C ∞ -Flows . . . . . . . . . . . 3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE . . . . 3.8 Forward–Backward Calculus for C ∞ -Flows of Jumps . . . . . . . . . . . . . .

77 77 81 86 96 100 101 104 109

xv

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Contents

3.9 3.10 4

Diffeomorphic Property and Inverse Flow for SDE with Jumps . . . . 116 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Diffusions, Jump-Diffusions and Heat Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Exponential Transformation and Backward Heat Equation . . . . . . . . . 4.3 Backward Diffusions and Heat Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dual Semigroup, Inverse Flow and Backward Diffusion . . . . . . . . . . . . 4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps . . . . . . . . 4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of Diffeomorphic Jumps. . . . . . . . . . . . . . . . . . . . . . 4.7 Volume-Preserving Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Jump-Diffusion on Subdomain of Euclidean Space . . . . . . . . . . . . . . . . .

125 126 129 137 140 146 155 161 164

5

Malliavin Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Derivative and Its Adjoint on Wiener Space . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sobolev Norms for Wiener Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Nondegenerate Wiener Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Difference Operator and Adjoint on Poisson Space. . . . . . . . . . . . . . . . . . 5.5 Sobolev Norms for Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Estimations of Two Poisson Functionals by Sobolev Norms . . . . . . . . 5.7 Nondegenerate Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Equivalence of Nondegenerate Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Product of Wiener Space and Poisson Space . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Sobolev Norms for Wiener–Poisson Functionals . . . . . . . . . . . . . . . . . . . . 5.11 Nondegenerate Wiener–Poisson Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Compositions with Generalized Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 168 174 183 190 196 201 209 214 222 226 233 239

6

Smooth Densities and Heat Kernels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 H -Derivatives of Solutions of Continuous SDE . . . . . . . . . . . . . . . . . . . . . 6.2 Nondegenerate Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Density and Fundamental Solution for Nondegenerate Diffusion. . . 6.4 Solutions of SDE on Wiener–Poisson Space . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Nondegenerate Jump-Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Density and Fundamental Solution for Nondegenerate Jump-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Short-Time Estimates of Densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Off-Diagonal Short-Time Estimates of Density Functions . . . . . . . . . . 6.9 Densities for Processes with Big Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Density and Fundamental Solution on Subdomain . . . . . . . . . . . . . . . . . .

245 246 250 253 259 265

Stochastic Flows and Their Densities on Manifolds . . . . . . . . . . . . . . . . . . . . . . 7.1 SDE and Stochastic Flow on Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold . . . . . . . . . . . 7.3 Brownian Motion, Lévy Process and Their Duals on Lie Group . . . .

303 303 311 317

7

273 277 284 288 295

Contents

7.4 7.5

xvii

Smooth Density for Diffusion on Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Density for Jump-Diffusion on Compact Manifold . . . . . . . . . . . . . . . . . . 328

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Chapter 1

Probability Distributions and Stochastic Processes

Abstract We introduce some basic facts on probability distributions and stochastic processes. Probability distributions and their characteristic functions are discussed in Sect. 1.1. Criteria for smooth densities of distributions are given by their characteristic functions. In Sect. 1.2, we consider Gaussian, Poisson and infinitely divisible distributions and give criteria for these distributions to have smooth densities. Concerning the density problem of an infinitely divisible distribution, we study the Lévy measure in detail. Regarding that the origin 0 is the center of the Lévy measure, we will give criteria for its smooth density by means of ‘the order condition’ at the center of the Lévy measure. Then we consider stochastic processes. In Sect. 1.4, we consider Wiener processes, Poisson processes, Poisson random measures and Lévy processes. Among them, Poisson random measures are exposed in detail. Next, in Sects. 1.5 and 1.6, we discuss martingales, semi-martingales and their quadratic variations. These are standard tools for the Itô calculus. In Sect. 1.7, we define Markov processes. The strong Markov property will be discussed. In Sect. 1.8, we study Kolmogorov’s criterion for a random field with multi-dimensional parameter to have a continuous modification.

1.1 Probability Distributions and Characteristic Functions Let Ω be a set and let F be a family of subsets of Ω with the following three properties: (a) Ω ∈ F, (b) if A ∈ F, then its complement set Ac ∈ F, and (c) if An ∈ F, n = 1, 2, . . ., then n An ∈ F. The set F is called a σ -field of Ω. An element of Ω is denoted by ω and is called a sample. Elements of F are called events. The pair (Ω, F) is called a measurable space. A map P : F → [0, 1] is called a measure if it satisfies the following: ≥ 0 for any A ∈ F, and (b)

(a) P (A)  ∞ if An , n = 1, 2, . . . are disjoint, then P ( ∞ A ) = n=1 n n=1 P (An ). Further, if (c) P (Ω) = 1 is satisfied, P is called a probability measure. The triple (Ω, F, P ) is called a probability space.

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_1

1

2

1 Probability Distributions and Stochastic Processes

Let (Ω, F, P ) be a probability space. A family of events {Aλ , λ ∈ Λ} is called independent,  if any finite subset {Aλ1 , . . . , Aλn } of {Aλ , λ ∈ Λ} satisfies P ( ni=1 Aλi ) = ni=1 P (Aλi ). Next, let G be a subset of F. If G is a σ -field, it is called a sub σ -field of F. Suppose that {Gλ , λ ∈ Λ} is a family of sub σ -fields of F. It is called independent, if any collection of events {Aλ , λ ∈ Λ}, where each Aλ runs in Gλ , is independent. Let A, B be events. Suppose that P (B) > 0. The conditional probability of A given B is defined by P (A|B) = P P(A∩B) (B) . Then P (A|B) = P (A) holds if and only if A and B are independent. Let Rd be a d-dimensional Euclidean space. Its elements are denoted by x = (x1 , . . . , xd ). Let B(Rd ) be the Borel field of Rd . A probability measure μ on d )) is called a probability distribution or simply a distribution. If integrals (Rd , B(R bi = Rd xi dμ(x) and aij = Rd (xi − bi )(xj − bj )μ(dx) exist, b = (b1 , . . . , bd ) is called the mean vector and A =  (aij ) is called the covariance matrix. For a vector b = (b1 , . . . , bd ), we set |b| = ( i=1 |bi |2 )1/2 and for a matrix A = (aij ), we set |A| = ( i,j aij2 )1/2 . When d = 1, R1 coincides with the space R of real numbers. b and A are called the mean and the variance, respectively. Let S be a Hausdorff topological space with the second countability and let B(S) be the Borel field of S. A map X; Ω → S is called measurable (or G-measurable) if the set {ω; X(ω) ∈ E} belongs to F (or G) for any Borel subset E in S. A measurable map X; Ω → S is called an S-valued random variable. If S = Rd , X is called a d-dimensional random variable. Further, if d = 1, it is a real (realvalued) random variable or simply a random variable. Let {Xλ ; λ ∈ Λ} be a family of S-valued random variables. We denote by σ ({Xλ }) the smallest sub σ -field of F with respect to which {Xλ } are measurable. Let {Yλ ; λ ∈ Λ } be another family of S  -valued random variables and let σ ({Yλ }) be the smallest σ field with respect to which {Yλ } are measurable. Then two families of random variables {Xλ } and {Yλ } are called mutually independent if these sub σ -fields are independent. Given an S-valued random variable X, we set μ(E) = P ({X ∈ E}), E ∈ B(S). It is a probability measure on (S, B(S)), called the law of X. The expectation of a real random variable X is defined as the Lebesgue integral by its law μ; E[X] = xμ(dx), if the integral is well defined. It is equal to the Lebesgue integral of X by the measure P ; E[X] = Ω X(ω)P (dω). If E[|X|] is finite, X is said to be an integrable random variable. Let A be an event. We denote by 1A the indicator function of the event A, i.e., 1A (ω) is a function taking the value 1 if ω ∈ A and the value 0 if ω ∈ / A. The expectation E[X1A ] is often denoted by E[X; A]. Let {Xn , n = 1, 2, . . .} be a sequence of real random variables. Let X be another real random variable. The sequence {Xn } is said to converge to X almost surely, ˜ = 1 and simply written as Xn → X, a.s, if there exists Ω˜ ∈ F such that P (Ω) ˜ for any ω ∈ Ω, {Xn (ω)} converges to X(ω). If P (|Xn − X| > ) converges to 0 as n → ∞ for any > 0, {Xn } is said to converge to X in probability. Let p ≥ 1. If E[|Xn |p ] < ∞ holds for any n and E[|Xn − X|p ] → 0 as n → ∞, {Xn } is said to converge to X in Lp . A sequence of random variables {Xn } is called uniformly integrable, if for any > 0 there is a positive constant c such

1.1 Probability Distributions and Characteristic Functions

3

that supn |Xn |>c |Xn | dP < . If {Xn } is a sequence of a uniformly integrable random variables converging to X in probability, then {Xn } converges to X in L1 , i.e., limn→∞ E[|Xn − X|] = 0. d E of Rd , we set μ(E) = Let μ1 and μ2 be distributions on R . For a Borel subset d Rd μ1 (E − x)μ2 (dx). Then μ is a distribution on R . We denote μ by μ1 ∗ μ2 and call it the convolution of two distributions μ1 and μ2 . It is immediate to see μ1 ∗ μ2 = μ2 ∗ μ1 . Let X, Y be independent Rd -valued random variables and let μX and μY be their laws, respectively. Let μX+Y be the law of X + Y . Then it holds that μX+Y = μX ∗ μY . Let X and Y be independent random variables. Then E[f (X)g(Y )] = E[f (X)]E[g(Y )] holds for any bounded continuous functions f, g. Let X be an S-valued random variable and let Z be a real integrable random variable. The weighted law of X with respect to Z is a bounded signed measure on (S, B(S)) defined by μZ (E) = E[Z; {X ∈ E}]. Its total variation measure |μZ | is given by |μZ |(E) = E[|Z|; {X ∈ E}]. If a bounded signed measure μ on Rd satisfies Rd |x|n |μ|(dx) < ∞, where |μ| is the total variation of the measure μ, then the measure μ is said to have an n-th moment. The characteristic function of a bounded signed measure μ on Rd is defined by  ψ(v) =

Rd

ei(v,x) μ(dx),

v = (v1 , . . . , vd ) ∈ Rd ,

(1.1)

√ d where i = −1 and (v, x) = j =1 vj xj . Let Z be a real integrable random variable, X = (X1 , . . . , Xd ) be an Rd -valued random variable and let μZ be its weighted law. Let ψ(v) be its characteristic function. Then we have  E[ei(v,X) Z] =

Rd

ei(v,x) μZ (dx) = ψ(v).

For a differentiable function f (x) on Rd , the partial derivatives ∂x∂ i f (x) are denoted by ∂xi f . We set ∂f = ∂x f = (∂x1 f, . . . , ∂xd f ). Further, if f is a  vector function (f 1 , . . . , f d ), we set ∂f = (∂xj f k )j =1,...,d,k=1,...,d  . It is called the Jacobian matrix. When d = d  , we will often denote it by ∇f . The set of positive integers is denoted by N and the set of nonnegative integers ¯ Let j be a multi-index of nonnegative integers of length d; j = is denoted by N. (j1 , . . . , jd ), where j1 , . . . , jd are nonnegative integers. We set |j| = j1 + · · · + jd . Let j = (j1 , . . . , jd ) be another multi-index of length d. If ji ≤ ji holds for any i = 1, . . . , d, then we denote the relation by j ≤ j . We define the higher order differential operator ∂ j and product x j for x = (x1 , . . . , xd ) ∈ Rd by j

j

∂ j = ∂xj = ∂x11 · · · ∂xdd ,

j

j

x j = x11 · · · xdd ,

4

1 Probability Distributions and Stochastic Processes

if |j| ≥ 1. If |j| = 0, i.e., j = (0, . . . , 0), we define ∂ 0 f = f and x 0 = 1. Let n0 ∈ N. If a function f (x) on Rd is n0 -times continuously differentiable and |∂ j f (x)| are bounded for any j with |j| ≤ n0 , f is said to be a function of Cbn0 -class, or a Cbn0 -function. ¯ If a bounded signed measure μ has n0 -th moment, Proposition 1.1.1 Let n0 ∈ N. then its characteristic function ψ(v) is a Cbn0 -function. Further, it holds that ∂vj ψ(v)

=i

|j|

 Rd

ei(v,x) x j μ(dx)

(1.2)

for any j with |j| ≤ n0 . ¯ The case n0 = 0 is trivial. Proof We prove (1.2) by the induction of n0 ∈ N. ˜ Suppose it is valid for j with |j| ≤ n0 − 1. Set ψ(v) := ei(v,x) x j μ(dx). We have for h = 0   i(v+hej ,x) ˜ + hej ) − ψ(v) ˜ ψ(v − ei(v,x)  j e x μ(dx), = h h where ej , j = 1, . . . , d are unit vectors in Rd . Let h tend to 0 in the above. Since

ei(v+hej ,x) − ei(v,x)

≤ |xj |

h holds for any h, the right-hand side converges to ei(v,x) (ixj )x j dμ, by the ˜ is differentiable with respect to Lebesgue convergence theorem. Therefore ψ(v) vj and we get ˜ =i ∂vj ψ(v)

 ei(v,x) xj x j μ(dx).

˜ = ∂ j+ej ψ(v) and ei(v,x) xj x j μ(dx) = ei(v,x) x j+ej μ(dx). It holds that ∂vj ψ(v)   Therefore we get the formula (1.2) for j + ej . From the above proposition, the moment of the measure μ can be computed as 

x j μ(dx) = i −|j| ∂vj ψ(v)

v=0

.

A Borel subset E of Rd is called μ-continuous, if μ(∂E) = 0 holds. Here ∂E is the boundary set of E defined by E¯ − E o . Theorem 1.1.1 (Lévy’s inversion formula) The characteristicfunction deterd mines the measure μ uniquely: If a rectangular set E = j =1 (aj , bj ] is μ-continuous, then

1.1 Probability Distributions and Characteristic Functions

5

d  1 d N  N  e−ivj bj − e−ivj aj μ(E) = lim ··· ψ(v1 , . . . , vd ) dv1 · · · dvd . N →∞ 2π −ivj −N −N j =1

(1.3) Proof Using Fubini’s theorem, we have 

 ···

N

−N

 =

N

d  e−ivj bj − e−ivj aj ψ(v1 , . . . , vd ) dv1 · · · dvd −ivj

−N j =1

    e−ivj bj − e−ivj aj  · · · ei( j vj xj ) μ(dx1 · · · dxd ) dv1 · · · dvd −ivj

d N  N

·

−N −Nj =1

   = ···

N

 ···

−N

   d  = ···

−N j =1 N

j =1 −N

Set the integral

N

d   eivj (xj −bj ) − eivj (xj −aj ) dv1 · · · dvd μ(dx1 · · · dxd ) −ivj

N

 eivj (xj −bj ) − eivj (xj −aj ) dvj μ(dx1 · · · dxd ). −ivj

· · · dvj of the above by J (N, xj , aj , bj ). It holds that

−N



N

J (N, x, a, b) = 2 0

Since limN →∞

N 0

sin t t

dt =

π 2

sin(x − a)v dv − 2 v



N 0

sin(x − b)v dv. v

holds, we have

⎧ ⎨ 0, if x < a or b < x, lim J (N, x, a, b) = π, if x = a or x = b, ⎩ N →∞ 2π, if a < x < b. The above convergence is a bounded convergence. Therefore, by Lebesgue’s bounded convergence theorem,  lim

N

N →∞ −N

 ···

N

d 

−N j =1

J (N, xj , aj , bj )μ(dx1 · · · dxd ) = (2π )d μ(E),

if μ(∂E) = 0. We have thus proved the inversion formula (1.3).

 

Corollary 1.1.1 Let X1 , . . . , Xd be real random variables. Denote by X its vector (X1 , . . . , Xd ). If E[ei(v,X) ] =

d  j =1

E[eivj Xj ]

(1.4)

6

1 Probability Distributions and Stochastic Processes

holds for all v, then X1 , . . . , Xd are mutually independent. Conversely, if X1 , . . . , Xd are mutually independent, (1.4) is valid. Further, the characteristic function of Y = X1 +·· ·+Xd is equal to the product of characteristic functions of Xj , j = 1, . . . , d, i.e., dj =1 E[eivXj ]. Proof Let μ be the law of X and let μj , j = 1, . . . , d be one-dimensional laws of Xj , j = 1, . . . , j respectively. Suppose that (1.4) holds. Then equation (1.4) is written as   ei(v,x) dμ(x1 , . . . , xd ) = ei(v,x) dμ1 (x1 ) · · · dμ(xd ). Therefore we have μ = μ1 × · · · × μd by the above theorem. Hence X1 , . . . , Xd are independent. Conversely, if X1 , . . . , Xd are independent, then eiv1 X1 , . . . , eivd Xd are also independent. Therefore  (1.4) holds. The formula implies that the characteristic function of Y is equal to dj =1 E[eivXj ].   We study the relation for the existence of the smooth density of the measure μ and the decay property of the characteristic function ψ(v) as |v| → ∞. Proposition 1.1.2 Let ψ(v) be the characteristic function of a bounded signed measure μ. 1. If |ψ(v)| dv < ∞, then μ has a bounded continuous density function f (x). Further, we have the inversion formula:  1 d  f (x) = e−i(v,x) ψ(v) dv. 2π Rd

(1.5)

2. If |ψ(v)||v|n0 dv < ∞ for some positive integer n0 , then the density function f (x) is of Cbn0 -class. Further, for any multi-index j with |j| ≤ n0 , we have ∂ j f (x) =

(−i)|j| (2π )d

 Rd

e−i(v,x) ψ(v)v j dv.

(1.6)

3. Let m0 be a positive integer. If ψ(v) is of Cbm0 -class and functions ∂vi (ψ(v)v j ) are integrable and converge to 0 as |v| → ∞ for any |j| ≤ n0 and |i| ≤ m0 , then the density function is of Cbn0 -class and satisfies (ix)i ∂ j f (x) =

(−i)|j| (2π )d

 Rd

e−i(v,x) ∂vi (ψ(v)v j ) dv

(1.7)

for any |j| ≤ n0 , |i| ≤ m0 . Furthermore, there exists a positive constant c such c that |∂ j f (x)| ≤ (1+|x|) m0 holds for all x.

1.1 Probability Distributions and Characteristic Functions

7

Proof 1. If |ψ(v)| is integrable, we can rewrite the inversion formula (1.3) as μ(E) =

d  1 d ∞  ∞  e−ivj bj − e−ivj aj ··· ψ(v1 , . . . , vd ) dv1 · · · dvd , 2π −∞ −∞ −ivj j =1

since the integrand is an integrable function on Rd with respect to the Lebesgue measure dv1 · · · dvd . We consider the right-hand side. For fixed a1 , . . . , ad , it is a function of (b1 , . . . , bd ), which we denote by F (b1 , . . . , bd ). Set φ(vj , aj , bj ) =

e−ivj aj − e−ivj bj . ivj

It holds that |φ(vj , aj , bj )| ≤ |bj − aj | and |∂bj φ(aj , vj , bj )| ≤ 1 for any vj , aj , bj . Then for any j = 1, . . . , d, we can change the order of the partial derivative ∂bj and the integral; we find that for any j , F (b1 , . . . , bd ) is partially differentiable with respect to bj . Repeating this argument for j = 1, . . . , d, we find that the function F is partially differentiable with respect to bj , j = 1, . . . , d and we get ∂b1 · · · ∂bd F (b1 , . . . , bd ) =

 1 d ∞  ∞ d ··· e−i j =1 vj bj ψ(v1 , . . . , vd ) dv1 · · · dvd . 2π −∞ −∞

Denote the above function by f (b1 , . . . , bd ). It is a bounded continuous function b b and we have a11 · · · add f (x) dx = μ(E) if E is a μ-continuous rectangular set Πjd=1 (aj , bj ]. Now we can choose an increasing sequence of μ-continuous

d rectangular sets En such that n En = R . Then we have μ(En ) → 1. This implies Rd f (x) dx = 1. Therefore, all rectangular sets should be μ-continuous and the assertion is proved. 2. If |ψ(v)||v|n0 is integrable on Rd , we can change the order of derivative operator ∂ j , |j| ≤ n0 and the integral operator in the inversion formula (1.5). Therefore we get the assertion. 3. Suppose that ψ(v) satisfies conditions of 3. The density function f (x) is of Cbn0 class by 2. By the formula of integration by parts, we have  Rd

e−i(v,x) ∂vk (ψ(v)v j ) dv = ixk

 Rd

e−i(v,x) ψ(v)v j dv.

Repeating this argument, we get the formula (1.7) for any |j| ≤ n0 and |i| ≤ m0 . Further, since the right-hand side of (1.7) is a bounded function, |x||i| |∂ j f (x)| is   a bounded function. Therefore |∂ j f (x)| ≤ c/(1 + |x|)m0 holds. Let f (x) be a (complex) C n0 -function on Rd . It is called rapidly decreasing if the function |∂ j f (x)|(1 + |x|2 )i converges to 0 as |x| → ∞ for any |j| ≤ n0 and

8

1 Probability Distributions and Stochastic Processes

nonnegative integer i. A C ∞ -function f (x) on Rd is said to be rapidly decreasing, if it is a rapidly decreasing C n0 -function for any n0 ∈ N. Theorem 1.1.2 Let ψ(v) be the characteristic function of a bounded signed measure μ. If ψ(v) is a rapidly decreasing continuous function, then the measure μ has a C ∞ -density function f . Further, if ψ(v) is a rapidly decreasing C ∞ -function, f is a rapidly decreasing C ∞ -function. Proof If ψ(v) is a rapidly decreasing continuous function, then |ψ(v)||v|n is an integrable function for any n. Therefore μ has a Cb∞ -density function. Further, if ψ(v) is a rapidly decreasing C ∞ -function, ∂vi (ψ(v)v j ) are integrable and converges to 0 as |v| → ∞ for any i and j. Then |x i ||∂ j f (x)| is a bounded function for any i and j by (1.7). Therefore f (x) should be rapidly decreasing.  

1.2 Gaussian, Poisson and Infinitely Divisible Distributions We give some distributions and their characteristic functions which will be used in this monograph. 1. Exponential distribution. Let μ be a distribution on R having the density function f (x) given by f (x) = λe−λx if x > 0, and f (x) = 0 if x ≤ 0, where λ > 0. Then μ is called the exponential distribution with parameter λ. Its mean is λ1 λ and the variance is λ12 . Further, its characteristic function is ψ(v) = λ−iv . 2. Gamma distribution. Let μ be a distribution on R having the density function k k−1 −λx f (x) given by f (x) = λ xΓ (k)e if x > 0, and f (x) = 0 if x ≤ 0, where λ > 0, k is a positive integer and Γ (k) = (k − 1)!. Then μ is called a Gamma distribution with parameters λ and k. Its mean is λk and variance is λk2 . Its characteristic function  k λ is ψ(v) = λ−iv . If k = 1, it coincides with the exponential distribution with parameter λ. Let σ1 , . . . , σk be mutually independent random variables with the same exponential distribution with parameter λ. Then the law of τk := σ1 + · · · + σk is the Gamma distribution with parameters λ and k. Indeed, since laws of σj are exponential distributions with the same parameter λ, their characteristic functions λ are equal to λ−iv . Sine σj are independent, the characteristic function of τk should k  λ λ be the k-fold product of λ−iv , i.e., it is equal to λ−iv (Corollary 1.1.1). Hence the law of τk is equal to the Gamma distribution with parameters λ and k.  3. Gaussian distribution. Let d  be a positive integer and let b = (b1 , . . . , bd ) be a d  -vector. Let A = (a ij ) be a d  × d  positive definite symmetric matrix and let det A be its determinant. Assuming that det A = 0, we define f (x) =

 1  −1 exp − (x − b)) . (x − b, A d 1 2 (2π ) 2 | det A| 2 1

(1.8)

1.2 Gaussian, Poisson and Infinitely Divisible Distributions

9

 Then μ(B) := B f (x) dx, B ∈ B, is a probability distribution on Rd , called a d  -dimensional Gaussian distribution. Its mean vector is equal to b and covariance matrix is equal to A. It is known that its characteristic function is equal to   1 ψ(v) = exp i(v, b) − (v, Av) , 2



v ∈ Rd .

(1.9)

Conversely, for a given d  -vector b and a nonnegative definite symmetric d  × d   matrix A, there exists a distribution μ on Rd , whose characteristic function is equal to (1.9). Such a distribution is unique. It is also called a Gaussian distribution. Its mean and covariance are again given by b and A, respectively. If the matrix A is positive definite, μ has a density function written as in (1.8). If the rank of A is less than d  , the measure μ is singular with respect to the Lebesgue measure. 4. Poisson distribution. Let μ be a distribution on R having point masses on the ¯ := {0, 1, 2, . . .} and set N μ({n}) =

λn −λ e , n!

n = 0, 1, 2, . . . ,

(1.10)

where λ is a positive constant. It is called a Poisson distribution. Its mean and the variance are equal to λ. λ is called the parameter of the Poisson distribution. Its characteristic function is computed as   ψ(v) = exp λ(eiv − 1) ,

v ∈ R.

(1.11) 

5. Compound Poisson distribution. Let μ be a distribution on Rd , whose characteristic function is given by ψ(v) = exp

 Rd0

 i(v,z) (e − 1)ν(dz) ,  



v ∈ Rd .

(1.12)



where ν is a finite measure on Rd0 = Rd \ {0}. The distribution μ is called a compound Poisson distribution (associated with measure ν). A Poisson distribution is a one-dimensional compound Poisson distribution associated with ν = λδ1 , where δ1 is the delta measure at the point 1.  6. Infinitely divisible distribution. A distribution μ on Rd is called infinitely divisible, if the characteristic function of μ is represented by    1 ψ(v) = exp i(v, b)− (v, Av)+  (ei(v,z) −1−i(v, z)1D (z))ν(dz) . 2 Rd0 



(1.13)

Here, v, b ∈ Rd , A is a matrix as in (1.9), D = {z ∈ Rd0 ; |z| ≤ 1} and ν is a measure   on Rd0 satisfying Rd  (|z|2 ∧1)ν(dz) < ∞. A measure ν on Rd0 satisfying the above 0 property is called a Lévy measure. The triple (A, ν, b) is called the characteristics

10

1 Probability Distributions and Stochastic Processes

of the infinitely divisible distribution. Occasionally, we regard the Lévy measure ν  is a measure on Rd satisfying ν({0}) = 0. We call the point 0 the center of the Lévy measure.  Let μ be a distribution on Rd . Suppose that for any n ∈ N, there exists a distribution μn such that μ = μnn , where μnn is the n-times convolution of the distribution μn . It is known that its characteristic function ψ(v) is represented by (1.13). Hence μ is an infinitely divisible law. The formula (1.13) is called the Lévy–Khintchine formula. Conversely, if the characteristic function of a distribution μ is written as (1.13), for any n ∈ N there are distributions μn such that μ = μnn . This is the origin of the word ‘infinitely divisible’. 7. Stable distribution. An infinitely divisible distribution μ is called stable if for  any a > 0, there are b > 0 and c ∈ Rd such that its characteristic function ψ(v) satisfies ψ(v)a = ψ(bv)ei(c,v) . Further, if the above holds for any a > 0, b = a 1/β with some 0 < β ≤ 2 and c = 0, then μ is called an β-stable distribution. β is called the index of the stable distribution. If the drift vector b and the Lévy measure ν are 0, the distribution μ satisfies the above property with β = 2. Hence a Gaussian distribution with mean 0 is 2-stable. If μ is β-stable with 0 < β < 2, then A = 0. Further, using the polar coordinate, the Lévy measure ν is represented by 



ν(B) =

∞

1B (rϑ)

λ(dϑ) Sd  −1

0

dr r 1+β

,

(1.14)



where Sd  −1 = {z ∈ Rd ; |z| = 1} and λ is a finite measure on Sd  −1 . In the one-dimensional case, S0 is equal to two points {−1, +1}. The Lévy measure of the one-dimensional stable distribution is given by  ν(dz) =

c1 z−1−β dz on (0, ∞), c2 |z|−1−β dz on (−∞, 0),

(1.15)

with c1 ≥ 0, c2 ≥ 0, c1 + c2 > 0. In particular if the distribution μ is symmetric, then it holds that c1 = c2 > 0. Its characteristic function is given by     πβ ψ(v) = exp − c|v|β 1 − iγ tan signv + iτ v , 2

(1.16)

where c > 0, γ ∈ [−1, 1] and τ ∈ R, if β = 1. If μ is a d  -dimensional rotation invariant β-stable distribution, its characteristic function is given by ψ(v) = exp{−c|v|β },



v ∈ Rd ,

(1.17)

where c > 0. For further details of infinitely divisible distributions and stable distributions, we refer to Sato [99].

1.2 Gaussian, Poisson and Infinitely Divisible Distributions

11

Let us consider whether or not an infinitely divisible distribution μ has a smooth density. If μ is rotation invariant and β-stable, its characteristic function is given by (1.17). It is a rapidly decreasing function of v and ψ(v)|v|n is an integrable function for any positive integer n. Therefore the measure μ has Cb∞ -density function by Proposition 1.1.2. We are interested in more general infinitely divisible distributions. We start with a one-dimensional infinitely divisible distribution with the second moment; its characteristic function is represented by   ivz  e − 1 − ivz   ψ(v) = exp ibv + ν (dz) , z2 R 

where ν  is a bounded measure on R. (At z = 0, we identify

(1.18) eivz −1−ivz z2

with

− 12 v 2 .) Setting ν  ({0}) = a and ν(dz) := z12 ν  (dz) for z = 0 (Lévy measure), we get the Lévy–Khintchine formula with characteristics (a, b, ν). Now, if a > 0, the distribution μ has a C ∞ -density, because μ is written as a convolution of two distributions μ1 , μ2 , where μ1 is a Gaussian measure with mean b and variance a. If a = 0, we consider the order of the concentration of the mass of ν around the center 0. Suppose that the mass of ν  is highly concentrated near the center and satisfies the order condition 0 < lim inf ρ→0

ν  (B0 (ρ)) ν  (B0 (ρ)) ≤ lim sup < ∞, α ρ ρα ρ→0

(1.19)

for some 0 < α < 2, where B0 (ρ) = {z; |z| ≤ ρ}. Then the Lévy measure ν is said to satisfy the order condition of exponent α at the center. If the Lévy measure ν is a finite measure, it does not satisfy the order condition, since lim inf of (1.19) is 0 for any 0 < α < 2. It was shown by Orey that if the one-dimensional Lévy measure satisfies the order condition of exponent 0 < α < 2 at the center, then the law μ has a C ∞ -density. Now, let μ be a d  -dimensional infinitely divisible distribution with characteristics (A, ν, b). Suppose that the matrix A is nondegenerate. Then μ is written as the convolution of two distributions μ1 and μ2 , where μ1 is a Gaussian distribution with mean b and covariance A, and μ2 is an infinitely divisible distribution with characteristics (0, ν, 0). Since μ1 has a C ∞ -density, the convolution μ1 ∗ μ2 has also a C ∞ -density. Suppose next that the matrix A is degenerate. Let {A0 (ρ); 0 < ρ < 1} be a  family of open neighborhoods of the center in Rd , which satisfies the following two properties for any 0 < ρ < 1: 

(i) A0 (ρ) ⊂ B0 (ρ) = {z ∈ Rd ; |z| < ρ},

(ii)

 ρ  1. Consequently Then the last term of the above dominates c (c∧1) 2 |v| 2−α   there exists c1 > 0 such that |ψ(v)| ≤ exp{−c1 |v| } if |v| > 1. From the above lemma, ψ(z) is a rapidly decreasing function. Further, if the Lévy measure ν has moments of any order, i.e., |z|n ν(dz) < ∞ for any n ∈ N, the characteristic function given by (1.13) is infinitely differentiable. Then the density function is a rapidly decreasing C ∞ -function. Therefore we have the following theorem from Theorem 1.1.2. Theorem 1.2.1 Let μ be an infinitely divisible distribution with characteristics (A, ν, b). 1. If the matrix A is nondegenerate, μ has a Cb∞ -density. 2. If at the center, the pair (A, ν) is nondegenerate and satisfies the order condition of exponent 0 < α < 2 with respect to a suitable family of star-shaped neighborhoods, then μ has a Cb∞ -density. Suppose further that the Lévy measure has moments of any order. Then the density is a rapidly decreasing C ∞ -function.

14

1 Probability Distributions and Stochastic Processes

We can check easily that the Lévy measure of a β-stable distribution satisfies the order condition of exponent 2−β at the center, since its Lévy measure is represented by (1.14). It should be noted that in the definition of the order condition at the center of the Lévy measure, any regularity for the Lévy measure ν is not assumed. It may or may not be absolutely continuous. It may be a measure with point mass only. However, the characteristic function of the infinitely divisible distribution has an upper estimate (1.22), which is compatible with the formula (1.17) for stable distribution. The nondegenerate condition at the center of the Lévy measure depends on the choice of the family of neighborhoods {A0 (ρ)}. We will consider an example. Let  X1 , . . . , Xd be mutually independent one-dimensional random variables, whose laws are stable with indices β1 , . . . , βd  , respectively. Then the vector X =  (X1 , . . . , Xd ) is subject to a d  -dimensional infinitely divisible distribution. Further, the Lévy measure of the law of X is concentrated on axis {zej ; z ∈ R}, j =  1, . . . , d  , where ej are unit vectors in Rd . On each axis {zej }, the Lévy measure cj takes the form νj (dz) = 1+βj dz. |z|

If β1 = · · · = βd  = β, the Lévy measure is nondegenerate at the center point 0 with respect to the family of open balls A0 (ρ) = {|z| < ρ}. It holds that Γρ = I and cj ϕ0 (ρ) = 2−β ρ 2−β . If βj , j = 1, . . . , d  are distinct, the Lévy measure is no longer nondegenerate at the center with respect to the family of open balls. We should then take {A0 (ρ)} as a family of star-shaped neighborhood such that    2−βj  A0 (ρ) ∩ zej ; z ∈ R = |z| α ej ; |z| < ρ holds for any j , where α = 2 − minj βj . Then the Lévy measure is nondegenerate at the center with respect to the above family of neighborhoods. This is a reason that we call A0 (ρ) a star-shaped neighborhood. See [46].

1.3 Random Fields and Stochastic Processes A family of S-valued random variables {Xλ ; λ ∈ Λ}, where Λ is a topological space, is called an S-valued random field (with parameter λ ∈ Λ). A random field {Xλ , λ ∈ Λ} is called measurable if Xλ (ω) is measurable with respect to the product σ -field B(Λ) × F, where B(Λ) is the topological Borel field of Λ. When S is Rd or R, it is called a d-dimensional or real-valued random field, respectively. ˜ = 1, we say that the statement holds If a statement holds for ω ∈ Ω˜ with P (Ω) for almost all ω or almost surely and it is written simply as a.s. Let X and X˜ be S˜ valued random variables on the same state space S. If P ({ω; X(ω) = X(ω)}) =1 ˜ holds, X is said to be equal to X almost surely and both are written as X = X˜ a.s. Let {Xλ , λ ∈ Λ} and {X˜ λ , λ ∈ Λ} be two S-valued random fields with the same

1.4 Wiener Processes, Poisson Random Measures and LévyProcesses

15

parameter set Λ. These two random fields are said to be equivalent provided that P (Xλ = X˜ λ ) = 1 holds for all λ ∈ Λ. Further, {X˜ λ , λ ∈ Λ} is called a modification of {Xλ , λ ∈ Λ}. If laws of (Xλ1 , . . . , Xλn ) and (X˜ λ1 , . . . , X˜ λn ) coincide for any finite subset λ1 , . . . , λn of Λ, then {Xλ } and {X˜ λ } are said to be equivalent in law. We denote by D the set of all maps x; Λ → S. The value of x at λ is denoted by xλ . For λ1 , . . . , λn ∈ Λ and Borel subsets B1 , . . . , Bn of S, we define a cylinder set of D by B = {x; xλ1 ∈ B1 , . . . , xλn ∈ Bn }. Let B(D) be the smallest σ -field of D, which contains all cylinder sets, where λ1 , . . . , λn run Λ and B1 , . . . , Bn run Borel sets of S. For a given random field {Xλ , λ ∈ Λ}, there is a unique probability measure P on B(D) satisfying P(B) = P (Xλ1 ∈ B1 , . . . , Xλn ∈ Bn ). The measure P is called the law of the random field {Xλ }. If the parameter set Λ is a time interval T such as [T0 , T1 ], [0, ∞) etc., then the random field {Xt , t ∈ T} is called an S-valued stochastic process. Stochastic processes are often written as {Xt }, or simply as Xt , t ∈ T, Xt etc. If Xt (ω) is continuous with respect to t for almost all ω, {Xt } is called a continuous process. Further, if Xt (ω) is right continuous with left-hand limits with respect to t for almost all ω, {Xt } is called a cadlag process. For a cadlag process {Xt }, we denote its left limit by Xt− , i.e., Xt− = lim ↓0 Xt− . If Xt (ω) is left continuous with right-hand limits with respect to t for almost all ω, {Xt } is called a caglad process. For a caglad process {Xt }, we denote its right limit by Xt+ , i.e., Xt+ = lim ↓0 Xt+ . Further, the process {Xt } is said to be continuous in probability if P (d(Xt+h , Xt ) > ) → 0 as h → 0 for any t ≥ 0 and > 0, where d is the metric of S. When we consider a stochastic process {Xt , t ∈ [T0 , T1 ]}, we usually assume that the process {Xt } describes the time evolution in the forward (positive) direction; at the initial time T0 , it starts from X0 and moves to Xt at time T0 < t ≤ T1 . We could call it a forward process. However, in later discussions, we will consider a process {Xˇ t , t ∈ [T0 , T1 ]} (with the same time parameter), which describes the time evolution in the backward (negative) direction; at the initial time T1 , it starts from Xˇ T1 and moves to Xˇ t at time T0 ≤ t < T1 . We will call it a backward process.

1.4 Wiener Processes, Poisson Random Measures and Lévy Processes 

An Rd -valued cadlag stochastic process {Xt , t ∈ [0, ∞)} is said to have independent increments, if {Xtm+1 − Xtm , m = 1, 2, . . . , n} are independent for any 0 ≤ t1 < t2 < · · · < tn+1 < ∞. It is called time homogeneous if the law of  Xt+h − Xs+h does not depend on h > 0. Let {Xt , t ∈ [0, ∞)} be a cadlag Rd valued process with X0 = 0. It is called a Lévy process if it is time homogeneous,

16

1 Probability Distributions and Stochastic Processes

continuous in probability and has independent increments. If {Xt } is a Lévy process, the d  -dimensional law of Xt is infinitely divisible for each t. Then the characteristic function of the law of Xt is given by the Lévy–Khintchine formula. Let {Xt } and {Xˆ t } be two d  -dimensional Lévy processes. Let μt and μˆ t be laws of random variables Xt and Xˆ t , respectively. If μt = μˆ t holds for all t > 0, these two Lévy processes are equivalent in law and hence the law P of the Lévy process is uniquely determined by {μt , t > 0}. Indeed, for any bounded continuous function  f on Rd n and 0 ≤ t1 < t2 < · · · < tn , E[f (Xt1 , . . . , Xtn )] is computed as 

 ···

f (x1 , . . . , xn )μt1 (dx1 )μt2 −t1 (dx2 − x1 ) · · · μtn −tn−1 (dxn − xn−1 ),

since Xt is time homogeneous and has independent increments. Hence if μt = μˆ t holds for any t, we have E[f (Xt1 , . . . , Xtn )] = E[f (Xˆ t1 , . . . , Xˆ tn )] for any bounded continuous function f . Then {Xt } and {Xˆ t } are equivalent. Let {Xt } be a d  -dimensional continuous Lévy process. If the law of each Xt is Gaussian, the process {Xt } is called a Brownian motion. Since {Xt } is time homogeneous, the mean vector of Xt is tb and covariance matrix is tA, where b and A are the mean and covariance of X1 . If b = 0 and A = I , then {Xt } is called a (d  dimensional) Wiener process or a standard Brownian motion. For Wiener processes {Xt }, laws of {Xt } are unique, since μt are Gaussian distributions with means 0 and covariance matrices tI . We will show the existence of a one-dimensional Wiener process, constructing it by the method of Wiener. We will take T = [0, 1]. Let {Zn ; n = 1, 2, . . .} be an independent sequence of Gaussian random variables with means 0 and variances 1. Let {φn (t), n = 1, 2, . . .} be a complete orthonormal system of the real L2 ([0, 1])t space. We set φ¯ n (t) = 0 φn (s) ds = (1(0,t] , φn ). Consider the infinite sum Xt =

∞ 

φ¯ n (t)Zn .

(1.23)

n=1

We show that it converges uniformly in t ∈ [0, 1] a.s., if we take a good orthonormal system and {φn } be a system of functions on [0, 1] defined by ⎧ m m+1 ⎪ ⎨ −1, if ≤t < , m = 0, 2, 4, . . . , n 2 2n φn (t) = m m+1 ⎪ ⎩ 1, if ≤t < , m = 1, 3, 5, . . . 2n 2n Then it is a complete orthonormal system and φ¯ n  ≡ supt |φ¯ n (t)| ≤ 1/2n+1 holds for any n. The system is called the Rademacher system.

1.4 Wiener Processes, Poisson Random Measures and LévyProcesses

17

Proposition 1.4.1 If {φn } is the Rademacher system, then the infinite sum (1.23) converges uniformly in t almost surely. Let {Xt , t ∈ [0, 1]} be the limit. It is a continuous process. For each t, Xt is Gaussian with mean 0 and covariance t. Further, {Xt , t ∈ [0, 1]} has independent increments and is time-homogeneous. n n−1 n ¯ Proof Set Xtn = | ≤ φ¯ n |Zn |, m=1 φm (t)Zm . Then, since supt |Xt − Xt we have ∞ 

E[sup |Xtn − Xtn−1 |] ≤ t

n=1

∞ 

φ¯ n E[|Zn |] ≤

n=1

∞ 

φ¯ n  < ∞.

n=1

 n−1 n | < ∞, so that Xtn converges uniformly to Xt This shows ∞ n=1 supt |Xt − Xt a.s. Therefore Xt is continuous in t a.s. Note that the law of n-vector (Z1 , . . . , Zn ) is n-dimensional Gaussian with mean 0 and covariance I . Then the linear sum j cj Zj is one-dimensional Gaussian with  mean 0 and variance j cj2 . Therefore for each t, Xtn is Gaussian with mean 0 n and variance j =1 φ¯ n (t)2 . Then the limit Xt is also Gaussian. Further, we have by Perseval’s equality E[Xt Xs ] =

∞ 

φ¯ n (t)φ¯ n (s) =

n=1

∞  (1(0,t] , φn )(1(0,s] , φn ) = (1(0,t] , 1(0,s] ) = t ∧ s. n=1

Therefore E[Xt2 ] = t holds for any t. Let 0 ≤ 0 < t1 < · · · < tn ≤ 1. Then (Xt1 − X0 , . . . , Xtn − Xtn−1 ) is an n-dimensional Gaussian random variable. It holds that E[(Xtm+1 − Xtm )2 ] = ti+1 − tm − tm + tm = tm+1 − tm . Further, if tl < tm , we have E[(Xtm+1 − Xtm )(Xtl+1 − Xtl )] = E[Xtm+1 Xtm+1 ] − E[Xtm+1 Xtl ] − E[Xtm Xtl+1 ] + E[Xtm Xtl1 ] = tl+1 − tl − tl+1 + tl = 0. Therefore its n × n-covariance matrix is diagonal. This means E[ei



m vm (Xtm −Xtm−1 )

  1 2 ] = exp − (tm − tm−1 )vm 2 m =

n  m=1

  E eivm (Xtm −Xtm−1 ) .

18

1 Probability Distributions and Stochastic Processes

Then Xtn − Xtn−1 , . . . , Xt1 − X0 are mutually independent (Corollary 1.1.1). Therefore {Xt } has independent increments and is time-homogeneous.   We can choose other orthonormal system {φn√ } and show that (1.23) converge uniformly in t a.s. √Take φ0 (t) = 1, φn (t) = 2 cos nπ t, n = 1, 2, . . .. Then φ¯ 0 (t) = t, φ¯ n (t) = nπ2 sin nπ t, n = 1, 2 . . .. It is known that the infinite sum (1.23) is also uniformly convergent. It is called Wiener’s construction. Let {Zn } be another independent sequence of Gaussian random variables with mean 0 and covariance 1, which is independent of {Zn }. We define a stochastic    ¯ process by Xt = ∞ φ n=1 n (t)Zn for t ∈ [0, 1]. Then {Xt } and {Xt } are equivalent  in law. Further, these two processes are independent. {Xt } is called an independent (n) copy of {Xt }. Now, take a sequence of independent copies {Xt }, n = 1, . . . of the process {Xt } of Proposition 1.4.1. Define another continuous process Xt by n (i) (n+1) i=1 X1 + Xt−n if t ∈ (n, n + 1]. Then {Xt , t ∈ [0, ∞)} is a Wiener process. Next, take an independent copy {Xt , t ∈ [0, ∞)} of the above one dimensional Wiener process {Xt , t ∈ [0, ∞)}. Then the pair {(Xt , Xt ), t ∈ [0, ∞)} is a 2-dimensional Wiener process. Repeating this argument, we can also define d  dimensional Wiener process. ¯ It is called a Next, let {Nt ; t ∈ [0, ∞)} be a Lévy process with values in N. Poisson process with intensity λ, if the law of each Nt is a Poisson distribution with parameter tλ. The law of a Poisson process {Nt } with intensity λ is unique, since {Nt } is a Lévy process. We shall construct a Poisson process with intensity λ. Let σn ; n = 1, 2, . . . be mutually independent positive random variables with a common exponential distribution with parameter λ. We set τ0 = 0 and τn = σ1 + · · · + σn for n ≥ 1. We ¯ by define a stochastic process Nt , t ∈ [0, ∞) with values in N Nt (ω) = n,

if τn (ω) ≤ t < τn+1 (ω).

(1.24)

Then P (Nt = n) = P (τn ≤ t < τn + σn+1 ). Note that the law of τn is the Gamma distribution with parameters λ and n. Then P (Nt = n) is computed as λn+1 (n − 1)!

 B

x n−1 e−λx e−λy dx dy = e−λt

(λt)n , n!

where B = {(x, y); 0 ≤ x ≤ t < x + y}. Therefore the law of Nt is Poisson with parameter λt. The random time τn defined above is called the n-th jumping time of the stochastic process Nt . The sequence of random times {τn } is called jumping times of the stochastic process {Nt }. Proposition 1.4.2 {Nt } defined by (1.24) is a Poisson process with intensity λ. Proof We want to show that {Nt } is time homogeneous and has independent increments. Our discussion is close to Sato [99]. We first show that the law of (τn+1 − t, σn+2 , . . . , σn+m ) under the condition Nt = n is the same as the law of (σ1 , . . . , σm ). Set a = P (Nt = n). Then we have

1.4 Wiener Processes, Poisson Random Measures and LévyProcesses

19

P (τn+1 − t > s1 , σn+2 > s2 , . . . , σn+m > sm |Nt = n) = P (τn ≤ t, τn+1 − t > s1 , σn+2 > s2 , . . . , σn+m > sm )/a = P (τn ≤ t, τn+1 − t > s1 )P (σn+2 > s2 , . . . , σm+n > sm )/a = P (τn+1 − t > s1 |Nt = n)P (σ2 > s2 , . . . , σm > sm ) = P (σ1 > s1 )P (σ2 > s2 , . . . , σm > sm ) = P (σ1 > s1 , σ2 > s2 , . . . , σm > sm ).

(1.25)

Here we used the equality P (τn+1 − t > s|Nt = n) = e−λs = P (σ1 > s), which is shown as follows. Note P (τn < t, τn +σn > t +s) =

λn+1 (n − 1)!

 B

x n−1 e−λx e−λy dx dy = e−λ(t+s)

(λt)n , n!

where B  = {(x, y); 0 < x < t + s < x + y}. Then we have P (Nt = n, τn+1 > t + s) P (τn ≤ t, τn+1 > t + s) = = e−λs . P (Nt = n) P (Nt = n) Now we will prove that {Nt } is time homogeneous and has independent increments. We have P (Nt1 − Nt0 = n1 , . . . , Ntm − Ntm−1 = nm |Nt0 = n0 ) = P (τn0 +n1 ≤ t1 < τn0 +n1 +1 , . . . , τn0 +···+nm ≤ tm < τn0 +···+nm +1 |Nt0 = n0 ) = P (τn1 ≤ t1 − t0 < τn1 +1 , . . . , τn1 +···+nm ≤ tm − t0 < τn1 +···+nm +1 ). In the last equality, we applied (1.25). Then we get P (Nt1 − Nt0 = n1 , . . . , Ntm − Ntm−1 = nm |Nt0 = n0 ) = P (Nt1 −t0 = n1 , . . . Ntm −t0 − Ntm−1 −t0 = nm ). Repeating this argument, we get the equality P (Nt0 = n0 , Nt1 − Nt0 = n1 , . . . , Ntm − Ntm−1 = nm ) = P (Nt0 = n0 ) · · · P (Ntm −tm−1 = nm ). We have in particular, P (Nt0 = n0 , Nt1 − Nt0 = n1 ) = P (Nt0 = n0 )P (Nt1 −t0 = n1 ).

(1.26)

20

1 Probability Distributions and Stochastic Processes

Summing up the above for n0 = 0, 1, 2, . . .. Then we get P (Nt1 − Nt0 = n1 ) = P (Nt1 −t0 = n1 ). This shows that Nt is time homogeneous. Further, we can rewrite (1.26) as P (Nt0 = n0 , Nt1 − Nt0 = n1 , . . . , Ntm − Ntm−1 = nm ) = P (Nt0 = n0 ) · · · P (Ntm − Ntm−1 = nm ).

(1.27)

This proves that {Nt } has independent increments.

 

Now, let {Nt } be a Poisson process with intensity λ. Then its law coincides with the law of {Nt } constructed above. Hence {Nt } is a cadlag nondecreasing process  is 0 or 1, almost surely. Set τ  = 0 and define with values in N such that Nt − Nt− 0  random times τn , n = 1, 2, . . . by induction as  τn

=

 ; N > N inf{t > τn−1 }, if the set {· · · } is non-empty, t τ n−1

∞,

otherwise.

Then τn is subject to the Gamma distribution with parameters λ and n, if we restrict the distribution to the interval [0, ∞). The sequence {τn } is called the jumping times of the Poisson process {Nt }.  Let T = [0, ∞) and U = T × Rd0 . Elements of U are denoted by u = (r, z) where r ∈ T, z ∈ Rd0 . Let (U, B(U), n) be a measure space, where n is a σ -finite ¯ measure on U. A family of N-valued random variables N (B), B ∈ B(U) is called the (abstract) Poisson random measure on U with intensity measure n, if the followings hold: ¯ 1. For every ω, N (·, ω) is an N-valued measure on U. 2. For each B, the law of N(B) is a Poisson distribution with parameter n(B). 3. If B1 , . . . , Bn are disjoint, N(B1 ), . . . , N(Bn ) are independent. We are interested in the Poisson random measure in the case that n is the product measure drν(dz), where dr is the Lebesgue measure on [0, ∞) and ν is a σ -finite  measure on Rd0 . We shall construct it. We first consider the case where ν is a finite measure. Let {Sn ; n = 1, 2, . . .} be a sequence of mutually independent random  variables with values in Rd with the common distribution μ(dz) = ν(dz)/λ, where  d λ = ν(R0 ). Let {Nt } be a Poisson process with intensity λ, which is independent  of {Sn }. For 0 ≤ s < t < ∞ and a Borel subset E of Rd0 , we set N ((s, t] × E) :=

 Ns

|z|>

zν(dz). 1>|z|>

t

0 |z|> zN(ds dz) is a

i(v,Y t ) ] is computed as E[e

with Lévy measure ν1|z|> ,



zN(dr dz) − t

compound Poisson process

   zN(dr dz)) exp − it (v,

|z|>

 zν(dz) 1≥|z|>

  (ei(v,z) − 1)ν(dz) exp it (v, 



zν(dz) 1≥|z|>

 (ei(v,z) − 1 − i(v, z)1D (z))ν(dz) .

Let → 0, then we get the formula (1.33) for Yt .

 

Let {Xt } be a Lévy process and let μt be the law of Xt . Since {Xt } is time homogeneous and has independent increments, for any n, μt is equal to n-times convolution of the law μt . Therefore μt is infinitely divisible. Then its characteristic n

function is represented by (1.33) by the Lévy–Khinchin formula. Therefore, for any given Lévy process {Xt }, the Lévy process {Xt } constructed by (1.32) is equivalent in law with {Xt }. Therefore we have the following theorem.

1.5 Martingales and Backward Martingales

25

Theorem 1.4.2 Let {Xt , t ∈ [0, ∞)} be a Lévy process. Then there exist a constant b, independent Brownian motion {Bt , t ∈ [0, ∞)} and Poisson random measure N(dr dz), and Xt is represented by (1.32). The formula (1.32) is called the Lévy–Itô decomposition or Lévy–Itô representation of a Lévy process. In some problems, it is convenient to use a Poisson point process instead of a  Poisson random measure. We will give its definition. By a point function on Rd0 ,  we mean a map q : Dq → Rd0 , where Dq is a countable set in [0, ∞). A counting measure of the point function is defined by N(B, q) = {t ∈ Dq ; (t, q(t)) ∈ B},

(1.34)



where B are Borel sets in U = [0, ∞) × Rd0 . A random variable q(ω) with values in the space of point functions is called a point process. If the counting measure of the point process q(ω) is a Poisson random measure with intensity dt dν, the point process is called a Poisson point process with intensity dtν(dz) and N of (1.34) is called the associated Poisson random measure. Given a Lévy measure ν, there exists a Poisson point process with intensity dtν(dz). Indeed, if ν is a finite measure, consider thecompound Poisson process with intensity dtν(dz). It is represented as Xt (ω) = 1≤n≤Nt (ω) Sn (ω) by (1.29). We define a point process q = q(ω) by setting Dq = {0 < τ1 < · · · < τn < ∞},

q(t) = (τi , Si ),

if τi = t.

Then it is a Poisson point process with intensity dtν(dz). We can show the existence of the Poisson point process for any Lévy measure ν by an argument similar to the proof of Theorem 1.4.1.

1.5 Martingales and Backward Martingales In this and the next section, we will study martingales. We will restrict our attentions to three topics of martingales, which are often used in later stochastic calculus. Three topics are the optional sampling theorem, Doob’s inequality and quadratic variation. Let (Ω, F, P ) be a probability space. Let G be a sub σ -field of F and let X be an integrable random variable. A G-measurable and integrable random variable Y is called the conditional expectation of X given G, if it satisfies E[X1G ] = E[Y 1G ] for any G ∈ G. Here, 1G is the indicator function of the set G. The conditional expectation Y exists uniquely a.s. For the proof, consider a signed measure P˜ (A) = E[X1A ], A ∈ G. It is absolutely continuous with respect to P , i.e., P˜ (A) = 0 holds for any A ∈ G with P (A) = 0. Then, in view of the Radon– Nikodym theorem, there exists an integrable G-measurable functional Y such that

26

1 Probability Distributions and Stochastic Processes

P˜ (A) = A Y dP holds for any A ∈ G. The uniqueness follows from the uniqueness of the Radon–Nikodym density. The conditional expectation of X with respect to G is denoted by E[X|G]. Further, if A is an element of F, the conditional expectation E[1A |G] is denoted by P (A|G) and is called the conditional probability of event A given G. Conditional expectations have following properties. Proposition 1.5.1 Let X, Y be integrable random variables. 1. 2. 3. 4. 5. 6.

E[c1 X + c2 Y |G] = c1 E[X|G] + c2 E[Y |G] holds a.s. for any constants c1 , c2 . E[|X||G] ≥ |E[X|G]| holds a.s. If G1 ⊂ G2 ⊂ F, then E[E[X|G2 ]|G1 ] = E[X|G1 ] holds a.s. If σ (X) and G are independent, then E[X|G] = E[X] a.s. If XY is integrable and X is G-measurable, then E[XY |G] = XE[Y |G]. Let f (x) be a convex function, bounded from below. If f (X) is integrable, we have f (E[X|G]) ≤ E[f (X)|G] a.s.

Proposition 1.5.2 Let X1 , . . . , Xn , . . . , X be integrable random variables. 1. (Fatou’s lemma) If Xn ≥ 0, lim inf E[Xn |G] ≥ E[lim inf Xn |G], n→∞

n→∞

a.s.

2. Let p ≥ 1. If Xn converges to X in Lp , then E[Xn |G] converges to E[X|G] in Lp . Proofs of these propositions are straightforward. These are omitted. We will define a martingale with discrete time parameter. Let N be the set of all positive integers. An increasing sequence {Fn , n ∈ N} of sub σ -fields of F is called a filtration. Given a filtration {Fn }, we will define a martingale. Let {Xn ; n ∈ N} be a sequence of real integrable random variables. We will assume that each Xn is Fn -measurable. It is called a martingale with discrete time if E[Xn |Fm ] = Xm holds for any 1 ≤ m < n. We denote by F0 the trivial σ -field {∅, Ω}, where ∅ is the empty set. A sequence of random variables fn , n ∈ N is called predictable if each fn is Fn−1 -measurable. Let {fn } be a predictable sequence. The martingale transform Yn , n ∈ N is defined by Yn =

n 

fm (Xm − Xm−1 ),

where X0 = 0.

m=1

If E[|fm ||Xm − Xm−1 |] < ∞ holds for any m ≥ 1, {Yn } is a martingale, since Yn is Fn -measurable and satisfies E[Yn+1 − Yn |Fn ] = fn+1 E[Xn+1 − Xn |Fn ] = 0.

1.5 Martingales and Backward Martingales

27

Let τ be a random variable with values in N ∪ {∞}. It is called a stopping time if {τ ≤ n} ∈ Fn holds for any n ∈ N. If τ is a stopping time, it is easily seen that n ∧ τ is also a stopping time for any positive integer n. For a stopping time τ , we set   Fτ = B ∈ F; B ∩ {τ ≤ n} ∈ Fn for any n . It is a sub σ -field of F. Proposition 1.5.3 Let {Xn , n ∈ N} be a martingale and let τ be a stopping time. 1. The stopped process {Yn = Xn∧τ , n ∈ N} is again a martingale. 2. Let σ be another stopping time. Then E[Xn∧τ |Fσ ] = Xn∧τ ∧σ holds for any n ∈ N. Proof We set fn = 1τ ≥n . It is a predictable sequence. Consider the martingale transform by fn ; Yn = nm=1 fm (Xm − Xm−1 ). It coincides with Xn∧τ . Therefore {Xn∧τ } is a martingale. Let σ be another stopping time. If B ∈ Fσ , we have E[Yn ; B] =

n 

n 

E[Yn ; B ∩ {σ = m}] =

m=0

E[Ym ; B ∩ {σ = m}] = E[Yn∧σ ; B].

m=0

Further, Yn∧σ is Fσ -measurable, because 

{Xn∧σ < a} ∩ {σ ≤ m} =

{Xl < a, σ = l} ∈ Fm

l≤m

holds for any m. Therefore we get the equality E[Yn |Fσ ] = Yn∧σ a.s. Substituting Yn = Xn∧τ , we get E[Xn∧τ |Fσ ] = Xn∧τ ∧σ a.s.   Proposition 1.5.4 Let {Xn , n ∈ N} be a martingale and let N ∈ N. Then we have for any a > 0  aP ( sup |Xn | > a) ≤ n≤N

supn≤N |Xn |>a

|XN | dP .

(1.35)

Proof For a given a > 0, we set τ = inf{n; |Xn | > a}

(= ∞ if the set {· · · } is empty).

Then τ is a stopping time. Indeed, we have {τ ≤ k} = i≤k {|Xi | > a} ∈ Fk for any k ∈ N. Then by Proposition 1.5.3, we have E[|XN ||Fτ ] ≥ |XN ∧τ |. Therefore,  aP (τ ≤ N) ≤

τ ≤N

 |XN ∧τ | dP ≤

τ ≤N

|XN | dP .

Since {τ ≤ N } = {supn≤N |Xn | > a}, we get the inequality of the Proposition.

 

28

1 Probability Distributions and Stochastic Processes

Proposition 1.5.5 Let {Xn , n ∈ N} be a martingale satisfying E[|Xn |p ] < ∞ for p any n, where p > 1. Set q = p−1 . Then we have for any N ∈ N,   E sup |Xn |p ≤ q p E[|XN |p ].

(1.36)

n≤N

Proof Set Y = supn≤N |Xn | and F (λ) = P (Y > λ). Then we have 



∞

E[Y ] = −

p

0

0

Since λP (Y > λ) ≤  E[Y p ] ≤ 0

≤

∞



λ F (λ) d(λ ) − lim λ F (λ) 0 ≤

∞

λ dF (λ) =

p

Y >λ |XN | dP

1 λ 

p p−1



p



∞

p

λ→∞

F (λ) d(λp ).

0

holds by Proposition 1.5.4, we have

   |XN |dP d(λp ) = |XN |

Y 0

Y >λ

 1 d(λp ) dP λ

1 p

1

|XN |Y p−1 dP ≤ qE[|XN |p ] E[Y (p−1)q ] q 1

1

≤ qE[|XN |p ] p E[Y p ] q . Therefore we have the inequality of the proposition.

 

We give the definition of a martingale with negative parameter. Let F−n , n ∈ N be a sequence of sub σ -fields of F satisfying F−n ⊂ F−(n−1) . Let {X−n , n ∈ N} be a sequence of integrable {F−n }-adapted random variables. It is called a martingale if E[X−m |F−n ] = X−n holds whenever −m > −n. A sub-martingale and a supermartingale are defined similarly. We will next define a martingale with continuous parameter T = [0, ∞). Let {Ft , t ∈ T} be a family of sub σ -fields of F satisfying Fs ⊂ Ft for  any s < t. We assume further that Ft contains all null sets of F and satisfies Ft = >0 Ft+ (right continuous). Then the family {Ft , t ∈ T} is called a filtration. A random variable τ with values in T ∪ {∞} is called a stopping time if {ω; τ (ω) ≤ t} ∈ Ft holds for any t ∈ T. For a stopping time τ , we set   Fτ = B ∈ F; B ∩ {τ ≤ t} ∈ Ft for any t ∈ T . It is a sub σ -field of F. Let {Xt } = {Xt , t ∈ T} be a stochastic process. {Xt } is called adapted for {Ft } (or {Ft }-adapted) and {Ft } is called admissible for {Xt } (or {Xt }-admissible), if Xt is Ft -measurable for each t ∈ T. An {Ft }-adapted cadlag process {Xt } is called a martingale or {Ft }-martingale, if Xt is integrable for any t and equalities E[Xt |Fs ] = Xs hold a.s. for any s < t. If equality signs are replaced by ≥ in the above, it is called a sub-martingale. Further, if {−Xt } is a sub-martingale {Xt } is called a super-martingale.

1.5 Martingales and Backward Martingales

29

Let p > 1. An {Ft }-adapted cadlag process {Xt } is called an Lp -martingale if it is a martingale and satisfies E[|Xt |p ] < ∞ for any t. An {Ft }-adapted cadlag process {Xt } is called a local martingale or local Lp -martingale, if there exists an increasing sequence of stopping times {τn } such that P (τn < T ) → 0 for (n) any positive constant T and each stopped process Xt := Xt∧τn , t ∈ T is a p martingale or L -martingale, respectively. A martingale is a local martingale and an Lp -martingale is a local Lp -martingale. Theorem 1.5.1 (Doob) Let {Xt } be a martingale. 1. For any stopping time τ , the stopped process {Yt = Xt∧τ } is a martingale. 2. For any two stopping times σ and τ , we have E[Xt∧σ |Fτ ] = Xt∧σ ∧τ for any t. 3. (Doob’s inequality) Suppose that {Xt } is an Lp -martingale for some p > 1. Set p q = p−1 . Then we have    sup |Xr |p ≤ q p E |Xt |p ,

 E

∀t ∈ T.

(1.37)

0≤r≤t (n) Proof We prove the first assertion. For n ∈ N, set tm = m/2n , m = 1, . . . , 2n T . Let τ be a stopping time. For n ∈ N, we define (n) τn (ω) = tm ,

(n)

(n) if tm−1 < τ (ω) ≤ tm .

(1.38)

Then τn , n = 1, 2, . . . is a sequence of stopping times decreasing to τ as n → ∞. We have E[Xt∧τn ; B] = E[Xs∧τn ; B] for any B ∈ Fs by Proposition 1.5.3. Therefore if limn→∞ E[Xt∧τn ; B] = E[Xt∧τ ; B] holds for any t, we get E[Xt∧τ ; B] = E[Xs∧τ ; B] and hence Xt∧τ is a martingale. In order to prove the above convergence, we will show that the family of random variables {Xt∧τn −Xt∧τ } is uniformly integrable, i.e., for any > 0 there exists c > 0 such that  sup n

|Xt∧τn −Xt∧τ |>c

|Xt∧τn − Xt∧τ | dP < .

(1.39)

Set Y−n = Xt∧τn − Xt∧τ and G−n = Ft∧τn . Then, in view of Proposition 1.5.3, 2, Y−n , n ∈ N is a {G−n }-martingale with negative parameter. Therefore |Y−n | is a sub-martingale. Since {|Y−n | > c} ∈ G−n , we have the inequality 

 |Y−n |>c

|Y−n | dP ≤

|Y−n |>c

|Y−1 | dP

for any n and c > 0. Set Q1 (A) := A |Y−1 | dP on G−1 . Then Q1 is absolutely continuous with respect to P . Therefore for any > 0, there exists δ > 0 such that for any A ∈ G−1 with P (A) < δ, the inequality Q(A) < holds. Further, for this δ, there exists c > 0 such that the inequality

30

1 Probability Distributions and Stochastic Processes

P (|Y−n | > c) ≤

1 1 E[|Y−n |] ≤ E[|Y−1 |] ≤ δ c c

holds for any n. We have thus shown that for any > 0, there exists c > 0 such that the inequality supn |Y−n |>c |Y−n | dP < holds. Therefore {Y−n } is uniformly integrable. Now we have 

 |Xt∧τn − Xt∧τ |1B dP ≤ +

|Yn |≤c

|Yn |1B dP .

Since {Yn } converges to 0 a.s., the last term converges to 0 as n → ∞. Therefore we have limn→∞ |Xt∧τn − Xt∧τ |1B dP = 0. This proves the equality limn→∞ E[Xt∧τn ; B] = E[Xt∧τ ; B]. The second assertion can be verified similarly, using Proposition 1.5.3, 2 and approximating both stopping times σ and τ from the above by sequences of stopping times with discrete values. Suppose next that {Xt } is an Lp -martingale for some p > 1. For n ∈ N, set (n) tm = m/2n ∧ t, m = 1, . . . , 2n (T + 1). Then E[supm0 Fˇ s− = Fˇ s for any s and each Fˇ s contains null sets of F. We will consider a backward semi-martingale. We consider left continuous rather than right continuous processes, since the time evolution is backward. Let {Fˇ s } be a backward filtration. Let {Xˇ s , s ∈ T} be an integrable caglad process, adapted to a backward filtration {Fˇ s }. It is called a backward martingale if E[Xˇ s |Fˇ t ] = Xˇ t holds a.s. for any s < t. A random variable σ with values in T∪{−∞} is called a backward stopping time, if {σ ≥ t} ∈ Fˇ t holds for any t ∈ T. An {Fˇ s }-adapted caglad process

1.6 Quadratic Variations of Semi-martingales

31

{Xˇ s } is called a backward local martingale if there exists a decreasing sequence of backward stopping times σn such that limn→∞ P (σn > 0) = 0 and each stopped process {Xˇ s∨σn } is a backward martingale. Further, if Xˇ s is written as the sum of a backward local martingale and a continuous process of bounded variation adapted to the backward filtration {Fˇ s }, {Xˇ s } is called a backward semi-martingale. Let {Fs,t , 0 ≤ s < t < ∞} be a family of sub σ -fields of F, each of which contains null sets of F. We assume that for any fixed s, Ft := Fs,t , t ≥ s is a filtration. We assume further that for any fixed t, Fˇ s := Fs,t is decreasing with  = Fˇ s respect to s, i.e., Fˇ s ⊂ Fˇ s  if s  ≤ s and left continuous, i.e., >0 Fˇ s− holds for any s < t. Then {Fs,t } is called a two-sided filtration. A backward local semi-martingale is defined similarly to a local semi-martingale. Let {Xt , t ∈ [0, ∞)} be a Wiener process and let N(dsdz) be a Poisson random  measure on U = [0, ∞) × Rd0 . Let Fs,t = σ (Xu − Xv , N(dr dz); s ≤ u < v ≤ t, dr ⊂ [s, t]). Wiener process and Poisson Then {Fs,t } is a two-sided filtration generated by t ˜ random measure. Then {Xt } and {Yt } = { 0 Rd f (z)N(dr dz)} are forward 0 martingales with respect to the filtration Ft = F0,t . Further, {Xˇ s := XT − Xs } and {Yˇs := YT − Ys } are backward martingales with respect to the backward filtration Fˇ s = Fs,T .

1.6 Quadratic Variations of Semi-martingales In this section, we will restrict time parameter to the finite interval T = [0, T ]. Martingales and stopping times with time parameter T are defined in the same way as those with time parameter [0, ∞). Let Π = {0 = t0 < t1 < · · · < tn = T } be a partition of T = [0, T ]. We set |Π | = maxm |tm − tm−1 |. We define the quadratic variation of a real cadlag process {Xt , t ∈ T} associated with the partition Π by XΠ t

=

n 

(Xtm ∧t − Xtm−1 ∧t )2 ,

t ∈ T.

(1.40)

m=1

We are interested in the limit of quadratic variations {XΠ t } as |Π | → 0. If it converges uniformly in t in probability to a process, the limit is denoted by Xt , t ∈ T and is called the quadratic variation of {Xt }. The quadratic variation of {Xt } is often written as {Xt }. In this section, we show that if {Xt , t ∈ T} is a semi-martingale with bounded jumps, the quadratic variation exists. We will first list properties of L2 martingales.

32

1 Probability Distributions and Stochastic Processes

Proposition 1.6.1 Let {Xt } be an L2 -martingale. 1. E[(Xt − Xs )2 |Fs ] = E[Xt2 − Xs2 |Fs ] holds for any 0 ≤ s < t ≤ T . 2. E[(Xv − Xu )(Xt − Xs )|Fs ] = 0 holds for any 0 ≤ s < t ≤ u < v ≤ T . 3. For any 0 ≤ s < t ≤ T , it holds that Π F [(Xt − Xs )2 |Fs ] = E[XΠ t − Xs |Fs ].

(1.41)

Proof In view of Proposition 1.5.1, 1 and 5, we have E[(Xt − Xs )2 |Fs ] = E[Xt2 − 2Xt Xs + Xs2 |Fs ] = E[Xt2 |Fs ] − 2E[Xt |Fs ]Xs + Xs2 = E[Xt2 |Fs ] − Xs2 . Then we get the equality of 1. Next, we have by Proposition 1.5.1, 3, E[(Xv − Xu )(Xt − Xs )|Fs ] = E[E[(Xv − Xu )|Fu ](Xt − Xs )|Fs ] = 0, proving the orthogonal property of 2. We have  (Xtm ∧t − Xtm−1 ∧t ) + (Xtl − Xs ), Xt − Xs = m;s N } (= ∞ if the set {· · · } is empty). It is called the first leaving time of the process Xt from the set {x; |x| ≤ N } or hitting time of

the process Xt to the set {x; |x| > N}. It is a stopping time, since {σN ≤ t} = 0≤r≤t {|Xr | > N } ∈ Ft holds for any t. Here r runs rational numbers. Then Xt∧τn ∧σN is a martingale by Theorem 1.5.1. Let M > 0 be a positive constant such that |ΔXt | ≤ M holds for any t a.s. Then |Xt∧τn ∧σN | is bounded by N + M for all n. Therefore the limit limn→∞ Xt∧τn ∧σN = Xt∧σN is also a martingale for any N . Since P (σN < T ) → 0 ans N → ∞, {Xt } is a local martingale with respect to stopping times {σN , N = 1, 2, . . .}. (N ) Set Xt = Xt∧σN . It is an L4 -martingale. By the definition the quadratic (n) variation with respect to the partition Πn , it holds that X(N ) (n) t = Xt∧σN . Further, (n) for any N , X(N ) t converges to a cadlag increasing X(N ) t uniformly in t in L2  sense as n → ∞. Then if N  ≥ N we have X(N ) t∧σN = X(N ) t . Therefore there exists a cadlag increasing process Xt such that Xt∧σN = X(N ) t . Consequently, (n) if {Πn } is regular, for any N Xt converges to Xt uniformly for t ≤ τN in L2 (n) sense. This shows that Xt converges to Xt uniformly in t ∈ T in probability. 2 Further, Xt − Xt is a local martingale.

1.6 Quadratic Variations of Semi-martingales

35

It remains to show that Xt does not depend on the choice of partitions {Πn }. Let Π {Πn } be another regular sequence of partitions. Let {Xt n } be the sequence of the associated quadratic variations and let XT be its limit as n → ∞. We can choose still another regular sequence of partitions {Πn } such that for any Πn ∈ {Πn } and Πn  ∈ {Πn }, there are Πn and Πn in {Πn } such that Πn ⊂ Πn and Πn  ⊂ Πn Π 

holds. The sequence {Xt n } converges to an increasing process Xt . It should coincide with Xt and Xt . Therefore the uniqueness of the quadratic variation follows.   Corollary 1.6.1 If {Xt } be a continuous local martingale, its quadratic variation {Xt } is a continuous increasing process. Further, if {At } is a continuous increasing process such that {Xt2 − At } is a local martingale, the equality At = Xt holds for any t ∈ T, a.s. Proof If {Xt } is a continuous local martingale, its quadratic variation {Xt } is a continuous process, since {XΠ t } are continuous processes. Suppose that {At } is a continuous increasing process such that {Xt2 − At } is a local martingale. Then {Mt := Xt − At } is a continuous local martingale. Since Mt is a process of bounded variation, we have Mt = 0. Therefore Mt2 is a continuous local martingale and hence Mt = 0 for any t. Therefore Xt = At holds for ant t.   Let {Xt } and {Yt } be cadlag processes. For a partition Π , we define the quadratic covariation of X, Y associated with Π by X, Y Π t =

n 

(Xtm ∧t − Xtm−1 ∧t )(Ytm ∧t − Ytm−1 ∧t ),

t ∈ T.

(1.43)

m=1

Theorem 1.6.2 Suppose that {Xt } and {Yt } are local martingales with bounded jumps. Then quadratic covariations {X, Y Π t } converge to a process of bounded covariation {X, Y t } uniformly in t in probability. Further, {Xt Yt − X, Y t } is a local martingale. The process X, Y t is called the quadratic covariation of Xt and Yt . Proof If Xt = Yt and it is a local martingale with bounded jumps, the existence of the quadratic variation Xt is a direct consequence of Lemma 1.6.1. We shall consider the case Xt = Yt . It holds that X, Y Π t =

 1 Π X + Y Π t − X − Y t . 4

Let |Π | tend to 0. Then the right-hand side converges to 14 (X + Y t − X − Y t ). Therefore X, Y Π t converges uniformly in t in probability. Denote the limit by X, Y t . It is a process of bounded variation. It satisfies

36

1 Probability Distributions and Stochastic Processes

X, Y t =

 1 X + Y t − X − Y t . 4

Then Xt Yt − X, Y t is a local martingale.

 

Theorem 1.6.3 Let {Xt } be a continuous semimartingale and let {Yt } be semimartingales with bounded jumps, decomposed as Xt = Mt + At and Yt = Nt + Bt respectively, where {Mt , Nt } are local martingales and {At , Bt } are continuous processes of bounded variation. Then {X, Y Π t } converges as |Π | → 0 to a continuous process of bounded variation {X, Y t , t ∈ T} uniformly in t in probability. Further, lim X, Y Π t = X, Y t = M, N t ,

|Π |→0

t ∈ T.

(1.44)

Proof It holds that Π Π Π Π X, Y Π t = M, N t + M, Bt + A, Nt + A, Bt . Π 1/2 (BΠ )1/2 → 0 and Let |Π | → 0. Then we have |M, BΠ t | ≤ (Mt ) t Π Π A, Bt → 0 since Bt → 0. Similarly, |A, NΠ t | → 0 holds. Since M, N Π t → M, N t , (1.44) holds. Next, note 1

1

|M, N t − M, N s | ≤ (Mt − Ms ) 2 (N t − N s ) 2 . The right-hand side converges to 0 uniformly a.s. as t → s, if Mt is continuous in t a.s. Therefore M, N t is a continuous process.   The quadratic covariation of Xt and Yt is often denoted by Xt , Yt , Xt , Yt t etc. Let [s, T ] be a sub-interval of [0, T ]. We shall consider the quadratic covariation of two semi-martingales Xt , Yt on the interval [s, T ]. Let Π = {t0 < · · · < tm } be a  =t partition of [s, T ]. We set tm tm ∧t and define X, Y Π s,t =

n 

  (Xtm − Xtm−1 )(Ytm − Ytm−1 ).

m=1

Assume the same conditions as in Theorem 1.6.2 for Xt and Yt . Then, as |Π | → 0, X, Y Π s,t , t ∈ [s, T ] converges to a continuous process of bounded variation. We denote the limit by X, Y s,t and call it the quadratic covariation of Xt and Yt on the interval [s, T ]. In particular, if both Xt , Yt are semi-martingales for t ∈ [0, T ], then it holds that X, Y s,t = X, Y t − X, Y s almost surely for any 0 ≤ s < t ≤ T . Note The quadratic covariation of two L2 martingales Xt , Yt was introduced by P.A. Meyer [85]. It is often denoted by [X, Y ]t . On the other hand, a bracket process X, Y t was introduced by Kunita–Watanabe [66], which is defined as a

1.7 Markov Processes and Backward Markov Processes

37

unique natural (predictable) process of bounded variation such that Xt Yt − X, Y t is a martingale. The existence of such bracket process is due to the Doob–Meyer decomposition theorem of submartinales [84]. If one of Xt and Yt is continuous, say Xt is continuous, then both of [X, Y ]t and X, Y  are continuous processes and coincide each other. In this monograph, we take X, Y t as the definition of the quadratic covariation of Xt and Yt . Theorem 1.6.2 is discussed in Kunita [59], Karatzas–Shreve [55] for continuous semi-martingales. We discussed this again, without using the Doob–Meyer decomposition of a sub-martingale. In this monograph, we do not follow difficult discussions concerning natural increasing processes and predictable increasing processes, which appear in the Doob–Meyer decomposition. See [55].

1.7 Markov Processes and Backward Markov Processes Let S be a Hausdorff topological space with the second countability. Let B(S) be its topological Borel field. We denote by Bb (S) the set of all real bounded B(S)measurable functions. Cb (S) is the set of all bounded continuous functions on S. A function P (x, E) of two variables x ∈ S, E ∈ B(S) is called a kernel on the space (S, B(S)) if for each x ∈ S, it is a finite measure in E and, for each E ∈ B(S), it is a B(S)-measurable function of x. We set for f ∈ Bb (S)  Pf (x) =

f (y)P (x, dy).

(1.45)

S

It is a B(S)-measurable function. If it is a bounded function for any f ∈ Bb (S), the kernel is called bounded. A bounded kernel P defines a linear transformation from Bb (S) into itself. Let T = [0, ∞). Let {Ps,t (x, E); 0 ≤ s < t < ∞} be a family of bounded kernels satisfying the following properties. 1. Semigroup property: Ps,t Pt,u f = Ps,u f holds for any s < t < u and f ∈ Bb (S). The semigroup property is equivalent to  Ps,t (x, dy)Pt,u (y, E) = Ps,u (x, E),

∀x ∈ S, ∀E ∈ B(S).

S

The equation is called the Chapman–Kolmogorov equation. 2. Continuity: limt↓s Ps,t f (x) = f (x) holds for all x for any f ∈ Cb (S). Then {Ps,t (x, E)} is called a transition function. The family of linear transformations {Ps,t } satisfying the above two properties is called a semi-group (of linear transformations).

38

1 Probability Distributions and Stochastic Processes

If Ps,t (x, S) = 1 holds for any x and s < t, the transition function is called conservative and if Ps,t (x, S) ≤ 1 holds for all x and s < t, the transition function is called Markovian. A conservative transition function is called a transition probability. If Ps,t (x, E) is Markovian, we adjoin a cemetery ∞ to S as a one point compactification if S is non-compact and as an isolated point if S is compact.  (x, E) on S  by We set S  = S ∪ {∞} and define a transition probability Ps,t  Ps,t (x, E) = Ps,t (x, E) if x ∈ S, E ⊂ S and  (x, {∞}) = 1 − Ps,t (x, S) Ps,t

if x ∈ S,

= 1 if x = ∞.

(1.46)

 (x, E) is a transition probability. We denote by B (S) the set of f ∈ Then Ps,t ∞  Bb (S ) such that f (∞) = 0. Further, C∞ (S) is the set of all f ∈ Cb (S  ) such that  f (x) = P f (x) for any x ∈ S and s < t if f (∞) = 0. Then it holds that Ps,t s,t f ∈ B∞ (S). Suppose that we are given a filtration {Ft ; t ∈ T} of sub σ -fields of F. Let t0 ∈ T. Let Xt , t ∈ [t0 , ∞) be an {Ft }-adapted S  -valued stochastic process, continuous in probability. It is called a Markov process of initial state (Xt0 , t0 ) with transition function {Ps,t (x, ·)} (or with semigroup {Ps,t }), if Xt0 ∈ S a.s. and  P (Xt ∈ E|Fs ) = Ps,t (Xs , E),

a.s.

holds for any Borel subset E of S  and for any t0 ≤ s < t < ∞. The Markov property is equivalent to that E[f (Xt )|Fs ] = Ps,t f (Xs ) holds for any f ∈ B∞ (S) and s < t. For each ω ∈ Ω, Xt (ω), t ∈ [t0 , ∞) is called a path of the Markov process. If paths of the Markov process are continuous a.s., the process is called a continuous Markov process. If paths are right (or left) continuous a.s., it is called a right continuous (or left continuous, respectively) Markov process. If Xt is an S-valued process, i.e., P (Xt ∈ S for all t) = 1, then Xt is called a conservative Markov process on S. The Markov process of the initial state (x, s) ∈ S × T is often denoted by Xtx,s and the family {Xtx,s } is called the system of Markov processes with transition function {Ps,t (x, ·)}. A transition function is time homogeneous if Ps+h,t+h (x, E) = Ps,t (x, E) holds for any 0 ≤ s < t < t + h < ∞), h > 0 and E. If it is time homogeneous, it holds that Ps+h,t+h f = Ps,t f for any f ∈ Bb (S). A time homogeneous transition function {Ps,t (x, E)} is often denoted by {Pt−s (x, E)} and its semigroup {Ps,t } is often denoted by {Pt−s }. Then it holds Ps Pt f = Ps+t f . A Markov process is time homogeneous if the associated transition function is time homogeneous. Let Xt , t ∈ [0, ∞) be a d  -dimensional Lévy process. Let Ft be the smallest σ -field containing all null sets of F, with respect to which Xu , 0 ≤ u ≤ t are  measurable. For a Borel subset E of Rd , we set Ps,t (x, E) = P (Xt − Xs + x ∈ E). Then, since Xt − Xs is independent of Fs , we have

1.7 Markov Processes and Backward Markov Processes

39

P (Xt ∈ E|Fs ) = P (Xt − Xs + Xs ∈ E|Fs ) = P (Xt − Xs + y ∈ E)|y=Xs = Ps,t (Xs , E). Therefore Xt is a time homogeneous Markov process with transition probability {Ps,t (x, E)} defined above. In particular, if Xt = Wt , t ∈ [0, ∞) is a Wiener process, it is a time homogeneous Markov process with transition probability Ps,t (x, E) =



1 (2π(t − s))

d 2

1

e− 2(t−s) |x−y| dy. 2

E

If Xt = Nt is a Poisson process, it is a time homogeneous Markov process with state space S = {0, 1, 2, . . .}. Its transition probability is given by Ps,t (i, i + n) =

(λ(t − s))n −λ(t−s) e , n!

n = 0, 1, 2, . . .

Jumping times τn , n = 1, 2, . . . of the Poisson process Nt are stopping times. We show this by induction of n. Suppose that τn−1 is a stopping time. Then {τn ≤ t} =

  {Nr > Nτn−1 , τn−1 < t} = {Nr > n − 1, τn−1 < t}. r≤t

r≤t

Since {τn−1 < t} ∈ Ft holds, the above set belongs to Ft . Since this is valid for any t, τn is a stopping time. Let τ be a stopping time with respect to the filtration {Ft }. We set Fτ = {A ∈ F; A ∩ {τ ≤ t} ∈ Ft holds for any t}.

(1.47)

It is a sub σ -field of F. A right continuous Markov process Xt , t ∈ [t0 , ∞) is said to have the strong Markov property, if the equality  P ({Xτ +t ∈ E} ∩ {τ + t < ∞}|Fτ ) = Pτ,τ +t (Xτ , E)1τ +t 0, E ∈ B(S) for any stopping time τ with τ ≥ t0 . The property is equivalent to that the equality E[f (Xτ +t )1τ +t0 holds for any backward stopping time τˇ with τˇ ≤ t1 and any bounded continuous function f on S. If the backward semigroup Pˇs,t maps Cb (S) into itself and is continuous in s, t, then the left continuous backward Markov process is a backward strong Markov process.

1.8 Kolmogorov’s Criterion for the Continuity of Random Field We shall introduce Kolmogorov’s criterion for a given random field to have a modification of a continuous random field. Let {X(x), x ∈ D} be a random field with values in a Banach space S with the norm  , where D is a bounded domain of Rd . Elements of D are denoted by x = (x1 , . . . , xd ), y = (y1 , . . . , yd ) etc. The distance of x, y is defined by |x − y| = max1≤i≤d |xi − yi |. The next theorem was shown first in the case d = 1 by Kolmogorov and Chentzov [20] and then it was extended to the case d > 1 by Totoki [114]. Theorem 1.8.1 (Kolmogorov–Totoki) Let {X(x), x ∈ D} be a random field with values in a separable Banach space S with norm  , where D is a bounded domain in Rd . Assume that there exist positive constants γ , C and α > d satisfying E[X(x) − X(y)γ ] ≤ C|x − y|α ,

∀x, y ∈ D.

(1.49)

¯ such that X(x) ˜ ˜ Then there exists a continuous random field {X(x), x ∈ D} = X(x) ¯ is the closure of D. holds a.s. for any x ∈ D, where D γ ] < ∞. ˜ Further, if 0 ∈ D and E[X(0)γ ] < ∞, we have E[supx∈D X(x) By a linear invertible transformation, the bounded domain D is transformed to a bounded domain included in the cube [0, 1]d . So we will discuss the case where D is included in the cube [0, 1]d . ∞ −k A real number 0 ≤ x ≤  1 has a dyadic expansion x = k=1 ak 2 , where n n −k ak are 0 or 1. We set x = k=1 ak 2 and call it the dyadic rational of length n defined from x. Let x = (x1 , . . . , xd ) be an element of [0, 1]d . For x, we set x n = (x1n , . . . , xdn ). If x = x n holds for some positive integer n, x is called a dyadic rational of length n. We denote by Δn the set of all dyadic rationals with length n in

d [0, 1]d and we set Δ := ∞ n=1 Δn . It is a dense subset of [0, 1] .

42

1 Probability Distributions and Stochastic Processes

Let β be a positive number less than or equal to 1. Given a map f : D → S, we define, for each positive integer n, the variation of continuity and the variation of β-Hölder continuity of f by Δn (f ) =

max

x,y∈Δn ∩D,|x−y|=2−n

f (x) − f (y),

Δβn (f ) = 2βn Δn (f ). Lemma 1.8.1 Suppose that a map f ; D → S satisfies the inequality

∞

β m=1 Δm (f )

< ∞. Then

∞   Δβm (f ) |x − y|β f (x) − f (y) ≤ 3

(1.50)

m=1

holds for any x, y ∈ D ∩ Δ. Further, the map f ; D ∩ Δ → S is extended to a ¯ → S such that f˜(x) = f (x) holds for x ∈ D ∩ Δ. β-Hölder continuous map f˜; D Proof For a given map f (x), x ∈ D and a positive integer n, define a step function ¯ by fn (x) = f (x n ) if x n ∈ D and fn (x) = f (x) if x n ∈ fn (x), x ∈ D / D, where x n is the dyadic rational of length n defined from x. Then it holds that fn (x) − fn−1 (x) = f (x n ) − f (x n−1 ) ≤ Δn (f ) ≤ Δβn (f ) for any x ∈ D. Therefore if n > k we have fn (x) − fk (x) ≤

n 

fm (x) − fm−1 (x) ≤

m=k+1

n 

Δβm (f )

m=k+1

for any x. Hence the sequence of functions {fn } converges uniformly. Denote the limit function by f˜. Then it holds that f˜(x) = f (x) for any x ∈ Δ = n Δn . Now take any two points x, y ∈ Δ. There exists k ∈ N such that 2−(k+1) ≤ |x − y| < 2−k . Further, there exists n ∈ N such that n > k and x = x n , y = y n ∈ Δn . Then we have n 

f˜(x) − fk (x) = f (x n ) − f (x k ) ≤

f (x m ) − f (x m−1 )

m=k+1

≤

n 

Δm (f ) ≤

m=k+1

≤

∞  m=1

n  m=k+1

 Δβm (f ) |x − y|β .

2−βm Δβm (f )

1.8 Kolmogorov’s Criterion for the Continuity of Random Field

43

f˜(y) − fk (y) is estimated in the same way. Since fk (x) − fk (y) ≤ Δk (f ), we get f˜(x) − f˜(y) ≤ f˜(x) − fk (x) + fk (x) − fk (y) + fk (y) − f˜(y) ∞   ≤3 Δβm (f ) |x − y|β . m=1

Therefore f satisfies the inequality (1.50) for any x, y ∈ Δ. Further f˜ is β-Hölder continuous by the above inequality. Since f˜(x) = f (x) holds for x ∈ D ∩ Δ, ¯ is a β-Hölder continuous extension of f (x), x ∈ D ∩ Δ. Hence we get f˜(x), x ∈ D the assertion of the lemma.   We shall apply the above lemma to the random field X(x). Observe that for each ω, X(·, ω) restricting x to D ∩ Δ can be regarded as a map form D ∩ Δ to S. Then we have ∞   β X(x, ω) − X(y, ω) ≤ 3 Δn (X(ω)) |x − y|β , (1.51) n=1

for any x, y ∈ D ∩ Δ. Lemma 1.8.2 Let β be a positive number satisfying βγ < α − d. Then, E

∞  

γ  1 γ

Δβn (X)

n=1

∞   − α−d−βγ γ 2 ≤

 n

1

· (2d C) γ < ∞,

(1.52)

n=1

where C is the positive constant in the inequality (1.49). Proof We will consider the case γ ≥ 1 only. Observe the inequality Δβn (X)γ ≤ ≤

 sup

X(x) − X(y)2nβ

γ

x,y∈Πn ∩D,|x−y|= 21n

 (X(x) − X(y)2nβ )γ ,

where the summation is taken for all x, y ∈ D ∩ Δn such that |x − y| = the number of summations is at most 2(n+1)d . Therefore E[Δβn (X)γ ] ≤ 2(n+1)d+nβγ E[X(x) − X(y)γ ] ≤ 2n(d+βγ −α) 2d C. In the last inequality, we applied the inequality (1.49). Therefore we get

1 2n .

Then

44

1 Probability Distributions and Stochastic Processes

E

∞  

∞ γ  1  1 γ Δβn (X) ≤ E[Δβn (X)γ ] γ

n=1

n=1

≤

∞ 

−n α−d−βγ γ

2



1

· (2d C) γ < ∞.

(1.53)

n=1

  Proof of Theorem 1.8.1 The random field X(x) restricting x on D ∩ Δ satisfies the  β inequality (1.50), where ∞ n=1 Δn (X) < ∞ a.s. Therefore {X(x), x ∈ D ∩ Δ} is uniformly β-Hölder continuous a.s. Then there exists a uniformly continuous ¯ such that X(x) = X(x) ˜ ˜ random field {X(x), x ∈ D} holds a.s. for any x ∈ D ∩ Δ. ˜ Since X(x) is continuous in probability, the equality X(x) = X(x) holds a.s for any ¯ ˜ x ∈ D. Consequently, the random field {X(x), x ∈ D} is a continuous modification of the random field {X(x), x ∈ D}. Now we have the inequality 

˜ ≤ X(0) + 3( sup X x∈D

Δβn (X))

n

γ ] < ∞. ˜ by (1.51). Therefore we have E[supx∈D X(x)

 

Chapter 2

Stochastic Integrals

Abstract We discuss Itô’s stochastic calculus, which will be applied in later discussions. In Sects. 2.1, 2.2, 2.3, and 2.4, we discuss stochastic calculus related to integrals by Wiener processes and continuous martingales. In Sect. 2.1, we define stochastic integrals based on Wiener processes and continuous martingales. In Sect. 2.2, we establish Itô’s formula. It will be applied for proving Lp -estimates of stochastic integrals, called the Burkholder–Davis–Gundy inequality, and Girsanov’s theorem. The smoothness of the stochastic integral with respect to parameter will be discussed in Sect. 2.3. Fisk–Stratonovitch symmetric integrals will be discussed in Sect. 2.4. In Sects. 2.5 and 2.6, we discuss stochastic calculus based on Poisson random measures. Stochastic integrals are defined in Sect. 2.5. The chain rules formula for jump processes and Lp -estimates of jump integrals will be discussed in Sect. 2.6. In Sect. 2.7, we discuss the backward processes and backward integrals. These topics are related to dual processes or inverse processes, which will be discussed in Chaps. 3 and 4.

2.1 Itô’s Stochastic Integrals by Continuous Martingale and Wiener Process Let (Ω, F, P ) be a probability space equipped with a filtration {Ft } of sub-σ -fields of F. In this chapter, we will restrict the time parameter to a finite interval T = [0, T ]. Let Xt , t ∈ T be a continuous local martingale with respect to the filtration {Ft } and let φ(r), r ∈ T be a real-valued process. Let 0 ≤ s < t ≤ T be a fixed time. We want to define the stochastic integral of φ(r) based on dXr , written as t φ(r) dXr . It is not a usual Lebesgue integral, since Xt is not a process of bounded s variation. At the beginning, we will introduce the class of integrand φ(r), for which the integrals can be defined. The predictable σ -field P (with respect to the filtration {Ft }) is the σ -field on T × Ω generated by left continuous {Ft }-adapted processes. A P-measurable stochastic process ϕ(t), t ∈ [0, T ] is called a predictable process (with respect to the filtration {Ft }). © Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_2

45

46

2 Stochastic Integrals

Let Xt , t ∈ T be a continuous L2 -martingale adapted to the filtration {Ft } and let Xt , t ∈ T be its quadratic variation. It is a unique continuous increasing process such that Xt2 − Xt , t ∈ T is a martingale (Sect. 1.6). Let φ(r), r ∈ T be a predictable process. We define its norm by φ = E



T

|φ(r)|2 dXr

1 2

.

0

A stochasticprocess φ(r), r ∈ T is called a simple predictable process if it is written as φ(r) = nm=0 φm 1(tm−1 ,tm ] (r), where 0 = t0 < t1 < · · · < tn = T and φm are bounded and Ftm−1 -measurable. We shall define the stochastic integral based on a continuous L2 -martingale Xt , t ∈ T. Let φ(r) be a simple predictable process. We define Mt =

n 

φm (Xtm ∧t − Xtm−1 ∧t ),

t ∈T

m=1

t and call it the stochastic integral of φ(r) by dXr and denote it by 0 φ(r) dXr . Then, Mt , t ∈ T may be regarded as a martingale transform with discrete time Π = {0 = t0 < · · · < tn = T }. Therefore we have the equality E[Mt −Mr |Fr ] = 0 a.s. for r ∈ Π . Let 0 ≤ s < t ≤ T . We may assume that s, t are adjoined in the partition Π . Then, using Propositions 1.5.1 and 1.6.1, we have 

E[(Mt − Ms )2 |Fs ] =

E[(Mtm − Mtm−1 )2 |Fs ]

m;tm−1 ≥s

= E[ =





2 φm (Xtm − Xtm−1 )2 |Fs ]

m 2 E[φm E[(Xtm − Xtm−1 )2 |Ftm−1 ]|Fs ]

m

=

 m

= E[

2 E[φm E[Xtm − Xtm−1 |Ftm−1 ]|Fs ]



2 φm (Xtm − Xtm−1 )|Fs ]

m

=E



s

t



φ(r)2 dXr Fs ,

a.s.

t Since the above holds for any s < t, Mt2 − 0 φ(r)2 dXr , t ∈ T is a martingale. t Then the quadratic variation of Mt , t ∈ T is given by 0 φ(r)2 dXr , t ∈ T in view of Corollary 1.6.1. We have thus shown

2.1 Itô’s Stochastic Integrals by Continuous Martingale and Wiener Process

!

t



"

t

φ(r) dXr =

0

|φ(r)|2 dXr ,

a.s.

47

(2.1)

0

for any 0 < t ≤ T . Taking expectations of both sides, we get

2   t    t

E φ(r) dXr = E |φ(r)|2 dXr ≤ φ2 0

0

for any t. Now let φ be a predictable process such that φ < ∞. Then there exists a sequence of simple predictable processes {φn } such that φ − φn  → 0 as n → ∞. Then, using Doob’s inequality for a martingale (Theorem 1.5.1), we have  t

 t

2    

E sup φn (r) dXr− φn (r) dXr ≤4E t∈T

0

0

T 0

2 

(φn (r) −φn (r)) dXr

= 4φn − φn 2 → 0, t as n, n → ∞. Hence limn→∞ 0 φn (r) dXr exists uniformly in t. We denote it by t 0 φ(r) dXr and call it the stochastic integral or Itô integral of φ(r) by dXr . The stochastic integral is a continuous L2 -martingale. It satisfies (2.1). We will extend the stochastic integral for a continuous local martingale Xt , t ∈ T. Let Xt , t ∈ T be its quadratic variation. Let L(X) be the set of all predictable T processes φ(t), t ∈ T satisfying 0 |φ(r)|2 dXr < ∞ a.s. Let φ ∈ L(X). For N > 0, define a stopping time by  t τN = inf{t ∈ T; |Xt | + |φ(r)|2 dXr > N }, 0

= ∞,

if the set {· · · } is empty. (N )

Then the stopped process Xt = Xt∧τN , t ∈ T is a bounded continuous martingale. See the proof of Theorem 1.6.1. It holds that X(N ) t = Xt∧τN . t (N ) Since E[ 0 |φ(r)|2 dX(N ) r ] < ∞, the family of stochastic integral Mt ≡ t (N ) 2 -martingale. Let N  > N and consider φ(r) dX , t ∈ T is well defined as an L r 0 t (N  ) (N  ) (N  ) (N ) ≡ 0 φ(r) dXr . Then it holds that Mt∧τN = Mt . Then there exists a Mt t (N ) continuous local martingale Mt , t ∈ T such that Mt∧τN = 0 φ(r) dXr holds for t any N = 1, 2, . . .. We set Mt = 0 φ(r) dXr and call it the stochastic integral or Itô t t integral of φ(r) by dXr . It holds that  0 φ(r) dXr  = 0 φ(r)2 dXr a.s. for any 0 < t ≤ T. Let Xt , t ∈ T and Yt , t ∈ T be continuous local martingales and let φ ∈ L(X) and ϕ ∈ L(Y ). In view of (2.1), we have the relation  t ! t "  t φ dX, ϕ dY = φ(r)ϕ(r) dX, Y r , a.s. (2.2) 0

for any 0 < t ≤ T .

0

0

48

2 Stochastic Integrals

Let φ1 , φ2 ∈ L(X) and let c1 , c2 be constants. Then we have 

t



t

(c1 φ1 + c2 φ2 )(r) dXr = c1

0



t

φ1 (r) dXr + c2

0

φ2 (r) dXr ,

a.s.

0

for any 0 < t ≤ T . Indeed, the equality holds if both φ1 , φ2 are simple predictable processes. Then the equality can be extended to arbitrary predictable processes φ1 , φ2 belonging to L(X). t For 0 ≤ s < t ≤ T , the integral s φ(r) dXr is defined as 

t



t

φ(r) dXr := 0

s

 φ(r)1[s,T ] (r) dXr =

t



s

φ(r) dXr −

0

φ(r) dXr ,

a.s.

0

Let Xt , t ∈ T be a continuous semi-martingale decomposed as Xt = Mt + At . Let |A|t be the process of the total variation of At . Let φ(r) be a predictable process T T satisfying 0 |φ(r)|2 dXr < ∞ and 0 |φ(r)| d|A|r < ∞, a.s. We define the integral  0

t



t

φ(r) dXr :=



t

φ(r) dMr +

0

φ(r) dAr , 0

t where 0 φ(r, ω) dA(r, ω) is the Lebesgue–Stieltjes integrals for each ω. Now, we shall consider stochastic integrals by a Wiener process. Let d  be a  positive integer. Let Wt = (Wt1 , . . . , Wtd ), t ∈ [0, ∞) be a d  -dimensional Wiener process. Suppose that Wt is adapted to a given filtration {Ft } and that Wu − Wt is independent of Ft for any u > t. Then Wt , t ∈ [0, ∞) is called an {Ft }-Wiener process. Components Wtk , k = 1, . . . , d  of the {Ft }-Wiener process Wt are L2 martingales with respect to {Ft }. Indeed, since Wtk − Wrk are independent of Fr , we have E[Wtk − Wrk |Fr ] = E[Wtk − Wrk ] = 0 a.s. Further, E[(Wtk − Wrk )(Wtl − Wrl )|Fr ] = E[(Wtk − Wrk )(Wtl − Wrl )] = δkl (t − r),

a.s.

Therefore we have W k , W l t = δkl (t − 0). We define a space of d  -dimensional predictable processes by    LT = φ(r) = (φ 1 (r), . . . , φ d (r)); predictable and

T

 |φ(r)|2 dr < ∞ ,

0

t   where |φ(r)|2 = dk=1 |φ k (r)|2 . Then for any φ(r) ∈ LT , integrals 0 φ k (r)dWrk , k = 1, . . . , d  are well defined for any 0 < t ≤ T . These are local martingales and satisfy

2.2 Itô’s Formula and Applications

!

49

 k

k

φ dW ,

l

φ dW

l

" t



t

= δkl

φ k (r)φ l (r) dr,

(2.3)

a.s.

0

We set 

t

It (φ) =

d   

(φ(r), dWr ) =

0

k=1 0

t

φ k (r) dWrk .

(2.4)

We next consider the stochastic integral where the integrand depends on parameter. Let Λ be a parameter set. We assume that Λ is an e-dimensional Euclidean space Re or its unit ball {λ ∈ Re ; |λ| < 1}. We denote by LT (Λ) the set of all d  -dimensional measurable random fields φλ (t), (t, λ) ∈ T × Λ such that for any λ, φλ (t) is predictable and satisfies Λ T |φλ (r)|2 dr dλ < ∞ a.s. For φλ (t) ∈ LT (Λ), t stochastic integrals 0 (φλ (r), dWr ) are well defined as continuous martingales for almost all λ. Further, the family of integrals has a modification such that it is (t, λ)measurable. We give a Fubini theorem for changing the order of integrals by dWr and dλ. Proposition 2.1.1 Let φλ (r) ∈ LT (Λ) and g(λ), λ ∈ Λ be a bounded measurable function of compact supports. Then we have for any t ∈ T,   Λ

0

t

 t    (φλ (r), dWr ) g(λ) dλ = φλ (r)g(λ) dλ, dWr , 0

a.s.

(2.5)

Λ

 Proof If φλ (t) is a simple functional written as m φλ (m)1(tm−1 ,tm ] (t) with bounded Ftm−1 ×B(Λ)-measurable functional φλ (m), the equality (2.5) can be shown directly. Since φλ (t) ∈ LT (Λ) can be approximated by a sequence of the above simple functionals, equality (2.5) is valid for any φλ (t) ∈ LT (Λ).  

2.2 Itô’s Formula and Applications A d-dimensional continuous stochastic process Xt = (Xt1 , . . . , Xtd ), t ∈ T is called a d-dimensional continuous semi-martingale, if all Xti , t ∈ T; i = 1, . . . , d are continuous semi-martingales. Let f (x1 , . . . , xd , t) be a smooth function. We consider the differential rule of the composite functional f (Xt , t) with respect to t. The rule is called the formula of the change of variables or Itô’s formula. Theorem 2.2.1 (Itô’s formula) Let f (x, t) be a C 2,1 -function on Rd × T and let Xt , t ∈ T be a d-dimensional continuous semi-martingale. Then f (Xt , t), t ∈ T is a continuous semi-martingale. Further, we have

50

2 Stochastic Integrals



t

∂f (Xr , r) dr (2.6) s ∂t d  ∂f 1  t ∂ 2f (Xr , r) dXri + (Xr , r) dXi , Xj r , ∂xi 2 ∂x ∂x i j s

f (Xt , t) = f (Xs , s) + +

d  

t

s

i=1

i,j =1

a.s. for any 0 ≤ s < t ≤ T . Proof It is sufficient to prove the case s = 0. Let Πn = {t0 < t1 < · · · < tn } be a partition of [0, t]. Then we have by the Taylor expansion of f (x, t), f (Xt , t)−f (X0 , 0) =

n    f (Xtm , tm ) − f (Xtm−1 , tm−1 ) m=1

=



f (Xtm , tm ) − f (Xtm , tm−1 )



m

+

d    ∂f (Xtm−1 , tm−1 )(Xtim − Xtim−1 ) ∂xi m i=1

d  1   ∂ 2 f j j (ξm , tm−1 )(Xtim −Xtim−1)(Xtm −Xtm−1) + 2 ∂xi ∂xj m i,j =1

= I1 + I2 + I3 , where ξm are random variables such that |ξm − Xtm−1 | ≤ |Xtm − Xtm−1 |. Let |Πn | tend to 0. Then we get 

t

∂f (Xr , r) dr, a.s. n→∞ 0 ∂t   t ∂f (Xr , r) dXri , lim I2 = n→∞ 0 ∂xi lim I1 =

a.s.

i

immediately from the definition of the stochastic integral. Further, applying Theorem 1.6.2, we get lim I3 =

n→∞

1 2 i,j

 0

t

∂ 2f (Xr , r) dXi , Xj r , ∂xi ∂xj

a.s.

Summing up these computations, we get the formula of the theorem.

 

2.2 Itô’s Formula and Applications

51

We set   LT = υ(r); predictable and

T

 |υ(r)| dr < ∞ .

0

Let φ(r) = (φ ik (r)) be a d × d  -dimensional process such that (φ i· (r)) ∈ LT for any i and let υ(r) = (υ 1 (r), . . . , υ d (r)) be a d-dimensional process such that υ i (r) ∈ LT for any i. We consider a d-dimensional semi-martingale represented by  Xt = X0 +

t



t

φ(r) dWr +

0

(2.7)

υ(r) dr, 0

where 

t

φ(r) dWr =

0

d   k=1 0

t

 φ ik (r) dWrk , i = 1, . . . , d .

Corollary 2.2.1 Let Xt = (Xt1 , . . . , Xtd ) be a continuous semi-martingale represented by (2.7). Then the formula (2.6) is rewritten as 

t

f (Xt , t) = f (Xs , s) + s d   d

+

t

i=1 k=1 s

∂f (Xr , r) dr ∂t

 ∂f (Xr , r)φ ik (r) dWrk + ∂xi d



t

∂f (Xr , r)υ i (r) dr ∂xi

s

i=1

d  d  1  t ∂ 2f (Xr , r)( φ ik (r)φ j k (r)) dr. + 2 s ∂xi ∂xj i,j =1

(2.8)

k=1

Proof We have X , X t = i

j

!

 φ

ik

(r) dWrk ,

k,l

φ

jl

(r) dWrl

" t

=

 k

t

φ ik (r)φ j k (r) dr

0

 

by (2.3). Therefore (2.6) implies (2.8).

We will discuss three problems by applying the above Itô’s formula. We first give Lévy’s characterization of a Wiener process. Proposition 2.2.1 Let Xt1 , . . . , Xtd be continuous {Ft }-martingales satisfying Xi , Xj t = δij t,

i, j = 1, . . . , d, a.s.

52

2 Stochastic Integrals

for any 0 < t < T . Then Xt = (Xt1 , . . . , Xtd ), t ∈ T is a d-dimensional {Ft }Wiener process. Proof We apply Itô’s formula (2.6) to the function f (x, t) = e √

e

√

−1(v,x) .

We have

−1(v,Xt −Xs )

=1+

√

−1



t √

e

−1(v,Xr −Xs )

s

j

1 j vj dXr − |v|2 2



t √

e

−1(v,Xr −Xs )

dr.

s

Let A ∈ Fs . Multiply the functional equation and then

above   √1A to each term of the j t take the expectation. Since E s e −1(v,Xr −Xs ) vj dXr Fs = 0 holds, we have √

E[e

−1(v,Xt −Xs )

1 1A ] = P (A) − |v|2 2



√

t

E[e

−1(v,Xr −Xs )

1A ] dr.

s √

Differentiating the above with respect to t, ϕt = E[e −1(v,Xt −Xs ) 1A ] satis1 2 t fies the differential equation, dϕ dt = − 2 |v| ϕt . Integrating it we obtain ϕt = 1

e− 2 |v|

2 (t−s)

P (A), or equivalently, √

E[e

−1(v,Xt −Xs )

1

1A ] = e− 2 |v|

2 (t−s)

P (A).

Consequently, the law of Xt − Xs is Gaussian with mean 0 and covariance (t − s)I . Further, Xt −Xs is independent of Fs . This proves that Xt is an {Ft }-Wiener process.   We next study Lp -estimations of stochastic integrals. Let p ≥ 2. We set   p LT = φ ∈ LT ; φp < ∞ , where φp = E

 T 0

|φ(r)|p dr

1

p

  p LT = φ ∈ LT ; φp < ∞ ,

.

Proposition 2.2.2 (Burkholder–Davis–Gundy inequality) Let p ≥ 2. p t Suppose that φ(r) ∈ LT . Then the integral It (φ) = 0 (φ(r), dWr ) is p-th integrable. There exists a positive constant Cp such that for any 0 < t ≤ T , the inequality 1

1

E[|It (φ)|p ] p ≤ Cp t 2

− p1

φp

(2.9)

p

holds for all φ ∈ LT . Proof We shall apply Itô’s formula to the function f (x, t) = |x|p and Xt = Mt := t 0 (φ(r), dWr ). Then,

2.2 Itô’s Formula and Applications



t

|Mt | = p p

|Mr |

p−1

0

53

1 sign(Mr ) dMr + p(p − 1) 2



t

|Mr |p−2 dMr ,

0

where sign(Mr ) is equal to 1 if Mr ≥ 0 and is equal to −1 if Mr < 0. The first term of the right-hand side is a martingale with mean 0. Therefore, taking the expectation for each term, we have   t 1 |Mr |p−2 dMr p(p − 1)E 2 0   p−2  p2 1 p p ≤ p(p − 1)E sup |Mr |p E Mt2 2 r n} (= ∞ if {· · · } is empty). Then the stopped process Zt∧τn , t ∈ T is a positive martingale. Therefore the equality E[Zt∧τn |Fr ] = Zs∧τn holds for any s < t and n, a.s. Then, using Fatou’s lemma for conditional measure P (·|Fr ) (Proposition 1.5.2), we have

E[Zt |Fs ] ≤ lim inf E[Zt∧τn |Fs ] = lim inf Zs∧τn = Zs . n→∞

n→∞

Therefore Zt , t ∈ T is a super-martingale. We give a sufficient condition that Zt , t ∈ T is an Lp -martingale.

(2.11)

54

2 Stochastic Integrals

T 2 Proposition 2.2.3 Let p > 1. If exp 0 |φ(r)|2 dr is in L2p −p , then Zt , t ∈ T T defined by (2.10) is an Lp -martingale. Further, if exp 0 |φ(r)|2 dr belongs to ∞− ∞− L , Zt , t ∈ T is an L -martingale. p

Proof We shall rewrite Zt as  1   p Zt = exp pMt − p2 Mt exp (2p2 − p)Mt . 2

(2.12)

p

By the Schwartz inequality, E[Zt ] is dominated by  E

2  1  2  1 1  2 2 E exp (2p2 − p)Mt . exp pMt − p2 Mt 2

Since 

 2   1 exp pMt − p2 Mt = exp 2pMt − (2p)2 Mt , 2

it is a positive local martingale and is a positive super-martingale, whose mean is less than or equal to 1. Therefore we have p E[Zt ]





≤ E exp (2p −p)Mt 2

 1 2

 t   1 2 2 = E exp (2p −p) |φ(r)|2 dr < ∞. 

0

It remains to show that Zt , t ∈ T is a martingale. Let {τn } be the sequence of stopping times defined above. By a similar argument, we have p E[Zt∧τn ]

  2 ≤ E exp (2p − p) 

t∧τn

|φ(r)|2 dr

 1 2

.

0

p

Therefore supn E[Zt∧τn ] < ∞. Consequently, the family of random variables {Zt∧τn , n = 1, 2, . . .} is uniformly integrable. Then the sequence Zt∧τn converges to Zt in L1 . Therefore, the inequality (2.11) can be replaced by the equality. This means that Zt , t ∈ T is a martingale. The second assertion of the proposition is immediate from the first assertion.   Now, suppose Zt , t ∈ T of (2.10) is a positive martingale. Define Q(B) = E[ZT 1B ],

∀B ∈ FT .

(2.13)

Then Q is a probability measure on (Ω, FT ). We denote it by ZT · P . Since Q is equivalent (mutually absolutely continuous) to P , stochastic processes on (Ω, FT , P ) can be regarded as stochastic processes on (Ω, FT , Q).

2.3 Regularity of Stochastic Integrals Relative to Parameters φ

Theorem 2.2.2 (Girsanov’s theorem) Wt := Wt − Wiener process with respect to Q.

55

t 0

φ(r) dr, t ∈ T is an {Ft }-

Proof We have the equality φ

Zt exp{i(v, Wt − Wsφ )}  t   t  1 t 2 = Zs exp (φ(r), dWr ) − |φ(r)| dr + i(v, Wt − Ws ) − φ(r) dr 2 s s s   t  t    1 1 ((φ + iv), dWr ) − |φ + iv|2 dr exp − |v|2 (t − s) . = Zs exp 2 s 2 s Take A ∈ Fs and multiply t above by 1A and then take t both sides of the expectations. Since E[exp{ s (φ + iv) dWr − 12 s |φ + iv|2 dr}|Fs ] = 1 holds by Proposition 2.2.3, we get for any A ∈ Fs ,  1  φ E[Zt exp{i(v, Wt − Wsφ )}1A ] = exp − |v|2 (t − s) E[Zs 1A ]. 2 Rewriting the above using the measure dQ = ZT dP , we get  1  φ EQ [exp{i(v, Wt − Wsφ )}1A ] = exp − |v|2 (t − s) Q(A). 2 φ

Therefore, Wt is an {Ft }-Wiener process with respect to Q.

 

2.3 Regularity of Stochastic Integrals Relative to Parameters In this section, we consider the continuity and the differentiability of stochastic integrals with respect to parameters. We assume that the parameter set Λ is equal to Re or a unit ball B1e = {λ ∈ Re ; |λ| < 1}. We denote the partial derivative ∂λ∂ i f by ∂λi f . We set ∂λ f = (∂λ1 f, . . . , ∂λe f ). For multi-index of nonnegative integers i = (i1 , . . . , ie ) we set ∂λi f = ∂λi11 · · · ∂λiee f . Let p > e ∨ 2. Let φλ (r), r ∈ T, λ ∈ Λ be a d  -dimensional measurable random field belonging to LT (Λ). If it satisfies  sup E λ



T

E 0

T

 |φλ (r)|p dr < ∞,

0

 |φλ (r) − φλ (r)|p dr ≤ cp |λ − λ |p

(2.14)

56

2 Stochastic Integrals

for any λ, λ ∈ Λ with a positive constant cp , then φλ (r) is said to belong to Lip,p n+Lip,p (Λ), the space LT (Λ) . Further, φλ (r) is said to belong to the space LT if it is n-times continuously differentiable with respect to λ for any s a.s. and for Lip,p any |i| ≤ n, derivatives φλ (r) = ∂λi φλ (r) satisfy (2.14). If φλ (r) ∈ LT (Λ), t the stochastic integral 0 (φλ (r), dWr ) is well defined a.s. for each fixed λ. We are interested in the problem of finding a modification of a family of random variables t { 0 (φλ (s), dWs ), λ ∈ Λ} which is continuous in (t, λ) a.s. and differentiable with respect to λ a.s. Proposition 2.3.1 Lip,p

1. Suppose that {φλ (r)} is a random field belonging to the class LT (Λ) where t p > e ∨ 2. Then the family of Itô’s stochastic integrals { 0 (φλ (r), dWr )} has a modification which is continuous in (t, λ). n+Lip,p (Λ) for some positive 2. Suppose further that {φλ (r)} belongs to the class LT integer n and p > e ∨ 2. Then the above family of stochastic integrals has a modification, which is n-times continuously differentiable with respect to λ a.s. and the derivatives are continuous in (t, λ) a.s. Further, for any |i| ≤ n, we have  ∂λi

t



t

(φλ (r), dWr ) =

0

0

(∂λi φλ (r), dWr ),

∀(t, λ) a.s.

(2.15)

Proof 1. For a real continuous function t f (t), t ∈ [0, T ], we define its norm by f  = sup0≤t≤T |f (t)|. Set Xtλ = 0 (φλ (r), dWr ), t ∈ T. Since it is a martingale, we have by Doob’s inequality and the Burkholder–Davis–Gundy inequality     E Xλ − Xλ p ≤ cE

T

 |φλ (r) − φλ (r)|p dr ≤ c |λ − λ |.

0

Apply the Kolmogorov–Totoki theorem (Theorem 1.8.1). Then we find that Xtλ has a modification which is continuous in λ with respect to the norm  . Then the modification is continuous in (t, λ) a.s. 1+Lip,p 2. Suppose φλ ∈ LT (Λ). For ∈ Θ = (−1, 0) ∪ (0, 1) and a positive integer λ, 1 ≤ i ≤ e, we set Yt = 1 (Xtλ+ i − Xtλ ), where i = ei and ei , i = 1, . . . , e are unit vectors in Re . It is a stochastic process with parameter (λ, ) ∈ Λ × Θ. It holds by the mean value theorem that  t  φλ+ i (r) − φλ (r) , dWr

0  t  1  = ∂λi φλ+θ i (r) dθ, dWr ,

Ytλ, =

0

0

(2.16)

2.3 Regularity of Stochastic Integrals Relative to Parameters

57

where |θ | ≤ 1. Therefore 



E[Y λ, − Y λ , p ] ≤ cE



T 0

 0

1

|∂λi φλ+θ i (r) − ∂λi φλ +θ i (r)| dθ

≤ C{|λ − λ |p + | −  |p }.

p

 dr

(2.17)

Consequently, Ytλ, has a modification which is continuous in t, λ, and is ¯ by the Kolmogorov–Totoki theorem. This continuously extended at = 0 ∈ Θ, means that Xtλ is continuously differentiable with respect to λ for any t and the derivative ∂λi Xtλ coincides with Ytλ,0 . It is continuous in t, λ a.s. We have as → 0, 

T

E 0



1 0

p 

∂λi φλ+θ i (r) dθ − ∂λi φλ (r) dr → 0.

t t Then we get the equality ∂λi 0 (φλ (r), dWr ) = 0 (∂λi φλ (r), dWr ) from (2.16). n+Lip,p Finally, if φλ (r) is of the class LT (Λ), we can repeat the above argument inductively. Then we find that the stochastic integral is n-times differentiable with respect to the parameter and we get the equality (2.15) for any |i| ≤ n.   For one dimensional predictable processes with parameter Λ, spaces LT (Λ), are defined similarly. When the parameter is a spatial parameter x, we will often write φx (r) as φ(x, r). Let (f k (x, r), k = 1, . . . , d  ) be a d  -dimensional predictable process with 2+Lip,p (Rd ) and let f 0 (x, r) be a spatial parameter x, belonging to the space LT 2+Lip,p predictable process with spatial parameter x belonging to LT (Rd ). We set Lip,p n+Lip,p LT (Λ), LT (Λ)

d   

F (x, t) = F (x, 0) +

k=0 0

t

f k (x, r) dWrk ,

∀(x, t) a.s.,

(2.18)

t t where F (x, 0) is a C 2 -function, Wt0 = t and 0 f 0 (x, r) dWr0 = 0 f 0 (x, r) dr. Then F (x, t) has a modification of C 2,0 -function of x, t by Proposition 2.3.1. We shall obtain a differential rule for the composite of the above stochastic process F (x, t) with spatial parameter x and a continuous semi-martingale Xt . The equation (2.19) below is called a generalized Itô’s formula or the Itô–Wentzell formula. Theorem 2.3.1 Let Xt be a d-dimensional continuous semi-martingale. Then the composite F (Xt , t) is a continuous semi-martingale. Further, for any s < t, we have

58

2 Stochastic Integrals d   

F (Xt , t) = F (Xs , s) +

t

k=0 s

+

d   i=1

t s

f k (Xr , r) dWrk d

 ∂F (Xr , r) dXri + ∂xi d



t ∂f k

∂xi

i=1 k=1 s

(Xr , r) dW k , Xi r

d  1  t ∂ 2F + (Xr , r) dXi , Xj r . 2 s ∂xi ∂xj

(2.19)

i,j =1

Proof It suffices to prove (2.19) in the case s = 0. Let Π = {t0 < t1 < · · · < tn } be a partition of [0, t]. Then we have n    F (Xtm−1 , tm )−F (Xtm−1 , tm−1 ) F (Xt , t)−F (X0 , 0) = m=1

+

n    F (Xtm , tm )−F (Xtm−1 , tm ) m=1

almost surely. Let |Π | → 0. Then we have n  d  n    F (Xtm−1 , tm ) − F (Xtm−1 , tm−1 ) = 

m=1

k=0

m=1 tm−1

d   

→

tm

k=0 0

t

f k (Xtm−1 , r) dWrk

f k (Xr , r) dWrk .

Further, by the Taylor expansion of F (tm , x), we have n    F (Xtm , tm ) − F (Xtm−1 , tm ) m=1

=

d  n   ∂F (Xtm−1 , tm−1 )(Xtim − Xtim−1 ) ∂xi i=1

+

m=1

d  n   ∂F i=1

+

m=1

∂xi

(Xtm−1 , tm ) −

  ∂F (Xtm−1 , tm−1 ) (Xtim − Xtim−1 ) ∂xi

d n  1    ∂ 2F j j (ξm , tm )(Xtim − Xtim−1 )(Xtm − Xtm−1 ) 2 ∂xi ∂xj i,j =1

m=1

= I1 + I2 + I3 ,



2.4 Fisk–Stratonovitch Symmetric Integrals

59

where ξm are random variables such that |ξm − Xtm−1 | ≤ |Xtm − Xtm−1 |. Let |Π | tend to 0. Then we get lim I1 =

|Π |→0

d   i=1

t 0

∂F (Xr , r) dXri ∂xi

immediately from the definition of the stochastic integral. Further, noting Theorem 1.6.2, we get 

lim I2 =

|Π |→0

d d   i=1 k=0

lim

|Π |→0

d   d

=

i=1 k=1 0

lim I3 =

|Π |→0

t

n   m=1

tm tm−1

  ∂f k (Xtm−1 , r) dWrk (Xtim − Xtim−1 ) ∂xi

∂f k (Xr , r) dW k , Xi r , ∂xi

d  1  t ∂ 2F (Xr , r) dXi , Xj r , 2 0 ∂xi ∂xj

a.s.

a.s.

i,j =1

Summing up these computations, we get the formula of the theorem.

 

2.4 Fisk–Stratonovitch Symmetric Integrals We will define the Fisk–Stratonovitch symmetric integral or simply the symmetric integral of φ(r) by a continuous semi-martingale Xt , t ∈ T by 

t 0

n   1    φ(tm ) + φ(tm−1 ) (Xtm − Xtm−1 ), |Π |→0 2

φ(r) ◦ dXr = lim

(2.20)

m=1

if the right-hand side exists. Here Π = {t0 < · · · < tn } are partitions of [0, T ] and  = t ∧ t. It holds that tm m   1      φ(tm ) + φ(tm−1 ) (Xtm − Xtm−1 )= φ(tm−1 )(Xtm − Xtm−1 ) + φ, XΠ t , 2 2 m

 1 m

where φ, XΠ t is the quadratic covariation of processes φ and X associated with the partition Π . If φ(t) is a semi-martingale with bounded jumps, then φ, XΠ t ,t ∈ [0, T ] converges to a continuous process of bounded variation, which we denote by φ, Xt (see Sect. 1.6). Then we have:

60

2 Stochastic Integrals

Proposition 2.4.1 Let {φ(r)} be a semi-martingale with bounded jumps. Then, the t symmetric integral 0 φ(r) ◦ dXr is well defined for any t ∈ T. Further, we have 

t



0

where

t 0

t

φ(r) ◦ dXr =

1 φ(r) dXr + φ, Xt , 2

0

a.s.,

(2.21)

φ(r) dXr is the Itô integral.

For 0 ≤ s < t ≤ T , we define the symmetric integral by  t  t  s φ(r) ◦ dXr = φ(r) ◦ dXr − φ(r) ◦ dXr . 0

s

0

Then it holds that  t  t 1 φ(r) ◦ dXr = φ(r) ◦ dXt + (φ, Xt − φ, Xs ). 2 s s Remark It is known that the symmetric integral is well defined if the martingale part of φ(r) is a local L2 -martingale. In such a case, the quadratic covariation φ, Xt exists and the equality (2.21) holds. For the proof, we need additional arguments for the quadratic covariation. See Protter [96]. A feature of the symmetric integral is that it works similarly as the usual differential calculus. Let Xt = (Xt1 , . . . , Xtd ), t ∈ T be a d-dimensional continuous semi-martingale represented by d   

Xti

=

k=1 0

t

 φ ik (r) ◦ dWrk +

t

υ i (r) dr.

(2.22)

0

Here, (φ ik (r)) is a d  -dimensional continuous semi-martingale written as d   

φ ik (r) = φ ik (0) +

l=1

r

0

 φ1ikl (s) dWsl +

r 0

φ2ik (s) ds,

where φ1ikl (r) ∈ LT , and φ2ik (r) ∈ LT . Then, using Itô integral, Xt is written as d   

Xti

=

k=1 0

t

 φ ik (r) dWrk +

t

υ i (r) dr +

0

1 2



t 0

φ¯ 1i (r) dr,

d  ikk ij where φ¯ 1i (r) = k=1 φ1 (r), i = 1, . . . , d. Set φ(r) = (φ (r)) and υ(r) = i (υ (r)). Using vector and matrix notations, we rewrite (2.22) as  t  t φ(r) ◦ dWr + υ(r) dr. Xt = 0

0

2.4 Fisk–Stratonovitch Symmetric Integrals

61

We shall reconsider the differential rule of the composite functional f (Xt , t), when Xt , t ∈ T is a semi-martingale represented by the symmetric integral. It will be seen that the differential rule for the symmetric integral is similar to the usual differential rule. Theorem 2.4.1 Let {Xt } be a d-dimensional continuous semi-martingale represented by (2.22). Let f (x, t) be a C 3,1 -function on Rd × T. Then we have for any 0 ≤ s < t ≤ T, 

t

f (Xt , t) = f (Xs , s) + s d  d   

+

i=1 k=1 s

t

∂f (Xr , r) dr ∂t

(2.23)

 ∂f (Xr , r)φ ik (r) ◦ dWrk + ∂xi d



t

s

i=1

∂f (Xr , r)υ i (r) dr. ∂xi

Proof It is sufficient to prove the case s = 0. We shall rewrite the term expressed by symmetric integral, using Itô integral. If f is a C 1,3 -function, Theorem 2.2.1 tells ∂f us that ∂x (Xt , t) is a continuous semi-martingale and it satisfies i  ∂f ∂f (Xt , t) = (X0 , 0) + ∂xi ∂xi d



j =1 0

t

∂ 2f j (Xr , r) dXr + Ct , ∂xi ∂xj

where Ct is a continuous process of bounded variation. Then the martingale part of ∂f the semi-martingale ∂x (Xt , t)φ ik (t) is i d  d   

t 0

j =1 l=1

d

 ∂ 2f (Xr , r)φ ik (r)φ j l (r) dWrl + ∂xi ∂xj



t

0

l=1

∂f (Xr , r)φ1ikl (r) dWrl . ∂xi

Therefore ! ∂f ∂xi =

(Xt , t)φ ik (t), Wtk

d   j =1 0

Then

t

" t

∂ 2f (Xr , r)φ ik (r)φ j k (r) dr + ∂xi ∂xj

d  t

d   

k=1 0

∂f ik k k=1 0 ∂xi (Xr , r)φ (r) ◦ dWr

t

∂f 1 (Xr , r)φ ik (r) dWrk+ ∂xi 2 d

+

1 2



t 0

0

t

∂f (Xr , r)φ1ikk (r) dr. ∂xi

is equal to

j =1 0





t

d   ∂ 2f (Xr , r) φ ik (r)φ j k (r) dr ∂xi ∂xj k=1

∂f (Xr , r)φ¯ 1i (r) dr. ∂xi

62

2 Stochastic Integrals

Therefore the right-hand side of (2.23) is equal to the right-hand side of (2.8) if we replace υ(r) in (2.8) by υ(r) + 12 φ¯ 1 (r). Therefore the formula (2.23) holds.   Let Λ be the e-dimensional parameter set. Let {φλ (r), λ ∈ Λ, t ∈ T} be a measurable random field such that for any λ, φλ (t) is a continuous semi-martingale. We assume     |φλ (r)|2 + |φλ (r), Xr | dλ dr < ∞. (2.24) Λ T

We give a Fubini theorem for the change of the order of the symmetric integrals ◦dWr and dλ. Proposition 2.4.2 Let {φλ (r), λ ∈ Λ, r ∈ T} be a measurable random field such that for any λ, φλ (r), r ∈ T is a continuous semi-martingales satisfying (2.24). Then we have for any t ∈ T   Λ

t

 t    φλ (r) ◦ dWr g(λ) dλ = φλ (r)g(λ) dλ ◦ dWr ,

0

0

a.s.

(2.25)

Λ

for any bounded measurable function g(λ) with compact supports. The proof is straightforward from Proposition 2.1.1. We shall study the regularity of the symmetric integrals with respect to parameters. 

Proposition 2.4.3 Let φλ (r) = (φλ1 (r), . . . , φλd (r)), λ ∈ Λ, r ∈ T be a d  dimensional measurable random field such that for any λ, φλ (r) is a continuous semi-martingale written as  φλ (r) =

r



r

φ1,λ (s) dWs +

0

φ2,λ (s) ds,

(2.26)

0

Lip,p

i (r) ∈ L where φλi (r), φ1,λ T

(Λ) for i = 1, . . . , d for some p > e ∨ 2. t 1. Then the symmetric integral 0 φλ (r) ◦ dWr has a modification which is continuous in (t, λ). i (r), i = 1, . . . , d belong to Ln+Lip,p (Λ) for 2. Assume further that φλi (r), φ1,λ T some positive integer n and p > e ∨ 2. Then the symmetric integral is n-times continuously differentiable with respect to λ a.s. and the derivative is continuous in (t, λ) a.s. Further, we have for any |i| ≤ n,  ∂λi

0

t



t

φλ (r) ◦ dWr = 0

∂λi φλ (r) ◦ dWr ,

∀(t, λ) a.s.

(2.27)

2.4 Fisk–Stratonovitch Symmetric Integrals

63

Proof

t 1. The symmetric stochastic integral 0 φλ (r) ◦ dWr is equal to the Itô integral t 1 t ¯ 0 φλ (r) dWr + 2 0 φ1,λ (r) dr. If φλ (r) satisfies (2.14), the first term of the righthand side has a modification which is continuous in (t, λ) by Propositions 2.3.1. The second term of the right-hand side is continuous in (t, λ) obviously. Therefore the symmetric integral has t a modification which t is continuous in (t, λ). 2. Under the assumption, integrals 0 φλ (r) dWr and 12 0 φ¯ 1,λ (r) dr are continu ously differentiable with respect to λ and we can change the order of ∂λ and , in view of Proposition 2.3.1. Then we get the formula (2.27).   We shall rewrite the generalized Itô’s formula (Theorem 2.3.1) in the previous section using symmetric integrals. We will assume slightly stronger conditions for f k (x, t). Let f k (x, t), k = 1, . . . , d  be predictable processes with spatial t t parameter x represented by f k (x, t) = 0 (f1k (x, r), dWr ) + 0 f2k (x, r) dr, where 3+Lip,p 3+Lip,p (Rd ) and f2k (x, r) ∈ LT (Rd ) for p > 2 ∨ d. Let f1k (x, r) ∈ LT 3+Lip,p f 0 (x, r) ∈ LT (Rd ). Let F (x, t), t ∈ T be a stochastic process with a spatial d parameter x ∈ R defined by d   

F (x, t) = F (x, 0) +

t

k=0 0

f k (x, r) ◦ dWrk ,

(2.28)

where F (x, 0) is a C 3 -function of x. Then F (x, t) is a C 3,0 -function of x, t a.s. and it is a continuous semi-martingale for any x by Proposition 2.4.3. Theorem 2.4.2 For any d-dimensional continuous semi-martingale Xt , t ∈ T, we have F (Xt , t) = F (Xs , s) d  t d    + f k (Xr , r) ◦ dWrk + k=0 s

i=1

(2.29) t

s

∂F (Xr , r) ◦ dXri , ∂xi

for any 0 ≤ s < t ≤ T . Proof It is sufficient to prove it in the case s = 0. Since f k (Xt , t) is a continuous semi-martingale, we have, similarly to the proof of Theorem 2.4.1,  0

t

 t d  1  t ∂f k f k (Xr , r) ◦ dWrk = f k (Xr , r) dWrk + (Xr , r) dW k , Xi r , 2 0 0 ∂xi

if k = 1, . . . , d  . Next,

i=1

∂F ∂xi (Xt , t)

is a continuous semi-martingale represented by

64

2 Stochastic Integrals

∂F ∂F (Xt , t) − (X0 , 0) ∂xi ∂xi d  t d  t   ∂f k ∂ 2F j = (Xr , r) dWrk + (Xr , r) dXr + Act , ∂x ∂x ∂x i i j 0 0 j =1

k=0

where Act is a continuous process of bounded variation, in view of Theorem 2.3.1. Therefore,  0

t

 t d  ∂F ∂F 1  t ∂f k i i (Xr , r) ◦ dXr = (Xr , r) dXr + (Xr , r) dW k , Xi r ∂xi 2 0 ∂xi 0 ∂xi k=1

+

1 2

d  t  j =1 0

∂ 2F (Xr , r) dXj , Xi r . ∂xi ∂xj  

Then (2.19) implies (2.29).

2.5 Stochastic Integrals with Respect to Poisson Random Measure We will define stochastic integrals based on Poisson random measures and compensated Poisson random measures. Let N(du) ≡ N(dr dz) be a Poisson random  measure on U = T × Rd0 with intensity n(du) = n(dr dz) = dr ν(dz), where ν  is a Lévy measure. Here, elements of T are denoted by r and elements of Rd0 are  denoted by z = (z1 , . . . , zd ). We define the compensated Poisson random measure N˜ by ˜ N(du) = N(du) − n(du).

(2.30)

Let {Ft , t ∈ T} be a filtration such that for any 0 ≤ s < t < t  ≤ T , N ((s, t] × E) is {Ft }-adapted and N ((t, t  ] × E) are independent of Ft . Then N (du) is called an {Ft }-Poisson random measure. Let ψ(u) = ψ(r, z) be a measurable random field with parameter u ∈ U. It is called predictable, if for any z, ψ(r, z) is a predictable process with respect to the filtration {Ft }. We set L2U



= ψ(u); predictable and E

 U

  |ψ(u)|2 n(du) < ∞ .

(2.31) 

For 0 < t ≤ T , we set Ut = {u = (r, z) ∈ U; 0 ≤ r ≤ t, z ∈ Rd0 }. It is a subdomain of U. We will define the stochastic integral Ut ψ(u)N˜ (du) for ψ ∈ L2U .

2.5 Stochastic Integrals with Respect to Poisson Random Measure

65

We first consider the integral for a simple predictable random field ψ(r, z) written as ψ(r, z) =

n 

ψm (z)1(tm−1 ,tm ] (r),

m=1 

where 0 = t0 < · · · < tn = T and ψm (z) are Ftm−1 ×B(Rd0 )-measurable functionals such that E[ (|ψm (z)| + |ψm (z)|2 )ν(dz)] < ∞. We define  Mt :=

Ut

˜ ψ(u)N(du) =

n   m=1

 Rd0

  ψm (z)N˜ ((tm−1 , tm ], dz),

(2.32)

 = t ∧ t and N ˜ ((s, t], A) = N ((s, t] × A) − (t − s)ν(A) are signed where tm m  d measures on R0 for almost all ω.

Lemma 2.5.1 Mt , t ∈ T defined by (2.32) is an L2 -martingale. Further,  Mt2

−

Ut

|ψ(u)|2 n(du),

t ∈T

(2.33)

is also a martingale. Proof We denote ψm (z)N˜ ((tm−1 , tm ], dz) by Zm . We want to show that for any m E[Zm |Ftm−1 ] = 0, 2 E[Zm |Ftm−1 ] = (tm − tm−1 )E



(2.34)



ψm (z)2 ν(dz) Ftm−1 .

 Suppose that ψm (z) is a deterministic step function ψm (z) = i ci 1Ei (z), where  ci are constants and Ei are disjoint subsets of Rd0 with ν(Ei ) < ∞. Since Zm and Ftm−1 are independent, we have E[Zm |Ftm−1 ] = E[Zm ] =



ci E[N˜ ((tm−1 , tm ] × Ei )] = 0,

i

2 2 E[Zm |Ftm−1 ] = E[Zm ]=



ci cj E[N ((tm−1 , tm ] × Ei )N ((tm−1 , tm ] × Ej )].

i,j

Since the law of N ((tm−1 , tm ] × Ei ) is Poisson with parameter (tm − tm−1 )ν(Ei ), its variance E[N˜ ((tm−1 , tm ], Ei )2 ] is equal to (tm − tm−1 )ν(Ei ). Further, if i = j , N˜ ((tm−1 , tm ], Ei ) and N˜ ((tm−1 , tm ], Ej ) are independent, so that the expectation of the product of these two random variables is equal to 0. Consequently, we get

66

2 Stochastic Integrals

2 E[Zm |Ftm−1 ] = (tm − tm−1 )



 ci2 ν(Ei ) = (tm − tm−1 )

i

 Rd0

ψm (z)2 ν(dz).

hold for any deterministic function ψm (z) satisfying Then, equalities (2.34) (|ψm (z)| + |ψm (z)|2 )ν(dz) < ∞. Next, if ψm (z) is Ftm−1 -measurable, we may regard that it is a deterministic function under the conditional measure P (·|Ftm−1 ) a.s. Therefore equalities (2.34) hold a.s. Further,  if m > l, E[Zm Zl |Ftl−1 ] = E[E[Zm |Ftm−1 ]Zl |Ftl−1 ] = 0. Then, since Mt = m;tm ≤t Zm , Mt − Ms satisfies

  

E Mt − M s F s = E



Zm Fs = 0,

 m;s t > τn−1 ; Jt − Jτn−1 > 0}, = ∞, otherwise. Then Xt has possible jumps at these stopping times. We have f (Xt , t) − f (X0 , 0) =



{f (Xτn − , τn ) − f (Xτn−1 , τn−1 )}

n;τn ≤t

+



n;τn ≤t

= I1 + I2 .

{f (Xτn , τn ) − f (Xτn − , τn )}

2.6 Jump Processes and Related Calculus

69

We shall calculate I1 . If τn−1 ≤ t, we set Xt = Xτn−1 +



t

 φ(r) dWr +

τn−1

t

υ  (r) dr,

τn−1



where υ  = υ − D ψν(dz). Then it holds that Xt = Xt for τn−1 ≤ t < τn and it has no jumps for t ≥ τn . We apply Itô’s formula for the continuous Itô process Xt . We have  τn ∂f f (Xτn − , τn ) − f (Xτn−1 , τn−1 ) = (Xr , r) dr τn−1 ∂t  τn  ∂f 1 τn ∂ 2 f (Xr− , r)φ(r) dWr + + (Xr , r)φ(r)2 dr 2 ∂x 2 (∂x) τn−1 τn−1  τn   τn ∂f ∂f (Xr , r)υ(r) dr − (Xr , r)ψ(r, z)n(dr dz). + τn−1 ∂x τn−1 D ∂x Summing up these for n such that τn ≤ t, we get 

t

I1 = 0



t

+ 0



 ∂f 1 t ∂ 2f (Xr− , r)φ(r) dWr + (Xr , r)φ(r)2 dr 2 ∂x 2 (∂x) 0 0  t ∂f ∂f (Xr , r)υ(r) dr − (Xr , r)ψ(r, z)n(dr dz). ∂x 0 D ∂x

∂f (Xr , r) dr + ∂t

t

Next, we have 

I2 =

{f (Xr , r) − f (Xr− , r)}

0 . Let Xt be the Itô process associated with α, υ and ψ . Then Xt has finite jumps, so that Itô’s formula is valid for Xt . Further, each term of (2.39) converges as → 0. Hence we get (2.39) for Xt .  

70

2 Stochastic Integrals

We will give Lp -estimations of the integral Ut ψ(u)N˜ (du). We transform the intensity measure n on U to a bounded measure m on U by setting m(du) = γ (u)2 n(du),

where γ (u) = |z| ∧ 1 for u = (r, z).

For p ≥ 2, we define   p LU = ψ ∈ LU ; ψp < ∞ ,

(2.40)



ψ(u)

p 1 2 p U γ (u) m(du) . In the case p = 2, the above LU coincides p with L2U defined by (2.31). We set L∞− = p>2 LU . U where ψp = E

p

Proposition 2.6.1 Let p ≥ 2. Let ψ(u) be an element of LU . Then the stochastic t ˜ ˜ integral Ut ψ(u)N(du) ≡ 0 Rd  ψ(r, z)N(dr dz) is p-th integrable. Further, 0 there exists a positive constant Cp such that  

E

Ut

p  1

p ˜ ψ(u)N(du) ≤ Cp ψp

(2.41)

p

holds for any 0 < t ≤ T and ψ ∈ LU . The equation (2.41) is rewritten as    t

E

0 Rd



p    t

˜ ψ(r, z)N(dr dz) ≤ Cp E

0 Rd



ψ(r, z) p 

drμ(dz) ,

γ (z)

where γ (z) = |z| ∧ 1 and μ(dz) = γ (z)2 ν(dz). t ˜ Proof We consider the process Yt = ψ N(dr dz). We apply the formula (2.39) 

0

for f (x, t) = |x|p and D = Rd0 . Then, |Yt |p =

  t 0

+



  t

 ˜ dz) |Yr− + ψ|p − |Yr− |p N(dr 

(2.42)

 |Yr + ψ|p − |Yr |p − p|Yr |p−2 Yr ψ n(dr dz).

0

Denote the first term of the right-hand side by Zt . Then Zt is a local martingale. We may choose an increasing sequence of stopping times τn such that P (τn < T ) → 0 as n → ∞, Zt∧τn is a martingale with mean 0 and the last term in (2.42) stopped at τn is integrable. To make the notation simple, we denote t ∧ τn by t  . Since |Yr + ψ|p − |Yr |p − p|Yr |p−2 Yr ψ =

1 p(p − 1)|Yr + θ ψ|p−2 ψ 2 2

≤ c1 |Yr |p−2 ψ 2 + c2 |ψ|p

2.6 Jump Processes and Related Calculus

71

holds for some |θ | < 1, we get from (2.42), E[|Yt  | ] ≤ c1 E p

  t

|Yr |

p−2



ψ n(dr dz) + c2 E 2

  t

0

 |ψ|p n(dr dz) .

0

(2.43)

We shall compute the first term of the right-hand side. It holds that  t

|Yr |

p−2

ψ n(dr dz) ≤ sup |Yr | 2

p−2

 t

r

0

0

≤ c3 sup |Yr |p + c4

ψ 2 n(dr dz)   t

r

p

p−2 p−2 , c4 p λ

where c3 =

we used the inequality ab 1 p

+

2 − p2 pλ p ≤ (λa) p

=

ψ 2 n(dr dz)

p 2

,

0

and λ is a positive constant. In the last inequality, +

(b/λ)q q



, where a, b > 0, λ > 0 p , q  > 1 and

= 1. Therefore,

1 q

  t E

|Yr |

p−2



ψ n(dr dz) ≤ c3 E 2

0



sup |Yr |

p



0 0. Let τ n be the first time t such that sup|x|≤M |∇Φs,t (x)−1 | exceeds n. These are stopping times and limn→∞ τ n = ∞ holds a.s. We consider 

dXtn = −

d 

∇Φs,t∧τ n (Xtn )−1 Vk (Xtn , t) ◦ dWtk .

k=0

Then coefficients satisfy the Lipschitz condition. Therefore it is a master equation. Let Xtn,x,s be the solution. It is a C ∞ -function of x. By the path-wise uniqueness of the solution, it holds that Xtn,x,s = Xtn+1,x,s for any |x| ≤ M if t < τ n . Therefore there exists Xtx,s such that Xtx,x = Xtn,x,s holds a.s. if t < τ n . Then Xtx,s is the unique solution of equation (3.59). Further, the solution Xtx,s of (3.59) has a modification of C ∞ -maps. We denote it by Ψˇ s,t (x). Lemma 3.7.2 It holds that Φs,t (Ψˇ s,t (x)) = Ψˇ s,t (Φs,t (x)) = x for any x a.s. for any s < t. Proof We denote Φs,t (x) and Ψˇ s,t (x) by Φt (x) and Ψˇ t (x), respectively. Then F (x, t) ≡ Φt (x) is a stochastic process with parameter x belonging to the class n+Lip,p LT (Rd ) for any n, p. Further, it satisfies conditions of Theorem 2.4.2. Then, setting F (x, t) = Φt (x) and Xt = Ψˇ t (x), we have by Theorem 2.4.2, 

◦dΦt (Ψˇ t ) =

d 

Vk (Φt ◦ Ψˇ t , t) ◦ dWtk + ∇Φt (Ψˇ t ) ◦ d Ψˇ t

k=0 

=

d  k=0



Vk (Φt ◦ Ψˇ t , t) ◦ dWtk −

d 

Vk (Φt ◦ Ψˇ t , t) ◦ dWtk = 0.

k=0

Therefore we have Φt (Ψˇ t (x)) = x for all x a.s. Apply Theorem 2.4.2 again for F (x, t) = Ψˇ t (x) and Xt = Φt (x). Then we have

3.7 Diffeomorphic Property and Inverse Flow for Continuous SDE

107



◦d Ψˇ t (Φt ) = −

d  (∇Φt )−1 Vk (Ψˇ t ◦ Φt , t) ◦ dWtk + ∇ Ψˇ t (Φt ) ◦ dΦt . k=0

Since Φt ◦ Ψˇ t (x) = x, we have ∇Φt (Ψˇ t )∇ Ψˇ t= I . Therefore we have ∇ Ψˇ t = (∇Φt )(Ψˇ t )−1 . Then we get ∇ Ψˇ t (Φt ) ◦ dΦt = k (∇Φt )−1 Vk (Ψˇ t ◦ Φt , t) ◦ dWtk . Consequently, we get ◦d Ψˇ t (Φt ) = 0. This proves Ψˇ t (Φt (x)) = x for all x a.s.   By the above two lemmas, maps Φs,t ; Rd → Rd are diffeomorphic a.s. and Ψˇ s,t −1 are inverse maps of Φs,t , i.e., Ψˇ s,t (x) = Φs,t (x) holds for any x a.s. We show that ˇ ˇ Xs := Ψs,t (x) satisfies the backward SDE (3.46) with coefficients −Vk (x, t). We have Φs,t (Ψˇ s,t (x)) = x for all x a.s. This implies ∇Φs,t (Ψˇ s,t (x))∇ Ψˇ s,t (x) = I . Therefore we have ∇Φs,t (Ψˇ s,t (x))−1 = ∇ Ψˇ s,t (x). Then equation (3.59) can be rewritten as d   

Ψˇ s,t (x) = x −

t

k=0 s

∇ Ψˇ s,r (x)Vk (Ψˇ s,r (x), r) ◦ dWrk .

(3.60)

Consequently, we have by Theorem 2.4.1 d   

f (Ψˇ s,t (x)) = f (x) −

t

k=0 s

Vk (f ◦ Ψˇ s,r )(x) ◦ dWrk ,

for any C ∞ -function f . Then Ψˇ s,t (x) should satisfy the backward differential rule d   

f (Ψˇ s,t (x)) = f (x) −

t

k=0 s

Vk (r)f (Ψˇ r,t (x)) ◦ dWrk ,

in view of Proposition 3.6.3. The above equation shows that Ψˇ s,t (x) satisfies the backward symmetric SDE; d   

Ψˇ s,t (x) = x −

k=0 s

t

Vk (Ψˇ r,t (x), r) ◦ dWrk .

(3.61)

It satisfies for any C ∞ -function f d   

f (Ψˇ s,t ) = f (x) −

t

k=0 s d   

= f (x) −

k=1 s

t

Vk (r)f (Ψˇ r,t ) ◦ dWrk Vk (r)f (Ψˇ r,t ) dWrk +

(3.62)  s

t

A(r)f (Ψˇ r,t ) dr, (3.63)

108

3 Stochastic Differential Equations and Stochastic Flows

where A(t) =

1 2

d 

k=1 (−Vk (t))

2f

+ (−V0 (t))f or simply d

1 A(t)f = Vk (t)2 f − V0 (t)f. 2

(3.64)

k=1

Summing up these facts, we get the following theorem. Theorem 3.7.1 Assume that coefficients Vk (x, t), k = 0, . . . , d  of a continuous symmetric SDE (3.1) are Cb∞,1 -functions. Then equation defines a continuous −1 (x) for any x and stochastic flow of diffeomorphisms {Φs,t }. Set Ψˇ s,t (x) = Φs,t s < t. Then {Ψˇ s,t } is a backward continuous stochastic flow and satisfies the backward SDE (3.61). It satisfies (3.62)–(3.63). Remark It is interesting to know that the equation for the forward flow given by (3.6) and the backward equation for the inverse flow Ψˇ s,t given by (3.7) are symmetric. If we rewrite equations using Itô integrals, the symmetric property disappears. In fact, the symmetric equation (3.6) is equivalent to the Itô equation d   

Φs,t (x) = x +

t

k=1 s

 Vk (Φs,r (x), r) dWrk

+

t

s

V0 (Φs,r (x), r) dr,

(3.65)

 where V0 (x, t) = V0 (x, t) + 12 k≥1 Vk (t)Vk (x, t). Further, equation (3.61) is equivalent to the backward Itô equation d   

Ψˇ s,t (x) = x −

k=1 s

t

Vk (Ψˇ r,t (x), r) dWrk −

where V0 (x, t) = V0 (x, t) −

1 2





t s

V0 (Ψˇ r,t (x), r) dr,

(3.66)

k≥1 Vk (t)Vk (x, t).

Remark Consider a continuous forward SDE with coefficients −Vj (x, t), j = 0, . . . , d  : The equation is written as d   

Xt = X0 −

t

j

Vk (Xr , r) ◦ dWr .

k=0 s

Theorem 3.7.1 tells us that the equation generates a stochastic flow of diffeomor−1 = Φˇ s,t . The theorem tells us further phisms, which we denote by {Ψs,t }. Set Ψs,t t ,x 1 that Xˇ s := Φˇ s,t1 (x) satisfies the backward SDE (3.46). Hence it coincides with the backward stochastic flow {Φˇ s,t } discussed in Sect. 3.5.

3.8 Forward–Backward Calculus for C ∞ -Flows of Jumps

109

Note The study of stochastic differential equations was initiated by Itô. In [50], stochastic differential equations driven by the Wiener process and Poisson random measure are discussed. Then continuous stochastic differential equations are studied extensively. The diffeomorphic property of maps x → Xtx received a lot of attention around 1980. We refer Elworthy [25], Malliavin [78], Bismut [8], Le Jan [75], Harris [36], Ikeda–Watanabe [41] and Kunita [59]. A method of proving the diffeomorphic property is to approximate SDEs by a sequence of stochastic ordinary differential equations. It is described as follows. Instead of a Wiener process Wt , we consider a suitable sequence of piecewise  smooth stochastic processes Wtn = (Wt1,n , . . . , Wtd ,n ) which converges to Wt , and consider a sequence of equations d

d n  Vk (xtn , t)W˙ tk,n + V0 (xtn , t). X = dt t k=1

dW k,n

where W˙ tk,n = dtt . Then its solution Xtn,x starting from x at time 0 defines a diffeomorphic map for any n. The sequence {Xtn,x , n = 1, 2, . . .} should converge to the solution Xtx of the SDE and it should be a diffeomorphic map. For detail, see Ikeda–Watanabe [41], Bismut [8], Malliavin [78]. Kunita [59] presented another method for proving the diffeomorphic property through a skillful use of the Kolmogorov–Totoki theorem. In this monograph, we took another method by constructing a backward flow Ψˇ s,t (x) which satisfies Lemma 3.7.2. An advantage of the new method is that it can be applied for proving the diffeomorphic property of solutions of SDE on a manifold as in Sect. 7.1.

3.8 Forward–Backward Calculus for C ∞ -Flows of Jumps We will consider stochastic flow of C ∞ -maps generated by a symmetric SDE with jumps (3.10), where V0 (x, t), . . . , Vd  (x, t) are Cb∞,1 -functions and g(x, t, z) satisfies Condition (J.1). For a slowly increasing C ∞ -function f , we define an integro-differential operator AJ (t) with time parameter t by AJ (t)f (x) = A(t)f +

  Rd0



(3.67) d

f (φt,z (x))−f (x) − 1D (z)



 zk V˜k (t)f (x) ν(dz),

k=1

where A(t) is the differential operator defined by (3.48) and φt,z (x) = g(x, t, z) + x   and D = {z = (z1 , . . . , zd ) ∈ Rd0 ; |z| ≤ 1}.

110

3 Stochastic Differential Equations and Stochastic Flows

Proposition 3.8.1 Let {Φs,t } be a stochastic flow of C ∞ -maps defined by symmetric SDE (3.10) with jumps. Let f be a slowly increasing C ∞ -function on Rd . Then, the stochastic flow satisfies the forward differential rule with respect to t: d   

f (Φs,t ) = f +

t

k=1 s

+

 t

  Rd0

s

d   

=f +

k=0 s

+ lim

 Vk (r)f (Φs,r− ) dWrk

s

t

AJ (r)f (Φs,r− ) dr s

 ˜ f (φr,z ◦ Φs,r− ) − f (Φs,r− ) N(dr dz)

t

(3.68)

Vk (r)f (Φs,r ) ◦ dWrk

  t

→0

+

  f (φr,z ◦ Φs,r− ) − f (Φs,r− ) N(dr dz)

|z|≥

   t V˜k (r)f (Φs,r− ) dr . − b k d

k=1

(3.69)

s

Proof The process Xt = Φs,t (x) is a solution of equation (3.10). Then it is an Itô process satisfying 

t

Xt = x +

 α(Xr , r) dWr +

s

t

β(Xr , r) dr +

 t

s

χ (Xr− , r, z)N˜ D (dr dz),

s

where χ (x, r, z) = g(x, t, z),

α(x, r) = (V1 (x, r), . . . , Vd  (x, r)), β(x, r) = V0 (x, r) +

1 Vk (r)Vk (x, r) + 2 k≥1

    g(x, r, z) − zk V˜k (x, r) ν(dz). D

k≥1

We apply Itô’s formula (Theorem 2.6.1) for the C 3 -function f . Then the diffusion part of the stochastic process f (Xt ) is equal to d  d   

i=1 k=1 s

t

d

 ∂f α (r) (Φs,r− ) dWrk = ∂xi



ik

k=1 s

t

Vk (r)f (Φs,r− ) dWrk .

3.8 Forward–Backward Calculus for C ∞ -Flows of Jumps

111

The jump part is equal to  t   f (φr,z (Φs,r− )) − f (Φs,r− ) N˜ D (dr dz) s

 t   ˜ dz) f (φr,z (Φs,r− )) − f (Φs,r− ) N(dr

=

s

+

 t s

|z|>1

  f (φr,z (Φs,r− )) − f (Φs,r− ) n(dr dz).

The drift part is equal to d  d d  t  ∂ 2f  1  t   ik ∂f α (r)α j k (r) (Φs,r− ) dr + β i (r) (Φs,r− ) dr 2 ∂xi ∂xj ∂xi s s i,j =1

+

  t s

k=1

i=1

   ∂f f (φr,z (Φs,r− )) − f (Φs,r− ) − gi (Φs,r− ) n(dr dz) ∂xi |z| 2. Set Y = We shall estimate the integral E[|Ys,t s,t t 

(x  ) − Y (x  ) is estimated as and Yt = Ys,t (x ). The drift term of Ys,t s,t

(x  ) Ys,t

 t



(β (Yr , r) − β(Yr , r)) dr s



≤ s

t

 |β (Yr , r) − β(Yr , r)| dr 

t

≤ cϕ0 ( ) s

+ s

 (1 + |Yr |) dr + c

t s

t

|β(Yr , r) − β(Yr , r)| dr

|Yr − Yr | dr.

Since E[|Yr |p ] are uniformly bounded with respect to 0 < s < t < T and x, , we have  t

p    t

p E[|Yr − Yr |p ] dr. E (β (Yr , r) − β(Yr , r)) dr ≤ c1 ϕ0 ( ) + c2 s

s

(x  ) − Similar estimates are valid for the diffusion term and jump term of Ys,t  Ys,t (x ). We have

 t

p    t 

 p  α (Yr , r) − α(Yr , r) dWr ≤ c1 ϕ0 ( ) + c2 E[|Yr − Yr |p ] dr, E s

s

p    t 

˜ dz) {χ (Yr , r, z)−χ (Yr , r, z)}N(dr E s

≤ c1 ϕ0 ( ) + c2



t s

E[|Yr −Yr |p ] dr.

Consequently, for any p ≥ 2 there exist c1 > 0, c2 > 0 such that  E[|Yt

− Yt | ] ≤ c1 (ϕ0 ( ) + ϕ0 ( )) + c2 p

p

s

t

E[|Yr − Yr |p ] dr.

Then Gronwall’s inequality implies E[|Yt − Yt |p ] ≤ C1 (ϕ0 ( )p + ϕ0 ( )). Since this is valid for any x  , s, we get (3.75) in the case |i| = 0, 1.

 

(x) with respect to x, s. uniform convergence of ∂ i Φs,t p Sobolev’s inequality in L (D, dx)-space, where D is

We are interested in the For this problem, we need a bounded domain in Rd . Let f (x) be a real function on D. We denote the weak derivative of f by ∂ i by ∂ i f . We denote by Dk,p the set of all f such that its weak derivatives ∂ i f , |i| ≤ k belong to Lp (D, dx). We quote Morrey’s Sobolev inequality without proof.

3.8 Forward–Backward Calculus for C ∞ -Flows of Jumps

115

Lemma 3.8.2 (Brezis, [14]) Let D be a bounded subdomain in Rd with smooth boundary. Let p > d. Then elements of Dk,p are C k−1 -functions. Further, there exists a positive constant ck,p such that 

sup |∂ i f (x)| ≤ ck,p



|∂ i f (x  )|p dx 

1

p

(3.76)

|i|≤k D

|i|≤k−1 x∈D

holds for any f ∈ Dk,p . Lemma 3.8.3 For any s < t, there exists a sequence { n } converging to 0 such

n

n that solutions {Φs,t (x)} and their derivatives {∂ i Φs,t (x)} converge to Φs,t (x) and i ∂ Φs,t (x), uniformly on compact sets with respect to x a.s. Proof By Lemma 3.8.2, we have

E[ sup |Φs,t (x) − Φs,t (x)|p ]

(3.77)

|x|≤M

≤ CE



|Φs,t (x  ) − Φs,t (x  )|p dx  +



|x  |≤M



|∂Φs,t (x  ) − ∂Φs,t (x  )|p dx  .

|x  |≤M

It converges to 0 as → 0 in view of Lemma 3.8.1. Therefore a subsequence

n {Φs,t (x)} converges to Φs,t (x) uniformly on compact sets with respect to x, t a.s.

n (x)} can be shown similarly.   The convergence of {∂ i Φs,t

n Proof of Proposition 3.8.2 Let us consider the sequence of {Φs,t (x)} obtained in

n Lemma 3.8.3. We saw that each Φs,t (x) is right continuous in s and satisfies (3.70) ˜ replacing N(dr dz) by N˜ n (dr dz). Therefore it is a right continuous backward semi-martingale with respect to s. Let n tend to ∞. Then terms d   

t

Vk (r)(f

k=1 s

n ◦ Φr,t )(x) dWrk ,



t s

n A Jn (r)(f ◦ Φr,t )(x) dr, etc.

should converge a.s. Therefore the limit Φs,t (x) is a backward semi-martingale with respect to s for any t, x, and it has a right continuous modification. We denote it by Φs,t (x), again. Then the above integrals should converge to d   

k=1 s



t

Vk (r)(f

◦ Φr,t )(x) dWrk ,

t

AJ (r)(f ◦ Φr,t )(x) dr, etc.

s

Therefore {Φs,t } satisfies (3.70). Finally, the equality (3.71) follows from (3.70), if we rewrite backward symmetric integrals using backward Itô integrals.  

116

3 Stochastic Differential Equations and Stochastic Flows

We shall next consider differential rules for the backward flow {Φˇ s,t } defined by the backward symmetric SDE (3.47). The following is the counterpart of Proposition 3.8.1, whose proof is omitted. Proposition 3.8.3 Let {Φˇ s,t } be the backward stochastic flow of C ∞ -maps defined by the backward symmetric SDE (3.47). Let f be a slowly increasing C ∞ -function on Rd . Then, we have the backward differential rule with respect to s: d   

f (Φˇ s,t ) = f +

t

Vk (r)f (Φˇ r,t ) dWrk +

k=1 s

 t

+ lim

→0 s

|z|≥

 d

=f +

t

k=0 s

−

→0

s





d 

b k

k=1

AJ (r)f (Φˇ r,t ) dr

(3.78)

s

Vk (r)f (Φˇ r,t ) ◦ dWrk

|z|≥ t

t

  ˜ f (φr,z ◦ Φˇ r,t ) − f (Φˇ r,t ) N(dr dz)

 t 

+ lim



(3.79)

{f (φr,z ◦ Φˇ r,t ) − f (Φˇ r,t )}N(dr dz)

 V˜k (r)f (Φˇ r,t ) dr .

s

3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps In this section, we assume further that the jump-map g(x, t, z) of the SDE (3.10) satisfies Condition (J.2) and hence φt,z (x) = g(x, t, z) + x are diffeomorphic for any t, z. We will discuss the diffeomorphic property of the solution of the SDE. Let

> 0. Associated with equation (3.10), we consider a forward SDE (3.72). The

} of diffeomorphisms. solution defines a right continuous stochastic flow {Φs,t

(x) = (Φ )−1 (x). Then X

(x), 0 ≤ s < t < ∞ ˇ s := Ψˇ s,t Lemma 3.9.1 Set Ψˇ s,t s,t satisfies the following backward equation with jumps: d   

Xˇ s = x −

−

k=0 s

t

Vk (Xˇ r , r) ◦ dWrk

  t s

(3.80) 



|z|≥

h(Xˇ r , r, z)N(dr dz) −

k=1

−1 where h(x, t, z) = x − φt,z (x) and

V˜k (x, t) = ∂zk g(x, t, z)

d 

z=0

b k

s

t

 V˜k (Xˇ r , r) dr ,

= ∂zk h(x, t, z)

z=0

.

3.9 Diffeomorphic Property and Inverse Flow for SDE with Jumps

117

is represented by (3.74), the inverse flow Ψ

(x) is ˇ s,t Proof Since the flow Φs,t represented by



0 (x), Ψˇ s,t if t < τ1 −1 −1 0 0 ˇ ˇ Ψs,τ1 ◦ φτ1 ,S1 ◦ · · · ◦ φτn ,Sn ◦ Ψτn ,t (x), if τn ≤ t < τn+1 .

(3.81)

(x) is cadlag with respect to s and t. It is cadlag with respect to s and t, since Φs,t 0 Note that Xˇ s = Ψˇ s,t satisfies the continuous backward SDE d   

Xˇ s = x −

t

k=1 s

Vk (Xˇ r , r) ◦ dWrk −



t

s

V0 (Xˇ r , r) dr.

(x) should satisfy the backward SDE Then the composite process Xˇ s = Ψˇ s,t d   

Xˇ s = x − −

k=1 s

 t s

|z|≥

t

Vk (Xˇ r , r) ◦ dWrk −



h(Xˇ r , r, z)N(dr dz).

s

t

V0 (Xˇ r , r) dr (3.82)

(We have shown a similar fact for a forward equation in Sect. 3.2.) The above equation can be rewritten as (3.80).   Theorem 3.9.1 Assume that diffusion and drift coefficients of symmetric SDE (3.10) are Cb∞,1 -functions and jump coefficients satisfy Conditions (J.1) and (J.2). Then the equation defines a right continuous stochastic flow of diffeomorphisms {Φs,t }. Let {Ψˇ s,t } be the backward flow of C ∞ -maps generated by a backward symmetric SDE (3.47) with coefficients (−Vk (x, t), k = 0, . . . , d  , −h(x, t, z)). Then it holds −1 that Ψˇ s,t (x) = Φs,t (x) for any x a.s. for any s < t. Proof We saw in Lemma 3.8.3 that for any s < t, there is a sequence { n }

n converging to 0 such that Φs,t (x) converges to Φs,t (x) uniformly on compact sets with respect to x almost surely. Then by the same reasoning, the sequence

n Ψˇ s,t (x) converges to Ψˇ s,t (x) uniformly on compact sets almost surely. Since

(Ψ

(x)) = Ψ

(Φ (x)) = x holds for all x almost surely, we get ˇ s,t ˇ s,t Φs,t s,t Φs,t (Ψˇ s,t (x)) = Ψˇ s,t (Φs,t (x)) = x for all x almost surely. This implies that maps Φs,t ; Rd → Rd are one to one and onto a.s. for any s < t. Further, since maps Ψˇ s,t are smooth a.s., maps Φs,t are diffeomorphic a.s. (x) exists for any t, a.s. Further, Now we will fix t0 . By the above argument, Φt−1 0 ,t −1 Φt0 ,t (x) should be cadlag with respect to t for any x. Finally, we define for any s < t,

118

3 Stochastic Differential Equations and Stochastic Flows

Φs,t = Φt0 ,t ◦Φt−1 . Then it is right continuous in t and further, a diffeomorphic map 0 ,s for any s < t a.s. Therefore {Φs,t } is a right continuous stochastic flow generated by the SDE (3.10).

satisfies the backward SDE (3.82), the limit Ψ ˇ s,t satisfies the Since Xˇ s = Ψˇ s,t   backward SDE with coefficients (−Vk (x, t), k = 0, . . . , d  , −h(x, t, z)). So far, we assumed that coefficients of the SDE are not big; we assumed that coefficients and their derivatives are bounded. Then we saw that the stochastic flow {Φs,t } belongs to L∞− (Theorem 3.4.1). The property is needed for proving that Φs,t (x) is a smooth Wiener–Poisson functional in the Malliavin calculus. See Chap. 6. We will relax conditions for jump-maps φt,z in SDE (3.10) for a while. Instead of Conditions (J.1) and (J.2), we introduce the following weaker conditions. Condition (J.1)K . (i) The function g(x, t, z) := φt,z (x)−x is of Cb∞,1,2 -class on Rd ×T×{|z| ≤ c} for some c > 0. (ii) It is of Cb∞ -class on Rd for any t ∈ T, |z| ≥ c and is piecewise continuous in (t, z) ∈ T × {|z| > c}. Condition (J.2)K . For any t, z, the map φr,z ; Rd → Rd is a diffeomorphism of −1 Rd . Further, h(x, t, z) := x − φt,z (x) satisfy Condition (J.1)K . Theorem 3.9.2 Assume Conditions (J.1)K and (J.2)K for jump -maps φt,z . Then the assertion of Theorem 3.9.1 is valid. Further, the integro-differential operator AJ (t) of (3.67) is well defined for any f ∈ Cb∞ , and further, the stochastic flow satisfies equations (3.68), (3.69), (3.70) and (3.71) for any f ∈ Cb∞ . Proof We truncate big jumps and consider g c (x, t, z) = g(x, t, z)1|z|≤c and c (x) = x + g c (x, t, z) instead of g(x, t, z), φ . Then the SDE with coefficients φt,z t,z (Vk (x, t), k = 0, . . . , d  , g c (x, t, z)) defines a stochastic flow of diffeomorphisms c (x)} in view of Theorem 3.9.1. We define a stochastic flow Φ {Φs,t s,t by  Φs,t (x) =

c (x), Φs,t

Φτcn ,t

c (x), ◦ φτn ,Sn ◦ · · · ◦ φτ1 ,S1 ◦ Φs,τ 1

if t < τ1 , if τn ≤ t < τn+1 .

Here τn , n = 1, 2, . . . are jumping times of the Poisson process Nt with intensity λc := ν({|z| > c}) with initial time s, and Sn , n = 1, 2, . . . are independent random variables with the identical distribution νc = ν1|z|>c /λc . Then {Φs,t } is a right continuous stochastic flow of diffeomorphisms generated by SDE with −1 coefficients (Vk (x, t), g(x, t, z)). Further, the inverse Ψˇ s,t = Φs,t satisfies the  backward equation with coefficients (−Vk (x, t), k = 0, . . . , d , −h(x, t, z)). The operator AJ (t) is well defined if we restrict f in Cb∞ . Equations (3.68)– (3.71) for f ∈ Cb∞ can be verified in the same way as in the proof of Propositions 3.8.1 and 3.8.2.   The stochastic flow Φs,t (x) of the above theorem may not be Lp -bounded. (3.45) may be infinite. We are also interested in the case where diffusion coefficients are unbounded. In such a case, the solution of the SDE may explode in finite time. In Chap. 7, we will

3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift

119

discuss SDEs on manifolds, where the solution may explode in finite time. Results in Chap. 7 can be applied to SDEs on Euclidean spaces with unbounded coefficients. Note The flow property of solutions for SDE with jumps was first studied by Fujiwara–Kunita [30]. Then, it was developed further by Fujiwara–Kunita [31] and Kunita [60]. Some technical conditions posed for jump coefficient g(x, t, z) in [30, 31] are removed in [60]. The latter is closed to the conditions in this book. However, concerning the proof of the regularity of the solution with respect to the initial data, we took another method introducing the master equation. Further, for the proof of the diffeomorphic property, a method different from these works is taken, approximating SDE by a sequence of SDE with finite Lévy measure.

3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift Let ν be a Lévy measure. If 0

ψ(r, z) drν(dz),

 t

→0 s

|z|>

ψ(r, z)N(dr dz),

if the right-hand sides exist and are finite almost surely, respectively. We introduce  L0U = ψ(r, z); predictable, C 2 -class in z, ψ(r, 0) = 0 for any r ∈ T,  ∂zk ψ(r, z), k ≤ 2 are bounded in (r, z) a.s. . (3.83) Then L0U is a subset of L∞− U .

120

3 Stochastic Differential Equations and Stochastic Flows

Proposition 3.10.1 Suppose that the Lévy measure ν has a weak drift. Let ψ(u) is an element of the functional space L0U defined by (3.83). Then the t t improper integrals s |z|>0+ ψ(r, z)N(dr dz) and s |z|>0+ ψ(r, z) drν(dz) exist. Further, both are integrable with respect to P and satisfy  t Rd0

s

˜ ψ(r, z)N(dr dz) 

 t

=

s

(3.84)

ψ(r, z)N(dr dz) −

|z|>0+

 t ψ(r, z) drν(dz).

s

|z|>0+

In particular, if ν has a strong drift, the above improper integrals are replaced by Lebesgue integrals. Proof We have ψ(r, z) = ∂z ψ(r, 0) · z + 12 (∂z2 ψ(r, θ z)z · zT , by Taylor’s expansion. Therefore, 

 t 1≥|z|>

s

t

ψ(r, z) dr dν =

 ∂z ψ(r, 0) dr ·

1 + 2

zν(dz) 1≥|z|>

s

  t

∂z2 ψ(r, θ z)z · zT dr dν.

(3.85)

s 1≥|z|>

The first term of the right-hand side converges as → 0, since ν has a weak drift. The second term of the right-hand side converges, since the function ∂z2 ψ(r, θ z)z·zT is integrable on {|z| ≤ 1} with respect to the measure t dr dν. Therefore (3.85) converges as → 0. The limit is the improper integral s 1≥|z|>0+ ψ(r, z) drν(dz). t Then improper integral s |z|>0+ ψ(r, z) drν(dz) is also well defined and it is integrable with respect to P . Further, in the equality  t s

˜ ψ(r, z)N(dr dz) =

 t

|z|>

ψ(r, z)N(dr dz) −

|z|>

s

 t ψ(r, z) drν(dz),

s

|z|>

t the left-hand side converges, since ψ ∈ L∞− U . Therefore s |z|> ψ(r, z)N(dr dz) t should converge to the improper integral s |z|>0+ ψ(r, z)N(dr dz) and (3.84) t t holds. Since s |z|>0+ ψ(r, z) drν(dz) is integrable, s |z|>0+ ψ(r, z)N(dr dz) is also integrable with respect to P .   Now, for the Itô process Xt of (2.38), we assume that φ(t) is a semi-martingale with bounded jumps and the symmetric integral by the Wiener process is well defined and that ψ(r, z) ∈ L0U . Then Itô process (2.38) is rewritten as  Xt = X0 +

t t0

 φ(r) ◦ dWr +

t t0

υ(r) dr +

 t t0

|z|>0+

ψ(r, z)N(dr dz).

(3.86)

3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift

121

For this Itô process, Itô’s formula is rewritten simply, which looks like a formula of usual differential-difference calculus. Formula (2.39) is rewritten as follows. Theorem 3.10.1 Assume that the Lévy measure has a weak drift. Let Xt , t ≥ t0 be an Itô process represented by (3.86). Let f (x, t) be a function of C 3,1 -class. Then we have for any t0 ≤ s < t < ∞, 

t

f (Xt , t) = f (Xs , s) +

∂f (Xr , r) dr ∂t

s d   d

+

 ∂f (Xr , r)φ ik (r) ◦ dWrk + ∂xi d

t

i=1 k=1 s

+

 t

|z|>0+

s



t s

i=1

∂f (Xr , r)υ i (r) dr ∂xi

{f (Xr− + ψ(r, z), r) − f (Xr− , r)}N(dr dz).

(3.87)

Proof Rewrite the symmetric integral of the above using the Itô integral. Next rewrite the last integral with Poisson random measure N , using the compensated Poisson random measure N˜ . Then we can show similarly to the proof of Theorem 2.6.1 that the right-hand side of the above is equal to the right-hand side of (2.39).   Next, we consider the backward Itô process written as Xˇ t = Xˇ 0 +



t1

ˇ ◦ dWr + φ(r)

t



t1

υ(r) ˇ dr +

t

 t1 t

|z|>0+

ˇ z)N(dr dz). ψ(r,

(3.88)

Formula (2.50) is rewritten as follows. Theorem 3.10.2 Assume that the Lévy measure has a weak drift. Let Xˇ t be a backward Itô process represented by (3.88). Let f (x, t) be a function of C 3,1 -class. Then we have for any t0 ≤ s < t < ∞, f (Xˇ s , s) = f (Xˇ t , t) −



t s

+

d  t d   i=1 k=1 s

+

 t s

|z|>0+

∂f ˇ (Xr , r) dr ∂t

 ∂f ˇ (Xr , r)φˇ ik (r) ◦ dWrk + ∂xi d

i=1



t s

∂f ˇ (Xr , r)υˇ i (r) dr ∂xi

ˇ z), r) − f (Xˇ r , r)}N(dr dz). {f (Xˇ r + ψ(r,

Let us reconsider SDE (3.10) on Rd . By Proposition 3.10.1, we have

(3.89)

122

3 Stochastic Differential Equations and Stochastic Flows

  t lim

→0

=

t0

g(Xr− , r, z)N(dr dz) −

|z|≥

 t t0





d 

b k 



g(Xr− , r, z)N(dr dz) −

|z|>0+

b0k

k=1

V˜k (Xr− , r) dr



t0

k=1 d 

t

t

V˜k (Xr− , r) dr.

t0

 Therefore, rewriting the vector V0 (t) − k b0k V˜k (t) by V0 (t) and setting φt,z (x) = g(x, t, z) + x, we can rewrite equation (3.10) simply as d   

Xt = X0 +

t

k=0 t0

+

 t

Vk (Xr , r) ◦ dWrk

{φr,z (Xr− ) − Xr− }N(dr dz).

t0 |z|>0+

(3.90)

If the Lévy measure has a strong drift, the last improper integral coincides with the Lebesgue integral. This expression of the SDE interprets the geometric meaning of the solution more than expression (3.10). Indeed, if the Lévy measure is of finite mass, the solution is given by the stochastic flow Φs,t represented by (3.18), i.e., the 0 generated by the continuous flow is written as composites of stochastic flow Φs,t  k SDE dXt = k Vk (Xt , t) ◦ dWt and diffeomorphic maps φt,z . If the Lévy measure has infinite mass, the solution should be the limit of flows given by (3.18). Further, for the operator AJ (t) the last term of (3.67) is equal to 



|z|>0+





f ◦ φt,z −f ν(dz) −

d 

b0k V˜k (t)f.

k=1

 Then, rewriting V0 (t) − k b0k V˜k (t) as V0 (t) again, the operator AJ (t) given by (3.67) is rewritten in a simple form:  AJ (t)f = A(t)f +

{f ◦ φt,z −f }ν(dz).

|z|>0+

(3.91)

If the Lévy measure has a strong drift, the last improper integral coincides with the Lebesgue integral. We call the triple (Vk (t), k = 0, . . . , d  , φt,z , ν) the characteristics of the SDE (3.90). Let {Φs,t } be the stochastic flow generated by the SDE (3.90). Then, rules of forward and backward calculus given in (3.69) and (3.71) are rewritten simply as

3.10 Simple Expressions of Equations; Cases of Weak Drift and Strong Drift d   

f (Φs,t ) = f +

t

k=0 s

+

 t s



|z|>0+ d   

=f +

t

k=1 s

+

 t s



|z|>0 d   

f ◦ Φs,t = f +

 t s

t



|z|>0+ d   

=f +

k=1 s

+

 t s



|z|>0

(3.92)

 f (φr,z ◦ Φs,r− ) − f (Φs,r− ) N(dr dz),  Vk (r)f (Φs,r− ) dWrk +

t

AJ (r)f (Φs,r ) dr s

 ˜ f (φr,z ◦ Φs,r− ) − f (Φs,r− ) N(dr dz),

k=0 s

+

Vk (r)f (Φs,r ) ◦ dWrk

t

123

(3.93)

Vk (r)(f ◦ Φr,t ) ◦ dWrk  (f ◦ Φr,t ) ◦ φr,z − f ◦ Φr,t N(dr dz)  Vk (r)(f ◦ Φr,t ) dWrk +

t

(3.94)

AJ (r)(f ◦ Φr,t ) dr

s

 ˜ dz). (f ◦ Φr,t ) ◦ φr,z − f ◦ Φr,t N(dr

(3.95)

Next, we will consider a backward SDE, which governs the inverse flow Ψˇ s,t =  Rewriting V0 (t) − k b0k Vk (t) as V0 (t) again, the backward SDE for Xˇ s = Ψˇ s,t1 (x) can be rewritten as

−1 Φs,t .

d   

Xˇ s = x −

k=0 s

t1

Vk (Xˇ r , r)◦dWrk −

 t1 s

−1 ˇ {Xˇ r −φr,z (Xr )}N(dr dz).

|z|>0+

(3.96)

We define an integro-differential operator AJ (t) by  AJ (t)f = A(t)f +

−1 {f ◦ φt,z −f }ν(dz),

|z|>0+

(3.97)

where A(t) is the differential operator defined by (3.64). Then the backward differential rule (3.78) for the inverse flow Ψˇ s,t is rewritten as

124

3 Stochastic Differential Equations and Stochastic Flows d   

f ◦ Ψˇ s,t = f −

t

k=0 s

+

 t



|z|>0+

s



t

=f +  t s

 −1 f (φr,z ◦ Ψˇ r,t ) − f (Ψˇ r,t ) N(dr dz) d   

AJ (r)f (Ψˇ r,t ) dr −

s

+

Vk (r)f (Ψˇ r,t ) ◦ dWrk

k=1 s



|z|>0

t

(3.98)

Vk (r)f (Ψˇ r,t ) dWrk

 −1 ˜ f (φr,z dz). ◦ Ψˇ r,t ) − f (Ψˇ r,t ) N(dr

(3.99)

Chapter 4

Diffusions, Jump-Diffusions and Heat Equations

Abstract We study diffusions and jump-diffusions on a Euclidean space determined by SDE studied in Chap. 3. We select topics which are related to the stochastic flow generated by the SDE; topics are concerned with heat equations and backward heat equations. In Sects. 4.1, 4.2, 4.3, and 4.4, we consider diffusion processes on a Euclidean space. In Sect. 4.1, we show that a stochastic flow generated by a continuous symmetric SDE defines a diffusion process. Its generator A(t) is represented explicitly as a second order linear partial differential operator with time parameter t, using coefficients of the SDE. Kolmogorov’s forward and backward equation associated with the operator A(t) will be derived. In Sect. 4.2, we discuss exponential transformation of the diffusion process by potentials. It will be shown that solutions of various types of backward heat equations will be obtained by exponential transformations by potentials. In Sect. 4.3, we study backward SDE and backward diffusions. We will see that Kolmogorov equations for backward diffusion will give the solution of heat equations. In Sect. 4.4, we present a new method of constructing the dual (adjoint) semigroup, making use of the geometric property of diffeomorphic maps Φs,t of stochastic flows. The method will present a clear geometric explanation of the adjoint operator A(t)∗ and the dual semigroup. We will see that the dual semigroup will be obtained by a certain exponential transformation (Feynman–Kac–Girsanov transformation) of a backward diffusion. In Sects. 4.5 and 4.6, we consider jump-diffusions on a Euclidean space. We will extend results for diffusion studied in Sects. 4.2, 4.3, and 4.4 to those for jumpdiffusions. We show that the dual semigroup of jump-diffusion is well defined if the jump coefficients are diffeomorphic. In Sect. 4.7, we return to a problem of the stochastic flow. We discuss the volume-preserving property of stochastic flows by applying properties of dual jump-diffusions. In Sect. 4.8, we consider processes killed outside of a subdomain of an Euclidean space and its dual processes.

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_4

125

126

4 Diffusions, Jump-Diffusions and Heat Equations

4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov Equations We will study diffusion processes on a Euclidean space determined by a continuous stochastic differential equations. Let us consider a continuous symmetric SDE on Rd : 

dXt =

d 

Vk (Xt , t) ◦ dWtk ,

(4.1)

k=0

where Vk (x, t), k = 0, . . . , d  are Cb∞,1 -functions of (x, t) ∈ Rd × T and Wt =  (Wt1 , . . . , Wtd ), t ∈ T is a d  -dimensional Wiener process. Here, T = [0, ∞), 0 Wt ≡ t and V0 (Xt , t) ◦ dWt0 ≡ V0 (Xt ) dt, conventionally. Let {Φs,t } be the stochastic flow of diffeomorphisms generated by the above continuous SDE. Then Xt = Xtx,s = Φs,t (x), t ≥ s is a solution of equation (4.1) starting from x at time s. Proposition 4.1.1 Laws of Φs,t (x): Ps,t (x, E) = P (Φs,t (x) ∈ E),

0 ≤ s < t < ∞, E ∈ B(Rd )

(4.2)

are determined uniquely from SDE (4.1). Further, {Ps,t (x, E)} is a transition probability. Proof We consider the equation (4.1) with the initial condition t 0 = s and X0 = x. t t Using the Itô integral it is written as Xt = x + s α(Xr , r) dWr + s β(Xr , r) dr. We saw in Sect. 3.3 that the equation has a pathwise unique solution (Theorem 3.3.1). Further, the solution Xt is approximated successively; setting Xs0 = x and  Xtn = Xsn−1 +

s

t

 α(Xrn−1 , r) dWr +

t s

β(Xrn−1 , r) dr,

n = 1, 2, . . . ,

Xtn converges to the solution Xt in Lp -sense, in view of Lemma 3.3.1. Since the law of Xtn is uniquely determined by coefficients α, β and x, the law of Xt is also determined uniquely from these coefficients and initial condition. For a bounded continuous function f , we set  Ps,t f (x) :=

Rd

Ps,t (x, dy)f (y) = E[f (Φs,t (x))].

(4.3)

Let {Fs,t , 0 ≤ s < t < ∞} be the two-sided filtration generated by the Wiener process Wt . Then Φt,u (y) is independent of Fs,t for any s < t < u. Therefore we have the semigroup property

4.1 Continuous Stochastic Flows, Diffusion Processes and Kolmogorov. . .

127

       Ps,u f (x) = E E f (Φt,u (Φs,t (x))) Fs,t = E E f (Φt,u (y)) y=Φ (x) s,t   = E Pt,u f (Φs,t (x)) = Ps,t Pt,u f (x), for s < t < u. We have further, limt↓s Ps,t f (x) = f (x), since Φs,t (x) → x as t ↓ s. Therefore, {Ps,t (x, E)} is a transition probability.   Now, for a given (x, t0 ) ∈ Rd × T, let us consider the stochastic process with parameter x, t0 given by Xtx,t0 = Φt0 ,t (x). In view of the flow property Φs,t ◦Φt0 ,s = Φt0 ,t , the equality Xtx,t0 = Φs.t ◦ Xsx,t0 holds a.s. for any t0 < s < t. Since Φs,t (x) and Ft0 ,s are independent, we have P (Xtx,t0 ∈ E|Ft0 ,s ) = Ps,t (Xsx,t0 , E)

a.s.

Therefore the stochastic process Xtx,t0 , t ∈ [t0 , ∞) is a continuous conservative Markov process on Rd with semigroup {Ps,t }. Further, if f is a bounded continuous function, Ps,t f (x) is also a bounded continuous function of x, since the flow Φs,t (x) is continuous in x a.s. Therefore, Xtx,t0 has the strong Markov property (Proposition 1.7.1). Then Xtx,t0 is a diffusion process of the initial state (x, t0 ) with semigroup {Ps,t }. Consequently, {Xtx,s } is a system of diffusion processes with semigroup {Ps,t }. A continuous function f on Rd is called slowly increasing if |f (x)|/(1 + |x|)m is a bounded function for some positive integer m. A C k -function f is called slowly increasing if for any |i| ≤ k, |∂ i f (x)|/(1 + |x|)m are bounded for some positive integer m. A C ∞ -function f is called slowly increasing if it is a slowly increasing C k -function for any k. We denote by O(Rd ) the set of all real slowly increasing C ∞ -functions on Rd . Let f (x) be a slowly increasing continuous function. Let T > 0. Since E[|Φs,t (x)|p ] ≤ cp (1 + |x|)p holds for any 0 ≤ s < t ≤ T , x ∈ Rd and p ≥ 2 (Theorem 3.4.1), the integral Ps,t f (x) = E[f (Φs,t (x))] is well defined for any s < t and x. Further, it is a slowly increasing function of x. Suppose further that f (x) is a slowly increasing C ∞ -function (or Cb∞ function, respectively). Since E[|∂ i Φs,t (x)|p ] is bounded with respect to x for any p ≥ 2 (Theorem 3.4.1), we can change the order of the derivative and expectation. In fact, E[f (Φs,t (x))] is a C ∞ -function of x and satisfies ∂ i E[f (Φs,t (x))] = E[∂ i (f ◦ Φs,t (x))] for any i. Therefore Ps,t f (x) is a slowly increasing C ∞ -function of x (or Cb∞ function of x, respectively) for each s < t. Hence {Ps,t } is the family of linear transformations on O(Rd ) (or on Cb∞ (Rd )), satisfying the semigroup property Ps,t Pt,u f = Ps,u f for any s < t < u. Associated with SDE (4.1), we  define differential operators (vector fields) ∂f Vk (t), k = 0, . . . , d  by Vk (t)f (x) = k Vki (x, t) ∂x and we define a second-order i partial differential operator A(t) with time parameter t by

128

4 Diffusions, Jump-Diffusions and Heat Equations d

1 A(t)f (x) = Vk (t)2 f (x) + V0 (t)f (x). 2

(4.4)

k=1

If f is a slowly increasing C ∞ -function (or Cb∞ -function), then both Vk (t)f and A(t)f (x) are slowly increasing C ∞ -functions (or Cb∞ -functions, respectively). Let ft,u (x), f (x, t) be C ∞ -functions of x with parameters t, u. Denote them by fλ (x), regarding {t, u}, t as parameter λ. Apply the differential operators Vk (t), A(t) to the function fλ (x). The functions Vk (t)fλ (x), A(t)fλ (x) are denoted by Vk (t)ft,u (x), A(t)ft,u (x), Vk (t)f (x, t), A(t)f (x, t), respectively. Theorem 4.1.1 Solutions of the continuous symmetric SDE (4.1) define a system of conservative diffusion process with semigroup {Ps,t }. If f is a slowly increasing C ∞ -function, Ps,t f (x) is also a slowly increasing C ∞ -function for any s < t. Further, it is continuously differentiable with respect to s, t for s < t and it satisfies the following forward and backward partial differential equations, respectively: ∂ Ps,t f (x) = Ps,t A(t)f (x), if t > s, ∂t ∂ Ps,t f (x) = −A(s)Ps,t f (x), if t > s. ∂s

(4.5) (4.6)

Equation (4.5) is called Kolmogorov’s forward equation and (4.6) is called Kolmogorov’s backward equation. Equation (4.6) shows that the function v(x, s) := Ps,t f (x), s < t is a solution of the backward heat equation ∂ v(x, s) = −A(s)v(x.s) ∂s with the final condition lims↑t v(x, s) = f (x). Proof Let f be a slowly increasing C ∞ -function. Then for any s < t and x, we have the forward differential calculus (Proposition 3.6.1): d   

f (Φs,t (x)) = f (x) +

k=1 s

t

 Vk (r)f (Φs,r (x)) dWrk +

t

A(r)f (Φs,r (x)) dr. s

(4.7) In equation (4.7), terms written in the stochastic integrals as dWrk are martingales with mean 0. Therefore, taking the expectation of each term of (4.7), we have  Ps,t f (x) = f (x) +

t

E[A(r)f (Φs,r (x))] dr s

 = f (x) +

t

Ps,r A(r)f (x) dr. s

(4.8)

4.2 Exponential Transformation and Backward Heat Equation

129

Therefore, the function Ps,t f (x) is continuously differentiable with respect to t. Differentiating both sides, we get the equality (4.5). We saw in Proposition 3.6.2 the rule of the backward differential calculus for the flow {Φs,t }; d   

f (Φs,t (x)) = f (x) +



t

Vk (r)(f

k=1 s

◦ Φr,t )(x) dWrk

+

t

A(r)(f ◦ Φr,t )(x) dr.

s

(4.9) Take the expectation of each term of (4.9). Since the second term of the right-hand side is a backward martingale, its expectation is 0. We shall consider the last term. We have   t   t  A(r)(f ◦ Φr,t )(x) dr = E A(r)(f ◦ Φr,t )(x) dr E s

s



  A(r)E f (Φr,t (x)) dr.

t

= s

In the last equality, we used ∂ i E[f (Φs,t (x))] = E[∂ i (f ◦ Φs,t (x))]. Then we get 

t

Ps,t f (x) = f (x) +

A(r)Pr,t f (x) dr.

(4.10)

s

Differentiate both sides with respect to s (< t) we get (4.6).

 

4.2 Exponential Transformation and Backward Heat Equation We are interested in partial differential equations (4.5) and (4.6) for more general types of operator. Let A(t) be the differential operator defined by (4.4). Let ck (x, t), k = 0, . . . , d  be Cb∞,1 -functions on Rd ×T. We will transform the operator A(t) by the functions c ≡ {c0 , . . . , cd  }. We define another differential operator Ac (t) by d

1 A (t)f = (Vk (t) + ck (t))2 f + (V0 (t) + c0 (t))f. 2 c

k=1

Here, (Vk (t) + ck (t)), k = 0, . . . , d  are linear operators defined by (Vk (t) + ck (t))f (x) =

 i

Vki (x, t)

∂f (x) + ck (x, t)f (x) ∂xi

(4.11)

130

4 Diffusions, Jump-Diffusions and Heat Equations

and (Vk (t) + ck (t))2 f (x) = (Vk (t) + ck (t)){(Vk (t) + ck (t))f }(x). Then Ac (t)f (x) is rewritten as 

Ac (t)f (x) = A(t)f (x) +

d 

ck (x, t)Vk (t)f (x) + c(x, ˜ t)f (x),

k=1

where d

 1  c(x, ˜ t) = c0 (x, t) + Vk (t)ck (x, t) + ck (x, t)2 . 2

(4.12)

k=1

We consider an exponential functional with coefficients c = {c0 , . . . , cd  }: Gs,t (x) =

Gcs,t (x)

= exp

d   k=0 s

t

 ck (Φs,r (x), r) ◦ dWrk .

(4.13)

Lemma 4.2.1 For any i, p ≥ 2 and T > 0, E[|∂ i Gs,t (x)|p ] is bounded with respect to 0 ≤ s < t ≤ T and x ∈ Rd . Proof Itô’s formula (Theorem 2.4.1) for the function f (x) = ex and Xt =  t Apply   k k s ck (r) ◦ dWr , where ck (r) := ck (Φs,r (x), r). Then we have d   

Gs,t = 1 +

t

k=0 s d   

= 1+

k=0 s

t

Gs,r ck (r) ◦ dWrk Gs,r ck (r) dWrk

d

1 + Gs,t ck (t), Wtk − Wsk s,t . 2 k=1

For the computation of the last term, note that the martingale part of the continuous semi-martingale Gs,t ck (t) is  k≥1 s

t

Gs,r Vk (r)ck (Φs,r , r) dWrk +

 k≥1 s

t

Gs,r ck (r)ck (r) dWrk

in view of Itô’s formula. Then we have for k = 1, . . . , d  Gs,t ck (t), Wtk − Wsk  =



t s

Gs,r {Vk (r)ck (Φs,r , r) + ck (Φs,r , r)2 } dr.

4.2 Exponential Transformation and Backward Heat Equation

131

Therefore, d   

Gs,t = 1 +

t

Gs,r ck (Φs,r , r) dWrk +

k=1 s

1 2



t

c(Φ ˜ s,r , r) dr.

s

Define ˆ r) = yc(Φ ˆ αx (y, s,r (x), r),

βx (y, ˆ r) = yˆ c(Φ ˜ s,r (x), r),

γx (x, ˆ r, z) = 0,

˜ r) is given by (4.12). These where c(x, r) = (c1 (x, r), . . . , cd  (x, r)) and c(x, functionals satisfy Conditions 1–3 of the master equation with parameter x. Further, Yt = Ytx,s := Gs,t (x) satisfies the equation  Ytx,s = 1 +

t s

 αx (Yrx,s , r) dWr +

t s

βx (Yrx,s , r) dr.

Then in view of Theorem 3.3.1, E[|Ytx,s |p ] is bounded with respect to x, s for any p ≥ 2. Since Ytx,s = Gs,t (x) satisfies the above equation, the derivative ∂Gs,t (x) = (∂x1 Gs,t (x), . . . , ∂xd Gs,t (x)) satisfies  t ∂Gs,t = (∂Gs,r c(Φs,r , r) + Gs,r ∂c(Φs,r , r)) dWr s



t

+

(∂Gs,r c(Φ ˜ s,r , r) + Gs,r ∂ c(Φ ˜ s,r , r)) dr.

s

˜ s,r (x), r) and ∂c(Φs,r (x), r), ∂ c(Φ ˜ s,r (x), r) are bounded Since c(Φs,r (x), r), c(Φ with respect to r and x (x is a parameter), coefficients of the equation satisfy Conditions 1–3 of the master equation. Therefore E[|∂Gs,t (x)|p ] should be bounded with respect to 0 ≤ s < t ≤ T and x ∈ Rd by Theorem 3.3.1. Repeating this argument inductively, we arrive at the assertion of the lemma.   Now, Yt = Gs,t satisfies the linear SDE: 

dYt =

d 

ck (Xt , t)Yt ◦ dWtk .

(4.14)

k=0

Let R+ be the open half line (0, ∞). We shall consider an SDE on the product space Rd × R+ for the process (Xt , Yt ) ∈ Rd × R+ defined by the pair of equations (4.1) and (4.14). Then the solution starting from (x, y) ∈ Rd × R+ at time s is given by (Φs,t (x), Gs,t (x)y). For any p ≥ 2, there exists cp > 0 such that E[|Φs,t (x)|p + |Gs,t (x)y|p ] ≤ cp (1 + |x| + |y|)p ,

∀0 ≤ s < t ≤ T .

132

4 Diffusions, Jump-Diffusions and Heat Equations

For a slowly increasing C ∞ -function f¯ on Rd × R+ , we can define the integral P¯s,t f¯(x, y) = E[f¯(Φs,t (x), Gs,t (x)y)] for any s, t, x and y. It is a slowly increasing C ∞ -function. We can apply the arguments of Sect. 4.1 to the diffusion process (Xt , Yt ) = (Φs,t (x), Gs,t (x)y). Then ¯ is written by its generator A(t) d

 ∂ 2 ¯  ∂ ¯ ¯ f¯ = 1 f. A(t) Vk (t) + ck (t)y f + V0 (t) + c0 (t)y 2 ∂y ∂y k=1

Now take f¯(x, y) = f (x)y, where f (x) is a slowly increasing C ∞ function on Rd . ¯ ¯ f¯(x, 1) holds. Apply it to the operator A(t). Then Ac (t)f (x) = A(t) d We will define another transition function on R and its transformation by c (x, E) := E[1E (Φs,t (x))Gcs,t (x)], Ps,t  c c Ps,t f (x) := Ps,t (x, dx  )f (x  ) = E[f (Φs,t (x))Gcs,t (x)].

(4.15)

It is the transformation of Ps,t f (x) by the exponential functional Gcs,t (x). Since {P¯s,t } satisfies the semigroup property P¯s,t P¯t,u = P¯s,u for any s < t < u, c } satisfies the semigroup property P c P c = P c . Further, for any bounded {Ps,t s,t t,u s,u c f converges to f as t → s. Hence {P c (x, E)} is a continuous function f , Ps,t s,t transition function. It is called the transition function of Φs,t (x) weighted by c or simply a weighted transition function. Since Kolmogorov’s forward and backward equations are valid for the diffusion process (Xt , Yt ) = (Φs,t (x), Gs,t (x)y) on Rd ×R+ , we have the following assertion in view of Theorem 4.1.1. c f (x) is also a Theorem 4.2.1 If f is a slowly increasing C ∞ -function of x, Ps,t ∞ slowly increasing C -function of x. Further, it is continuously differentiable with respect to s, t for s < t and satisfies the following forward and backward partial differential equations, respectively:

∂ c c P f (x) = Ps,t Ac (t)f (x), ∂t s,t ∂ c c P f (x) = −Ac (s)Ps,t f (x). ∂s s,t

(4.16) (4.17)

Equation (4.16) is called Kolmogorov’s forward equation and (4.17), Kolmogorov’s backward equation. As an application of the above theorem, we will consider the final value problem for a backward heat equation. Let 0 < t1 < ∞ be a fixed time, called the final time. Let f1 (x) be a slowly increasing continuous function on Rd . We want to find

4.2 Exponential Transformation and Backward Heat Equation

133

a function v(x, s) of C 2,1 -class defined on Rd × (0, t1 ) satisfying the following backward equation: ⎧ ⎨ ∂ v(x, s) = −Ac (s)v(x, s), ∀0 < s < t1 , ∂s ⎩ lim ∀x ∈ Rd . s↑t1 v(x, s) = f1 (x),

(4.18)

If the above function v(x, s) exists, it is called a solution of the final value problem for the backward heat equation associated with the operator Ac (t). Theorem 4.2.2 If f1 is a slowly increasing C ∞ -function, the final value problem for the backward heat equation (4.18) has a unique slowly increasing solution v(x, s). The solution is of C ∞,1 -class. It is represented by d    v(x, s) = E f1 (Φs,t1 (x)) exp

k=0 s

t1

ck (Φs,r (x), r) ◦ dWrk

 .

(4.19)

c f (x). Since Proof The function v(x, s) given by (4.19) coincides with Ps,t 1 1 c Ps,t1 f1 (x) satisfies (4.17), v should be a solution of equation (4.18). It is a C ∞ function of x, since v(x, s) = E[f1 (Φs,t1 (x))Gcs,t1 (x)] is infinitely differentiable t  c f (x) dr . with respect to x. Further, we have ∂xi v(x, s) = ∂xi f1 (x) + s 1 Ac (r)Pr,t 1 1 i Therefore ∂x v(x, s) is continuously differentiable with respect to s. We give the uniqueness of the solution. Let v  (x, s) be any slowly increasing solution of equations (4.18) of C 2,1 -class. Set v0 (x, s) = v(x, s) − v  (x, s). Then we have by Itô’s formula, d   

v0 (Φs,t , t)Gcs,t

= v0 (x, s)+

k=1 s

t

(Vk (r) + ck (r))v0 (Φs,r , r)Gcs,r dWrk

 t  ∂v0 (Φs,r , r) Gcs,r dr. Ac (r)v0 (Φs,r , r) + + ∂r s ∂ v0 (x, s) = 0 holds. Since v0 (x, s) is a solution of (4.18), Ac (s)v0 (x, s) + ∂s c Therefore Mt = v0 (Φs,t (x), t)Gs,t (x) is a local martingale. Further, since v0 is of polynomial growth, E[|Mt |p ] < ∞. Therefore Mt is actually a martingale. Then we have Mt = E[MT |Ft ] = 0, because MT = 0 holds a.s. Therefore v0 (Φs,t (x), t) = 0 holds for all x a.s. Let Ψˇ s,t (x) be the inverse map of Φs,t (x). Then we have v0 (Φs,t ◦ Ψs,t (x), t) = 0 a.s. for any x. Therefore we get v0 (x, t) = 0 for any x. Since this is valid for any t, v0 (x, t) should be identically 0.  

A semigroup {Ps,t } is said to be of C 2,1 -class if for any Cb2 -function f , Ps,t f (x) is a C 2,1 function of x, s for any t. The above theorem shows that for a given partial differential operator Ac (t), the semigroup of C 2,1 -class satisfying Kolmogorov’s

134

4 Diffusions, Jump-Diffusions and Heat Equations

backward equation exists uniquely. The semigroup is said to be generated by the operator Ac (t) and Ac (t) is called the generator of the semigroup. In particular, when c = 0, a system of diffusion processes {Xtx,s = Φs,t (x); (x, s) ∈ Rd ×T} with the semigroup {Ps,t } is called a diffusion process with the generator A(t)(= A0 (t)) or a system of diffusion processes generated by the operator A(t). Further, since the law of Φs,t (x) is uniquely determined by the operator A(t), {Φs,t } is called the stochastic flow associated with the operator A(t). Remark It is not obvious whether the semigroup satisfying Kolmogorov’s forward equation is unique or not. However, if the semigroup is defined by a diffusion process, is of C 2,1 -class and satisfies Kolmogorov’s forward equation, then it is unique. We will prove this fact in the case c = 0. Let Xt is the diffusion process with the semigroup Ps,t mentioned above. Then Ns = Ps,t f (Xs ) is a martingale, because for s < u < t, E[Nu |Fs ] = E[Pu,t f (Xu )|Fs ] = Ps,u Pu,t f (Xs ) = Ps,t f (Xs ) = Ns . Note that the operator A(t) is written as A(t)f (x) =

 ∂ 2f ∂f 1  ij α (x, t) (x) + β i (x, t) (x), 2 ∂xi ∂xj ∂xi i,j

(4.20)

i

where 

α (x, t) = ij

d 

d

j Vki (x, t)Vk (x, t),

k=1

1 β(x, t) = V0 (x, t) + Vk (t)Vk (x, t). 2 k=1

Set βri = β i (Xr , r) and βr = (βr1 , . . . , βrd ), Since f (Xt ) − martingale, both  Xt −

t t0



j

br dr and Xti Xt −

t

t0



j

Xri βr dr −

t t0

j

t t0

Xr βri dr −

A(r)f (Xr ) dr is a



t

α ij (Xr , r) dr

t0

are martingales. On the other hand, by Itô’s formula (Theorem 2.2.1), j

j

Xti Xt = Xti0 Xt0 +



t t0



j

Xri dXr +

t t0

j

Xr dXri + Xi , Xj t0 ,t .

Comparing these equations, we find that  Xi , Xj t0 ,t =

t

α ij (Xr , r) dr.

t0

Now, apply Itô’s formula to the function f (x, s) = Ps,t f (x) and x = Xs . Then we have

4.2 Exponential Transformation and Backward Heat Equation

135

 t  ∂ f (Xr , r) + A(r)f (Xr , r) dr + a local martingale. Nt = f (Xt0 , t0 ) + t0 ∂r t Therefore the integral t0 {· · · } dr is a continuous martingale, so that it should be 0 a.s. Therefore the integrand {· · · } is 0 a.s. Since this is valid for any Xt , we have ∂ { ∂r + A(r)}f (x, r) = 0. This shows that f (x, s) = Ps,t f satisfies Kolmogorov’s backward equation. Then the semigroup is unique by Theorem 4.2.1. Now, we will rewrite the exponential functional Gs,t (x) using the Itô integral. Setting ck (r) = ck (Φs,r (x), r), k = 0, . . . , d  and c˜ (r) = c(Φ ˜ s,r (x), r), we have d   

d   

ck (r) ◦ dWrk =

k=0

k=1 s d   

=

t

k=1 s

d

1  ck (t), Wtk s,t + 2

ck (r) dWrk +

 s

k=1

t

1 2

ck (r) dWrk −

 t d s k=1

ck (r)2 dr + (0)

t



t

c0 (r) dr c˜ (r) dr.

s (1)

Therefore, Gcs,t (x) can be decomposed as the product of Gs,t and Gs,t , where (0)

Gs,t = exp G(1) s,t = exp (0)



t

 c(Φ ˜ s,r , r) dr ,

s d  t  k=1 s

d

ck (Φs,r , r) dWrk − (0)

1 2



t

 ck (Φs,r , r)2 dr .

k=1 s

(0)

Set Ps,t f (x) = E[f (Φs,t (x))Gs,t (x)]. Then {Ps,t } is a semigroup and satisfies (0) Ps,t f (x) = f (x) +



t

s

(0) Ps,r (A(r) + c(r))f ˜ (x). (0)

˜ is added to Hence by the transformation Ps,t → Ps,t , the potential term c(t) the generator. The generator is changed from A(t) to the operator A(0) (t) = (1) A(t) + c(t). ˜ It is called the Feynman–Kac transformation. Next, Gs,t is a positive (1) martingale with mean 1. In fact, apply Proposition 2.2.3 to Zt ≡ Gs,t . Since d  t 2 ∞− , Z is a positive L∞− -martingale with t k=1 s ck (Φs,r (x), r) dr belongs to L (1) mean 1. Then we can define another probability measure P (1) by dP (1) = Gs,t dP . t With respect to P (1) , W˜ tk = Wtk − s ck (Φs,r , r) dr, k = 1, . . . , d  is a Wiener process by Girsanov’s theorem (Theorem 2.2.2). Then Xt = Φs,t (x) satisfies d   

Xti = xi +

k=0 s

t

d   

Vki (Xr , r) ◦ d W˜ rk +

k=1 s

t

ck (Xr , r)Vki (Xr , r) dr,

a.s. P (1) .

136

4 Diffusions, Jump-Diffusions and Heat Equations

Therefore, with respect to P (1) , Xt = Φs,t (x) is a diffusion process with the   generator A(1) (t)f = A(t)f + dk=1 ck (t)Vk (t)f. Hence by the transformation  (1) Ps,t → Ps,t , the drift term (first-order differential operator) k ck (t)Vk (t) is added to the generator. The generator is changed from A(t) to the operator A(1) (t). It is called the Girsanov transformation. Consequently, our exponential transformation coincides with the product of the Feynman–Kac transformation and the Girsanov transformation. We will call it the Feynman–Kac–Girsanov transformation. Equation (4.19) is called the Feynman–Kac–Girsanov formula. We are also interested in the initial value problem for a heat equation associated with the operator Ac (t). Let 0 ≤ t0 < ∞ be a fixed time, called the initial time. Let f0 (x) be a slowly increasing continuous function on Rd . We want to find a function u(x, t) of C 2,1 -class defined on Rd × (t0 , ∞) satisfying the equation ⎧ ⎨ ∂ u(x, t) = Ac (t)u(x, t), ∀t0 < t < ∞, ∂t ⎩ lim ∀x ∈ Rd . t↓t0 u(x, t) = f0 (x),

(4.21)

If such a function exists, it is called a solution of the initial value problem for the heat equation associated with the operator Ac (t). The problem can be solved in time homogeneous case, using the Feynman–Kac– Girsanov formula. Theorem 4.2.3 For the operator Ac (t), we assume that Vk (t), k = 1, . . . , d  and ck (t), k = 0, . . . , d  are time homogeneous (not depending on t). Then for any slowly increasing C ∞ -function f0 , the initial value problem for the heat equation (4.21) has a unique slowly increasing C ∞ -solution. It is given by the Feynman–Kac–Girsanov formula d  t    ck (Φt0 ,r (x)) ◦ dWrk . u(x, t) = E f0 (Φt0 ,t (x)) exp

(4.22)

k=0 t0

c depends on t − s only, Proof In the time homogeneous case, the semigroup Ps,t ∂ c ∂ c c c f. Pt−s f = − ∂t∂ Pt−s which we denote by Pt−s . Then we have ∂s Ps,t f = ∂s Therefore equation (4.17) is written as ∂t∂ Ptc f = Ac Ptc f. Consequently, u(x, t) = c f (x) is the solution of the heat equation (4.21). Further, u(x, t) is represented Pt−t 0 0 by (4.22).  

The above argument can be applied to complex-valued potentials √ ck , k = 0, . . . , d  . Suppose that these are pure imaginary, written as ck (x) = −1Θk (x). We define a complex-valued exponential functional by d  t  √   Gt0 ,t (x) = exp −1 Θk (Φt0 ,r (x)) ◦ dWrk k=0 t0

(4.23)

4.3 Backward Diffusions and Heat Equations

137

and define a semigroup by Ptc0 ,t f (x) = E[f (Φt0 ,t (x))Gt0 ,t (x)]. Then u(x, t) = Ptc0 ,t φ0 (x) is a solution of the heat equation associated with the operator AΘ defined by d

√ √ 1 A f = (Vk + −1Θk )2 f + (V0 + −1Θ0 )f. 2 Θ

k=1

If A is a Laplacian, i.e., Vk = ∂xk for k = 1, . . . , d  where d  = d, the operator AΘ is called a Schrodinger operator with magnetic field Θ = (Θ0 , . . . , Θd  ). (Matsumoto–Taniguchi [83]). The solution of the heat equation associated with the operator AΘ is represented by Feynman–Kac–Girsanov formula (4.22) with √ ck = −1Θk , k = 0, . . . , d. In the time-dependent case, discussions are complicated. It will be studied in the next section. Remark Backward heat equations are used in mathematical finance. Consider the equation on (0, ∞) × [0, t1 ] defined by  σ (s)2  ∂ ∂2 ∂ v(x, s) + x 2 2 + rx − r v(x, s) = 0, 0 < s < t1 , ∂s 2 ∂x ∂x (final condition) lim v(x, s) = (x − K) ∨ 0. s→t1

where σ (t) > 0 and r, K are positive constants, called the volatility, the interest rate and the exercise price, respectively. The equation is called the Black–Scholes partial differential equation. The solution v(x, s) is the pricing function of the call option of exercise price K. For details, see Karatzas–Shreve [56] and Lamberton–Lapeyre [72]. The function f1 (x) = (x − K) ∨ 0 is a continuous function but it is not smooth. Then it is not clear whether the possible solution v(x, s) given by (4.19) is smooth or not with respect x. This problem will be solved by constructing the fundamental solution for the Black–Scholes partial differential equation, which will be done in Sect. 6.3, using the Malliavin calculus.

4.3 Backward Diffusions and Heat Equations Let us consider a continuous backward symmetric SDE on Rd with coefficients Vk (x, t), k = 0, . . . , d  , which are C ∞,1 -functions. For a given x, t, suppose that there exists a d-dimensional continuous backward semi-martingale Xˇ s , s < t which satisfies

138

4 Diffusions, Jump-Diffusions and Heat Equations d   

Xˇ s = x +

k=0 s

t

Vk (Xˇ r , r) ◦ dWrk ,

(4.24)

where the stochastic integrals ◦dWrk , k = 1, 2, . . . , d  are backward symmetric integrals and ◦dWr0 = dr. Then Xˇ s is called a solution of the equation starting from x at time t. The equation has a unique solution, which we denote by Xˇ sx,t . Then, there exists a continuous backward stochastic flow of diffeomorphisms {Φˇ s,t } such that Xˇ sx,t = Φˇ s,t (x) holds for any x, t, as we studied in Chap. 3. We set Pˇs,t (x, E) = P (Φˇ s,t (x) ∈ E),

0 ≤ s < t < ∞, E ∈ B(Rd ).

Let {Fs,t , 0 ≤ s < t < ∞} be the two-sided filtration generated by the Wiener process Wt . Then Φˇ t,u (x) is independent of Fs,t for any s < t < u. Hence transformations Pˇs,t f (x) := f (y)Pˇs,t (x, dy) have the backward semigroup ˇ ˇ ˇ property Ps,u f (x) = Pt,u Ps,t f (x) for any s < t < u. The stochastic process Xˇ s = Xˇ sx,t , s ∈ [0, t] has the backward Markov property P (Xˇ s ∈ E|Ft,T ) = Pˇs,t (Xˇ t , E). It has the (backward) strong Markov property. Hence Xˇ sx,t is a backward diffusion process with the backward transition probabilities Pˇs,t (x, ·). The system of processes {Xˇ sx,t , (x, t) ∈ Rd ×T} is called a system of backward diffusion process with transition probabilities Pˇs,t (x, E). By Proposition 3.6.3, the backward flow {Φˇ s,t } satisfies the rules of the following backward and forward differential calculus, respectively: f ◦ Φˇ s,t = f +



t s



t

Vk (r)f ◦ Φˇ r,t dWrk

k=1 s t

=f +

d   

A(r)f ◦ Φˇ r,t dr +

d   

A(r)(f ◦ Φˇ s,r ) dr +

s

t

k=1 s

(4.25)

Vk (r)(f ◦ Φˇ s,r ) dWrk .

Expectations of the last terms of the above two equations are both equal to 0. Therefore, taking expectations for each terms of the above, we have Pˇs,t f = f +

 s

t

Pˇr,t A(r)f dr,

Pˇs,t f = f +



t

A(r)Pˇs,r f dr.

(4.26)

s

Consequently, the backward semigroup {Pˇs,t } is generated by the same differential operator A(t) given by (4.4). Consider a backward exponential functional with coefficients c = {c0 , . . . , cd  }:

4.3 Backward Diffusions and Heat Equations

ˇ cs,t (x) = exp ˇ s,t (x) = G G

d   k=0 s

139

t

 ck (Φˇ r,t (x), r) ◦ dWrk .

(4.27)

For any slowly increasing continuous function f on Rd , we can define the integral   c ˇ cs,t (x) f (x) := E f (Φˇ s,t (x))G Pˇs,t

(4.28)

c } satisfies the for any s, t and x. It is a slowly increasing C ∞ -function. Further, {Pˇs,t c Pˇ c f = Pˇ c f for any u > t > s. The following backward semigroup property Pˇt,u s,t s,u theorem corresponds to Theorem 4.2.1 for forward diffusion. c f (x) is also a Theorem 4.3.1 If f is a slowly increasing C ∞ -function of x, Pˇs,t ∞ slowly increasing C -function of x. Further, it is continuously differentiable with respect to s, t for s < t and satisfies the following backward and forward partial differential equations, respectively:

∂ ˇc c Ac (s)f (x), P f (x) = −Pˇs,t ∂s s,t ∂ ˇc c f (x). P f (x) = Ac (t)Pˇs,t ∂t s,t

(4.29) (4.30)

c f satisfies Proof We can show similarly to equations (4.26) that Pˇs,t c f =f + Pˇs,t

 s

t

c c A (r)f dr, Pˇr,t

c Pˇs,t f =f +

 s

t

c Ac (r)Pˇs,r f dr.

c f is differentiable with respect to t (> s) and s(< t). Differentiating Therefore Pˇs,t the first equation by s, we get (4.29) and differentiating the second equation by t, we get (4.30).   c } is generated by the differential Consequently, the backward semigroup {Pˇs,t operator Ac (t) given by (4.11). Equation (4.30) tells us that u(x, t) = Pˇtc0 ,t f0 (x) is a solution of the heat equation (4.21). Therefore we have the followings:

Theorem 4.3.2 If f0 is a slowly increasing C ∞ -function, the heat equation (4.21) has a unique slowly increasing solution u(x, t). The solution is a C ∞,1 -function of x, t. It is represented by the backward Feynman–Kac–Girsanov formula d  t    ˇ ck (Φˇ r,t (x), r) ◦ dWrk , u(x, t) = E f0 (Φt0 ,t (x)) exp

(4.31)

k=0 t0

where {Φˇ s,t } is the backward flow defined by a backward symmetric SDE with coefficients Vk (x, t), k = 0, . . . , d  .

140

4 Diffusions, Jump-Diffusions and Heat Equations

So far, we have discussed the existence of solutions for a heat equation with a smooth initial function f0 and for a backward heat equation with a smooth final function f1 . If the function f0 is not smooth, it is not easy to show that u(x, t) defined by (4.31) is a solution of the heat equation, since it is not clear whether the function u(x, t) may be smooth or not with respect to x. We will show in Chap. 6 that if the operator Ac (t) is ‘nondegenerate’, the function u(x, t) given by (4.31) is smooth and satisfies the heat equation. An approach to the problem is to construct the fundamental solution for the equation (4.21). Another method is to extend smooth functions f0 in formulas (4.31) to a generalized function f0 . It will be completed by using the Malliavin calculus, which will be studied in Chap. 5.

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion We will denote by C0∞ or C0∞ (Rd ) the set of all real-valued C ∞ -functions of compact supports. Let A(t) be the differential operator defined by (4.4). It is a linear map from C0∞ (Rd ) into itself. We shall study its dual (adjoint) operator A(t)∗ with respect to the Lebesgue measure dx. A linear map A(t)∗ ; C0∞ (Rd ) → C0∞ (Rd ) is called a dual of A(t) if it satisfies   A(t)f (x) · g(x) dx = f (x) · A(t)∗ g(x) dx (4.32) Rd

Rd

for any f, g ∈ C0∞ (Rd ). Proposition 4.4.1 The dual A(t)∗ is well defined and is given by d

1 (Vk (t) + div Vk (t))2 g − (V0 (t) + div V0 (t))g, A(t) g = 2 ∗

(4.33)

k=1

where div V (x, t) :=



∂V i i ∂xi (x, t)

Proof For a vector field V = parts: 



is a Cb∞,1 -function.

V i ∂x∂ i , we have a formula of the integration by

 Vf (x) · g(x) dx = −

 f (x) · V g(x) dx −

f (x) · div V (x)g(x) dx,

for f, g of C0∞ (Rd ). Therefore we have V ∗ g = −V g − div V g. Then the adjoint of     A(t) is given by A(t)∗ g = dk=1 12 (Vk (t)∗ )2 g + V0 (t)∗ g. The geometric meaning of terms div Vk (x, t) in the expression (4.33) is not clear. We want to define directly the dual semigroup of {Ps,t }, making use of the inverse

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion

141

of the stochastic flow {Φs,t }. It is interesting to know how the expression of the dual operator A(t)∗ is related to the geometric property of diffeomorphic maps Φs,t of the stochastic flow. Let {Φs,t } be the stochastic flow of diffeomorphisms generated by SDE (4.1) and let {Ps,t } be the semigroup defined by (4.3). Given 0 ≤ s < t < ∞, a linear operator ∗ ; C ∞ (Rd ) → C ∞ (Rd ) is called the dual of P , if the equality Ps,t s,t b b 

 Ps,t f (x) · g(x) dx =

Rd

Rd

∗ f (x) · Ps,t g(x) dx

∗ exists uniquely for any s, t holds for all f, g ∈ C0∞ (Rd ). We will show the dual Ps,t ∗ and the family {Ps,t } is a backward semigroup with the generator A(t)∗ . Let Ψˇ s,t ; Rd → Rd be the inverse map of the diffeomorphism Φs,t . Let ∇ Ψˇ s,t be the Jacobian matrix of the diffeomorphism Ψˇ s,t and let det ∇ Ψˇ s,t be the Jacobian (Jacobian determinant). We will first remark that it is positive a.s. for any s < t. If t is fixed, Ψˇ s,t is continuous in s a.s., since it satisfies the backward SDE with coefficients −Vk (x, t), k = 0, . . . , d  by Theorem 3.7.1. Then its Jacobian det ∇ Ψˇ s,t is also continuous with respect to s. Note that it does not take the value 0, since Ψˇ s,t is diffeomorphic for any s. Further, it holds that det ∇ Ψˇ t,t = 1, since Ψˇ t,t is the identity map. Consequently, det ∇ Ψˇ s,t should be positive for all s < t a.s. ∗ }. We need the formula of the Now we will define the dual semi-group {Ps,t d change of variables on R . Let f be an element of C0∞ and let φ be a diffeomorphic map from Rd onto itself. Then we have the formula of the change of variables;



 Rd

f (x) dx =

Rd

f (φ(x))| det ∇φ(x)| dx,

(4.34)

where ∇φ is the Jacobian matrix of the map φ; Rd → Rd and det ∇φ is the Jacobian of the map φ. Now take another g ∈ C0∞ and replace f by f ◦ φ · g and φ by φ −1 in the above formula. Then we obtain the formula   f (φ(x))g(x) dx = f (x)g(φ −1 (x))| det ∇φ −1 (x)| dx. Rd

Rd

Setting φ(x) = Φs,t (x) in the above formula, we have 

 Rd

f (Φs,t (x)) · g(x) dx =

Rd

f (x) · g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x) dx,

(4.35)

almost surely. Now applying Theorem 3.4.2 to the backward SDE, we find that for any p ≥ 2, E[|∇ Ψˇ s,t (x)|p ] are bounded with respect to x and 0 ≤ s < t ≤ T . Therefore we can define a bounded kernel by ∗ (x, E) := E[1E (Ψˇ s,t (x)) det ∇ Ψˇ s,t (x)], Ps,t

142

4 Diffusions, Jump-Diffusions and Heat Equations

and a linear transformation C0∞ (Rd ) → Cb∞ (Rd ) by ∗ Ps,t g(x)

 :=

Rd

∗ Ps,t (x, dy)g(y) = E[g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x)].

(4.36)

Taking the expectation of each side of (4.35), we have the formula of the duality: 

 Rd

Ps,t f (x) · g(x) dx =

Rd

∗ f (x) · Ps,t g(x) dx.

(4.37)

∗ of (4.36) is the dual of P with respect to the Lebesgue measure Consequently, Ps,t s,t dx. ∗ } satisfy the backward Since {Ps,t } satisfy Ps,u = Ps,t Pt,u , dual operators {Ps,t ∗ ∗ ∗ semigroup property Ps,u g = Pt,u Ps,t g for u > t > s (Sect. 1.7). Further, the dual semigroup and the dual operator A(t)∗ are related by ∗ Ps,t g(x)



t

= g(x) + 

s t

= g(x) + s

∗ Pr,t A(r)∗ g(x) dr ∗ A(r)∗ Ps,r g(x) dr

∗ }. for any g ∈ C0∞ (Rd ). Therefore, A(t)∗ is the generator of the dual semigroup {Ps,t ∗ We will study further the relation between the dual semigroup {Ps,t } and the inverse flow {Ψˇ s,t }. Recall that the generator of the diffusion defined by the stochastic flow {Φs,t } is given by (4.4). Recall further that Xˇ s = Ψˇ s,t (x) is the solution of the continuous backward SDE with coefficients −Vk (x, t), k = 0, . . . , d  starting from x at time t. Hence Xˇ s = Ψˇ s,t is a backward diffusion. We define its backward transition probabilities by

Pˇ s,t (x, E) := E[1E (Ψˇ s,t (x))]. It defines the backward semigroup {Pˇ s,t }. We saw in Theorem 3.7.1 that the backward flow {Ψˇ s,t } satisfies f (Ψˇ s,t (x)) = f (x) +

 s

t

d   

A(r)f (Ψˇ r,t (x)) dr −

k=1 s

t

Vk (r)f (Ψˇ r,t (x)) dWrk

 t d  t  =f (x)+ A(r)(f ◦ Ψˇ s,r )(x) dr − Vk (r)(f ◦ Ψˇ s,r )(x) dWrk , s

where

k=1 s

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion

A(t) =

143

1 Vk (t)2 f − V0 (t)f. 2

(4.38)

k≥1

Take expectations for each term of the above. Then the backward semigroup {Pˇ s,t } satisfies equations Pˇ s,t f = f (x)+



t

Pˇ r,t A(r)f dr,

Pˇ s,t f = f (x)+

s



t

A(r)Pˇ s,r f dr.

(4.39)

s

Therefore A(t) is the generator of the backward semigroup {Pˇ s,t }. We define a backward exponential functional with coefficients c associated with the inverse flow {Ψˇ s,t } by c Gˇs,t (x) = exp

d  

t

k=0 s

ck (Ψˇ r,t (x), r) ◦ dWrk

 (4.40)

−div V Lemma 4.4.1 Jacobian det ∇ Ψˇ s,t (x) is represented as Gˇs,t (x), where −div V = −(div V0 (x, t), . . . , div Vd  (x, t)).

Proof We will prove that det ∇ Ψˇ s,t (x) satisfies the following backward linear equation for all x a.s.: d   

det ∇ Ψˇ s,t (x) = 1 −

k=0 s

t

div Vk (Ψˇ r,t (x), r) det ∇ Ψˇ r,t (x) ◦ dWrk .

(4.41)

Let us recall the rule of the backward differential calculus (3.52). It is written as d   

f (Φs,t (x)) − f (x) =

k=0 s

t

Vk (r)(f ◦ Φr,t )(x) ◦ dWrk ,

using the backward symmetric integrals, where f is a function of C0∞ (Rd ). Since f (Φs,t (x)) dx = det ∇ Ψˇ s,t (x)f (x)dx holds by the formula of the change of variables, we have 

d   

Rd

f (x)(det ∇ Ψˇ s,t (x) − 1) dx =

d k=0 R



t s

 Vk (r)(f ◦ Φr,t )(x) ◦ dWrk dx.

We shall compute the right-hand side. Use the Fubini theorem (Proposition 2.4.2) for integrals ◦dWrk and dx. Then apply the formula of the integration by parts and the formula of the change of variables. Then we have for any k = 0, . . . , d  ,

144

4 Diffusions, Jump-Diffusions and Heat Equations



 Rd

=

t s

 Vk (r)(f ◦ Φr,t )(x) ◦ dWrk dx

 t  s

=− =−

Rd

 t  s

Rd

s

Rd

 t  

=−

 Vk (r)(f ◦ Φr,t )(x) dx ◦ dWrk

Rd

 divVk (x, r)f (Φr,t (x)) dx ◦ dWrk  f (x)divVk (Ψˇ r,t (x), r) det ∇ Ψˇ r,t (x) dx ◦ dWrk 

t

f (x) s

 divVk (Ψˇ r,t (x), r) det ∇ Ψˇ r,t (x) ◦ dWrk dx.

Since this is valid for any f ∈ C0∞ (Rd ), we get the equality d   

det ∇ Ψˇ s,t (x) − 1 = −

k=0 s

t

div Vk (Ψˇ r,t (x), r) det ∇ Ψˇ r,t (x) ◦ dWrk ,

a.e. x, almost surely. We know that each symmetric integral of the right-hand side is continuous in x (Proposition 2.4.3). Therefore the above equality holds for all x almost surely. This proves (4.41).  t Apply Theorem 3.10.2 for f (x) = ex and Xt = − k s div Vk (Ψˇ r,t , r) ◦ dWrk . −div V Then we find that Gˇs,t (x) satisfies the following backward linear SDE: d   

−div V Gˇs,t (x) = 1 −

k=0 s

t

−div V div Vk (Ψˇ r,t (x), r)Gˇr,t ◦ dWrk .

(4.42)

−div V . Then we The equation is of the same form as (4.41) replacing det ∇ Ψˇ s,t by Gˇs,t −div V ˇ ˇ have Gs,t (x) = det ∇ Ψs,t (x), by the uniqueness of the solution of the SDE.  

We transform the operator A(t) by c = −div V. Then we get the relation A(t)∗ g = A−div V (t)g

(4.43)

Theorem 4.4.1 Let {Ps,t } be the semigroup defined by the stochastic flow {Φs,t } ∗ } be its dual semigroup. generated by a continuous SDE (3.1) and let {Ps,t ∞ ∗ Then if g is a slowly increasing C -function, Ps,t g(x) is a slowly increasing C ∞ -function for any s < t. Further, the dual semigroup is obtained from the backward semigroup {Ps,t } defined by the inverse flow {Ψˇ s,t }, through the backward exponential transformation with coefficients −div V, i.e., for any s < t and g, it holds that ∗ −div V Ps,t g(x) = E[g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x)] = Pˇ s,t g(x).

Its generator is given by (4.43).

(4.44)

4.4 Dual Semigroup, Inverse Flow and Backward Diffusion

145

So far, we have studied the dual of the operator A(t) and the dual of the semic } which were defined group {Ps.t }. We will extend these to the dual of Ac (t) and {Ps,t c in Sect. 4.2. Let A (t) be the differential operator defined by (4.11). Then its dual (adjoint) Ac (t)∗ is given by Ac (t)∗ g=

d

1 (Vk (t) + ck (t) + div Vk (t))2 g − (V0 (t) + c0 (t) + div V0 (t))g 2 k=1

= Ac−div V (t)g.

(4.45)

c }. For any f, g ∈ C ∞ (Rd ), we Next we will define the dual of the semigroup {Ps,t 0 have the formula of the change of variables:

 Rd

f (Φs,t (x))Gs,t (x)g(x) dx 

=

Rd

f (x)Gs,t (Ψˇ s,t (x))g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x) dx,

almost surely. Since Gs,t (Ψˇ s,t (x)) = exp Gs,t (Ψˇ s,t (x)) det ∇ Ψˇ s,t (x) = exp



d   k=0 s

t

d t k ˇ k=0 s ck (Ψr,t (x)) ◦ dWr

(4.46)  , we have

 (ck − div Vk )(Ψˇ r,t (x), r) ◦ dWrk .

It is Lp -bounded for any p ≥ 2. Then d  t    c−div V ˇ ˇ Ps,t g = E g(Ψs,t ) exp (ck − div Vk )(Ψˇ r,t , r) ◦ dWrk

(4.47)

k=0 s

is well defined. We can take the expectation for both sides of (4.46). Then we get 

 c f (x)g(x) dx = Ps,t

c−div V f (x)Pˇ s,t g(x) dx.

(4.48)

c−div V c . Therefore Pˇ s,t is the dual of Ps,t c } be the semigroup defined by (4.15), which is an exponenTheorem 4.4.2 Let {Ps,t c } exists and it is tial transformation of the semigroup {Ps,t }. Then the dual of {Ps,t given by (4.47), where {Ψˇ s,t } is the inverse flow of {Φs,t }. Further, its generator is given by (4.45).

Finally, we will discuss briefly the dual of time homogeneous diffusions. We assume that coefficients Vk , k = 0, . . . , d  of the SDE do not depend on t. Then its generator A is given by Af = 12 k≥1 Vk2 +V0 . Let Pt∗ be the dual of Pt . Then these

146

4 Diffusions, Jump-Diffusions and Heat Equations

∗ for any s, t > 0. Hence the dual semigroup is again stationary. satisfy Ps∗ Pt∗ = Pt+s We can construct the dual diffusion as a forward process. Indeed, consider an SDE on Rd given by 

dXt = −

d 

Vk (Xt ) ◦ dWtk .

(4.49)

k=0

Let {Φs,t } be the stochastic flow generated by the above SDE. It is a diffusion  process with the generator A = 12 k≥1 Vk2 − V0 . In particular, if div Vk (x) = 0 holds for k = 0, . . . , d  , A∗ coincides with A. Hence the dual Pt∗ is a conservative semigroup defined by the forward diffusion Φs,t (x). Then Φ0,t (x) coincides with the dual process of Φ0,t (x) in Hunt’s potential theory for Markov processes. Note There are many studies of diffusion processes and their generators. For diffusion determined by SDE, we refer to Itô [50], Stroock–Varadhan [109], Dynkin [24], Ikeda–Watanabe [41] etc. Backward equations for diffusion determined by SDEs are studied in Freidlin–Wentzell [27] and Oksendal [90]. Our method using the rule of the backward differential calculus might be new. Feynman–Kac transformation of the diffusion process is widely known. Girsanov transformation (Theorem 2.2.1) is also widely used. See Ikeda–Watanabe [41] for the application to the problem of solving SDE. See Karatzas–Shreve [56] and Kunita [62] for the application to mathematical finance. Our exponential transformation is the product of the above two transformations. Its application to the Schrodinger operator is discussed in Matsumoto–Taniguchi [83]. Feynman–Kac formula for the solution of time-homogeneous heat equations is well known. However, we should remark that the formula should not hold for solutions of non-homogeneous (non-stationary) heat equation. In such a case we have to consider a similar formula for a backward diffusion. Extensions to the Feynman–Kac–Girsanov formula might be new. ∗ g(y) = The dual semigroup is defined usually as Ps,t Rd ps,t (x, y)g(x) dx, if the transition function Ps,t (x, dy) has a density function ps,t (x, y) with respect to the Lebesgue measure dy. However, in our definition, we do not assume the existence of the density ps,t (x, y).

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps Let us consider a stochastic differential equation with jumps on a Euclidean space. To avoid complicated notations, we assume that the Lévy measure ν has a weak drift. We shall consider the equation studied in Sect. 3.10. It is given by  t d  t  Vk (Xr , r) ◦ dWrk+ Xt =X0+ k=0 t0

t0

{φr,z (Xr− )−Xr− }N(dr dz),

|z|>0+

(4.50)

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps

147

where the last integral by the Poisson random measure is the improper integral. For diffusion and drift coefficients, we assume that these are Cb∞,1 -functions and for jump coefficients, we assume that g(x, t, z) ≡ φt,z (x) − x satisfies Condition (J.1) in Sect. 3.2. Let {Φs,t } be the stochastic flow of C ∞ -maps generated by equation (4.50). Then, similarly to the proof of Proposition 4.1.1, we can show that its law Ps,t (x, E) = P (Φs,t (x) ∈ E),

0 ≤ s < t < ∞, E ∈ B(Rd ).

is determined uniquely from SDE (4.50) and {Ps,t (x, E)} is a transition probability. Further, the solution Xt , t ≥ t0 is a conservative strong Markov process on Rd with the transition probability. The fact can be verified as shown in Sect. 4.1 for the diffusion process. It is called a jump-diffusion process. In particular, if there is no diffusion part and no drift part, i.e., Vk (x, t) ≡ 0, k = 0, . . . , d  , the solution Xt , t ≥ t0 is called a jump process. For a slowly increasing continuous function f (x) on Rd , we define Ps,t f (x) as in Sect. 4.1. Ps,t f (x) is again a slowly increasing continuous function for any s < t, since E[|Φs,t (x)|p ]1/p ≤ cp (1 + |x|) holds for any x. It satisfies the semigroup property Ps,t Pt,u f = Ps,u f for any s < t < u and slowly increasing continuous function f . Next, if f (x) is a slowly increasing C ∞ -function (or Cb∞ -function), we can change the order of the derivative and expectation and we have the equality ∂ i E[f (Φs,t (x))] = E[∂ i (f ◦ Φs,t (x))] for any i. Therefore Ps,t f (x) is a slowly increasing C ∞ -function of x (or Cb∞ -function) for each s < t. Now, consider the rule of the forward differential calculus (3.93) for the flow {Φs,t }. Take expectations for each term of (3.93). Since expectations of the second and fourth terms of the right-hand side are 0, we get 

t

Ps,t f (x) = f (x) +

Ps,r AJ (r)f (x) dr,

(4.51)

s

for any slowly increasing C ∞ -function f . Here AJ (t) is an integro-differential operator defined by  AJ (t)f = A(t)f +

{f ◦ φt,z −f }ν(dz),

|z|>0+

(4.52)

where A(t) is the differential operator defined by (4.4) and the last integral by the Lévy measure ν is the improper integral. If the Lévy measure has a strong drift, it coincides with the Lebesgue integral. Furthermore, using the rule of the backward differential calculus (3.95), we can show that Ps,t f satisfies  Ps,t f (x) = f (x) +

t

AJ (r)Pr,t f (x) dr, s

(4.53)

148

4 Diffusions, Jump-Diffusions and Heat Equations

for any slowly increasing C ∞ -function f . Therefore the semigroup {Ps,t } satisfies Kolmogorov’s forward and backward equations associated with the operator AJ (t) of (4.52). We will summarize the result. Theorem 4.5.1 The solution Xt , t ≥ t0 of (4.50) is a conservative jump-diffusion process. If f (x) is a slowly increasing C ∞ -function, Ps,t f (x) is a slowly increasing C ∞ -function of x. Further, it is continuously differentiable with respect to s, t for s < t and satisfies the Kolmogorov forward equation (4.5) and the Kolmogorov backward equation (4.6) associated with the operator AJ (t) given by (4.52). The integro-differential operator AJ (t) of (4.52) is called the generator of the jump-diffusion process Xt . Further the flow {Φs,t } is called the stochastic flow associated with the operator AJ (t). Let us next consider an exponential transformation of the jump-diffusion. We will define an exponential Wiener–Poisson functional Gs,t (x), which is an extension of the exponential Wiener functional defined by (4.13). Let ck (x, t), k = 0, . . . , d  be Cb∞,1 -functions on Rd × T and let dt,z (x) be a positive Cb∞,1,2 -function of (x, t, z) such that dt,0 (x) = 1 holds for any x, t. Then both dt,z (Φs,r (x)) − 1 and log dr,z (Φs,r (x)) belong to the class L0U defined by (3.83). Then in view of Proposition 3.10.1, improper integrals  t s

{dr,z (Φs,r (x)) − 1} drν(dz),

|z|>0+

 t log dr,z (Φs,r− (x))N (dr dz)

s

|z|>0+

etc. are well defined for any s < t and x, since the Lévy measure has a weak drift. For a given x and s < t, we define an exponential functional with coefficients d = (dt,z (x)) by Gds,t (x)

= exp

  t

 log dr,z (Φs,r− (x))N (dr dz) .

|z|>0+

s

(4.54)

c,d We set Gs,t (x) = Gs,t (x) = Gcs,t (x)Gds,t (x), where Gcs,t (x) is defined by (4.13) for the stochastic flow {Φs,t }.

Lemma 4.5.1 For any i, T > 0 and p ≥ 2, E[|∂ i Gs,t (x)|p ] is bounded with respect to 0 ≤ s < t ≤ T and x ∈ Rd . Proof We first show that Gs,t satisfies a linear SDE d   

Gs,t = 1 +

k=1 s

+

 t s

Rd0



t

Gs,r− ck (r) dWrk



t

+

Gs,r− c¯ (r) dr

s

˜ Gs,r− {dz (r) − 1}N(dr dz),

(4.55)

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps

149

where ck (r) = ck (Φs,r− (x), r), c¯ (r) = c(Φ ¯ s,r− (x), r), dz (r) = dr,z (Φs,r− (x)). Here, d

c(x, ¯ r) : = c0 (x, r) +

1 {Vk (r)ck (x, r) + ck (x, r)2 } 2 k=1

 +

{dr,z (x) − 1}ν(dz).

|z|>0+

(4.56)

For the proof, apply the rule of the forward differential calculus (3.87) (Theorem 3.10.1) to the function f (x) = ex and the Itô process Xt := log Gs,t , t > s. Then eXt − 1 is equal to d   

t

k=0 s d   

=

eXr ck (r) ◦ dWrk +

 t s

 X +log d  (r)  z e r− − eXr− N(dr dz)

|z|>0+

 t  t ˜ eXr− ck (r) dWrk+ eXr− c˜ (r) dr + eXr− {dz (r) − 1}N(dr dz).

t

k=1 s

s

s

|z|>0

This proves (4.55). We may regard Gs,t as a solution of the master equation (3.21). In fact, note that coefficients ci (x, r), c(x, ¯ r) and dr,z (x) − 1 are bounded functions. Then, regarding x as a parameter, we set c = (c1 , . . . , cd  ), αx (y, r) = yc(Φs,r− (x), r) and ¯ s,r− (x), r), βx (y, r) = y c(Φ

χx (y, r, z) = y(dr,z (Φs,r− (x), r) − 1).

These functionals satisfy Conditions 1–3 of the master equation and y = Gs,r (x) is the solution of the master equation with the above coefficients. Then we find that E[|Gs,t (x)|p ] is bounded with respect to x ∈ Rd , 0 ≤ s < t ≤ T for any p ≥ 2. Next, the derivative ∂Gs,t (x) satisfies  t ∂Gs,t = (∂Gs,r− c (r) + Gs,r− ∂c (r)) dWr s



t

+

(∂Gs,r− c¯ (r) + Gs,r− ∂ c¯ (r)) dr

s

 t   ˜ + dz), ∂Gs,r− (dz (r) − 1) + Gs,r− ∂(dz (r) − 1) N(dr s

since Gs,t satisfies (4.55). We will apply Theorem 3.3.1 again. Since E[|Gs,r− ∂c (r)|p ],

   E |Gs,r− |p |∂(dz (r) − 1)|p ν(dz)

150

4 Diffusions, Jump-Diffusions and Heat Equations

are bounded with respect to s < r and x (x is a parameter), E[|∂Gs,t (x)|p ] should be bounded with respect to s < t, x. Repeating this argument inductively, we arrive at the assertion of the lemma.   We can define   c,d c,d Ps,t f (x) = E f (Φs,t (x))Gs,t (x)

(4.57)

c,d for any slowly increasing continuous function f . Then {Ps,t } has the semigroup property. We define further an integro-differential operator with parameter c, d by

 AJc,d (t)f (x) = Ac (t)f (x) +

{dt,z (x)f (φt,z (x)) − f (x)}ν(dz)

|z|>0+

(4.58)

for slowly increasing C ∞ -function f , where Ac (t) is the differential operator defined by (4.11). If the Lévy measure has a strong drift, the last integral coincides with the Lebesgue integral. It is rewritten by a direct calculation as 



AJc,d (t)f = AJ (t)f +

d 

ck (t)Vk (t)f +

k=1

(dt,z −1){f ◦φt,z −f }ν(dz)+ c(t)f, ¯

|z|>0+

where c(x, ¯ t) is defined by (4.56). c,d f (x) is also a Theorem 4.5.2 If f is a slowly increasing C ∞ -function of x, Ps,t slowly increasing C ∞ -function of x. Further, it is continuously differentiable with respect to s, t for s < t and satisfies the following forward and backward integrodifferential equations, respectively:

∂ c,d c,d c,d P f (x) = Ps,t AJ (t)f (x), ∂t s,t ∂ c,d c,d P f (x) = −AJc,d (s)Ps,t f (x). ∂s s,t

(4.59) (4.60)

Proof The first assertion follows from Lemma 4.5.1, immediately. We prove the second assertion. Using a symmetric integral, we can show that equation (4.55) is rewritten as d   

Gs,t = 1+

k=0 s

t

Gs,r ck (r)◦dWrk +

 t s

Gs,r− {dz (r)−1}N(dr dz),

|z|>0+

(4.61)

similarly to the diffusion case (Sect. 4.2). The pair (Φs,t (x), Gs,t (x)y) may be regarded as a (d + 1)-dimensional jump-diffusion process determined by the pair of SDEs (4.50) and (4.61). Let f¯(x, y) be a slowly increasing C ∞ -function on Rd+1 . Then, using the rule of the forward differential calculus (3.93) for the pair process,

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps

151

we find that f¯(Φs,t (x), Gs,t (x)y) − f¯(x, y)  t A¯ J (r)f¯(Φs,r (x), Gs,r (x)y) dr + a martingale with mean 0. = a

Here A¯ J (t) is the operator given by 

d    ∂ 2 ¯ ∂ ¯ f (x, y) Vk (t) + ck (t)y A¯ J (t)f¯(x, y) = f (x, y) + V0 (t) + c0 (t)y ∂y ∂y k=1    + f¯(φt,z (x), ψt,z (x, y)) − f¯(x, y) ν(dz), |z|>0+

¯ ¯ ¯ where ψt,z (x, y) = dt,z (x)y. Hence t Ps,t f (x, y) := E[f (Φs,t (x), Gs,t (x)y)] ¯ ¯ ¯ ¯ ¯ ¯ satisfies Ps,t f (x, y) = f (x, y) + s Ps,r AJ (r)f (x, y) dr. Further, from the rule ¯ ¯ ¯ of t the backward differential calculus (3.95), we have Ps,t f (x, y) = f (x, y) + ¯ J (r)P¯r,t f¯(x, y) dr. Therefore A¯ J (t) is the generator of the semigroup {P¯s,t }. A s Now take f¯(x, y) = f (x)y, where f (x) is a slowly increasing smooth function on Rd . Apply it to the operator A¯ J (t). Then we have Ac,d (t)f (x) = A¯ J (t)f¯(x, 1). c,d Consequently, we can show that Ps,t f (x) satisfies the forward and backward equations (4.59) and (4.60), similarly to the case of diffusion discussed in Sect. 4.2.   t d  t  k  Set Xt = k=0 s ck (r)◦dWr + s |z|>0+ dz (r)N (dr dz). Using the Itô integral, t (0) (1) (2) (0) it is rewritten as Xt = Xt + Xt + Xt , where Xt = s c¯ (r) dr and d   

(1) Xt

=

(2)

=

Xt

k=1 s

 t s

t

ck (r) dWrk

d

1 − 2



k=1 s

log dz (r)N (dr dz) −

|z|>0+

t

ck (r)2 dr,

 t s

{dz (r) − 1} drν(dz).

|z|>0+

c,d Hence exponential functional Gs,t (x) = Gs,t (x) is decomposed as the product of (0)

(0)

(1)

(1)

three exponential functionals Gs,t (x) = eXt , Gs,t (x) = eXt (2)

(0)

(2)

and Gs,t (x) =

eXt . We saw in Sect. 4.2 that the transformation by Gs,t (x) is the Feynman– Kac transformation. By this transformation, the potential term c(t)f ¯ is added (1) to the operator AJ (t). Further, the transformation by  Gs,t (x) is the Girsanov transformation. By this transformation, the drift term ck Vk (t)f is added to (2) AJ (t). We shall consider the transformation by Gs,t (x). We denote it by Zt . Apply Theorem 3.10.1 by setting F (x) = ex . Then we get

152

4 Diffusions, Jump-Diffusions and Heat Equations

Zt = 1 −

 t

Zr− (dz (r) − 1) drν(dz) +

s

= 1+

 t

 t s

Zr− (dz (r) − 1)N (dr dz)

˜ dz). Zr− (dz (r) − 1)N(dr

s

(4.62) p

Hence Zt is a positive local martingale. Further, E[Zt ] < ∞ holds for any p ≥ 2. Then Zt is a positive martingale with mean 1. Then we can define another (2) probability measure P (2) by dP (2) = Gs,t (x) dP . Lemma 4.5.2 Set N˜ d (dr dz) = N(dr dz) − dz (r) drν(dz) and  t

Yt =

s

Rd0



ψ(r, z)N˜ d (dr dz),

(4.63)

where ψ(r, z) is a bounded predictable random field. Then Yt is a martingale with respect to P (2) . (2)

Proof Let Zt = eXt . If the product Zt Yt is a martingale with respect to the measure P , Yt is a martingale with respect to P (2) . In fact, if Zt Yt is a martingale with respect to P , we have E[Zt Yt ; B] = E[Zs Ys ; B] for any t > s and B ∈ Fs . Therefore, E (2) [Yt ; B] = E[Zt Yt ; B] = E[Zs Ys ; B] = E (2) [Ys ; B],

∀B ∈ Fs ,

showing that Yt is a martingale with respect to the measure P (2) . Apply Theorem 3.10.1 to the product of Zt and Yt . Then we have Zt Yt =

 t s

−

(Zr− + dz (r) − 1)(Yr− + ψ(r, z) − Zr− Yr− )N (dr dz)

 t s

=

 t s

{Yr− (dz (r) − 1) + Zr− ψ(r, z)} drν(dz)

˜ dz). (Yr− (dz (r) − 1) + Zr− ψ(r, z))N(dr  

Therefore Zt Yt is a martingale.

The solution Xt = Φs,t (x) of equation (4.50) starting from x at time s satisfies (3.93). We can rewrite it as d   

f (Φs,t ) = f +

t

k=1 s

+

 t s

|z|>0+

 Vk (r)f (Φs,r ) dWrk +

s

t

AdJ (r)f (Φs,r ) dr

{f ◦ φr,z ◦ Φs,r− − f ◦ Φs,r− }N˜ d (dr dz),

4.5 Jump-Diffusion and Heat Equation; Case of Smooth Jumps

153

where  AdJ (t)f = AJ (t)f +

(dt,z − 1)(f ◦ φt,z − f )ν(dz).

|z|>0+

(4.64)

(2)

Define the semigroup {Ps,t } by 

(2)

Ps,t f (x) =

(2)

f (Φs,t (x)) dP (2) = E[f (Φs,t (x))Gs,t (x)].

Then it satisfies (2)

Ps,t f (x) = f (x) +



t s

(2) d Ps,r AJ (r)f (x) dr. (2)

Therefore the generator of the semigroup {Ps,t } is AdJ (t)f . Consequently, by the (2) , the jump term (dt,z − 1){f ◦ φt,z − f }ν(dz) is added transformation Ps,t → Ps,t to AJ (t)f . It may be considered as a Girsanov transformation for a jump process. The final value problem for the backward heat equation associated with the integro-differential operator AJc,d (t) is defined in the same way as the problem associated with the differential operator Ac (t) defined by (4.18). Theorem 4.5.3 The final value problem for the backward heat equation associated with operator AJc,d (t) has a unique slowly increasing solution for any slowly increasing C ∞ -function f1 . Further, the solution v is represented by v(x, s) = c,d E[f1 (Φs,t1 (x))Gs,t (x)]. 1 The proof, which is carried out similarly to the proof of Theorem 4.2.2 will be omitted here. Consequently, for a given integro-differential operator AJc,d (t) of (4.58), there exists a unique semigroup of C 2,1 -class satisfying (4.59) and (4.60). c,d The operator AJc,d (t) is called the generator of the semigroup {Ps,t }. In particular, if c = 0 and d = 1, the operator AJc,d (t) coincides with the operator AJ (t). Then the semigroup {Ps,t } satisfying Kolmogorov’s backward equation associated with AJ (t) is unique. The semigroup is said to be generated by the operator AJ (t). The jumpdiffusion process with the semigroup {Ps,t } is called the jump-diffusion process with the generator AJ (t). The initial value problem for the heat equation associated with the integrodifferential operator AJc,d (t) is defined in the same way as the problem associated with the differential operator Ac (t) defined by (4.11). In order to get the solution of the initial value problem, we consider the backward SDE with characteristics (Vk (x, t), k = 0, . . . , d  , g(x, t, z), ν). It is written as

154

4 Diffusions, Jump-Diffusions and Heat Equations d   

Xˇ s = x +

t

k=0 s

Vk (Xˇ r , r) ◦ dWrk +

 t

g(Xˇ r , r, z)N(dr dz).

(4.65)

s

The solution exists uniquely, which we denote by Xˇ sx,t . There exists a backward stochastic flow {Φˇ s,t } of C ∞ -maps such that Xˇ sx,t = Φˇ s,t (x) holds a.s. for any t, s, x. Set Pˇs,t (x, E) = P (Φˇ s,t (x) ∈ E). It is the transition function of the backward Markov process Xˇ s . Further, similarly to the case of diffusion in Sect. 4.3, we can show that the backward semigroup {Pˇs,t } satisfies Pˇs,t f = f +



t

Pˇs,t f = f +

Pˇr,t AJ (r)f dr,



s

t

AJ (r)Pˇs,r f dr,

s

where AJ (t) is the integro-differential operator given by (4.52). Therefore, the generator of the backward semigroup {Pˇs,t } is the same as that of the forward semigroup {Ps.t }. Let c = (c0 (x, t), . . . , cd  (x, t)) and d = (dt,z ) be functions satisfying the same conditions as those for Gs,t of (4.54). We define backward exponential functionals by ˇ cs,t G

= exp

d   k=0 s

ˇ ds,t = exp G

  t s

t

ck (Φˇ r,t , r) ◦ dWrk ,  log dr,z (Φˇ r,t )N (dr dz) .

|z|>0+

(4.66)

c,d c,d c,d ˇ s,t ˇ cs,t (x)G ˇ ds,t (x). We set Pˇs,t ˇ s,t and G (x) = G f (x) = E[f (Φˇ s,t (x))G (x)]. Then c,d c,d ˇ {Ps,t } is a backward semigroup. Its generator is equal to AJ (t). Hence we have the following theorem.

Theorem 4.5.4 If f0 is a slowly increasing C ∞ -function, the initial value problem for the heat equation associated with the operator AJc,d (t) has a unique slowly increasing solution u(x, t), (x, t) ∈ Rd × (t0 , ∞). It is of C ∞,1 -class and is represented by   ˇ tc,d,t (x) , u(x, t) = E f0 (Φˇ t0 ,t (x))G 0

(4.67)

where {Φˇ s,t } is the right continuous backward stochastic flow generated by the symmetric backward SDE with characteristics (Vk (x, t), k = 0, . . . , d  , g(x, t, z), ν). In particular, if Vk (t), k = 1, . . . , d  and ck (t), k = 0, . . . , d  , dz (t) are time homogeneous (not depending on t), the solution is given by the Feynman–Kac– (x)], making use of the forward Girsanov formula u(x, t) = E[f0 (Φt0 ,t (x))Gtc,d 0 ,t stochastic flow {Φs,t } with characteristics (Vk , k = 0, . . . , d  , g(·, z), ν).

4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of. . .

155

4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of Diffeomorphic Jumps In this section, we assume that jump-maps φt,z ; Rd → Rd defined by φt,z (x) = g(x, t, z) + x are diffeomorphic i.e., they satisfy Condition (J.2) in Chap. 3. Then the stochastic flow {Φs,t } is a flow of diffeomorphisms by Theorem 3.9.1. Proposition 4.6.1 Let AJ (t) be an integro-differential operator given by (4.52). We assume that coefficients Vk (x, t), k = 0, . . . , d  are Cb∞,1 -functions and that jump coefficient g(x, t, z) = φt,z (x) − x satisfies Conditions (J.1) and (J.2) in Chap. 3. Then the dual operator AJ (t)∗ ; C0∞ (Rd ) → Cb∞ (Rd ) is well defined. It is represented by d

1 (Vk (t) + div Vk (t))2 g − (V0 (t) + div V0 (t))g AJ (t) g = 2 k=1    −1 −1 (det ∇φt,z + )g ◦ φt,z − g ν(dz), ∗

|z|>0+

(4.68)

−1 −1 is the inverse map of the diffeomrphic map φt,z and det ∇φt,z is its where φt,z Jacobian determinant.

Before we proceed to the proof of the proposition, let us check that the last improper integral of (4.68) is well defined and is a Cb∞ -function of x, if g ∈ −1 (x) is written as x −h(x, t, z), where h(x, t, z) has the same C0∞ (Rd ). Note that φt,z property as that of g(x, t, z). Indeed, we have ∇x h(x, t, z) = O(|z|) uniformly in x. Therefore, −1 det ∇φt,z (x) = det(I − ∇h(x.t, z)) = 1 + O(|z|) −1 uniformly in x. Then det ∇φt,z > 0 holds for sufficiently small |z|. Since it is continuous in z and does not take the value 0 (because of diffeomorphic maps), it should be positive for all z. Further, we get −1 −1 −1 −1 −1 det ∇φt,z g ◦ φt,z − g = (det ∇φt,z − 1)g ◦ φt,z + (g ◦ φt,z − g) = O(|z|).

uniformly in x. Then the improper integral of the above function with respect to the Lévy measure with weak drift is well defined, and in fact, a Cb∞ -function. Proof of Proposition 4.6.1 The operator AJ (t) is written as A(t) + A2 (t), where A2 (t)f = |z|>0+ {f ◦ φt,z −f }ν(dz). We have computed the dual of the operator A(t) in Sect. 4.4. So we will consider the operator A2 (t). We want to show that the dual of the operator A2 (t) is represented as

156

4 Diffusions, Jump-Diffusions and Heat Equations

A2 (t)∗ g =



 |z|>0+

 −1 −1 det ∇φt,z · g ◦ φt,z − g ν(dz).

(4.69)

Let φ be a diffeomorphic map on Rd such that det ∇φ(x) is a bounded function. Then from formula of change of variables (4.34), we have  f (φ 

−1

 (x)) dx =

f (φ −1 (x))g(x) dx =

f (x)| det ∇φ(x)| dx,  f (x)g(φ(x))| det ∇φ(x)| dx,

for any functions f, g of C0∞ (Rd ). Therefore,    =

|z|>0+

 {f (φt,z (x)) − f (x)}ν(dz) · g(x) dx

 

|z|>0+

 =

f (x) ·

  −1 −1 f (x)g(φt,z (x))| det ∇φt,z (x)| − f (x)g(x) dx ν(dz)

 |z|>0+

   −1 −1 | det ∇φt,z (x)|g(φt,z (x)) − g(x) ν(dz) dx.

−1 (x) is positive, A2 (t)f (x) · g(x) dx = f (x) · A2 (t)∗ g(x) dx Since det ∇φt,z holds. These computations yield the dual formula.   ∗ }, making use of the inverse flow. Now we will define the dual semigroup {Ps,t Discussions are similar to the diffusion case in Sect. 4.4. Let f, g ∈ C0∞ (Rd ) −1 . Then we have the formula (4.35). We saw in Sect. 3.9 that and Ψˇ s,t = Φs,t the inverse flow {Ψˇ s,t } is a solution of a backward SDE with characteristics −1 (−Vk (t), k = 0, . . . , d  , φt,z (x) − x, ν). Applying Theorem 3.38 to the backward SDE, we find that for any p > 1, E[|∂ Ψˇ s,t (x)|p ] are bounded with respect to x and 0 < s < t < T . Then E[| det ∇ Ψˇ s,t (x)|p ] is also bounded for any p ≥ 2. Therefore ∗ g(x) by (4.36). It is a function of C ∞ (Rd ) for g ∈ C0∞ (Rd ), we can define Ps,t b ∗ and satisfies (4.37). Therefore Ps,t defined by (4.36) is the dual of Ps,t . The dual ∗ } and the dual operator A (t)∗ are related by semigroup {Ps,t J ∗ Ps,t g(x) = g(x) +



t

s

 = g(x) +

s

t

∗ Pr,t AJ (r)∗ g(x) dr

(4.70)

∗ AJ (r)∗ Ps,r g(x) dr

for any g ∈ C0∞ (Rd ). Therefore, AJ (t)∗ is the generator of the dual semi-group ∗ }. {Ps,t

4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of. . .

157

For a fixed x, t1 , Xˇ s = Xˇ sx,t1 = Ψˇ s,t1 (x), s ∈ [0, t1 ] is a right continuous backward stochastic process satisfying the backward SDE (3.96). It is a backward Markov process associated with the backward transition probability Pˇ s,t (x, E) = P (Ψˇ s,t (x) ∈ E). Since Ψˇ s,t satisfies (3.99), the backward semigroup Pˇ s,t satisfies Pˇ s,t f = f +



t

Pˇ r,t AJ (r)f dr,

s

where  AJ (t)f = A(t)f +

|z|>0+

−1 {f ◦ φt,z − f }ν(dz),

(4.71)

where A(t) is the differential operator defined by (4.38). The dual operator AJ (t)∗ can be regarded as a transformation of the operator AJ (t). It is rewritten as 

∗

AJ (t) g = AJ (t)g +

div Vk (t)Vk (t)g

k=1

 +

d 

|z|>0+

−1 −1 (det ∇φt,z − 1){g ◦ φt,z − g}ν(dz) + c∗ (t)g,

(4.72)

where d

 1  c (t) = −div V0 (t) + div Vk (t)Vk (t)(div Vk (t)) + div Vk (t)2 2 k=1  −1 + (det ∇φt,z − 1)ν(dz). (4.73) ∗

|z|>0

We will next consider | det ∇ Ψˇ s,t (x)|. We set Jˇs,t (x) = det ∇ Ψˇ s,t (x). Lemma 4.6.1 |Jˇs,t | satisfies the following backward equation: d   

|Jˇs,t | = 1 −

k=0 s

+

 t s

t

div Vk (Ψˇ r,t , r)|Jˇr,t | ◦ dWrk 

|z|>0+

 −1 ˇ det ∇φr,z (Ψr,t ) − 1 |Jˇr,t |N(dr dz).

(4.74)

Proof For f ∈ C0∞ (Rd ), we have a rule of the backward differential calculus with respect to r:

158

4 Diffusions, Jump-Diffusions and Heat Equations d   

f (Φs,t (x)) − f (x) = +

k=0 s

 t |z|>0+

s

t

Vk (r)(f ◦ Φr,t )(x) ◦ dWrk

  f (Φr,t (φr,z (x))) − f (Φr,t (x)) N(dr dz).

See (3.94). In the proof of Lemma 4.4.1, we showed the equality   d  t   f (Φs,t (x)) − f (x) − Vk (r)(f ◦ Φr,t )(x) ◦ dWrk dx k=0 s



f (x) |Jˇs,t (x)| − 1 +

=

d   



k=0 s

t

 div Vk (Ψˇ r,t (x), r)|Jˇr,t (x)| ◦ dWrk dx.

Further, we have by the Fubini theorem and formula of change of variables,   t

 Rd

=

s

 t

Rd



|z|>0+

s

  f (Φr,t ◦ φr,z (x)) − f (Φr,t (x)) N(drdz) dx



 t

Rd

   −1 f (Φr,t (x))| det ∇(φr,z (x))| − f (Φr,t (x)) dx N(dr dz)  −1 ˇ f (x)(| det ∇φr,z (Ψr,t (x))| − 1)|Jˇr,t (x)| dx N(dr dz)

  t

 =

|z|>0+

|z|>0+

s

=



Rd

f (x) s

|z|>0+

 −1 ˇ (| det ∇φr,z (Ψr,t (x))| − 1)|Jˇr,t (x)|N(dr dz) dx.

Since these two equalities are valid for any f ∈ C0∞ (Rd ), we get the equality d   

|Jˇs,t | − 1 =

t

k=0 s

+

−div Vk (Ψˇ r,t , r)|Jˇr,t | ◦ dWrk

 t s

|z|>0+

  −1 ˇ | det ∇φr,z (Ψr,t )| − 1 |Jˇr,t |N(dr dz).

This proves the equality of the lemma.

 

Associated with the inverse flow {Ψˇ s,t }, we set d (x) = exp Gˇs,t

  t s

 log dr,z (Ψˇ r,t (x))N (dr dz) .

|z|>0+

c,d c (x)Gˇ d (x), where Gˇ c (x) is given by (4.40). and Gˇs,t (x) = Gs,t s,t s,t

(4.75)

4.6 Dual Semigroup, Inverse Flow and Backward Jump-Diffusion; Case of. . .

159

Proposition 4.6.2 Assume the same condition as in Proposition 4.6.1. Then −div V ,det ∇φ −1 det ∇ Ψˇ s,t is positive and is represented by det ∇ Ψˇ s,t = Gˇs,t , where −1 (x). −div V = (−div V0 (x, r), . . . , −div Vd  (x, r)) and det ∇φ −1 = det ∇φr,z

Proof Set d   

Yˇs = −

k=0 s

t

div Vk (Ψˇ r,t , r) ◦ dWrk +

  t

−1 ˇ log det ∇φr,z (Ψr,t )N (dr dz)

s |z|>0+

and apply the rule of the backward differential calculus (3.89) (Theorem 3.10.2) for the exponential function f (x) = ex . Then we get d   

e

Yˇs

= 1−

k=0 s

+

t

 t s

ˇ

eYr div Vk (Ψˇ r,t ) ◦ dWrk

|z|>0+

ˇ −1 ˇ eYr {det ∇φr,z (Ψr,t ) − 1}N(dr dz),

ˇ similarly to the proof of Lemma 4.5.1. Hence both eYs and |Jˇs,t | satisfy the same ˇ backward SDE (4.72). Then we have eYs = |Jˇs,t |, by the uniqueness of the solution of the SDE. It remains to show that Jˇs,t (y) is positive a.s. Instead of {Ψˇ s,t }, we will consider

}, > 0, which are obtained from {Ψ ˇ s,t } by cutting off all a family of flows {Ψˇ s,t −1

= Ψ 0 if ˇ s,t jumps φt,z , |z| < , as in the proof of Lemma 3.8.1. It is written as Ψˇ s,t t < τ1 and

0 = Ψˇ s,τ ◦ φτ−1 ◦ · · · ◦ φτ−1 ◦ Ψˇ τ0n ,t , Ψˇ s,t 1 n ,Sn 1 ,S1

if τn ≤ t < τn+1 ,

0 } is a continuous flow determined by a continuous backward SDE, where {Ψˇ s,t  and {τi , Si ; i = 1, 2, . . .} is a sequence of stopping times and Rd0 -valued random

satisfies det Ψ

= det Ψ 0 > 0 if t < τ and ˇ s,t ˇ s,t variables. Then Jacobian of Ψˇ s,t 1

0 det ∇ Ψˇ s,t = det ∇ Ψˇ s,τ det ∇φτ−1 · · · det ∇φτ−1 det ∇ Ψˇ τ0n ,t > 0, 1 n ,Sn 1 ,S1 0 , det ∇φ −1 etc. are all positive. Now a if τn ≤ t < τn+1 , since det ∇ Ψˇ s,τ τi ,Si i

n

n subsequence Ψˇ s,t (x) and ∇ Ψˇ s,t (x) converges to Ψˇ s,t (x) and ∇ Ψˇ s,t (x) in probability. Therefore det ∇ Ψˇ s,t (x) is nonnegative. Then it is strictly positive a.s.   ∗ g(x) is obtained from the By the above proposition, the dual semigroup Ps,t backward semigroup Pˇ s,t g(x) = E[g(Ψˇ s,t (x))] by the exponential transformation by (4.75). Then the operator AJ (t) is transformed to the operator AJ (t)∗ of (4.68). Consequently, we have the following theorem.

160

4 Diffusions, Jump-Diffusions and Heat Equations

Theorem 4.6.1 Assume the same condition as in Proposition 4.6.1. 1. Let {Ψˇ s,t } be the inverse flow of {Φs,t }. Then Xˇ s = Xˇ sx,t = Ψˇ s,t (x) is a backward Markov process with the backward semigroup {Pˇ s,t } having the generator AJ (t) defined by (4.71). −1 2. The Jacobian of Ψˇ s,t (x) is represented as G −divV,det ∇φ ∗ } is obtained by a backward exponential transformation 3. The dual semigroup {Ps,t of the Ψˇ s,t ;   −div V,det∇φ −1 ∗ Ps,t g(x) = E g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x) = Pˇ s,t g(x).

(4.76)

∗ g(x) is also a Further, if g is a slowly increasing C ∞ -function of x, then Ps,t slowly increasing C ∞ -function of x. 4. The generator of the dual semi-group is given by (4.68).

So far, we have studied the dual of the operator A(t) and the dual of the semic,d group {Ps.t }. We will extend these to duals of AJc,d (t) and Ps,t which were defined c,d in Sect. 4.5. Let AJ (t) be the integro-differential operator defined by (4.58). Then, by a direct computation, we find that its dual AJc,d (t)∗ is expressed by c−div V,d·det∇φ −1

AJc,d (t)∗ g = AJ

(t)g.

(4.77)

c,d,∗ c,d }. Let Gs,t (x) = Gs,t (x) be an We will define the dual semigroup {Ps,t ∞ exponential functional with coefficients c, d. For any C -functions of compact supports f, g, we have the formula of the change of variables:

 Rd

=

f (Φs,t (x))Gs,t (x)g(x) dx  Rd

f (x)Gs,t (Ψˇ s,t (x))g(Ψˇ s,t (x)) det ∇ Ψˇ s,t (x) dx,

(4.78)

c,d ˇ c,d almost surely. Note Gs,t (Ψs,t ) = Gˇs,t . Then E[|Gˇs,t (x)|p ] is also bounded for any p. Therefore,

  c,d,∗ c,d g(x) = E g(Ψˇ s,t (x))Gˇs,t (x) det ∇ Ψˇ s,t (x) Ps,t is well defined for any g ∈ C0∞ (Rd ). Taking expectations for both sides of (4.78), c,d,∗ c,d is the dual of Ps,t . Further, it is written as we find that Ps,t   c,d,∗ ˇ ∗s,t (x) , g(x) = E g(Ψˇ s,t (x))G Ps,t c,d ˇ ∗s,t (x) = Gˇs,t where G (x) det ∇ Ψˇ s,t (x).

(4.79)

4.7 Volume-Preserving Flows

161

c,d Theorem 4.6.2 Assume the same condition as in Proposition 4.6.1. Let {Ps,t } be the simigroup defined by (4.57), which is an exponential transformation of the c,d semigroup {Ps,t }. Then the dual of Ps,t exists and it is given by (4.79), where {Ψˇ s,t } c,d ∗ ˇ s,t = Gˇs,t is the inverse flow of {Φs,t } and G det ∇ Ψˇ s,t . Further, its generator is given by (4.77). −1 Remark The backward process Xˇ s = Ψˇ s,t (x) = Φs,t (x) has the left limit Xˇ s− = lim ↓s Xˇ s− . It is a modification of Xˇ s and has the backward strong Markov property. See Sect. 1.7.

Remark For the existence of the dual operator AJ (t)∗ and the dual semigroup ∗ , we assumed that jump-maps φ Ps,t t,z are diffeomorphic. If jump-maps are not diffeomorphic, it should be difficult to consider dual processes.

4.7 Volume-Preserving Flows In this section, we will discuss a different topic. We are interested how the volume of a set is changed through the transformation by stochastic flows. Let B be a Borel subset of Rd . We denote its volume (=Lebesgue measure) by |B|. Suppose that a map φ : Rd → Rd is diffeomorphic. It is called volumepreserving if the volume of the set φ −1 (B) = {φ −1 (x); x ∈ B} is equal to |B| for any Borel set B. Note that φ is volume-preserving if and only if φ −1 is volumepreserving. Next, φ is called volume-gaining (or volume-losing) if |φ −1 (B)| ≤ |B| (or |φ −1 (B)| ≥ |B|) holds for any Borel set B. Note that φ is volume-losing if and only if φ −1 is volume-gaining. By the formula of the change of the variable (4.34) we have the equality 

 Rd

f (φ(x)) dx =

Rd

f (x)| det ∇φ −1 (x)| dx.

For a bounded Borel subset B of Rd , we set f = 1B . It holds that 1B ◦ φ = 1φ −1 (B) . Therefore we have  | det ∇φ −1 (x)| dx, ∀B (Borel subset of Rd ). (4.80) |φ −1 (B)| = B

Hence φ is volume-preserving if and only if | det ∇φ −1 (x)| = 1 holds for any x. Further, φ is volume-gaining if and only if | det ∇φ −1 (x)| ≤ 1 holds for any x. Given a vector field V (x) on Rd of Cb∞ -class, let φt (x), t ≥ 0, x ∈ Rd be the t deterministic flow such that φ0 (x) = x and satisfies dφ dt (x) = V (φt (x)) for any t ∈ R. Then maps φt , t ≥ 0; Rd → Rd are diffeomorphic for all t. If maps φt are volume-preserving for all t ≥ 0, the flow φt is called volume-preserving. Volumelosing and volume-gaining flows are defined similarly. It is known that a flow φt is

162

4 Diffusions, Jump-Diffusions and Heat Equations

volume-preserving if and only if the divergence of the associated vector field V is identically 0, and further, the flow φt is volume-gaining if and only if divV (x) ≥ 0 for any x. We shall study the similar problem for stochastic flows. Let {Φs,t } be the stochastic flow generated by SDE (4.50) with jumps. It is called volume-preserving or volume-gaining if maps Φs,t ; Rd → Rd are volume-preserving a.s. for any s < t or maps Φs,t are volume-gaining a.s. for any s < t. Theorem 4.7.1 Assume the same condition as in Proposition 4.6.1. 1. The flow {Φs,t } is volume-preserving if and only if either one of the following holds: −1 (a) det ∇ Ψˇ s,t (x) = 1 holds a.s. for any s < t and x, where Ψˇ s,t = Φs,t .  (b) div V0 (x, t) = 0, div Vk (x, t) = 0, k = 1, . . . , d and det ∇φt,z (x) = 1 hold for all x, t, z. ∗ coincides with the (c) AJ (t)∗ = AJ (t), namely, the dual semigroup Ps,t semigroup of the inverse flow {Ψˇ s,t }.

2. The flow {Φs,t } is volume-gaining if and only if either one of the following holds: (a’) For any s < t and x det ∇ Ψˇ s,t (x) is less than or equal to 1 a.s. (b’) div V0 (x, t) ≥ 0, div Vk (x, t) = 0, k = 1, . . . , d  and det ∇φt,z (x) ≥ 1 holds for all x, t, z. 3. The flow {Φs,t } is volume-losing if and only if either one of the following holds: (a”) For any s < t and x, det ∇ Ψˇ s,t (x) is greater than or equal to 1 a.s. (b”) div V0 (x, t) ≤ 0, div Vk (x, t) = 0, k = 1, . . . , d  and det ∇φt,z (x) ≤ 1 holds for all x, t, z. Proof We give the proof of assertions 1 and 2. Assertion 3 can be verified similarly. Since maps Φs,t ; Rd → Rd are diffeomorphic, we have (4.80). Therefore the flow Φs,t is volume preserving (or gaining) if and only if | det ∇ Ψˇ s,t (x)| = 1 a.s. (or ≤ 1 a.s., respectively). Now, | det ∇ Ψˇ s,t (x)| satisfies the following backward equation: d   

| det ∇ Ψˇ s,t | − 1 = −

k=0 s

+

t

div Vk (Ψˇ r,t , r)| det ∇ Ψˇ r,t | ◦ dWrk

 t s

|z|>0+

−1 ˇ {| det ∇φr,z (Ψr,t )| − 1}| det ∇ Ψˇ r,t |N(dr dz).

See Lemma 4.6.1. If | det ∇ Ψˇ s,t (x)| = 1 (or ≤ 1) holds, the right-hand side (semimartingale) is identically 0 (or nonpositive, respectively). This implies that the martingale part is 0 and the process of bounded variation part is 0 (or nonpositive, −1 (φ (x)). Then (a) implies (b) respectively). Note that det ∇φr,z (x) = 1/ det ∇φr,z r,z and (a’) implies (b’). Conversely if (b) (or (b’)) holds, then | det ∇ Ψˇ s,t (x)| = 1 (or

4.7 Volume-Preserving Flows

163

≤ 1) a.s. for any s, t, x. Therefore we have proved (b) → (a) (or (b ) → (b )). Next, −div V,det∇φ −1

since AJ (t)∗ g = AJ

(t)g, the equivalence of (b) and (c) is obvious.

 

∗ = P ˇ s,t for any Corollary 4.7.1 If the flow is volume-preserving, we have Ps,t s < t. Further, consider time homogeneous continuous flow. If the flow is volume∗ = P preserving and V0 = 0, then Ps,t is self adjoint, i.e., Ps,t s,t holds for any s < t. ∗ } is conservative or We are interested in the case where the dual semigroup {Ps,t Markovian. The dual semigroup is conservative if and only if the Lebesgue measure dx is an invariant measure of the transition probabilities, i.e., Rd Ps,t (x, E) dx = |E| holds for any s < t. The dual semigroup is Markovian if and only if dx is an excessive measure of the transition probabilities, i.e., Rd Ps,t (x, E) dx ≤ |E| holds for any s < t. We will give their geometric characterization. The stochastic flow {Φs,t } is said to be volume-preserving in the mean or volume−1 −1 gaining in the mean, if E[|Φs,t (B)|] = |B| or E[|Φs,t (B)|] ≤ |B| holds for any t > s and Borel sets B, respectively.

Theorem 4.7.2 Assume the same condition as in Proposition 4.6.1. 1. The following assertions are equivalent: (a) The flow {Φs,t } is volume-preserving in the mean. (b) E[| det ∇ Ψˇ s,t (x)|] = 1 for any s < t and x. ∗ } is conservative, i.e., P ∗ 1(x) = 1 holds for any x, s < (c) Dual semigroup {Ps,t s,t t. (d) The function c∗ (t) defined by (4.73) satisfies c∗ (x, t) = 0 for any x, t. 2. The following assertions are equivalent: The flow {Φs,t } is volume-gaining in the mean. E[| det ∇ Ψˇ s,t (x)|] ≤ 1 for any s < t and x. ∗ } is Markovian, i.e., P ∗ 1(x) ≤ 1 holds for any x, s < t. Dual semigroup {Ps,t s,t ∗ The function c (t) defined by (4.73) satisfies c∗ (x, t) ≤ 0 for any x, t. Proof We have from (4.80) the equality |Ψˇ s,t (B)| = B | det ∇ Ψˇ s,t (y)| dy. Taking expectations for both sides, we have (a’) (b’) (c’) (d’)

E[|Ψˇ s,t (B)|] =



E[| det ∇ Ψˇ s,t (x)|] dx,

∀B.

(4.81)

B

Therefore, the flow is volume-preserving in the mean (or volume-gaining in the mean), if and only if E[| det ∇ Ψˇ s,t (x)|] = 1 (or ≤ 1, respectively) holds for any s < t and x. The equivalence of assertions (b), (c) and (d) (or (b’), (c’) and (d’), respectively) will be obvious.   ∗ } is nonconFinally, we will consider the case where the dual semigroup {Ps,t servative. Then the flow is not volume-preserving. We will study the speed of the volume-gaining (or volume-losing). It should be related to the potential function c∗ (x, t) of the dual semigroup defined by (4.73).

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4 Diffusions, Jump-Diffusions and Heat Equations

Theorem 4.7.3 For any Borel subset B of Rd with finite volume, we have ∃ lim

t→s

1 (E[|Ψˇ s,t (B)|] − |B|) = t −s



c∗ (x, s) dx.

(4.82)

B

Proof We have from (4.81)  ∗ 1(x) − 1 Ps,t E[| det ∇ Ψˇ s,t (x)|] − 1 dx = dx t −s t −s B B  t ∗  ∗ s Pr,t A(r) 1(x)dr = c∗ (x, s) dx, dx → t −s B B

E[|Ψˇ s,t (B)|] − |B| = t −s



as t → s.

  Rd

Rd ; c∗ (x, s)

For a given s ∈ T, define a subset of by C+ (s) = {x ∈ > 0}, C− (s) = {x ∈ Rd ; c∗ (x, s) < 0} and C0 (s) = {x ∈ Rd ; c∗ (x, s) = 0}. Then the above theorem tells us that if t − s is small, Ψˇ s,t (B) is volume-gaining (or volumelosing) in the mean if B ⊂ C− (s) (or B ⊂ C+ (s)). Further, it is volume-preserving in the mean if B ⊂ C0 (s). Note We discussed the volume-preserving problem with respect to the Lebesgue measure. It may be interesting to consider the problem with respect to other measures such as the Gaussian measure. We refer to Kunita–Oh [65].

4.8 Jump-Diffusion on Subdomain of Euclidean Space Let {Φs,t (x)} be the right continuous stochastic flow defined by SDE (4.50) and let c,d (x) be an exponential functional defined by (4.54). Let D be a bounded open Gs,t subset of Rd . Let τ (x, s) be the first leaving time of the process Xt = Xtx,s = Φs,t (x), t ∈ [s, ∞) from the set D. We shall consider the killed process at time τ (x, s). It is defined by Xtx,s = Φs,t (x) if t < τ (x, s) and by Xtx,s = ∞ if t ≥ τ (x, s). We define the weighted law of the killed process by  c,d c,d Qs,t (x, E) = E 1E (Xtx,s )Gs,t (x)]    c,d = E 1E (Φs,t (x))Gs,t (x) 1D (Φs,r (x)) ,

(4.83)

 where the product is taken for all rationals r such that s < r < t. We set c,d c,d Qs,t f (x) = D Qs,t (x, dy)f (y). Then it is written as    c,d c,d f (x) = E f (Φs,t (x))Gs,t (x) 1D (Φs,r (x)) . Qs,t

(4.84)

4.8 Jump-Diffusion on Subdomain of Euclidean Space

165

c,d c,d c,d c,d Proposition 4.8.1 {Qs,t } has the semigroup property: Qs,t Qt,u = Qs,u holds for c,d f is differentiable with respect any s < t < u. Further, for any f ∈ C0∞ (D), Qs,t to t and satisfies

∂ c,d c,d (AJc,d (t)f )(x) Q f (x) = Qs,t ∂t s,t

(4.85)

for any s < t and x ∈ D. Proof For any s < t < u, we have {τ (x, s) > u} = {τ (x, s) > t} ∩ {τ (Xt , t) > u}. c,d f (x) is equal to Then Qs,u c,d (x)1τ (x,s)>u ] E[f (Φs,u (x))Gs,u

   

c,d c,d =E E f (Φt,u (Φs,t (x)))Gt,u (Φs,t (x))Gs,t (x)1τ (Φs,t (x),t)>u Fs,t 1τ (x,s)>t

 c,d  c,d c,d c,d = E Qt,u f (Φs,t (x))Gs,t (x)1τ (x,s)>t = Qs,t Qt,u f (x). c,d Therefore Qs,t has the semigroup property. We have by Itô’s formula

 c,d f (Φs,t (x))Gs,t (x)

= f (x) + s

t

c,d AJc,d (r)f (Φs,r (x))Gs,r (x) dr + Mt ,

where Mt is a martingale with mean 0. Consider the stopped process of each term of the above by the stopping time τ = τ (x, s). Then we have  c,d f (Φs,t∧τ (x))Gs,t∧τ (x) = f (x) +

t∧τ

s t

 =f (x)+ s

c,d AJc,d (r)f (Φs,r (x))Gs,r (x) dr + Mt∧τ

c,d AJc,d (r)f (Φs,r (x))Gs,r (x)1τ >r dr + Mt∧τ .

It holds that f (Φs,t∧τ ) = f (Φs,t ) if t < τ and f (Φs,t∧τ ) = 0 if t ≥ τ . Therefore, taking the expectation of each term of the above equality, we get c,d E[f (Φs,t (x))Gs,t (x)1τ >t ]

= f (x) + E



t s

 c,d AJc,d (r)f (Φs,r (x))Gs,r (x)1τ >r dr ,

since Mt∧τ is a martingale with mean 0 by Doob’s optional sampling theorem (Theorem 1.5.1). Therefore we get the equality

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4 Diffusions, Jump-Diffusions and Heat Equations

 c,d Qs,t f (x)

t

= f (x) + s

c,d Qs,r (AJc,d (r)f )(x) dr.

 

This proves (4.85).

c,d (x, ·) by Qs,t (x, ·). Then the killed In the case c = 0 and d = 1, we denote Qs,t x,s process Xt is a strong Markov process with transition function Qs,t (x, ·). c,d f with respect to dx. We are interested in the dual of the above semigroup Qs,t We define    c,d,∗ c,d g(x) = E g(Ψˇ s,t (x))Gˇs,t (x) 1D (Ψˇ r,t (x))| det ∇ Ψˇ s,t (x)| , (4.86) Qs,t −1 where Ψˇ r,t (x) = Φr,t (x) and det ∇ Ψˇ s,t (x) is its Jacobian determinant. (We set dr,z = 1 if Xt is a diffusion.) It satisfies the backward semigroup property c,d,∗ c,d,∗ c,d,∗ Qs,t g = Qs,u g for any s < t < u. Qt,u c,d,∗ c,d Proposition 4.8.2 Qs,t is the dual of Qs,t ; It holds that



 D

c,d Qs,t f (x) · g(x) dx

= D

c,d,∗ f (x) · Qs,t g(x) dx

(4.87)

for any continuous function f, g supported by compact subsets of D. Further, it satisfies  c,d,∗ Qs,t g(x) = g(x) +

s

t

c,d,∗ Qr,t (AJc,d (r)∗ g)(x) dr,

(4.88)

where AJc,d (r)∗ is the dual operator of Ac,d (r). −1 Proof Since Φs,r ◦ Φs,t = Ψˇ r,t holds for 0 < r < t, we have

 D

c,d f (Φs,t (x))g(x)Gs,t (x)



= D



c,d f (x)g(Ψˇ s,t (x))Gˇs,t (x)

 1D (Φs,r (x)) dx



 1D (Ψˇ r,t (x)) | det ∇ Ψˇ s,t (x)| dx.

Taking the expectation for each term of the above, we get the formula (4.87). Equation (4.88) is proved similarly to the proof of (4.85).   c,d c,d c,d c,d The semigroup {Qs,t } satisfies ∂t∂ Qs,t f (x) = Qs,t AJ (t)f (Kolmogorov’s ∞ forward equation) for any C -function f satisfying f (x) = 0 on D c , in view c,d of (4.85). However, Kolmogorov’s backward equation for {Qs,t } is not evident, c,d since it is not clear whether Qs,t f (x) is a smooth function of x. In Sect. 6.10, we show that it is true if the jump-diffusion is ‘pseudo-elliptic’, making use of the Malliavin calculus.

Chapter 5

Malliavin Calculus

Abstract We will discuss the Malliavin calculus on the Wiener space and on the space of Poisson random measures, called the Poisson space. There are extensive works on the Malliavin calculus on the Wiener space. Also, there are some interesting works on the Malliavin calculus on the Poisson space. These two types of calculus have often been discussed separately, since the Wiener space and that of the Poisson space are quite different. Even so, we are interested in the Malliavin calculus on the product of these two spaces; we want to develop these two theories in a compatible way. In Sects. 5.1, 5.2, and 5.3, we study the Malliavin calculus on the Wiener space. We define ‘H -derivative’ operator Dt and its adjoint δ (Skorohod integral by Wiener process) on the Wiener space. Then, after introducing Sobolev norms for Wiener functionals, we get a useful estimate of the adjoint operator with respect to Sobolev norms (Theorem 5.2.1). In Sect. 5.3, we give Malliavin’s criterion for which the law of a given Wiener functional has a smooth density. It will be stated in terms of the Malliavin covariance. In Sects. 5.4, 5.5, 5.6, and 5.7, we study the Malliavin calculus on the space of Poisson random measure. Difference operator D˜ u and its adjoint δ˜ (Skorohod integral by Poisson random measure) are defined for Poisson functionals following Picard [92]. We will define a family of Sobolev norms conditioned to a family of star-shaped neighborhoods {A(ρ), 0 < ρ < 1}. Then a criterion for the smooth density of the law of a Poisson functional will be given using this family of Sobolev norms. In Sects. 5.8, 5.9, and 5.10, we study the Malliavin calculus on the product of the Wiener space and Poisson space. Sobolev norms for Wiener–Poisson functionals are studied in Sect. 5.9. In Sect. 5.10, we study criteria for the smooth density. Finally in Sect. 5.11, we discuss the composition of a ‘nondegenerate’ Wiener–Poisson functional and Schwartz’s distribution. Results of this chapter will be applied in the next chapter for getting the fundamental solution of heat equations discussed in Chap. 4.

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_5

167

168

5 Malliavin Calculus

5.1 Derivative and Its Adjoint on Wiener Space The Malliavin calculus on the Wiener space has been studied extensively since the 1980s. It is not our purpose to present the theory in full. We will restrict our attention to the problem of finding smooth density of the law of a given random variable. Let us define a Wiener space. In this chapter, it is convenient to take a finite  interval T = [0, T ] for the time parameter. Let d  be a positive integer. Let W = Wd0  be the set of all continuous maps w; T → Rd such that w(0) = 0. We denote by B(W) the smallest σ -field with respect to which w(t) are measurable for any t ∈ T.  We denote w(t) by Wt (w) = (Wt1 (w), . . . , Wtd (w)). A probability measure P on (W, B(W)) is called a Wiener measure if Wt is a d  -dimensional Wiener process. The triple (W, B(W), P ) is called a (classical) Wiener space. Let S be a complete metric space. An S-valued B(W)-measurable function F (w) (random variable) is called an S-valued Wiener functional. If S is the space of real numbers, it is called simply a Wiener functional. In our discussion of the Malliavin calculus, we will often study Lp -estimates of Wiener functionals for large p instead of L2 -estimates. Let p > 1. We denote by Lp , the set of all real Wiener functionals X such that E[|X|p ] < ∞. The norm of 1 X is defined by XLp = E[|X|p ] p . We set L∞− = p>1 Lp . It is a vector space of Wiener functionals, and is an algebra, i.e., if X, Y ∈ L∞− , then XY ∈ L∞− . Further, the set L∞− is an F -space with respect to a countable family of norms  n =  Ln , n ∈ N, where N is the set of all positive integers. Here a vector space L is called an F -space (a version of Frechet space) if it is equipped with a countable family of norms  n , n ∈ N and is a complete metric space by the metric d(X, Y ) =

∞  1 X − Y n . 2n 1 + X − Y n n=1

Hence a sequence {Xm } of L∞− converges to X ∈ L∞− with respect to the metric d if and only if Xm − Xn converges to 0 for any positive integer n. Note that if the family of norms are ordered in a different way,  n , n ∈ N, the corresponding metric d  defines the equivalent topology for L∞− . Let H be the totality of d  -dimensional real measurable functions h(t) =  (h1 (t), . . . , hd (t)) on T such that their L2 -norms |h|H := ( T |h(t)|2 dt)1/2 are finite. Then H is a real Hilbert space with the inner product f, g =

  d T

 f k (t)g k (t) dt.

k=1

The space H is called a Cameron–Martin space.

5.1 Derivative and Its Adjoint on Wiener Space

169

Given h ∈ H , we want to define a linear transformation Th of a Wiener functional F of L∞− by ¯ Th F (w) = F (w + h), t ¯ where h(t) := 0 h(s) ds, t ∈ T is an element of W. In order that it is well defined, the transformation should be absolutely continuous, i.e., if F (w) = 0 ¯ = 0 holds a.s. The fact can be verified by Girsanov’s a.s., then F (w + h) theorem (Theorem 2.2.2), to be shown below. We will show that Th is a continuous linear transformation from L∞− into itself. For h ∈ H , we set W (h) = d  k k k=1 T h (t) dWt . Then {W (h), h ∈ H } is a Gaussian system with mean 0 and covariance E[W (h)W (k)] = h, k. Now, set for t ∈ T, Zh (t) = exp

d   k=1 0

t

h

k

(s) dWsk

1 − 2



t

 |h(s)|2 ds .

0

It is a positive Lp -martingale with mean 1 for any p ≥ 2 (Proposition 2.2.3). Then   1 Zh = Zh (T ) = exp W (h) − h, h 2

(5.1)

is an element of L∞− . Define another probability measure Qh by dQh = Zh dP . ¯ is a Wiener process with Then by Girsanov’s theorem (Theorem 2.2.2), w(t) − h(t) ¯ with respect to P coincides with the respect to Qh . Therefore, the law of F (w + h) law of F with respect to Qh . Then we have 1



1

E[|Th F |p ] = EQh [|F |p ] = E[|F |p Zh ] ≤ E[F |pr ] r E[Zhr ] r  , 1

where r  is the conjugate of r. Consequently, we get Th F Lp ≤ Zh  p r  F Lpr , L showing that Th is a continuous linear transformation from Lpr to Lp . In particular, if F = 0 a.s., then Th F = 0 a.s., so that Th is an absolutely continuous transformation. Since this is valid for any r > 1 and p > 1, Th is a linear transformation from L∞− into itself and further, the transformation is continuous with respect to the metric d. Hence Th is a continuous linear transformation on the F -space L∞− . Let λ ∈ R. For a given h ∈ H , we consider a family of linear transformations {Tλh , λ ∈ R} on L∞− . These satisfy the group property Tλh Tλ h = T(λ+λ )h for any λ, λ ∈ R. We will define the derivative operator as the infinitesimal generator of the one-parameter group of linear transformations {Tλh }. Let F ∈ L∞− . Suppose that there exists an H -valued Wiener functional F  with |F  |H ∈ L∞− satisfying lim d

λ→0

T F − F  λh , F  , h = 0, λ

(5.2)

170

5 Malliavin Calculus

for any h ∈ H (0) , which is a dense subset of H . Then F is said to be H -differentiable and F  is called the H - derivative. H -derivative of a given F is at most unique. In the following, we denote it as DF . We denote by D the set of all H -differentiable Wiener functionals F of L∞− . (1) (d  ) The H -valued functional DF is often written as Dt F or (Dt F, . . . , Dt F ),  t ∈ T. It may be regarded as an Rd -valued stochastic process. However, we should note that Dt F may not be well defined for any t a.s. P , but it is well defined a.e. t, w with respect to the product measure dt dP . We will show differential rules for composite functionals. Proposition 5.1.1 Let F1 , . . . , Fd ∈ D. Let f (x1 , . . . , xd ) be a C 1 -function on Rd such that f and ∂xj f := ∂x∂ j f, j = 1, . . . , d are of polynomial growth. Then f (F1 , . . . , Fd ) belongs to D. Further, we have Df (F1 , . . . , Fd ) =

d 

∂xj f (F1 , . . . , Fd ) · DFj ,

a.s.

(5.3)

j =1

Proof We have by the mean value theorem,  1 f (Tλh F1 , . . . , Tλh Fd ) − f (F1 , . . . , Fd ) λ  Tλh Fj − Fj , ∂xj f (F1 + θ (Tλh F1 − F1 ), . . . , Fd + θ (Tλh Fd − Fd )) × = λ j

a.s., where |θ | ≤ 1. Let λ tend to 0. Since Fi are H -differentiable, Tλh Fj Tλh Fj −Fj converge to F converge to DFj , h in L∞− . Therefore, the above j and λ converges to j ∂xj f (F1 , . . . , Fd ) · DFj , h in L∞− . Therefore f (F1 , . . . , Fd ) is H -differentiable and the equality (5.3) holds.   For a positive integer i ≥ 2, we denote by H ⊗i the i-times tensor product of H . 1 It is again a Hilbert space with norm | |H ⊗i defined by |f |H ⊗i = ( Ti |f (t)|2 dt) 2 , where Ti is the i-times product of the set T and t = (t1 , . . . , ti ) ∈ Ti . Suppose that F is H -differentiable and that the functional DF, h1  is H -differentiable for any h1 ∈ H . If there is an H ⊗ H -valued functional D 2 F such that |D 2 F |H ⊗H ∈ L∞− and D 2 F, h1 ⊗ h2  = DDF, h1 , h2 ,

∀h1 , h2 ∈ H,

then we say that DF is H -differentiable and D 2 F is the second H -derivative. For i ≥ 3, we can define the i-th H -derivative D i F , successively. Suppose that the (i − 1)-th H -derivative D i−1 F is well defined as an H ⊗(i−1) -valued Wiener functional such that |D i−1 F |H ⊗(i−1) ∈ L∞− and D i−1 F, h1 ⊗ · · · ⊗ hi−1  is H -

5.1 Derivative and Its Adjoint on Wiener Space

171

differentiable. If there is an H ⊗i -valued functional D i F such that |D i F |H ⊗i ∈ L∞− and D i F, h1 ⊗ · · · ⊗ hi−1 ⊗ hi  = DD i−1 F, h1 ⊗ · · · ⊗ hi−1 , hi , for all h1 , . . . , hi ∈ H , then D i F is called the i-th H -derivative of F . Note that for any i, the i-th H -derivative of a given F is at most unique. D i F is an element of H ⊗i . It is written as Dt1 · · · Dti F , Dt F or Dti F , where t = (t1 , . . . , ti ) ∈ Ti . It is well defined a.e. with respect to the product measure dt dP . We may regard it as a (d  )i -dimensional random field with parameter t = (t1 , . . . , ti ) ∈ Ti . The totality of Wiener functionals F in L∞− , such that i-th H -derivative D i F l ∞ exists for any i ≤ l, is denoted by D and we set D = l Dl . Elements of Dl and those of D∞ are called l-times H -differentiable and infinitely H -differentiable, respectively. We will give examples of H -differentiable Wiener functionals. Let f ∈ H and   let F = W (f ) = dk=1 T f k (t) dWtk . It is a linear Wiener functional. We have Tλh F = W (f ) + λf, h a.s. Therefore F is H -differentiable and DF, h = f, h holds a.s. This proves Dt F = f (t) a.e. dt dP . Since it is a constant function, its higher-order derivatives are 0, i.e., Dti F = 0 holds if i ≥ 2. Let f1 , . . . , fd ∈ H and let p; Rd → R be a polynomial. A functional F given by p(W (f1 ), . . . , W (fd )) is called a polynomial Wiener functional or simply a polynomial. We denote by P the set of all polynomials. Note that any F ∈ P has a finite moment of any order, i.e., E[|F |p ] < ∞ for any p > 1. Therefore, P is a dense subset of L∞− . We will show that the domain D∞ contains the set of polynomials P and therefore it is a dense subset of L∞− . If F is a polynomial of the form F = p(W (f1 ), . . . , W (fd )), then as in Proposition 5.1.1, F is in D∞ and satisfies Dt F =



∂xj p(W (f1 ), . . . , W (fd ))fj (t),

a.e. dt dP .

(5.4)

j

Further, the polynomial F is i-times H -differentiable for any i and we have D t1 · · · D ti F =



∂xj1 · · · ∂xji p(W (f1 ), . . . , W (fd ))fj1 (t1 ) · · · fji (ti ),

j1 ,...,ji

a.e. dtdP . Hence P ⊂ D∞ . If F is a polynomial of degree d, then Dtd F is a constant function and Dtd+1 F = 0. Another interesting nonlinear Wiener functional, which is H -differentiable, is a solution of a continuous SDE. In Sect. 6.1, we will show the fact and will derive its derivative explicitly. Concerning the differential operator D, we have a formula of the integration by parts.

172

5 Malliavin Calculus

Proposition 5.1.2 If F ∈ D∞ , it holds that E[DF, h] = E[F W (h)]

(5.5)

for any h ∈ H . Further, if F, G ∈ D∞ , it holds that E[F DG, h] = E[−GDF, h + GF W (h)]

(5.6)

for any h ∈ H . Proof Let λ ∈ R and let Zλh be the functional defined by (5.1) for λh ∈ H . Then we have E[Tλh F ] = E[F Zλh ]. Differentiate both sides with respect to λ at λ = 0. d Since dλ E[Tλh F ]|λ=0 = E[DF, h] and d

Z

 −1 , W (h) → 0 λ

λh

as λ → 0, we get (5.5). Next, if F, G are in D∞ , then the product F G is in D∞ and satisfies D(F G) = DF · G + F · DG. Hence we get the equality (5.6).   We will define the adjoint operator of the derivative operator D. Let  Fl , l = 1, . . . , n be elements of D∞ and hl , l = 1, . . . , n beelements of H . Then l Fl hl is an H -valued random variable. We may regard Fl hl (t) as a d  -dimensional stochastic process with parameter t ∈ T, a.e. dt dP . It is called a simple functional with parameter t ∈ T. We denote by ST∞ the set of all simple functionals with parameter T. For Xt = nl=1 Fl hl (t) ∈ ST∞ , we set δ(X) =

n  l=1

Fl W (hl ) −

n 

DFl , hl .

(5.7)

l=1

Then δ is a linear map from ST∞ to L∞− . It is called the adjoint operator of D, or the Skorohod integral of Xt by the Wiener process. The followings are known as properties of Skorohod integrals. See Nualart [88]. Lemma 5.1.1 1. For any X = {Xt } ∈ ST∞ , δ(X) satisfies the adjoint property E[Gδ(X)] = E[DG, X]

(5.8)

for any G ∈ D∞ . 2. For any X = {Xt } ∈ ST∞ , δ(X) is H -differentiable and satisfies the following commutation relation: Dt  δ(X) = δ(Dt  X) + Xt  ,

a.e. dt  dP .

(5.9)

5.1 Derivative and Its Adjoint on Wiener Space

173

3. The isometric property: E[δ(X)δ(Y )] is equal to d d         (k  ) E Xtk Ytk dt + E Dt Xsk Ds(k ) Ytk ds dt k=1

T

T T

k,k  =1

(5.10)

for any X, Y ∈ ST∞ . Equality (5.10) is called the Shigekawa–Nualart–Pardoux energy identity (see [89, 101]).   Proof Since Gδ(X) = l GFl W (hl ) − l GDFl , hl , we have E[Gδ(X)] = E



GFl W (hl ) −



l

 GDFl , hl  .

l

Apply Proposition 5.1.2 to the above. Then the right-hand side is written as 

E[DG, hl Fl ] = E[DG, X].

l

Therefore we get (5.8). Further, δ(X) is H -differentiable, since each term of the right-hand side of (5.7) is H -differentiable. We have indeed,    Dt Fl W (hl ) + Fl hl (t) − Dt (DFl , hl ) Dt δ(X) = l

l

= δ(Dt X) + Xt ,

l

a.e. dt dP .

Therefore the equality of the commutation relation (5.9) holds. Suppose X, Y ∈ ST∞ . Then we have E[δ(X)δ(Y )] = E[X, Dδ(Y )] = E[X, Y ] + E[X, δ(DY )]         (k  ) = E Xtk Ytk dt + E Ds(k ) Xtk Dt Ysk ds dt . k

T

Here we used (5.8), (5.9) and (5.8) in turn.

k,k 

T T

 

Note The derivative operator D is a basic tool in stochastic analysis on the Wiener space. It has been discussed in various contexts. Cameron [15] introduced the derivative D on the Wiener space and showed a version of the adjoint formula (5.8). It was extended and applied to potential theory on the Wiener space by Gross [34, 35]. Malliavin [77] studied the derivative operator D through the Ornstein–Uhlenbeck operator on the Wiener space. Stroock [105] discussed the

174

5 Malliavin Calculus

finite dimensional approximation of the operator D. Ikeda–Watanabe [40, 41] and Watanabe [116] developed Malliavin’s theory by introducing norms of Sobolev-type through the Ornstein–Uhlenbeck operator. Our definition of the derivative D is close to Cameron [15], Gross [34, 35] and Shigekawa [100]. It can be applied directly to solutions F of stochastic differential equations driven by Wiener processes and further we can compute the derivatives DF explicitly. It will be discussed in Sect. 6.1. The absolute continuity of the transformation Th on the Wiener space was shown by Cameron–Martin [16]. It was extended to a wider class of transformations by Maruyama [82] and Girsanov [33], and is called the Girsanov transformation (Theorem 2.2).

5.2 Sobolev Norms for Wiener Functionals Let us define Sobolev norms of Lp -type for Wiener functionals, making use of derivative operator D. It is convenient to set T0 = H 0 = {∅} (empty set), ¯ be the set of all nonnegative D 0 F = D∅0 F = F and |D 0 F |H 0 = |F |. Let N ¯ and p ≥ 2, we will define a norm for F belonging to Dm integers. For m ∈ N p

1

1

by F 0,p = E[|D 0 F |H 0 ] p = E[|F |p ] p if m = 0 and F m,p := E

m 

|D

i

p F |H ⊗i

1

p



= E |F | + p

i=0

m   i=1

Ti

|Dti F |2 dt

p  1 2

p

,

(5.11)

¯ and if m ≥ 1. It is straightforward to prove that  m,p is a norm for any m ∈ N ¯ p ≥ 2} is called Sobolev norms for p ≥ 2. The system of norms { m,p , m ∈ N, m m,p Wiener functionals. We denote the completion of D by the above norm by D . We set D∞ = m,p Dm,p . Then D∞ ⊂ L∞− . It is an F -space with countable ¯ p = 2, 3, . . .}. Further, P ⊂ D∞ ⊂ D∞ . Elements of D∞ norms { m,p , m ∈ N, are called a smooth Wiener functional.  Next, let X = {Xt } be an element of ST∞ . It is written as X = l Fl hl ∞ and h ∈ H . Its H -derivatives are defined by (finite sum), where F ∈ D l l  D i X = l D i Fl hl . It is an H ⊗(i+1) -valued functional. Its norm is |D i X|H ⊗(i+1) =

 Ti+1

|



Dti Fl hl (t)|2 dt dt

1 2

.

l

¯ and p ≥ 2, we We will define Sobolev norms for X ∈ ST∞ as follows. For m ∈ N p

1

set X0,p = E[|X|H ] p if m = 0 and

5.2 Sobolev Norms for Wiener Functionals

Xm,p := E

m 

175

p

|D i X|H ⊗(i+1)

1

p

(5.12)

i=0

= E

 

|Xt | dt 2

T

p 2

+

m   i=1

Ti+1

|Dti Xt |2 dt dt

p  1 2

p

,

if m ≥ 1. If X ∈ ST∞ does not depend on t, denote it by F . Then the above norm √ Xm,p coincides with T F m,p . Hence these two norms are compatible with each other. For any X ∈ ST∞ , its norms Xm,p are finite for all m, p. The completion of m,p m,p ∞ ∞ ST by the above norm is denoted by DT . We set D∞ m,p DT . DT is an T = ¯ p = 2, 3, . . .. F -space with respect to countable norms  m,p , m ∈ N, Proposition 5.2.1 The derivative operator D is extended as a continuous linear operator from F-space D∞ to F-space D∞ T . Further, for any positive integer m and p ≥ 2, it holds that DF m,p ≤ F m+1,p for any F ∈ D∞ . Proof Let {hl } be a complete orthonormal system in H . If F ∈ D∞ , DF is ∞ expanded as l Gl hl , where Gl = DF, hl  and Gl ∈ D . Its finite sum n ∞ . Then its limit DF belongs to D∞ . Further, it satisfies G h belongs to S l=1 l l T T DF m−1,p ≤ F m,p for any m and p ≥ 2.   The extended DF is again called the H -derivative of F ∈ D∞ . ∞ Lemma 5.2.1 (Hölder’s inequality) If F, G ∈ D∞ and X ∈ D∞ T , then F G ∈ D ∞   ¯ and F X ∈ DT . Let m ∈ N, p ≥ 2 and r, r > 1 with 1/r + 1/r = 1. Then there exists a positive constant c such that

F Gm,p ≤ cF m,pr Gm,pr  , F Xm,p ≤ cF m,pr Xm,pr 

(5.13)

hold for any F, G, X with the above properties. Proof We give the proof of the latter inequality only. We first consider the case m = 0. We have by Hölder’s inequality p

p

1

pr 

1

p

p

F X0,p = E[|F X|H ] ≤ E[|F |pr ] r E[|X|H ] r  ≤ F 0,pr X0,pr  . We next consider the case m = 1. Since D(F X) = DF · X + F · DX, we have |D(F X)|H ⊗2 ≤ |DF |H |X|H + |F ||DX|H ⊗2 .

176

5 Malliavin Calculus

Therefore, using Hölder’s inequality, we get   1 p pr 1 pr  1 pr  1 E[|D(F X)|H ⊗2 ] ≤ 2p E[|DF |H ] r E[|X|H ] r  + E[|F |pr ] r E[|DX|H ⊗2 ] r    p p p p ≤ 2p F 1,pr X0,pr  + F 0,pr X1,pr  . Therefore we get (5.13) in the case m = 1. Repeating this argument inductively, we get (5.13) for any positive integer m.   We will consider the L2 -extension of the adjoint operator δ, making use of Sobolev norms. The adjoint operator δ is extended to a continuous linear operator 1,2 0,2 and satisfies the inequality δ(X) from D1,2 0,2 ≤ X1,2 for any X ∈ DT , T to D in view of the isometric property (5.10). δ(X) is called again the Skorohod integral of Xt (by Wiener process). We show that if Xt is a predictable process, the Skorohod integral coincides with the Itô integral. Proposition 5.2.2 Let {Ft } be the filtration generated by the Wiener process Wt and let Xt be a predictable functional with respect to the filtration {Ft }, belonging to D1,2 T . Then the Skorohod integral δ(X) coincides with the Itô integral  k dW k . Further, the isometric equality is written for short as X s s k T d    E[δ(X) ] = E (Xtk )2 dt . 2

k=1

T

(5.14)

d  n k Proof Suppose that Xt := m=1 Fm 1(tm−1 ,tm ] (t)ek is a simple process, k=1 k where Fm are real Ftm−1 -measurable for any m and ek , k = 1, . . . , d  are unit vectors   in Rd . Then the k-th component of Xt is Xtk = m Fmk 1(tm−1 ,tm ] (t). It holds that Dt Fmk = 0 for t > tm−1 . Therefore we have DFmk , 1(tm−1 .tm ]  = 0. Consequently, we have from the formula (5.7),   k k k δ(X) = Fm (Wtm − Wtm−1 ) = Xsk dWsk . k

m

k

T

The fact can be extended to any predictable element X belonging to D1,2 T . If Xs is predictable then we have Dt Xs = 0 if t ≥ s, as we have shown above.   Therefore we have Dt(k ) Xsk Ds(k ) Xtk = 0 a.e. ds dt for any k, k  . Consequently, the in (5.10) is 0, and we have the isometric property E[δ(X)2 ] =  last integral k 2   k E[ T (Xt ) dt]. If Xt is not predictable, the Skorohod integral is called the anticipating stochastic integral.

5.2 Sobolev Norms for Wiener Functionals

177

Remark We shall consider the anticipating stochastic differential equation, or Skorohod SDE. Let Wt be a one-dimensional Wiener process. We denote the t anticipating stochastic integral of Xs 1(0,t] (s) by 0 Xs δWs . We consider stochastic differential equations represented by anticipating stochastic integrals;  Xt = X0 +

t

 σ (Xs )δWs +

0

t

b(Xs ) ds, 0

where X0 ∈ D1,2 and σ, b are smooth functions on R and Xt ∈ D1,2 T . The equation is called the Skorohod equation. If we replace the above Skorohod integral by the t Itô integral 0 σ (Xs ) dWs in the above equation, the equation coincides with the Itô equation discussed in Chap. 3. We are interested in the existence and the uniqueness of the solution of the Skorohod equation. If X0 is F0 -measurable, the Itô equation has a unique solution. Further, it is also the solution of the Skorohod t equation, since tthe solution Xt of the Itô equation is predictable and hence 0 σ (Xs )δWs = 0 σ (Xs ) dWs holds (Proposition 5.2.2). Then the Skorohod equation also has a unique solution if X0 is F0 -measurable. If X0 is not F0 -measurable, Itô equation has no solution (not well defined). However, the Skorohod equation has a solution. Indeed, let Φ0,t (x) be the stochastic flow generated by the Itô equation. Then Φ0,t (x) ∈ D1,2 T for any x (see Sect. 6.1) and the composite process Xt = Φ0,t (X0 ) belongs to D1,2 T . Hence t the Skorohod integral 0 σ (Xs )δWs is well defined and Xt satisfies the Skorohod equation. The uniqueness of the solution of the Skorohod equation appears to be a difficult problem. For details, see Nualart [88]. By the isometric property of the Skorohod integral δ(X), we have the inequality δ(X)0,2 ≤ X1,2 . The inequality can be extended to higher derivatives. ¯ there exists a positive constant cm such that the Proposition 5.2.3 For any m ∈ N, inequality δ(X)m,2 ≤ cm Xm+1,2 holds for any X ∈ D∞ T . Proof To avoid complicated notations, we will consider the case of a onedimensional Wiener space (d  = 1), where T is equal to the interval [0, 1]. We will prove the inequality of the lemma for X ∈ ST∞ . Let i be a positive integer. Since Ds δ(X) = Xs + δ(Ds X) holds, we have by induction Dsi δ(X)

=

δ(Dsi X) +

i 

Dsi−1 Xsh , h

(5.15)

h=1

where s = (s1 , . . . , si ) ∈ Ti and sh = s − {sh }. We will compute L2 (ds dP )-norm of each term of the above. Using the adjoint formula (5.8) and the commutation relation (5.9), we have

178

5 Malliavin Calculus

 E = = =

Ti

  δ(Dsi X)2 ds = E[δ(Dsi X)2 ] ds Ti

  Ti

T

Ti

T

 

 E[Dt δ(Dsi X)Dsi Xt ] dt ds  {E[δ(Dt Dsi X)Dsi Xt ] + E[|Dsi Xt |2 ]} dt ds

   Ti

T T

E[Dt Dsi Xt 

·D

t

Dsi Xt ] dt  dt



  ds +

Ti

T

E[|Dsi Xt |2 ] dt ds

≤ X2i+1,2 . Further, we have  E

Ti

 2 ≤ X2i−1,2 . |Dsi−1 X | ds ds s h i h h

Summing up these inequalities for i = 1, . . . , m, we get the inequality of the proposition.   We will extend the inequality for arbitrary Sobolev norms. The following is an analogue of the Burkholder–Davis–Gundy inequality (Proposition 2.2.2) for Skorohod integral. ¯ and a Theorem 5.2.1 Let δ be the Skorohod integral operator. For any m ∈ N positive even number p, there exists a positive constant cm,p such that δ(X)m,p ≤ cm,p Xm+p−1,p

(5.16)

holds for any X ∈ D∞ T . Proof It is sufficient to prove the inequality for X ∈ ST∞ , since ST∞ is dense in D∞ T . We will consider the case of a one-dimensional Wiener space (d  = 1), where T is equal to the interval [0, 1]. Let p ≥ 2 be a positive even number. Let X ∈ ST∞ . Then δ(X) ∈ L∞− . By the adjoint property of δ (Lemma 5.1.1), we get E[δ(X) ] = E[δ(X)δ(X) p

p−1

]=E

 T

 Xt1 Dt1 (δ(X)p−1 ) dt1 .

We will decrease the power p − 1 of δ(X)p−1 , step by step. By the commutation relation for δ, we have Dt1 (δ(X)p−1 ) = (p − 1)δ(X)p−2 (Xt1 + δ(Dt1 X)). Therefore, the above is equal to

5.2 Sobolev Norms for Wiener Functionals

(p − 1)E

179

      Xt21 dt1 δ(X)p−2 + (p − 1)E Xt1 δ(Dt1 X) dt1 δ(X)p−2 .

  T

T

Apply Lemma 5.1.1 again to the last term. Then it is written as (p − 1)E

  T T

= (p − 1)E

Dt1 Xt2 Dt2 (Xt1 δ(X)p−2 ) dt1 dt2

  T T



 Dt1 Xt2 Dt2 Xt1 δ(X)p−2

  + (p − 2)Xt1 Dt2 δ(X) · δ(X)p−3 dt1 dt2    Dt1 Xt2 Dt2 Xt1 dt1 dt2 δ(X)p−2 = (p − 1)E T T

+ (p − 1)(p − 2)E + (p − 1)(p − 2)E

  T T

 

T T

Dt1 Xt2 Xt1 Xt2 dt1 dt2 δ(X)p−3



 Dt1 Xt2 Xt1 δ(Dt2 X) dt1 dt2 δ(X)p−3 .

Further, if p ≥ 3 the last term is rewritten as (p − 1)(p − 2)E

   T T T

 Dt3 (Dt1 Xt2 Xt1 δ(X)p−3 )Dt2 Xt3 dt1 dt2 dt3 .

Repeating this argument, we find that E[δ(X)p ] is equal to sums of the following terms:   p j i j ≤ q ≤ p. Dt11 Xti11 · · · Dtqq Xtqq dt , E q 2 T Here il are nonnegative integers satisfying il = 1, 2 and i1 + · · · + iq = p; jl are nonnegative integers satisfying 0 ≤ jl ≤ p − 1 and j1 + · · · + jq ≤ p. Further, t = (t1 , . . . , tp ) and tl are subsets of t \ {tl }. If jl = 0, tl is an empty set and it holds j that Dtll Xtill = Xtill . Then, by Hölder’s inequality, the above integral is dominated by  



E

 T jl T

l;il =0,jl =0

j

2

|Dtll Xtill | il dtl dtl

 p  il 2

p

 



×

E

l;il =0,jl =0

2

T

|Xtill | il dtl

 p  il 2

p

.

(5.17) It holds that for il = 1, 2 and jl = 0,   E

T jl

 T

2 j |Dtll Xtill | il

 p  il dtl dtl

2

p

≤E

  T jl

j |Dtll X|2H

 p  il dtl

l ≤ Xijll ,p ≤ Xip−1,p .

2

p

180

5 Malliavin Calculus

Further, we have   E

 p  il

2

T

|Xtill | il

2

dtl

p

l l ≤ Xi0,p ≤ Xip−1,p .

 p l = Xp−1,p . Consequently, there Then (5.17) is dominated by l;il =0 Xip−1,p p exists a positive constant cp such that E[δ(X)p ] ≤ cp Xp−1,p . This proves the inequality δ(X)0,p ≤ c0,p Xp−1,p . We will next consider the case where m ≥ 1 and p is a positive even number. By the definition of Sobolev norms, we have p

δ(X)m,p = E[δ(X)p ] +

m   p   2 . E |Dsi δ(X)|2 ds

(5.18)

Ti

i=1

The first term of the right-hand side has been computed. We shall consider other terms. Note the equality (5.15). Then there is a positive constant cp such that   E

Ti

|Dsi δ(X)|2 ds

p  2

(5.19)

i    p     2 i 2 ≤ cp E + |δ(Ds X)| ds E Ti

Ti−1 ×T

h=1

|Dsi−1 Xsh |2 dsh dsh h

 p  2

.

We have    p    2 =E |δ(Dsi X)|2 ds ··· δ(Dsi1 X)2 · · · δ(Dsi p X)2 ds1 · · · ds p . E Ti

Ti

Ti

2

2

By a computation similar to that for E[δ(X)p ], we find that the above is equal to the sum of terms    j j Dt11 (Dsi1 Xt1 )i1 · · · Dtqq (Dsi p Xtp )iq ds1 · · · ds p dt , E ip T

2

Tq

where t = (t1 , . . . , tq ) and q = dominated by 

  E

T jl

l;il =0,jl =0

×

2

2

 l;il =0,jl =0

p 2 , . . . , p.

  Ti

   E

Tl T

By Hölder’s inequality, the above is

il

j

T

|Dtll Dsii Xtl | 2 dtl dsi dtl il

|Dsii Xtl | 2 dsi dtl

 p  il 2

p

 p  il 2

p

.

(5.20)

5.2 Sobolev Norms for Wiener Functionals

181

It holds that   E

  T jl

Ti

il

j

T

|Dtll Dsii Xtl | 2 dtl dsi dtl

  

E

Ti

il

|Dsii Xtl | 2 dsl dtl

T

 p  il 2

p

 p  il 2

p

l l ≤ Xii+j ≤ Xii+p−1,p , l ,p

l l ≤ Xii,p ≤ X|ii+p−1,p .

 p l = Xi+p−1,p . ConseTherefore, (5.20) is dominated by l;il =0 Xii+p−1,p quently there exists a positive constant c such that   E

Ti

|δ(Dsi X)|2 ds

p  2

≤ c Xi+p−1,p . p

(5.21)

We shall consider the last term of (5.19). It holds that   E

Ti−1 ×T

|Dsi−1 Xsh |2 dsi dsh i

p  2

p

≤ Xi−1,p .

(5.22)

Substitute (5.21) and (5.22) in (5.19). Then we get   E

Ti

|Dsi δ(X)|2 ds

p  2

p

≤ cXi+p−1,p .

 Xm+p−1,p . This Then we obtain from (5.18) the inequality δ(X)m,p ≤ cm,p proves δ(X)m,p ≤ cm,p Xm+p−1,p .   p

p

We will summarize properties of the adjoint operator. For a real number r, we denote by [r] the largest integer which is dominated by r. If r > 0, 2[r/2] is equal to the largest even number which is dominated by r. Further, if r > 0, −2[−r/2] is the smallest even number which is bigger than or equal to r. We set r = −2[−r/2]. Theorem 5.2.2 1. Skorohod integral operator δ is extended to a continuous linear operator from ∞ ¯ F-space D∞ T to F-space D . Further, for any m ∈ N and p ≥ 2, there exists a positive constant cm,p such that δ(X)m,p ≤ cm,p Xm+p−1,p holds for any X ∈ D∞ T . ∞ 2. Adjoint formula (5.8) holds for any X ∈ D∞ T and G ∈ D . ∞ 3. Commutation relation (5.9) holds for any X ∈ DT . 4. Isometric property (5.10) holds for any X, Y ∈ D∞ T .

(5.23)

182

5 Malliavin Calculus

Proof We will prove (5.23). Since the norm Xm,p is nondecreasing with respect to p, we have by Theorem 5.2.1, δ(X)m,p ≤ δ(X)m,p ≤ cXm+p−1,p . Assertions (2)–(4) are immediate from Lemma 5.1.1.

 

Derivative operator D and its adjoint δ can be applied to complex-valued functionals. Let F √ = F1 + iF2 , where Fk , k = 1, 2 are real smooth Wiener functionals and i = −1. Then we define DF by DF1 + iDF2 . Let X = X1 + iX2 , where Xk , k = 1, 2 are real smooth H -valued functionals. Then we define δ(X) by δ(X1 ) + iδ(X2 ). Norms F m,p and Xm,p for these complex-valued functionals are defined in the same way. F is called smooth if F1 and F2 are smooth. The set of smooth functionals is denoted again as D∞ . Complex H -valued smooth functionals are defined similarly and we denote by D∞ T the set of all complex H -valued smooth functionals. Then Theorems 5.2.1 and 5.2.2 are valid for complex functionals. Note For arguments of the Malliavin calculus based on the Ornstein–Uhlenbeck semigroup, we refer to Malliavin [77], Ikeda–Watanabe [40], Watanabe [116], and further, books of Shigekawa [102], Nualart [88] and Matsumoto–Taniguchi [83]. Sobolev-type norms for Wiener functional were defined in [40] and [116] in a different context. In these works, norms of a Wiener functional F are defined for 1 m all real m and p > 1 by E[|(I − L) 2 F |p ] p , where L is the Ornstein–Uhlenbeck  operator. To avoid confusion of notations, we  will denote their norms as F m,p . They showed that if m is a positive integer, 0≤m ≤m F m ,p (j  are nonnegative integers) is equivalent to our norm F m,p . The fact is called Meyer’s equivalence [102]. Further, with respect to the estimate of the adjoint operator δ, they assert the inequality ||δ(X)m,p ≤ c Xm+1,p . In the case where m is a positive integer, it is equivalent to ||δ(X)m,p ≤ cXm+1,p ,

(5.24)

which is stronger than the inequality of our Theorem 5.2.1. However, its proof is not clear to the author. In this monograph, we have given a new direct proof for a slightly weaker assertion (Theorem 5.2.1). Studies of the Malliavin calculus for non-classical Wiener spaces (curved Wiener space etc.) may be found in Driver’s survey work on Wiener space [23].

5.3 Nondegenerate Wiener Functionals

183

5.3 Nondegenerate Wiener Functionals Let F = (F 1 , . . . , F d ) be a d-dimensional Wiener functional. If its components F i , i = 1, . . . , d are infinitely H -differentiable, F is called infinitely H -differentiable. We denote by (D∞ )d the set of all such d-dimensional functionals. If each component F i belongs to D∞ , F is said to belong to (D∞ )d . Its Sobolev norm is defined by the sum of Sobolev norms of components F j , j = 1, . . . , d. It holds that (D∞ )d ⊂ (D∞ )d . Elements of the latter are called smooth Wiener d 1 d functional. The space (D∞ T ) and the norm of its element Xt = (Xt , . . . , Xt ) are defined similarly. Let F = (F 1 , . . . , F d ) be an element of (D∞ )d . Let G be an element of D∞ . We consider the law of F weighted by G: μG (dx) = E[1dx (F )G]. In this section, we study the existence of the smooth density of the law μG (dx) with respect to the Lebesgue measure. For this purpose, we consider its characteristic function. It is given by  ψG (v) =

Rd

ei(v,x) μG (dx) = E[ei(v,F ) G],

(5.25)

√ where i = −1 is pure imaginary. It is a C ∞ -function of v. Indeed, let F j = (F 1 )j1 · · · (F d )jd , where j = (j1 , . . . , jd ) is a multi-index of nonnegative integers jl , l = 1, . . . , d. Since |F | ∈ L∞− , the functional ei(v,F ) F j G is integrable for any j j j j and we can change the order of the derivatives ∂v ≡ ∂v11 · · · ∂vdd and integral. Then we have ∂vj E[ei(v,F ) G] = i |j| E[ei(v,F ) F j G]. This shows that the characteristic function ψG (v) is a C ∞ -function and ∂v ψG (v) coincides with the above function. In Chap. 1, we discussed the existence of the smooth density of a weighted law μG . Proposition 1.1.2 tells us that if |ψG (v)||v|n is integrable for some positive integer n, then the weighted law μG (dx) has a C n -density function fG (x). Now, in order to show the integrability of |ψG (v)||v|n , we need to assume that F is nondegenerate, which will be defined using the Malliavin covariance. For a smooth Wiener functional F = (F 1 , . . . , F d ) we define a matrix R F called the Malliavin covariance by j

   Dt F (Dt F )T dt. R F = DF i , DF j  = T

(5.26)

184

5 Malliavin Calculus

Let Sd−1 = {v ∈ Rd ; |v| = 1}. For θ ∈ Sd−1 , we set R F (θ ) = (θ, R F θ ). The functional F is called nondegenerate, if R F (θ ) are invertible a.s. for any θ ∈ Sd−1 and inverses satisfy sup E[R F (θ )−p ] < ∞

(5.27)

θ

for any p, where the supremum for θ is taken in the set Sd−1 . Since the norm  0,p coincides with Lp -norms, the above condition is equivalent to that the inequality supθ R F (θ )−1 0,p < ∞ holds for any p ≥ 2. Remark We give a simple example of a nondegenerate Wiener functional. Let F = (F 1 , . . . , F d ) be a linear functional of the Wiener process Wt given by Fi =

 k

T

hik (s) dWsk ,

i = 1, . . . , d,

where hik (s), i = 1, . . . , d, k = 1, . . . , d  are square integrable functions on T. We saw in Sect.  5.1 that the above F is H -differentiable and its derivative is computed as Dt F = hi,k (t) = H (t) (d × d  -matrix). Then its Malliavin covariance is equal to a constant (non-random) matrix    RF = hik (t)hj k (t) dt = H (t)H (t)T dt. k

T

T

Therefore F is nondegenerate if and only if the above matrix R F is invertible. The above F is Gaussian distributed and its covariance matrix is computed as T H (t)H (t)T dt. Therefore, the Malliavin covariance R F coincides with the covariance matrix of F . It is well known that a Gaussian distribution has a C ∞ density if its covariance matrix is nondegenerate (positive definite). The density function is given by (1.8) with A = R F and b = 0. Now, we want to show that the law of any nondegenerate Wiener functional has a C ∞ -density. We first discuss the inverse of the Malliavin covariance. We need a lemma. Lemma 5.3.1 Suppose that G ∈ D∞ is invertible and the inverse G−1 belongs to L∞− . Then G−1 belongs to D∞ . Further, for any m, p, there exists cm,p > 0 such that the inequality G−1 m,p ≤ cm,p (1 + Gm,2mp )m (1 + G−1 0,2(m+1)p )m+1 holds for any G with the above property.

(5.28)

5.3 Nondegenerate Wiener Functionals

185

Proof If G−1 ∈ L∞− , then Tλh (G−1 ) ∈ L∞− . Then Tλh (G) is invertible and satisfies Tλh (G−1 ) = 1/Tλh (G) a.s. Hence we have −1 Tλh (G) − G Tλh (G−1 ) − G−1 = , λ Tλh (G)G λ

a.s.

Let λ tend to 0. The right-hand side converges to −1 DG, h in Lp for any p > 1. G2 −1 −1 This means that G is H -differentiable and D(G ) = −1 DG. Then, repeating G2 this argument inductively, we find that the i-th H -derivative of G−1 exists and is equal to Dti G−1 =



(−1)r

Gi−r Dti11 G · · · Dtirr G Gi+1

t = (t1 , . . . , tr ),

,

where the sum is taken for positive integers r and ik such that r ≤ i and i1 +· · ·+ir = i. Using Hölder’s inequality, we get   Gi−r D i1 G · · · D ir G 2  p  2

t1 tr

dt

Gi+1   1 p  1 

2 2

Gi−r D i1 G · · · D ir G 2 dt ≤E E |G|−2(i+1)p

E

t1

(5.29)

tr

≤ c Gi,2ip G−1 0,2(i+1)p ip

(i+1)p

≤ c (1 + Gi,2ip )ip (1 + G−1 0,2(i+1)p )(i+1)p . Now for a given m ∈ N, the above inequality is valid for any i = 0, 1, 2, . . . , m. Further, the last term of the above is nondecreasing with respect to i and p. Therefore, summing up (5.29) for i = 0, . . . , m, we get the inequality G−1 m,p ≤ cm,p (1 + Gm,2mp )mp (1 + G−1 0,2(m+1)p )(m+1)p . p

 

This proves the inequality of the lemma. (D∞ )d

R F (θ )−1

Proposition 5.3.1 If F of is nondegenerate, belongs to any θ . Further, for any m, p, there exists a positive constant cm,p such that

D∞

for

sup R F (θ )−1 m,p ≤ cm,p (1 + DF 2m,4mp )m (1 + sup R F (θ )−1 0,2(m+1)p )m+1 θ

θ

(5.30) holds for any nondegenerate F .

186

5 Malliavin Calculus

Proof We apply Lemma 5.3.1 for G = R F (θ ). It is sufficient to show that any m, p, there exists a positive constant cm,p such that sup R F (θ )m,p ≤ cm,p DF 2m,2p .

(5.31)

θ

The inequality is immediately verified if m = 0. We consider the case m ≥ 1. Let i ≤ m. Since R F (θ ) = T (θ, Dt F )2 dt, and 

Dti (θ, Dt F )2 =



i  ,i  ∈N,i  +i  =i



Dti (θ, Dt F ) · Dti (θ, Dt F ),

we have |Dti R F (θ )| ≤ ≤





i  ,i  T

 i  ,i 



|Dti (θ, Dt F ) · Dti (θ, Dt F )| dt 

T

|Dti (θ, Dt F )|2 dt

1  



2

T

|Dti (θ, Dt F )|2 dt

1 2

.

Therefore we get p

R F (θ )m,p ≤

p     2 E |Dti R F (θ )|2 dt i≤m

≤



  E

i  ,i  ≤m

≤



i  ,i  ≤m

T

Ti

 |Dti (θ, Dt F )|2 dt i+1

p  1   2 dt E



|Dti (θ, Dt F )|2 dt dt

p  1 2

Ti+1

DF i  ,2p DF i  ,2p ≤ c DF m,2p , p

p

2p

for any θ . Then the assertion of the lemma follows.

 

Now, we want to show that if F is nondegenerate, its weighted characteristic function ψG (v) is rapidly decreasing. Theorem 5.3.1 For any N ∈ N, there exist m ∈ N, p ≥ 2 and c > 0 such that the inequality

 

E ei(v,F ) G ≤

N c  F −1 DF  sup R (θ )  Gm,p m,p m,p |v|N θ

(5.32)

holds for all |v| ≥ 1 for any nondegenerate Wiener functional F of (D∞ )d and G ∈ D∞ .

5.3 Nondegenerate Wiener Functionals

187

Proof For a nondegenerate smooth functional F , we will define a complex H valued Wiener functional with parameter v ∈ Rd \ {0} by Xt = XtF,v =

−i(v, Dt F ) · R F (θ )−1 , |v|2

where θ = v/|v|. Then Xt ∈ D∞ T by Proposition 5.3.1 and Lemma 5.2.1. Further, using the adjoint equality (5.8) for the Skorohod integral, we have    E ei(v,F ) δ(XG) = E =E





= E[e

T

T

Dt (ei(v,F ) )Xt G dt



ei(v,F ) · i(v, Dt F )Xt G dt

i(v,F )

(5.33) 

G],

since i(v, Dt F )Xt dt = 1. We will repeat this argument. Setting L(G) = δ(XG), we have the iteration formula E[ei(v,F ) G] = E[ei(v,F ) L(G)] = · · · = E[ei(v,F ) LN (G)] for N = 1, 2, . . .. Consequently, using Theorem 5.2.1 and Hölder’s inequality, we get |E[ei(v,F ) G]| ≤ LN (G)0,2 ≤ δ(XLN −1 (G))0,2 ≤ c1 XLN −1 (G)1,2 ≤ c2 X1,4 LN −1 (G)1,4 . Repeat this argument for LN −1 (G)1,4 . Then for any N , there exist increasing sequences ml , pl , l = 1, . . . , N and a positive constant cN such that |E[ei(v,F ) G]| ≤ cN Xm1 ,p1 · · · XmN ,pN GmN ,pN ≤ cN XN mN ,pN GmN ,pN . By Hölder’s inequality, we have cm,p i(v, D)F m,2p R F (θ )−1 m,2p |v|2 cm,p DF m,2p sup R F (θ )−1 m,2p . ≤ |v| θ

Xm,p ≤

Therefore we get the inequality of the theorem.

 

Theorem 5.3.2 Suppose that a d-dimensional smooth Wiener functional F is nondegenerate. Then for any G ∈ D∞ , the law of F weighted by G has a rapidly decreasing C ∞ -density. Further, the density function fG (x) and its derivatives are

188

5 Malliavin Calculus

given by the Fourier inversion formula; for any multi-index j = (j1 , . . . , jd ) of nonnegative integers, we have (−i)|j| ∂ fG (x) = (2π )d



j

Rd

e−i(v,x) ψG (v)v j dv,

(5.34)

where ψG (v) = E[ei(v,F ) G]. Proof We saw in Theorem 5.3.1 that the function ψG (v) is a rapidly decreasing C ∞ -function of v ∈ Rd . Therefore |ψG (v)||v|n0 is integrable for any n0 ∈ N. Then the weighted law μG (dx) has a C n0 -density fG (x). Since this is valid for any n0 , fG (x) is actually a C ∞ -function. It satisfies the formula (5.34) by Proposition 1.1.2. Further, ψG (v) is a C ∞ -function of v. Then the density fG (x) is a rapidly decreasing function. Therefore fG (x) is a rapidly decreasing C ∞ -function.   We can replace the nondegenerate condition of a smooth Wiener functional F by Malliavin’s condition using the determinant of the Malliavin covariance R F . Let λ1 ≥ λ2 > · · · ≥ λd ≥ 0 be eigen-values of the matrix R F . If λd > 0 a.s and it −p satisfies E[λd ] < ∞ for all p > 1, then we have supθ∈Sd−1 E[(θ, R F θ )−p ] < ∞ and hence F is nondegenerate. Note that the above is equivalent to that E[(λ1 · · · λd )−p ] < ∞ holds for all p > 1. Therefore, if | det R F | > 0 a.s. and E[| det R F |−p ] < ∞ holds for all p > 1, then F is nondegenerate. Therefore we get a cerebrated result in the Malliavin calculus for the Wiener space. Theorem 5.3.3 (Malliavin [77], Malliavin–Thalmaier [80]) Let F be a ddimensional smooth Wiener functional. Assume that | det R F | > 0 a.s. and (det R F )−1 ∈ L∞− . Then F is nondegenerate and the law of F weighted by G ∈ D∞ has a rapidly decreasing C ∞ -density. The Malliavin criterion for the smooth density (Theorem 5.3.3) has been applied to diffusion processes whose generator is elliptic or further, hypo-elliptic. See Malliavin [77], Kusuoka–Stroock [69, 70] etc. We will discuss the problem in Chap. 6. Remark (Formula of integration by parts) In literatures on Malliavin calculus, the formula of the integration by parts is often obtained and it is applied to the proof of the smooth density. We will derive the formula directly, using the property of adjoint operator δ stated in Theorem 5.2.2. Let F = (F 1 , . . . , F d ) be a nondegenerate smooth functional. Let f be a Cb∞ function on Rd . Then f (F ) is a one-dimensional  smooth functional. By Proposition 5.1.1, we have D(f (F )) = di=1 ∂xi f (F )DF i . Let (γij ) be the inverse matrix of the Malliavin covariance R F . Then we have ∂xi f (F ) = D(f (F )),

 j

γij DF j T

5.3 Nondegenerate Wiener Functionals

189

for any i = 1, . . . , d. Let G ∈ D∞ . Then G Proposition 5.3.1. Then we get

 j

∞ γij DF j ∈ D∞ T , since γij ∈ D by

E[∂xi f (F ) · G] = E[D(f (F )), G



γij DF j ]

j

= E[f (F ) · δ(G



γij DF j )].

j

 Here we use the adjoint formula (5.8). Set li (G) = δ(G j γij DF j ). It is a continuous linear operator from D∞ into itself by Theorem 5.2.2 and Proposition 5.3.1. Then we get the formula of the integration by parts:     E ∂xi f ◦ F · G = E f ◦ F · li (G) .

(5.35)

Further, by induction, the formula is extended as     E ∂xi1 · · · ∂xin f ◦ F · G = E f ◦ F · li1 ,...,in (G) ,

(5.36)

where li1 ,...,in is a continuous linear operator from D∞ into itself by Theorem 5.2.2. We will give an alternative proof of the smooth density for the law of F , by using the above formula of the integration by parts. Set f (x) = ei(v,x) , where v = (v1 , . . . , vd ) and x = (x1 , . . . , xd ). Then it holds that ∂xi1 · · · ∂xin f (x) = i n vi1 · · · vin ei(v,F ) . Therefore, we get by (5.36) i n vi1 · · · vin E[ei(v,F ) G] = E[ei(v,F ) li1 ,...,in (G)]. Therefore if |vi1 · · · vin | = 0, we have

 

E ei(v,F ) G ≤

1 li ,...,i (G)0,2 |vi1 · · · vin | 1 n

for any n. Therefore the weighted characteristic function ψG (v) = E[ei(v,F ) G] is a rapidly decreasing function. Further, since F has finite moments of any order, ψG (v) is a C ∞ -function. Then weighted law of F by G has a rapidly decreasing C ∞ -density. In this monograph, we did not use the formula of the integration by parts for the proof of Theorem 5.3.3, since the formula should not hold for the Poisson space, as we will discuss shortly.

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5 Malliavin Calculus

5.4 Difference Operator and Adjoint on Poisson Space We are also interested in the problem whether the law of a functional of a Poisson random measure has a smooth density. For the problem, we will study the Malliavin calculus on the space of Poisson random measures.    Let T = [0, T ] and Rd0 = Rd \ {0}. Let U = T × Rd0 and B(U) be its Borel field. Let Ω be the set of all integer-valued measures ω on U and let B(Ω) be the smallest σ -field of Ω with respect to which ω(E), E ∈ B(U) are measurable. Let P be a probability measure on (Ω, B(Ω)) such that N(dt dz) := ω(dt dz) is a Poisson random measure with intensity measure n(dt dz) = dtν(dz), where ν is a  Lévy measure on Rd0 . The triple (Ω, B(Ω), P ) is called a space of Poisson random measure with the intensity n. We will define two transformations Ω → Ω following Picard [92]. For each u = (t, z) ∈ U, we define two maps εu− , εu+ ; Ω → Ω by εu− ω(E) = ω(E ∩ {u}c ),

∀E ∈ B(U),

εu+ ω(E) = ω(E ∩ {u}c ) + 1E (u),

∀E ∈ B(U).

Then equality εuθ1 ◦ εuθ2 = εuθ1 holds for any u and θ1 = ± and θ2 = ±. For any ω ∈ Ω, εu+ ω = ω holds for almost all u with respect to the measure N , in view of the definition of the measure N . Further, for almost all ω ∈ Ω, εu− ω = ω holds for almost all u with respect to the measure n, since the set of u such that εu− ω = ω is at most countable and countable sets are null sets with respect to the measure n(du) = dtν(dz). Let us remark that the Poisson random measure ω ∈ Ω has no point mass almost surely, i.e., P (ω({u}) = 0) = 1 for any {u}, since E[ω({u})] = E[N({u})] = n({u}) = 0. Then the equality εu+ ω(E) = ω(E) + δu (E) holds for almost all ω. Let S be a complete metric space. An S-valued B(Ω)-measurable function F (ω) is called an S-valued Poisson functional. If S is the space of real numbers, it is called simply a Poisson functional. Transformations εuθ , θ = ±1 induce transformations for a Poisson functional F (ω). We denote F (εuθ ω) by F ◦ εuθ (ω). Let Zu , u ∈ U be a family of real valued Poisson functionals. We assume that it is measurable with respect to (u, ω) (random field). From the definitions of transformations εu+ and εu− , equalities  U

Zu ◦ εu+ N(du) =



U

Zu ◦ εu− n(du)

 

=

U

U

Zu N (du),

(5.37)

Zu n(du)

(5.38)

hold for almost all ω, for any positive random field Zu , u ∈ U.

5.4 Difference Operator and Adjoint on Poisson Space

191

Lemma 5.4.1 (Picard [91]) If a positive random field Zu , u ∈ U satisfies Zu ◦εu+ = Zu ◦ εu− for any u almost surely, then  E

U

   Zu N(du) = E Zu n(du) . U

(5.39)

Proof Let I be a sub σ -field of B(U) × B(Ω) generated by sets A × B, where A ∈ B(U) and B ∈ BAc (Ω). Here BAc (Ω) is the σ -field generated by N (C), C ⊂ Ac . If Zu satisfies Zu ◦ εu+ = Zu ◦ εu− for all u almost surely, then Zu is I-measurable. In fact, let Zu be a step functional given by Zu (ω) =

n 

φm (ω)1Am (u),

m=1

where {A1 , . . . , An } is a measurable partition of U and φm , m = 1, . . . , n are bounded positive B(Ω)-measurable functionals. If Zu ◦ εu+ = Zu ◦ εu− holds for any u ∈ U, then φm ◦εu+ = φm ◦εu− holds for any u ∈ Am , m = 1, . . . , n. Hence each φm c (Ω)-measurable. Therefore, Zu is I-measurable. Further, since is B Zu N (du) A m = m φm N (Am ) and, φm and N(Am ) are independent for any m, we have  E

U

=E

  Zu N(du) = E[φm ]E[N (Am )]





m

φm n(Am ) = E

m

 U

 Zu n(du) .

Now, for a given I-measurable positive random field Zu , there exists a sequence n of positive I-measurable step functionals Zun such for nthat Zu ↑ Zu as n ↑ ∞ n all u almost surely. Since E[ Zu N(du)] = E[ Zu n(du)] holds for any Zun , the equality E[ Zu N (du)] = E[ Zu n(du)] holds for the limit random field Zu .   Corollary 5.4.1 For any positive random field Zu , we have  E

U

   Zu N(du) = E Zu ◦ εu+ n(du) . U

(5.40)

Proof The functional Yu := Zu ◦ εu+ satisfies Yu ◦ εu+ = Yu ◦ εu− for all u almost surely. Indeed, we have (Zu ◦ εu+ ) ◦ εu− (ω) = Zu (εu+ ◦ εu− ω) = Zu (εu+ ω) = Zu ◦ εu+ (ω), (Zu ◦ εu+ ) ◦ εu+ (ω) = Zu (εu+ ◦ εu+ ω) = Zu (εu+ ω) = Zu ◦ εu+ (ω).

192

5 Malliavin Calculus

Therefore equality (5.39) holds for Yu . Then, using (5.37) we get  E

U

     Zu N (du) = E Zu ◦ εu+ N(du) = E Zu ◦ εu+ n(du) . U

U

  We should remark that transformations εu+ may not be absolutely continuous, i.e., P (A) = 0 may not imply P (εu+ A) = 0 for some A ∈ F. Therefore, if a Poisson functional F is defined a.s. P , F ◦ εu± may not be defined for any u a.s. However, F ◦ εu+ is well defined a.e. (u, ω) with respect to n(du)P (dω). Indeed, if F = 0 holds a.s., then E[ I F

N(du)] = 0 holds, where 1 (u) = 1|z|> (u). Then we have E[ F ◦ εu+ 1 n(du)] = 0 in view of Corollary 5.4.1. Therefore F ◦ εu+ = 0 holds a.e. n(du)P (dω). For a bounded Poisson functional F , we define difference operator {D˜ u , u ∈ U} by D˜ u F = F ◦ εu+ − F.

(5.41)

It is well defined a.e. n(du)P (dω). If u = (t, z), we denote D˜ (t,z) F by D˜ t,z F . j For u = (u1 , . . . , uj ) ∈ Uj , we set εu+ = εu+1 ◦ · · · εu+j and D˜ u = D˜ u = j D˜ u1 · · · D˜ uj . D˜ u F is well defined a.e. dnj dP , where nj is the j -fold product of the measure n given by nj (du1 · · · duj ) = n(du1 ) × · · · × n(duj ). The functional j D˜ u F is invariant by the permutation; let σ be a permutation of {1, . . . , j }. For u = (u1 , . . . , uj ), set uσ = (uσ1 , . . . , uσj ). Then it holds that F ◦ εu+ = F ◦ εu+σ a.e. j j dnj dP . Further, D˜ u F = D˜ uσ F holds a.e. dnj dP . Further, we should remark that if ui = uj (i = j ) holds for some u = (u1 , . . . , uj ), then D˜ u F = 0 a.e. If j = 0, we regard U0 is an empty set ∅, ε∅+ is the identity transformation and we set D˜ ∅ F = F . For u = (r, z) ∈ U, we set γ (u) = |z| ∧ 1. It is a positive function of u = (r, z), since |z| > 0 holds for any u. For u = (u1 , . . . , uj ) ∈ Uj , we set γ (u) = γ (u1 ) · · · γ (uj ) and |u| = max1≤i≤j γ (ui ). A bounded Poisson functional F is called smooth if E[|D˜ u F |p ] s. Then D˜ v Zu D˜ u Zv = 0 holds for any u = (t, z), v = (s, z ) such that t = s a.e. Therefore the last term in (5.45) is equal to 0 if Zu is predictable.   Finally, we show that δ˜ has a property, similar to the definition of δ given by (5.7).  Proposition 5.4.2 Let Zu = nl=1 Fl hl (u), where Fl ∈ D˜ ∞ and hl are bounded measurable function on U with compact support. Then we have ˜ δ(Z) =

n  l=1

˜ l) − Fl N(h

n   l=1

D˜ u Fl hl (u)n(du),

a.s.

(5.47)

196

5 Malliavin Calculus

Proof We have ˜ l hl ) = δ(F





˜ Fl ◦εu− hl (u)N(du) = Fl

 ˜ (Fl −Fl ◦εu− )hl (u)N˜ (du). hl (u)N(du)−

We shall compute the last term. It holds by (5.37) that 

(Fl − Fl ◦ εu− )hl (u)N(du) =



(Fl ◦ εu+ − Fl ◦ εu− ◦ εu+ )hl (u)N(du) = 0.

Since Fl − Fl ◦ εu− = Fl ◦ εu+ − Fl ◦ εu− holds by definitions of εu+ and εu− , we have 

(Fl − Fl ◦ εu− )hl (u) n(du) =

 

=  =

(Fl ◦ εu+ − Fl ◦ εu− )hl (u)n(du) (Fl ◦ εu+ − Fl )hl (u)n(du) D˜ u Fl hl (u)n(du).

Summing up these computations for l = 1, 2, . . . , n, we get (5.47).

 

5.5 Sobolev Norms for Poisson Functionals In Sect. 5.2, we defined Sobolev norms for Wiener functionals, making use of differential operator Dt , t ∈ T. In this section, we will define Sobolev norms for Poisson functionals, making use of the difference operator D˜ u F, u ∈ U. We should be careful about the definition of Sobolev norms, since we want to apply it for proving the existence of the smooth density of the law of ‘nondegenerate’ Poisson functionals. Its definition will become more complicated than that of a Wiener functional, because the smooth density of a Poisson functional F will possibly be related to the asymptotic property of D˜ u F as γ (u) converges to 0. Then we should define Sobolev norms for D˜ u F , restricting u to subdomains A of U, where A are open neighborhoods of the center of the intensity measure n. Discussions in this section and those in Sects. 5.6, 5.7, and 5.8 are close to works by Ishikawa–Kunita–Tsuchiya [46], though notations for Sobolev norms are changed. We begin with a preliminary observation. Let F be s smooth Poisson functional. For a positive integer n, an L2 -type Sobolev norm for F , restricted to a domain A ⊂ U such that n(A) > 0 should be given by n    2 E[|F | ] + j =1

Aj

j E[|D˜ u F |2 ]nj (du)

1 2

.

(5.48)

5.5 Sobolev Norms for Poisson Functionals

197

Here Aj is the j -product set of A, included in the j -product set Uj . Further, nj is the j -product measure of the intensity measure n defined on Uj . The norm is well j defined, since functionals D˜ u F are invariant by the permutation of u. We could write it as F n,2;A , but in order to avoid the confusion with Sobolev norm F n,2 for a Wiener functional, we will denote (5.48) by F 0,n,2;A . (The first suffix 0 indicates that the derivative operator DF is not involved. In Sect. 5.10, we will define Sobolev norms for Wiener–Poisson functionals denoted by F m,n,p;A .) It is compatible with Sobolev norms of L2 -type F m,2 for Wiener functionals. In order to define Sobolev norms of Lp (p > 2) type, we will transform infinite j measures nj on Aj to a probability measure mA on Aj in the following way. Let m be a bounded measure on U defined by  m(E) =

γ (u)2 n(du). E j

We define a probability measure mA on A and its product measure mA on Aj by mA (E) =

m(A ∩ E) , m(A)

j

mA (du) = mA (du1 ) · · · mA (duj ),

respectively, where u = (u1 , . . . , uj ) ∈ Uj . Further, we define positive functions on A and Aj by γ (u) , γA (u) = √ m(A)

γA (u) = γA (u1 ) · · · γA (uj ).

1 is the normalizing constant for the function γ (u), because we have Note that √m(A) 2 2 A γ (u) n(du) = m(A). Then we can rewrite the L -Sobolev norm (5.48) as



F 0,n,2;A = E[|F | ] + 2

n   j j =1 A

1 E[|D˜ u F |2 ] j 2 m (du) . A 2 γA (u) j

Let A(1) = {u ∈ U; |z| ≤ 1}. Suppose that A is a domain in A(1) which includes the center of the Lévy measure ν and satisfies 0 < m(A) ≤ 1. For given j ∈ N, p ≥ 1 2, we define a norm of F ∈ D˜ ∞ by F 0,0,p;A = E[|F |p ] p if n = 0 and if n ≥ 1, F 0,n,p;A

n    p = E[|F | ] +

j j =1 A

j 1 E[|D˜ u F |p ] j p m (du) . A γA (u)p

(5.49)

Remark The Sobolev norm for Wiener functionals given in (5.11) is of L2,p type; it is L2 -type with respect to t and is Lp -type with respect to w. But the Sobolev

198

5 Malliavin Calculus

norm  0,n,p;A for Poisson functionals given in (5.49) is of Lp,p -type. We defined Sobolev norms of Lp.p -type, because the difference rule of D˜ u and the differential rule of Dt are distinct. In fact, a difference rule of the operator D˜ u is given by D˜ u (F G) = (D˜ u F )G + GD˜ u F + D˜ u F D˜ u G,

(5.50)

while the differential rule of the operator Dt is given by a simpler form Dt (F G) = (Dt F )G + F Dt G. by

∞ . We define its norms Let {Zu , u ∈ U} be a smooth Poisson random field of D˜ U 0

  E[|Z |p ] 1 p u m (du) , (5.51) A p γ (u) A A n  j  1  E[|D˜ u Zu |p ] j p p = Z0,0,p;A+ m (du)m (du) . A A p γ (u)p j +1 γ (u) A A A

Z0,0,p;A = Z0,n,p;A

j =1

When a set A is fixed in discussions, we will often drop ‘; A’ in the definitions of Sobolev norms and write them simply as F 0,n,p := F 0,n,p;A ,

Z0,n,p := Z0,n,p;A .

Lemma 5.5.1 (Hölder’s inequality) For any n ∈ N and p ≥ 2, there is a positive constant c not depending on the set A such that F G0,n,p;A ≤ cF 0,n,pr;A G0,n,pr  ;A ,

(5.52)

F Z0,n,p;A ≤ cF 0,n,pr;A Z0,n,pr  ;A

(5.53)

∞ , and r, r  > 1 with 1/r + 1/r  = 1. hold for any F, G ∈ D˜ ∞ , Z ∈ D˜ U 0

Proof We will prove (5.53) only. By Hölder’s inequality, we have 

p

F Z0,0,p =

1 E[|F Zu |p ] mA (du) ≤ E[|F |pr ] r p γA (u)

p





1

E[|Zu |pr ] r  mA (du) γA (u)p p

≤ F 0,0,pr Z0,0,pr  . ˜ We can show similarly to the above Note the difference rule (5.50) for D.  

E[D˜ u F Zu |p ] p p mA (du )mA (du) ≤ F 0,1,pr Z0,0,pr  , γA (u )p γA (u)p E[F D˜ u Zu |p ] p p mA (du )mA (du) ≤ F 0,0,pr Z0,1,pr  . γA (u )p γA (u)p

5.5 Sobolev Norms for Poisson Functionals

Note that

m(A)1/2 γA (u )

=

1 γ (u )

199

≥ 1 for u ∈ A. Then



E[D˜ u F D˜ u Zu |p ] mA (du )mA (du) γA (u )p γA (u)p  p E[D˜ u F D˜ u Zu |p ] 2 ≤ m(A) mA (du )mA (du) γA (u )p · γA (u )p γA (u)p p

p

≤ F 0,1,pr Z0,1,pr  . Repeating this argument for j = 2, 3, . . ., we get (5.53) for any n.

 

We will define another norms that are stronger than Sobolev norms defined above. For n ∈ N and p ≥ 2, new norms are defined by n j   E[|D˜ u F |p ]  p1  F 0,n,p = E[|F |p ] + sup , γ (u)p u∈A(1)j

(5.54)

n j  E[|Zu |p ]  E[|D˜ u Zu |p ]  p1  Z0,n,p = sup + sup . p p p u∈A(1) γ (u) (u,u)∈A(1)j +1γ (u) γ (u)

(5.55)

j =1

j =1

Here, supu∈A(1) |ϕ(u)| etc. means the essential supremum of the function ϕ(u) on the measurable space (A(1), n), which is defined by   inf c ∈ R; |ϕ(u)| ≤ c for almost every u ∈ A(1) with respect to measure n .    Norms F 0,0,p and Z0,0,p are defined by cutting off terms nj=1 in the above definitions. We may regard them as Sobolev norms of L∞,p -type, i.e., the essential supremum is taken for variable u ∈ U with respect to the measure n and the Lp norm is taken with respect to the measure P . Thus these norms should be bigger than Sobolev norms of Lp,p -type. Indeed, we have the inequalities; 

F ∈ D˜ ∞ ,



∞ Z ∈ D˜ U , 0

F 0,n,p;A ≤ F 0,n,p , Z0,n,p;A ≤ Z0,n,p ,

(5.56)

for any n, p and A. ˜ 0,n,p the completion of D˜ ∞ by the norm   We denote by D 0,n,p and we set ∞ 0,n,p ∞ ˜ ˜ ˜ . Elements of D are called smooth Poisson functionals. Let D = n,p D 0,n,p ∞ by the norm   ˜∞ ˜ ˜ 0,n,p . be the completion of D˜ U DU n,p DU 0,n,p . We set DU = 0 ˜ ∞ and D ˜ ∞ are algebras and are F -spaces with respect to countable Then both D 

U

norms  0,n,p , n, p ∈ N.

200

5 Malliavin Calculus

˜ ∞ and Z ∈ D ˜ ∞ , norms F 0,n,p;A and Z0,n,p;A are well Then for any F ∈ D U defined and are finite for any n, p and A ⊂ A(1). It is not difficult to prove the following proposition. Proposition 5.5.1 The operator D˜ u is extended to a continuous linear operator  ˜ ∞ to F-space D ˜ ∞ . Further, it holds that DF ˜  from F-space D 0,n,p ≤ F 0,n+1,p U ˜ ∞ for any n, p. for any F ∈ D Later, we will use another norms for Poisson functionals. For n ∈ N, p ≥ 2, we will define norms of F by  |F |0,n,p

n  1  p p = E[|F | ] + sup E[|F ◦ εu+ |p ] . j =1 u∈A(1)

(5.57)

j

We will study relations of these norms. 



Proposition 5.5.2 For any n, p, the norm | |0,n,p is weaker than the norm  0,n,p ; there is a positive constant cn,p such that 

˜ ∞. ∀F ∈ D



|F |0,n,p ≤ cn,p F 0,n,p ,

(5.58)

Proof Let u ∈ A(1)j . We can rewrite F ◦ εu+ as F ◦ εu+ =



D˜ v F = F +

v⊂u

j 



D˜ v F.

(5.59)

i=1 v⊂u,v=i

Therefore we have sup E[|F u

◦ εu+ |p ]

j    p ≤ 2 E[|F | ] + sup E[|D˜ vi F |p ] . j

v

i=1

Then, introducing another norms by n  1  p  sup E[|D˜ vi F |p ] , |F |0,n,p = E[|F |p ] + j i=1 v∈A(1)





we have the inequality (|F |0,n,p )p ≤ (n + 1)2n (|F |0,n,p )p . We can show similarly 





the inequality (|F |0,n,p )p ≤ (n+1)2n (|F |0,n,p )p . Therefore two norms |F |0,n,p and 







|F |0,n,p are equivalent. Further, since |F |0,n,p ≤ F 0,n,p holds, the norm |F |0,n,p 

is weaker than the norm F 0,n,p for any n, p.

 

5.6 Estimations of Two Poisson Functionals by Sobolev Norms

201

 Let L˜ 0,n,p be the completion of D˜ ∞ by the norm | |0,n,p . We set L˜ ∞ = ˜ ∞ ⊂ L˜ ∞ holds. ˜ 0,n,p . Then L˜ ∞ is an F -space and the relation D n,p L For a Poisson functional Xt with parameter t ∈ [0, 1), we set 

|X|0,n,p =

 T



(|Xt |0,n,p )p dt

1

p

.

(5.60)

Note Sobolev norms for Poisson functionals are defined in several ways. Ishikawa et al. defined Sobolev norms for Poisson functionals in [46]. Notations of their Sobolev norms are changed in this monograph. Norms  n,p;A ,  ∗n,p and | |∗n,p 



in [46] correspond to our norms  0,n,p ,  0,n,p and | |0,n,p , respectively. Hayashi–Ishikawa [37] defined another Sobolev norms. Their norm  n,0,p,(ρ) is close to our norm  0,n,p;A(ρ) , but these two are not equivalent norms.

5.6 Estimations of Two Poisson Functionals by Sobolev Norms Let n(dt dz) = dtν(dz) be the intensity measure of the Poisson random measure. The set {u = (t, z); z = 0} is called the center of the intensity measure n. For a  given family of star-shaped neighborhoods {A0 (ρ)} of 0 ∈ Rd (see Sect. 1.2), we set A(ρ) = {u = (t, z) ∈ U; z ∈ A0 (ρ)} and call {A(ρ)} the family of star-shaped neighborhoods of the center of the intensity n. We define the function ϕ(ρ) by  ϕ(ρ) := m(A(ρ)) = T

|z|2 ν(dz) = T ϕ0 (ρ).

(5.61)

A0 (ρ)

Now, suppose that the Lévy measure ν satisfies the order condition of exponent α, then the function ϕ(ρ) satisfies the order condition of exponent α. We will fix 0 < ρ0 ≤ 1 such that ϕ(ρ0 ) ≤ 1. We are interested in estimations of Poisson functionals by Sobolev norms { 0,n,p;A(ρ) ; 0 < ρ < ρ0 } and their dependence on the parameter ρ. In this section, we will give estimations of two Poisson functionals. The first one is the ˜ ∞ and χρ is the indicator function IA(ρ) of ˜ Skorhod integral δ(Zχ ρ ), where Z ∈ D U0 the set A(ρ). We first study the L2 -estimate. If p = 2, the Sobolev norm is written simply as Z0,n,2;A(ρ) =



E[|Zu | ]n(du) + 2

A(ρ)

n  

1 2 j E[|D˜ u Zu |2 ]nj (du)n(du) .

j +1 j =1 A(ρ)

Then, using the isometric property (5.45), we have

202

5 Malliavin Calculus 2 ˜ E[δ(Zχ ρ) ] = E



   Zu2 n(du) + E

A(ρ)2

A(ρ)

 D˜ u Zv D˜ v Zu n(du)n(dv)

≤ Z20,1,2;A(ρ) ˜ we can for any 0 < ρ < ρ0 . Further, using the commutation relation for D˜ u and δ, show easily that for any positive integer n there is a positive constant cn such that 2 ˜ δ(Zχ ρ )0,n,2;A(ρ) ≤ cn Z0,n+1,2;A(ρ)

(5.62)

holds for all 0 < ρ < ρ0 . We want to extend the above inequality for any p ≥ 2. The following is the counterpart to Theorem 5.2.1, where we obtained an estimation of the Skorohod integral for Wiener functionals. Theorem 5.6.1 ([46]) Given n ∈ N and a positive even number p ≥ 4, there exists a positive constant cn,p such that the inequality ˜ δ(Zχ ρ )0,n,p;A(ρ) ≤ cn,p Z0,n+p−1,p;A(ρ)

(5.63)

∞ , where the constant c holds for any Z ∈ D˜ U n,p does not depend on the sets 0 {A(ρ), 0 < ρ < ρ0 }.

The proof of the theorem is similar to that for Theorem 5.2.1, but details are more p ˜ complicated. We shall first consider the estimation of E[δ(Zχ ρ ) ] for positive even number p ≥ 4. We prepare a lemma. Let 1 ≤ q ≤ p be a positive integer. An element u = (u1 , . . . , uq ) of the product set Uq is written as {u1 , . . . , uq }. For ε = (ε1 , . . . , εq ) ∈ {0, 1}q and u = {u1 , . . . , uq }, we set D˜ uε = D˜ uε11 · · · D˜ uεlq (= D˜ uεlq · · · D˜ uε11 ). ∞ . Then E[δ(Z) p ] is ˜ Lemma 5.6.1 Let p ≥ 4 be an even number. Let Z ∈ D˜ U 0 q written as sums of the following multiple integrals by n (du), q = 1, . . . , p, where nq (du) = n(du1 ) · · · n(duq ).

 Uq

E

p1   j =1

pq    ε ε · · · nq (du). Z D˜ u1j D˜ uqj u1 q Zuq 1

(5.64)

j =1

Here, pi ∈ N satisfy p1 + · · · + pq = p and εij , i = 1, . . . , q, j = 1, . . . , pi are given by εij := (εij (1), . . . , εij (q − 1)) ∈ {0, 1}q−1 . Further, ui ; i = 1, . . . , q are given by ui := u − {ui } = {ui (1), . . . , ui (q − 1)}, which satisfy q  pi q−1   {ui (k); εij (k) = 1}. u = {u1 , . . . , uq } ⊂ i=1 j =1 k=1

(5.65)

5.6 Estimations of Two Poisson Functionals by Sobolev Norms

203

Proof Our discussion is close to the proof of Theorem 5.2.1. We have by the adjoint formula (5.43)   p p−1 ˜ ˜ E[δ(Z) ] = E Zu1 D˜ u1 δ(Z) n(du1 ) . U

p−1 step by step. Using the difference rule ˜ We will decrease the power p − 1 of δ(Z) ˜ of Du and the commutation relation (5.44), we have p−1 p−2 p−2 p−2 ˜ ˜ ˜ ˜ ˜ ˜ ˜ δ(Z) D˜ u1 δ(Z) = D˜ u1 δ(Z) + δ(Z) D˜ u1 δ(Z) + D˜ u1 δ(Z) D˜ u1 δ(Z) p−2 p−2 p−2 ˜ ˜ ˜ D˜ u1 Z))(δ(Z) ˜ ˜ = (Zu1 + δ( + D˜ u1 δ(Z) ) + δ(Z) D˜ u1 δ(Z) . p] ˜ Substitute the above and apply the adjoint formula again. Then we see that E[δ(Z) is written as sums of terms   j j j j p−2 ˜ n(du1 )n(du2 ) , E D˜ u12 Zui11 D˜ u21 Zui22 D˜ u22 D˜ u11 δ(Z) U2

where i1 , i2 , j1 , j2 , j1 , j2 are nonnegative integers satisfying i1 + i2 = 2 and j1 , j2 , j1 , j2 ≤ 1. Here D˜ u0 is the identity transformation. p ] is written as sums ˜ Repeating this procedure inductively, we find that E[δ(Z) of terms stated in (5.64). For further details, see [46].   Proof of Theorem 5.6.1 We will first prove the inequality (5.63) in the case n = 0. p ˜ By Lemma 5.6.1, E[δ(Zχ ρ ) ] is written as a sum of terms (5.64), replacing Z by Zχρ . In this case, (5.64) is written as  E

p1  

A(ρ)q

j =1

$ =E

(

pq    ε1j ε ˜ nq (du) Z Du1 Zu1 · · · D˜ uqj u q q

p1

A(ρ)q

j =1

pq ˜ ε1j ˜ εqj j =1 Du1 Zu1 ) · · · ( j =1 Duq Zuq ) γA(ρ) (u1 )2 · · · γA(ρ) (uq )2

% q mA(ρ) (du) ,

q

since

mA(ρ) (du) γA(ρ) (u1 )2 ···γA(ρ) (uq )2

= 1A(ρ)q nq (du). Let us consider the nominator in the

above expression. For u = (u1 , . . . , uk ) and ε ∈ {0, 1}k , we set ε (u) = γA(ρ) (u1 )ε1 · · · γA(ρ) (uk )εk . γA(ρ)

We will show that there is a positive constant C = Cp (not depending on 0 < ρ < 1) such that the inequality

204

5 Malliavin Calculus

1 γA(ρ) (u1 )2 · · · γA(ρ) (uq )2 ≤ 



γA(ρ) (u1 )ε1j · · ·

j

(5.66) C

 j

 γA(ρ) (uq )εqj γA(ρ) (u1 )p1 · · · γA(ρ) (uq )pq

holds for any u ∈ A(ρ)q . Note first that by the order condition of the Lévy measure, ϕ(ρ) ≥ cρ α holds for 0 < ρ < ρ0 . Then, since m(A(ρ)) = ϕ(ρ), we have for any |u| ≤ ρ < ρ0 , γA(ρ) (u) =

γ (u) ϕ(ρ)

1 2

≤

1 c

1 2

α

1

ρ 1− 2 ≤ c− 2 .

Now, denote by {εij (k) = 1} the total number of triples (i, j, k) such that εij (k) = 1. We set r = {εij (k) = 1} + p − 2q. Since {εij (k) = 1} ≥ q holds by the relation (5.65), r is a nonnegative integer. Then there are ui1 , . . . , uir ⊂ u such that  j

   εqj γ p1 pq γA(ρ) (u1 )ε1j · · · A(ρ) (u1 ) · · · γA(ρ) (uq ) j γA(ρ) (uq ) γA(ρ) (u1 )2 · · · γA(ρ) (uq )2

= γA(ρ) (ui1 ) · · · γA(ρ) (uir ). The last term of the above is less than or equal to c−r/2 for any u ∈ A(ρ)q . Therefore, setting C = c−r/2 , we get the inequality (5.66). The inequality (5.66) implies $

|

E

p1

pq % ˜ ε1j ˜ εqj j =1 Du1 Zu1 | · · · | j =1 Duq Zuq | q m (du) A(ρ) γA(ρ) (u1 )2 · · · γA(ρ) (uq )2

A(ρ)q

$ ≤ CE

p1 

A(ρ)q j =1

(5.67)

ε % pq ε  |D˜ uqj |D˜ u1j q Zuq | q 1 Zu1 | · · · m (du) γA(ρ) (u1 )ε1j γA(ρ) (u1 ) γA(ρ) (uq )εqj γA(ρ) (uq ) A(ρ) j =1

∞ . In the formula (5.67), the total number of for any 0 < ρ < ρ0 and any Z ∈ D˜ U 0 product terms is equal to p. Apply Hölder’s inequality to (5.67) with respect to the q product measure dmA(ρ) dP . Then (5.67) is dominated by

 $ C E i,j

A(ρ)q



ε D˜ uiji Zui ij

γA(ρ) (ui )ε γA(ρ) (ui )

%1

p p

q p ≤ CZ0,q−1,p;A(ρ) .

mA(ρ) (du)

Therefore, the absolute value of (5.64) is dominated by C  Z0,q−1,p;A(ρ) for all p ˜ 0 < ρ < ρ0 and Z ∈ D˜ ∞ . Note that E[δ(Zχ ρ ) ] is written as a finite sum of p

U0

5.6 Estimations of Two Poisson Functionals by Sobolev Norms

205

terms (5.64) for q = 1, . . . , p. Since norms  0,n,p;A(ρ) are nondecreasing with p p  ˜ respect to n, |E[δ(Zχ ρ ) ]| is dominated by C Z0,p−1,p;A(ρ) for all 0 < ρ < ρ0 and Z ∈ D˜ ∞ with another constant C  . Consequently, we get U0

p p p  ˜ ˜ δ(Zχ ρ )0,0,p = E[δ(Zχ ρ ) ] ≤ C Z0,p−1,p;A(ρ) .

(5.68)

Next, we will consider the case n ≥ 1. It holds, by the definition of Sobolev norms that p p ˜ ˜ δ(Zχ ρ )0,n,p;A(ρ) = E[|δ(Zχ ρ )| ]

+

(5.69)

p   j

˜ E D˜ v δ(Zχ ρ)

n   j j =1 A(ρ)

γA(ρ) (v)p

j

mA(ρ) (dv).

The first term of the right-hand side has been computed at (5.68). We shall consider ˜ other terms. Take any integer 1 ≤ j ≤ n. Note the commutation relation of δ; ˜ ˜ ˜ D˜ v δ(Zχ ) = δ( D Zχ ) + Z χ . Then we have ρ v ρ v ρ j˜ ˜ D˜ vj Zχρ ) + D˜ v δ(Zχ ρ ) = δ(



j −1 D˜ vi Zvi χρ ,

i

where v = (v1 , . . . , vj ) and vi = v − {vi }, i = 1, . . . , j . Therefore, there is a positive constant cp such that

p   j

˜ E D˜ v δ(Zχ ρ) j mA(ρ) (dv) (5.70) γA(ρ) (v)p A(ρ)j

 j  j −1   E 

δ(  ˜ D˜ vj Zχρ ) p j  E[|D˜ vi Zvi χρ |p ] j m (dv) + m (dv) . ≤ cp A(ρ) A(ρ) γA(ρ) (v)p γA(ρ) (v)p



i=1

It holds that ˜ D˜ v Zχρ )p ] ≤ c D˜ v Zχρ  E[δ( 0,p 0,p−1,p;A(ρ) j

p

=

p c0,p

j

p

     E D˜ uj D˜ vj Zv χρ p j  m (du)mAρ (dv), γA(ρ) (u)p γA(ρ) (v)p A(ρ) 

j ≤p−1

206

5 Malliavin Calculus

in view of (5.68) for the case n = 0. Then we have   ˜ D˜ uj Zχρ )p j E δ( mA(ρ) (dv) γA(ρ) (v)p A(ρ)j

p   j  j   E D˜ u D˜ v Zv χρ p j j ≤ c0,p mA(ρ) (du)mA(ρ) (dv)mA(ρ) (dv) p p p γ (u) γ (v) γ (v) A(ρ) A(ρ) A(ρ) 



j ≤p−1

p

≤ c0,p



p

j  ≤p−1

p

Zχρ 0,j +j  ,p;A(ρ) ≤ cZχρ 0,j +p−1,p;A(ρ) .

(5.71)

Therefore, the first term of the right-hand side of (5.70) is dominated by p CZχρ 0,j +p−1,p;A(ρ) . We have further,  A(ρ)j

p   j −1 E D˜ vi Zvi χρ p mkA(ρ) (dv) ≤ Zχρ 0,j −1,p;A(ρ) . γA(ρ) (v)p

(5.72)

Since the fact is valid for any j = 1, . . . , n, we obtain from (5.70), ˜ δ(Zχ ρ )0,n,p;A(ρ) ≤ cn,p Zχρ ||0,n+p−1,p;A(ρ) . p

p

p

This proves the assertion of the theorem.

 

We will summarize properties of Skorohod integrals by Poisson random measure. Theorem 5.6.2 1. Skorohod integral operator δ˜ is extended to a continuous linear operator from ˜ ∞ to F-space D ˜ ∞ . Further, for any n ∈ N and p ≥ 2, there exists a F-space D U positive constant cn,p such that for any 0 < ρ < ρ0 ˜ δ(Zχ ρ )0,n,p;A(ρ) ≤ cn,p Z0,n+p−1,p:A(ρ)

(5.73)

˜ ∞ , where p = −2[−p/2]. holds for any Z ∈ D U ˜ ∞ and G ∈ D ˜ ∞. 2. Adjoint formula (5.43) holds for any Z ∈ D U ˜ ∞. 3. Commutation relation (5.44) holds for any Z ∈ D U ˜ ∞. 4. The isometric property (5.45) holds for any Y, Z ∈ D U ˜ ∞ and Z ∈ D ˜ ∞. 5. Hölder’s inequalities (5.52), (5.53) hold for any F, G ∈ D U  Further, similar Hölder’s inequalities are valid for norms  0,n,p . A d-dimensional Poisson functional F = (F 1 , . . . , F d ) is said to belong to ˜ ∞ . Sobolev ˜ ∞ )d , if all components F j , j = 1, . . . , d belong to D the space (D  norms F 0,n,p;A and F 0,n,p are defined as sums of the corresponding norms for components F j , j = 1, . . . , d.

5.6 Estimations of Two Poisson Functionals by Sobolev Norms

207

Difference operator D˜ and its adjoint δ˜ can be extended to complex-valued functionals. The space of complex-valued smooth Poisson functionals is denoted ˜ ∞ and the space of complex U-valued smooth functionals is denoted by again by D ∞ ˜ . Proposition 5.5.1 and Theorem 5.6.1 are valid for complex-valued functionals. D U ˜

We will next study the estimation of the exponential functional ei(v,DF ) − 1 with respect to Sobolev norms. In the next lemma, we are interested in the dependence of the estimation with respect to parameter ρ. Lemma 5.6.2 For any n ∈ N and p ≥ 2 there exists a positive constant cn,p such ˜ ∞ )d , v ∈ Rd and 0 < ρ < ρ0 , that for any F ∈ (D ˜

ei(v,DF ) − 10,n,p;A(ρ)

(5.74)

n ˜  ˜  2 ≤ cn,p |v|ϕ(ρ) 2 DF 0,n,(n+1)p (1 + |v|ρϕ(ρ) DF 0,n,(n+1)p ) . 1

1

˜

Proof The norm ei(v,DF ) − 10,n,p;A(ρ) is dominated by the sum of the following terms for j = 0, . . . , n: $ %1 & ˜ j i(v,D˜ u F ) ' p |Du (e − 1)| p j E mA(ρ) (du)mA(ρ) (du) . (5.75) γA(ρ) (u)γA(ρ) (u) A(ρ)j +1 ˜ j We will compute the functional D˜ u (ei(v,Du F ) − 1). It is not simple, since we have to use the difference rule (5.50). If j = 1, we have +

˜ ˜ ˜ D˜ u (ei(v,Du F ) − 1) = ei(v,Du F ◦εu ) − ei(v,Du F ) ˜

˜

˜

˜

˜

˜

˜

= ei(v,Du Du F )+i(v,Du F ) − ei(v,Du F ) = ei(v,Du F ) (ei(v,Du Du F ) − 1). If j = 2, we have by the difference rule (5.50) ˜ ˜ ˜ ˜ ˜ ˜ D˜ u D˜ u (ei(v,Du F ) − 1) = ei(v,Du F ) (ei(v,Du Du F ) − 1)(ei(v,Du Du F ) − 1) ˜

˜

˜

˜

˜

˜

+ ei(v,Du F ) ei(v,Du Du F ) (ei(v,Du Du Du F ) − 1) ˜

˜

˜

˜

˜

+ ei(v,Du F ) (ei(v,Du Du F ) − 1)ei(v,Du Du F ) ˜

˜

˜

(ei(v,Du Du Du F ) − 1). Repeating this argument, we have ˜ j D˜ u (ei(v,Du F ) − 1) =



j0 ,...,jj

eiZu,u

j 

˜ jl ˜

(ei(v,Dul Du F ) − 1),

l=0

where jl , l = 0, . . . , j are nonnegative integers satisfying jl ≤ j and j0 +· · ·+jj ≥

j0 ,...,jj are sums j . ul , l = 0, . . . , j are subsets of u satisfying l ul = u. Further, Zu,u

208

5 Malliavin Calculus j

of real functionals written as (v, D˜ ul D˜ u F ), jl ≤ jl − 1, ul ⊂ ul . The summation l

j0 ,...,jj

is taken for all such j0 , . . . , jj and u0 , . . . , uj . Then it holds that |eiZu,u Therefore, (5.75) is dominated by a sum of terms & j

$ E

l=0 |e

A(ρ)j +1

≤ ϕ(ρ)

j

i(v,D˜ ull D˜ u F )

− 1| γA(ρ) (u)γA(ρ) (u) $& j

j +1 2

sup

l=0 |e

E

'p

%1

p

j

mA(ρ) (du)mA(ρ) (du) j

i(v,D˜ ull D˜ u F )

− 1|

In the above we used the equality γAρ (u)γA(ρ) (u) = ϕ(ρ)− we have the inequality j

l=0 |e

j

i(v,D˜ ull D˜ u F )

− 1|

γ (u)γ (u)

'p % 1

p

.

γ (u)γ (u)

(u,u)∈A(ρ)j +1

| = 1.

˜ j0 ˜

j +1 2

(5.76)

γ (u)γ (u). Further,

˜ jl ˜

|ei(v,Du0 Du F ) − 1|  ρ|ei(v,Dul Du F ) − 1| ≤ , γ (u0 )γ (u) γ (ul )γ (u) j

l=1

j j if γ (u) ≤ ρ. In fact, since u ⊂ l=0 ul , we have γ (u) ≥ l=0 γ (ul ) and further, ρ γ (u) ≥ 1 holds for u ∈ A(ρ). Therefore, applying Hölder’s inequality, (5.76) is dominated by

ϕ(ρ)

j +1 2

sup

E

%1 $ i(v,D˜ uj0 D˜ u F ) j ˜ jl ˜ 0 − 1| p   ρ|ei(v,Dul Du F ) − 1| p p |e γ (u0 )γ (u)

(u,u)∈A(ρ)j +1

≤ ϕ(ρ)

j +1 2

γ (ul )γ (u)

l=1

I0 · I1 · · · Ij ,

(5.77)

where I0 :=

Il :=

$ sup

E

(u0 ,u)∈A(ρ)j +1

sup

E

|ei(v,Du0 Du F ) − 1| (j +1)p γ (u0 )γ (u) ˜ j0 ˜

%

1 (j +1)p

,

% 1 $ i(v,D˜ ujl D˜ u F ) l − 1| (j +1)p (j +1)p ρ|e

(ul ,u)∈A(ρ)j +1

γ (ul )γ (u)

˜ jl ˜

,

l = 1, . . . , j.

Since |ei(v,Du0 Du F ) − 1| ≤ |(v, D˜ u00 D˜ u F )| holds, we have ˜  I0 ≤ |v|DF 0,j0 ,(j +1)p , Therefore, (5.76) is dominated by

j

˜  Il ≤ ρ|v|DF 0,jl ,(j +1)p ,

l = 1, . . . , j.

5.7 Nondegenerate Poisson Functionals

209

c|v|(j +1) ρ j ϕ(ρ)

j +1 2

(j +1) ˜  (DF . 0,j,(j +1)p )

(5.78)

Hence (5.75) is also dominated by (5.78) (with a different constant c). Finally, summing up the above inequality for j = 0, . . . , n, we get the inequality of the lemma.  

5.7 Nondegenerate Poisson Functionals ˜ ∞ )d . We are interested in the existence of the smooth Let F be an element of (D density of the law of F . Conditions needed could be related to ‘Malliavin covariance’. The Malliavin covariance of a smooth d-dimensional Poisson functional ˜ ∞ )d conditioned to the set A(ρ) is defined by F ∈ (D R˜ ρF =

1 ϕ(ρ)



D˜ u F (D˜ u F )T n(du),

(5.79)

A(ρ)

where ϕ(ρ) = n(A(ρ)). Set R˜ ρF (θ ) = (θ, R˜ ρF θ ). F is called nondegenerate if R˜ ρF (θ ) are invertible a.s., inverses belong to L˜ ∞ for any θ ∈ Sd−1 and satisfy  sup |R˜ ρF (θ )−1 |0,n,p < ∞,

∀n, p,

(ρ,θ)

where the supremum is taken for all 0 < ρ < ρ0 and θ ∈ Sd−1 . For a Wiener functional G, we saw in Lemma 5.3.1 that if G ∈ D∞ is invertible and the inverse G−1 belongs to L∞− , then G−1 is a smooth Wiener functional belonging to D∞ . We remark the similar fact for Poisson functionals G. ˜ ∞ is invertible and G−1 Lemma 5.7.1 Suppose that a Poisson functional G ∈ D ∞ −1 ∞ ˜ . Then G ∈ D ˜ . Further, for any n, p, there is a positive constant belongs to L cn,p such that G−1 0,n,p ≤ cn,p (1 + G0,n,2qp )q (1 + |G−1 |0,n,2qp )q 





(5.80)

holds for any G with the above property, where q = 2n − 1. The proof is not simple. It will be given in a more general form in Theorem 5.10.2. By the above lemma, the nondegenerate condition for a Poisson functional F ∈ ˜ ∞ )d is equivalent to the condition (D sup R˜ ρF (θ )−1 0,n,p < ∞, 

(ρ,θ)

∀n, p.

210

5 Malliavin Calculus

In applications, however, it is not easy to handle the above nondegenerate condition. We are interested in the limit of the Malliavin covariances R˜ ρF as ρ → 0. For this, let us introduce a regularity condition for a Poisson functional. It is convenient to set D˜ t,0 F = 0. Then the operator D˜ t,z F (= D˜ (t,z) F ) is defined for any (t, z). A Poisson functional F is called regular if it satisfies the following. Condition (R) D˜ t,z F is twice continuously differentiable with respect to z =   1 (z , . . . , zd ) ∈ Rd a.s. Set 

Λt (F ) =

d 



sup |∂zi D˜ t,z F | +

i=1 |z|≤1

d 

sup |∂zi ∂zj D˜ t,z F |.

i,j =1 |z|≤1

(5.81)



Then |Λ(F )|0,n,p < ∞ holds for any n and p.

˜ ∞ )d We shall rewrite the Malliavin covariance of F in the case where F ∈ (D is regular. Since the measure n(dt dz) is equal to dtν(dz), the covariance can be written as    F ˜ Rρ = D˜ t,z F D˜ t,z F T νA0 (ρ) (dz) dt, T

A0 (ρ)

where νA0 (ρ) is a measure on A0 (ρ) given by ϕ(ρ)−1 ν(dz). Set ∂ D˜ t,0 F = (∂z1 D˜ t,z F, . . . , ∂zd  D˜ t,z F )|z=0 .

(5.82)

  We assume that the family of matrices Γρ = A0 (ρ) zi zj νA0 (ρ) (dz) converges to a matrix Γ0 as ρ → 0. Then R˜ ρF converges to K F , where  KF =

T

∂ D˜ t,0 F Γ0 (∂ D˜ t,0 F )T dt.

(5.83)

The functional K F is called the Malliavin covariance of F at the center. Further, F is called nondegenerate at the center if K F (θ ) ≡ (θ, K F θ ) are invertible for any θ ∈ Sd−1 and inverses satisfy sup |K F (θ )−1 |0,n,p < ∞, 

∀n, p.

(5.84)

θ

Here the supremum for θ is taken in the set Sd−1 . Example We give a simple example where the Malliavin covariance at the center is computed directly. Let F = (F 1 , . . . , F d ) be a linear functional of Poisson random measure written as F i = U hi (s, z)N˜ D (ds dz). We saw in Sect. 5.4 the equality D˜ (t,z) F = h(t, z) a.e. Therefore,

5.7 Nondegenerate Poisson Functionals

R˜ ρF =

  T

211

 h(t, z)h(t, z)T νA0 (ρ) (dz) dt.

If h(s, z) is differentiable in z, we have ∂ D˜ t,0 F = ∂h(t, z)|z=0 = (hij (t)) ≡ H (t) a.e. Consequently, the Malliavin covariance of F at the center is given by  KF =

T

H (t)Γ0 H (t)T dt.

It is a non-random matrix. Then K F ◦ εu+ = K F holds a.s. for any u. Therefore,  if the matrix K F is positive definite (invertible), then |K F (θ )−1 |0,n,p = K F (θ )−1 . Hence F is nondegenerate at the center The law of F is infinitely divisible and its characteristic function is given by ψ(v) = exp

  T

Rd



  ei(v,h(s,z)) − i(v, h(s, z)1D (z))ν(dz) ds .

We can show similarly to the proof of Lemma 1.2.1 the inequality −R

  T

 Rd0



    v v j i ij ei(v,h(s,z))−1−i(v, h(s, z))1D (z) ν(dz) ds ≥ α α K , 2 |v| 2 |v| i,j

where K = T H (s)Γ H (s)T ds. Since {K } are uniformly positive definite by our assumption, inequality |E[ei(v,F ) ]| ≤ c2 exp{−c1 |v|2−α } holds with positive constants c1 and c2 , as in Lemma 1.2.1. Then the law of F has a C ∞ -density. ˜ ∞ )d is regular and is nondegenerate at the center, We want to show that if F ∈ (D its law has a smooth density: However, a direct proof of the fact is difficult. As a first step, we will introduce another technical nondegenerate condition. Associated ˜ ∞ )d , we define a family of Poisson functionals with parameter (ρ, v) ∈ with F ∈ (D (0, 1) × (Rd \ {0}): 1 Q (ρ, v) = 2 |v| ϕ(ρ)



˜

|ei(v,Du F ) − 1|2 n(du).

F

(5.85)

A(ρ)

Let δ be a constant satisfying 1 < δ < 2/α, where α is the exponent of the Lévy measure. We fix it and define for (ρ, θ ) ∈ (0, ρ0 ) × Sd−1 , 1

QFρ (θ ) = QF (ρ, ρ − δ θ ).

(5.86)

˜ ∞ )d is called nondegenerate with respect to QFρ or A Poisson functional F of (D simply δ-nondegenerate if {QFρ (θ )} are invertible a.s. for any (ρ, θ ) and inverses satisfy

212

5 Malliavin Calculus

sup |QFρ (θ )−1 |0,n,p < ∞, 

∀n, p,

(5.87)

(ρ,θ)

where the supremum is taken for 0 < ρ < ρ0 and θ ∈ Sd−1 . We have defined three types of nondegenerate condition for a d-dimensional smooth Poisson functional F . In the remainder of this section, we will show that if F is δ-nondegenerate for some 1 < δ < 2/α, its law has a rapidly decreasing C ∞ density. In the next section, we will show that these three nondegenerate conditions are equivalent. The next one corresponds to Proposition 5.3.1 for Wiener functionals. Proposition 5.7.1 For any n, p, there exists a positive constant cn,p such that for ˜ ∞ )d , any δ-nondegenerate F of (D QFρ (θ )−1 0,n,p 

(5.88)

 (n+1)2n+2  2n  ˜  1 + |QFρ (θ )−1 |0,n,2n+2 p ≤ cn,p 1 + DF 0,n,(n+1)2n+2 p holds for all (ρ, θ ). Proof We will show that for any n ∈ N, p ≥ 2, there exists a positive constant c such that 2(n+1) ˜  sup QFρ (θ )0,n,p ≤ c(1 + DF . 0,n,2(n+1)p ) 



(5.89)

(ρ,θ)

Note that QFρ (θ ) is rewritten as QFρ (θ ) =

1 |v|2



˜

A(ρ)

|ei(v,Du F ) − 1|2 mA (du), γ (u)2

(5.90)

˜ where ρ = |v|−δ . Since |ei(v,Du F ) − 1|2 ≤ |(v, D˜ u F )|2 , we have 1

 ˜  . QFρ (θ )0,0,p = E[QFρ (θ )p ] p ≤ DF 0,0,p

(5.91)

Further, if 1 ≤ j ≤ n and p ≥ 2,

p  1  j E D˜ u QFρ (θ ) p γ (u)

1 1 ≤ 2 |v| ϕ(ρ)

 A(ρ)

p  1  j ˜ E D˜ u (|ei(v,Du F ) − 1|2 ) p mA (du), γ (u)γ (u)2

where ρ = |v|−δ . We can show similarly to the proof of Lemma 5.6.2 that there is a positive constant c such that the above is dominated by

5.7 Nondegenerate Poisson Functionals

c

213

|v|2(j +1) ρ 2j 2(j +1) 2(j +1) ˜  ˜  (1 + DF ≤ c (1 + DF , 0,j,2(j +1)p ) 0,j,2(j +1)p ) |v|2

since |v|ρ ≤ 1. Therefore we get (5.89). Now we will apply (5.80) for G = QFρ (θ ). Then (5.88) follows.

 

We are now in the position of establishing the smooth density of the law of a δ-nondegenerate Poisson functional. Theorem 5.7.1 ([46]) For any N ∈ N, there exist n ≥ 1, p ≥ 2 and a positive ˜ ∞ )d , constant c such that for any δ-nondegenerate F of (D

 

E ei(v,F ) G ≤

c |v|

(1− αδ 2 )N

(5.92)  N   n+1 ˜  (1 + DF sup QFρ (θ )−1 0,n,2p G0,n,p 0,n,2(n+1)p ) (ρ,θ)

˜ ∞. hold for all |v| ≥ 1 and G ∈ D Proof We define a complex valued random field {Zu = ZuF,v , u ∈ U} associated with a given δ-nondegenerate functional F = (F 1 , . . . , F d ) and v ∈ Rd with |v| ≥ 1: Zu = ZuF,v =

1 ˜ · χρ (u)(e−i(v,Du F ) − 1) · QF (ρ, v)−1 , |v|2 ϕ(ρ)

(5.93)

˜ ∞ for any F, v. We will apply the adjoint where ρ = |v|−δ . It is an element of D T equation (5.43) for Zu = Zu G and ei(v,F ) . Then we have    ˜ =E E ei(v,F ) δ(ZG) =E





 D˜ u (ei(v,F ) )Zu Gn(du)

(5.94)

 ˜ ei(v,F ) (ei(v,Du F ) − 1)Zu n(du) · G

   ˜ = E ei(v,F ) G (ei(v,Du F ) − 1)Zu n(du) = E[ei(v,F ) G], ˜ ˜ since (ei(v,Du F ) − 1)Zu n(du) = 1. Therefore, setting L(G) = δ(ZG), we have the iteration formula E[ei(v,F ) G] = E[ei(v,F ) L(G)] = · · · = E[ei(v,F ) LN (G)],

(5.95)

for N = 1, 2, . . .. Consequently, we get by using Theorem 5.6.1 and Hölder’s inequality

214

5 Malliavin Calculus N −1 ˜ |E[ei(v,F ) G]| ≤ LN (G)0,0,2 = δ(ZL (G))0,0,2

≤ cZLN −1 (G)0,1,2 ≤ c Z0,1.4 LN −1 (G)0,1,4 , where  0,n,p =  0,n,p;A(ρ) . Repeating this argument, there are increasing sequences cN , nN , pN , N = 1, 2, . . . such that the inequality |E[ei(v,F ) G]| ≤ cN Z0,n1 ,p1 · · · Z0,nN ,pN G0,nN ,pN ≤ cN ZN 0,nN ,pN G0,nN ,pN

(5.96)

holds, since norms  0,n,p are nondecreasing with respect to n, p. We will compute Z0,n,p . Apply Hölder’s inequality to (5.93). Then we have Z0,n,p ≤

c

˜

|v|2 ϕ(ρ)

χρ (e−i(v,DF ))−1)0,n,2p QF (ρ, v)−1 0,n,2p .

Set ρ = |v|−δ . Then we have from Lemma 5.6.2 ˜

n+1 ˜  . χρ (e−i(v,DF ) − 1)||0,n,2p ≤ c|v|ϕ(ρ) 2 (1 + DF 0,n,2(n+1)p ) 1



Further, we have QF (ρ, v)−1 0,n,2p ≤ sup QFρ (θ )−1 0,n,2p . 

(ρ,θ)

Furthermore,

1 1 |v|ϕ(|v|−δ ) 2

≤

c |v|

(1− αδ 2 )

holds for any |v| ≥ 1, since ϕ(|v|−δ ) ≥ c|v|−αδ

by the order condition. Therefore we get Z0,n,p ≤

c |v|

(1− αδ 2 )

 n+1 ˜  (1 + DF sup QFρ (θ )−1 0,n,2p . 0,n,2(n+1)p ) (ρ,θ)

Substitute the above in (5.96), then we get the inequality of the theorem.

 

As an immediate consequence of Theorem 5.7.1, we have the next corollary. ˜ ∞ )d is δ-nondegenerate, the law of F weighted by Corollary 5.7.1 If F of (D ˜ ∞ has a rapidly decreasing C ∞ -density. The density function and its G ∈ D derivatives are given by the Fourier inversion formula (5.34).

5.8 Equivalence of Nondegenerate Conditions In Sect. 5.7, we defined three types of nondegenerate Poisson functionals. In this section, we show the equivalence of these nondegenerate conditions.

5.8 Equivalence of Nondegenerate Conditions

215

˜ ∞ )d . Theorem 5.8.1 ([46]) Let F be a regular functional of (D 1. If F is δ-nondegenerate for some 1 < δ < 2/α, then F is nondegenerate at the center. Further, for any n, p the inequality sup |K F (θ )−1 |0,n,p ≤ sup |QFρ (θ )−1 |0,n,p 



θ

(5.97)

(ρ,θ)

˜ ∞ )d . holds for all regular δ-nondegenerate functionals F ∈ (D 2. If F is nondegenerate at the center, F is δ-nondegenerate for any 1 < δ < 2/α. Further, for any n, p there is a positive constant c such that the inequality 

δ |QFρ (θ )−1 |0,n,p ≤ cCn,p (F )|K F (θ )−1 |0,n,2p 



(5.98)

holds for any ρ, θ and F mentioned above. Here 



1

δ Cn,p (F ) ≤ (1 + |Λ(F )|0,n,8(2+δ  )p )2(2+δ ) (1 + |K F (θ )−1 |0,n,8p ) 2 , 



(5.99)

where Λt (F ) is the Poisson functional given by (5.81) and δ  is the conjugate of δ. Proof of the first half. Suppose that F is δ-nondegenerate. Consider the integrand of QFρ (v). It holds that ˜ |ei(v,Du F ) − 1|2 ≤ |(v, D˜ u F )|2 = v∂ D˜ t,0 F zzT ∂ D˜ t,0 F T v T + O(|z|3 ),

a.e. dndP . Here we used the Taylor expansion D˜ t,z F = ∂ D˜ t,0 F z + O(|z|2 ). Integrating the above by n(du) on the set T × A0 (ρ), we get   1 ˜ |ei(v,Dt,z F ) − 1|2 dtν(dz) |v|2 ϕ(ρ) T A0 (ρ)  v v = (∂ D˜ t,0 F )Γρ (∂ D˜ t,0 F )T dt ( )T + O(ρ), |v| T |v|

a.s.

Therefore setting θ = v/|v|, we have lim sup QFρ (θ ) ≤ K F (θ ), ρ→0

or

lim inf QFρ (θ )−1 ≥ K F (θ )−1 , ρ→0

a.s.

Operating the transformation εu+ on both sides of the above, we get again lim inf QFρ (θ )−1 ◦ εu+ ≥ K F (θ )−1 ◦ εu+ , ρ→0

a.e. dnj dP .

216

5 Malliavin Calculus

Therefore, we have |K F (θ )−1 |0,n,p ≤ supρ |QFρ (θ )−1 |0,n,p for all θ ∈ Sd−1 . Then the first assertion of the theorem follows.   



The proof of the latter half of the theorem is long. Our discussion is close to [46]. It will be completed after discussing Lemmas 5.8.1, 5.8.2, 5.8.3, and 5.8.4. We will introduce two other matrix-valued Poisson functionals. We will take any δ satisfying ˜ ∞ )d . For 0 < ρ < ρ0 and v = 0 1 < δ < 2/α. Let F be a regular functional of (D F we define Qρ (θ ) by (5.86) and further, SρF

1 = ϕ(ρ)

KρF

1 = ϕ(ρ)



D˜ 1u F (D˜ u F ) A(ρ)∩{|D˜ u F |≤ρ δ }

 

T A0 (ρ)

T

(5.100)

n(du).

(∂ D˜ t,0 F )z · zT (∂ D˜ t,0 F )T n(dt dz).

(5.101)

We set further, SρF (θ ) = (θ, SρF θ ),

KρF (θ ) = (θ, KρF θ ).

Discussions will be divided into the following three steps: 1. We compare two norms |QFρ (θ )−1 |0,n,p and |SρF (θ )−1 |0,n,p . See Lemma 5.8.1. 



2. We compare two norms |KρF (θ )−1 |0,n,p and |K F (θ )−1 |0,n,p . See Lemma 5.8.2. 



3. We compare two norms |SρF (θ )−1 |0,n,p and |Kρ (θ )−1 |0,n,p . See Lemmas 5.8.3 and 5.8.4. 



Lemma 5.8.1 Suppose that SρF (θ ) are invertible a.s. for any ρ, θ . Then QFρ (θ ) are also invertible a.s. Further, for any n, p there is a positive constant c1 such that the inequality |QFρ (θ )−1 |0,n,p ≤ c1 |SρF (θ )−1 |0,n,p 



(5.102)

holds for any ρ, θ . ˜

Proof Note that the inequality |ei(v,Du F ) − 1|2 ≥ c|(v, D˜ u F )| holds with some constant c > 0 on the set |D˜ u F | < |v|−1 . Then we have inequalities QFρ (θ ) ◦ εu+ ≥ cSρF (v) ◦ εu+ ,

a.e. dnj dP .

Therefore, if SρF (v) ◦ εu+ are invertible a.e., then QFρ (θ ) ◦ εu+ are also invertible a.e. and satisfy     E (QFρ (θ ) ◦ εu+ )−p ≤ c−p E (SρF (θ ) ◦ εu+ )−p ,

a.e. dnj .

Taking the essential supremum with respect to u and summing up these for j = 0, . . . , n, we get the inequality of the lemma.  

5.8 Equivalence of Nondegenerate Conditions

217

Lemma 5.8.2 Suppose that K F (θ ) are invertible a.s. for any θ ∈ Sd−1 . Then KρF (θ ) are also invertible a.s. for any 0 < ρ < ρ0 , θ ∈ Sd−1 . Further, for any n, p there exist two positive constants c2 < c3 such that the inequality c2 |K F (θ )−1 |0,n,p ≤ |KρF (θ )−1 |0,n,p ≤ c3 |K F (θ )−1 |0,n,p 





(5.103)

holds for any ρ, θ . Proof Let λρ and Λρ be the minimum and the maximum eigenvalues of Γρ , respectively. Since lim infρ→0 Γρ ≥ Γ0 and Γ0 is nondegenerate, there exists 0 < ρ1 < ρ0 such that 0 < inf0 0. Then, for any N ∈ N, there exist m, n ∈ N, p > 4 and a positive constant C such that for any d-dimensional δ¯ ∞ )d , nondegenerate Wiener–Poisson functional F in (D

5.11 Nondegenerate Wiener–Poisson Functionals

 

E ei(v,F ) G ≤

235

C  n+1 ˜  (DF m,n,p + (1 + DF ) m,n.(n+1)p ) |v|N γ0 N ¯ Fρ (θ )−1 m,n,p Gm,n,p (5.150) × sup Q (ρ,θ)

¯ ∞. hold for all |v| ≥ 1 and G ∈ D Proof We will define a d  -dimensional stochastic process Xt and a random field Zu associated with a given δ-nondegenerate smooth Wiener–Poisson functional F : ˜

(e−i(v,Du F ) − 1)1A(ρ) (u) , |v|2 ϕ(ρ)Q¯ F (ρ, v) (5.151) where ρ = |v|−δ . Adjoint formulas (5.117) and (5.118) are extended to complex valued Wiener–Poisson functionals. Then we have   ˜ E ei(v,F ) {δ(XG) + δ(ZG)} Xt = XtF,v =

=E

Zu = ZuF,v =

   D˜ u (ei(v,F ) )Zu Gn(du) Dt (ei(v,F ) )Xt G dt + E



=E

−i(v, Dt F ) , ¯ F (ρ, v) |v|2 Q



   ˜ ei(v,F ) i(v, Dt F )Xt G dt + E ei(v,F ) (ei(v,Du F ) − 1)Zu Gn(du)

  ei(v,F ) G 2 F ) dt (v, D t ¯ F (ρ, v) |v|2 Q    ei(v,F ) G i(v,D˜ u F ) 2 +E |e − 1| n(du) |v|2 ϕ(ρ)Q¯ F (ρ, v) A(ρ)  i(v,F )  =E e G. =E



˜ Therefore, setting L(G) = δ(XG) + δ(ZG), we have the iteration formula; E[ei(v,F ) G] = E[ei(v,F ) L(G)] = · · · = E[ei(v,F ) LN (G)],

(5.152)

N −1 (G)). Then, ˜ for N = 1, 2, . . .. Now, note LN (G) = δ(XLN −1 (G)) + δ(ZL using Theorem 5.10.1 and Hölder’s inequality, we have N −1 ˜ (G))0,0,2 LN (G)0,0,2 ≤ δ(XLN −1 (G))0,0,2 + δ(ZL

≤ c1 (XLN −1 (G)1,0,2 + ZLN −1 (G)0,1,2 ) ≤ c2 (X1,0,4 + Z0,1,4 )LN −1 (G)1,1,4 ,

236

5 Malliavin Calculus

where  m,n,p =  m,n,p;A(ρ) . Repeating this argument inductively, there exist mN , nN , pN and cN such that LN (G)0,0,2 ≤ cN (XmN ,nN ,pN + ZmN ,nN ,pN )N GmN ,nN ,pN . Further, by Hölder’s inequality Xm,n,p + Zm,n,p   1 1 ˜ ) −i(v,DF χ ≤ i(v, DF ) + (e − 1) m,n,2p ρ m,n,2p |v|2 |v|2 ϕ(ρ) ¯ F (ρ, v)−1 m,n,2p . × Q We have 

i(v, DF )m,n,2p ≤ c|v|DF m,n,2p , ˜

(5.153)

n+1 ˜  e−i(v,DF ) − 1m,n,2p ≤ c|v|ϕ(ρ) 2 (1 + DF , m,n,2(n+1)p ) 1



¯ F (ρ, v)−1 m,n,2p ≤ sup Q ¯ Fρ (θ )−1  Q m,n,2p . (ρ,θ)

The first inequality is immediate from the definition of the norm. The second inequality follows from Lemma 5.10.2. The third one is obvious. Then we get Xm,n,p + Zm,n,p   1 1  ˜  DF m,n,2p + (1 + DF )n+1 ≤c 1 m,n,2(n+1)p |v| |v|ϕ(ρ) 2

(5.154)

¯ Fρ (θ )−1  × sup Q m,n,2p , (ρ,θ)

where ρ = |v|−δ . Since dominated by

1 |v|

≤

c1 |v|ϕ(|v|−δ )

≤

c2 |v|γ0 ,

the last term of the above is

c  n+1 ˜  ¯ Fρ (θ )−1  (DF m,n,2p + (1 + DF ) sup Q m,n,2p . m,n,2(n+1)p ) |v|γ0 (ρ,θ) Then we get the inequality of the theorem, rewriting 2p as p.

 

¯ ∞ )d is δ-nondegenerate. Then for any G ∈ Corollary 5.11.1 Suppose that F in (D ∞ ¯ D , the law of F weighted by G has a rapidly decreasing C ∞ -density. Further, the density function fG (x) is given by the Fourier inversion formula (5.34). Now we will discuss two other nondegenerate conditions. A Wiener–Poisson functional F is called regular if it satisfies Condition (R) introduced in Sect. 5.7. ¯ ∞ )d . We will define its Malliavin covariance at Let F be a regular functional of (D the center by

5.11 Nondegenerate Wiener–Poisson Functionals

K¯ F =





T

Dt F (Dt F )T dt +

T

237

(∂ D˜ t,0 F )Γ0 (∂ D˜ t,0 F )T dt,

(5.155)

where ∂ D˜ t,0 F is defined by (5.82). For θ ∈ Sd−1 , we set K¯ F (θ ) = (θ, K¯ F θ ). If {K¯ F (θ ), θ ∈ Sd−1 } are invertible and inverses satisfy  sup |K¯ F (θ )−1 |0,n,p < ∞,

∀n, p,

(5.156)

θ

then F is called nondegenerate at the center. ¯ ∞ )d . Theorem 5.11.2 Let F be a regular functional belonging to (D 1. If F is δ-nondegenerate for some 1 < δ < 2/α, then F is nondegenerate at the center. Further, it holds for any n, p that ¯ Fρ (θ )−1 | sup |K¯ F (θ )−1 |0,n,p ≤ sup |Q 0,n,p . 



θ

(5.157)

(ρ,θ)

2. If F is nondegenerate at the center, then F is δ-nondegenerate for any 1 < δ < 2/α. Further, for any n, p there is a positive constant c such that the inequality 

δ ¯ Fρ (θ )−1 | ¯ F −1 sup |Q 0,n,p ≤ cCn,p (F ) sup |K (θ ) |0,n,2p , 



(5.158)

θ

(ρ,θ)

¯ ∞ )d , which is regular and nondegenerate at the center, holds for any F ∈ (D  δ where Cn,p (F ) is given by (5.99) replacing K F (θ ) by K¯ F (θ ). Proof The first assertion and the inequality (5.157) can be verified in the same way as in the first assertion of Theorem 5.8.1. We will prove the second assertion by a method similar to the proof of Theorem 5.8.1. We set S¯ρF = R F + SρF and K¯ ρF = R F + KρF , where SρF and KρF are defined by (5.100) and (5.101), respectively. Then we can verify the followings: ¯ Fρ (θ ) are also 1. Suppose that S¯ρF (θ ) are invertible a.s. for any ρ, θ . Then Q invertible a.s. for any ρ, θ . Further, there is a positive constant c1 such that the inequality ¯ F −1  ¯ Fρ (θ )−1 | |Q 0,n,p ≤ c1 |Sρ (θ ) |0,n,p

(5.159)

holds for any ρ, θ and n, p (Lemma 5.8.1). 2. Suppose that K¯ F (θ ) are invertible a.s. for any θ ∈ Sd−1 . Then K¯ ρF (θ ) are also invertible a.s. for any (ρ, θ ). Further, there are two positive constants c2 , c3 such that for any (ρ, θ ), the inequalities    c2 |K¯ F (θ )−1 |0,n,p ≤ |K¯ ρF (θ )−1 |0,n,p ≤ c3 |K¯ F (θ )−1 |0,n,p

hold for any n, p (Lemma 5.8.2).

(5.160)

238

5 Malliavin Calculus

¯ ∞ )d is regular and nondegenerate at the center, functionals {S¯ρF (θ )} are 3. If F ∈ (D invertible. Further, it holds for any p > 1 that   1   E (S¯ρF (θ ) ◦ εu+ )−p ≤ c5 Bpδ (F )E (K¯ ρF (θ ) ◦ εu+ )−2p 2 ,

∀ρ, θ, u,

(5.161)



where Bpδ (F ) is given by (5.108). Indeed, since the equality S¯ρF (θ ) − K¯ ρF (θ ) = SρF (θ ) − KρF (θ ) holds, discussions of Lemma 5.8.3 can be applied to the present case and we get 

P (!F (θ ) ◦ εu+ < ) ≤ c4 Aδp (F ) p ,

∀0 < < 1, θ ∈ Sd−1 , u ∈ Uk (5.162)  where Aδp (F ) is given by (5.105) (replacing K F by K¯ F ). Then we get (5.161) similarly to the proof of Lemma 5.8.4. Now we get from (5.161),   δ |S¯ρF (θ )−1 |0,n,p ≤ c6 Cn,p (F )|K¯ ρF (θ )−1 |0,n,2p .

(5.163)  

Combining this with (5.160), the inequality of the theorem follows. Let us consider another family of Malliavin covariances conditioned to A(ρ).

R¯ ρF =



1 Dt F (Dt F ) dt + ϕ(ρ) T



T

˜ u (D˜ u F )T n(du). DF

(5.164)

A(ρ)

Suppose that R¯ ρF (θ ) are invertible and inverses satisfy  sup |R¯ ρF (θ )−1 |0,n,p < ∞,

∀n, p.

(5.165)

(ρ,θ)

Then F is said to be nondegenerate. The following can be verified similarly to the case of Poisson functionals discussed in Sect. 5.8, Theorem 5.8.2. ¯ ∞ )d . Theorem 5.11.3 Let F be a regular functional belonging to (D 1. If F is nondegenerate, it is nondegenerate at the center. Further, we have for any n, p,   sup |K¯ F (θ )−1 |0,n,p ≤ sup |R¯ ρF (θ )−1 |0,n,p . θ

(ρ,θ)

(5.166)

5.12 Compositions with Generalized Functions

239

2. If F is nondegenerate at the center, it is nondegenerate. Further, for any n, p there is a positive constant c such that 0 sup |R¯ ρF (θ )−1 |0,n,p ≤ cCn,p (F ) sup |K¯ F (θ )−1 |0,n,2p , 

(ρ,θ)



(5.167)

θ

0 (F ) is given by (5.99) with δ  = 0 replacing for all nondegenerate F , where Cn,p F F ¯ K (θ ) by K (θ ).

From Theorems 5.11.2 and 5.11.3, we have the following, ¯ ∞ )d : The following Theorem 5.11.4 Let F be a regular functional belonging to (D statements are equivalent. (i) (ii) (iii) (iv)

F F F F

is δ-nondegenerate for some 1 < δ < 2/α. is δ-nondegenerate for any 1 < δ < 2/α. is nondegenerate at the center. is nondegenerate.

Remark We will discuss briefly the density problem in the case where the Lévy measure ν may not satisfy the order condition. A typical case is that the Lévy ¯ ∞ )d , we consider measure is of finite mass. For a Wiener–Poisson functional F ∈ (D F T the Malliavin covariance R = T Dt F · DFt dt. The functional F is called strongly nondegenerate if supθ E[R F (θ )−p ] < ∞ holds for any p ≥ 2. Then we can show that for any N ∈ N, there exist m ∈ N, p ≥ 2 and c > 0 such that the inequality



E[ei(v,F ) G] ≤

N c  DF m,0,p sup R F (θ )−1 m,0,p Gm,0,p N |v| θ

(5.168)

holds for all |v| ≥ 1 for any strongly nondegenerate Wiener–Poisson functional ¯ ∞ )d and G ∈ D ¯ ∞ . The proof can be carried out similarly to the proof of F of (D Theorem 5.3.1, replacing norms  m,p by  m,0,p . Consequently, if F is a strongly nondegenerate Wiener–Poisson functional in the above sense, its law weighted by G has a rapidly decreasing C ∞ -density.

5.12 Compositions with Generalized Functions ¯ ∞ )d and let f (x) be a smooth Let F be a smooth Wiener–Poisson functional of (D d function on R . Then the composite f (F ) is a smooth Wiener–Poisson functional. In this section, we will define the composite of a generalized function f and a nondegenerate smooth Wiener–Poisson functional F . The composite f (F ) will no longer be a smooth Wiener–Poisson functional. It should be a generalized Wiener– Poisson functional.

240

5 Malliavin Calculus

Let us recall the Schwartz distribution or generalized function on a Euclidean space. Let S = S(Rd ) be the space of complex valued rapidly decreasing C ∞ functions on Rd . It is called the Schwartz space. We define semi-norms by |f |n =

   

(1 + |x|2 )j |∂ k f (x)|

2

1 dx

2

.

(5.169)

|k|+j ≤n

Denote the completion of S with respect to the norm | · |n by S n . For a negative integer −n, we define the norm by |f |−n =

sup

g∈S n ,|g|n ≤1

|(f, g)|,

(5.170)

where (f, g) = f (x)g(x) dx. The completion

of S by the norm | · |−n is denoted by S −n . We set S ∞ = n>0 S n and S −∞ = n>0 S −n . Then it holds that S ∞ = S and S −∞ = S  is the space of tempered distributions. Let f (x) be an integrable function on Rd . We define the Fourier transform and the inverse Fourier transform of f by  1 d  2 e−i(v,x) f (x) dx, d 2π R  1 d  2 ˇ (v) = fˇ(v) = Ff ei(v,x) f (x) dx, d 2π R

Ff (v) = fˆ(v) =

respectively. Proposition 5.12.1 The Fourier transform F; S → S is a linear, one to one and onto map. Further, it satisfies the following properties: ˇ = FFf ˇ 1. It holds that F Ff = f for any f ∈ S. 2. (Parseval’s equality) We have f g¯ dx = Ff Fg dv for any f, g ∈ S. 3. (Fourier inversion formula) Let i and j be multi-indexes of nonnegative integers. Then we have ˇ ))(x) (ix)i ∂ j f (x) = F(∂ i (v j Ff

(5.171)

for any f ∈ S. Proof The linear property of the Fourier transform is obvious from the definition of the transform. Then it is sufficient to prove (1)–(3) for a real function f . If f is a ˇ = f in density function of a signed measure μ, we have the inversion formula F Ff view of (1.5). Next, take the complex conjugate for both sides. Then we have f = ˇ = FFf ˇ ˇ = F Ff . Therefore the first assertion follows. Further, equalities F Ff ˇ FFf = f show that the map F : S → S is one to one and onto.

5.12 Compositions with Generalized Functions

241

By the Fubini theorem, we have 

  1 d   2 ei(v,x) g(v) f (x)g(x) dx = ˆ dv dx f (x) 2π    1 d    2 ˆ dv = fˆ(v)g(v) ˆ dv. = e−i(v.x) f (x) dx g(v) 2π  

Finally equality (5.171) follows from (1.7). 

Next, we will define generalized Wiener–Poisson functionals. Let  m,n,p be Sobolev norms on the Wiener–Poisson space defined in Sect. 5.10. We shall define p ¯ m,n,p , we set X, Y  = E[XY ]. The their dual norms. For F ∈ L¯ p−1 and G ∈ D  dual norm of  m,n,p is defined by F ∗m,n,p :=

sup ¯ m,n,p ,Gm,n,p ≤1 G∈D

|F, G|.



p

¯ m,n,p We denote by D the completion of L¯ p−1 by the norm  ∗m,n,p and we set ∗

∞ ¯∗ = ¯ m,n,p . We have D ¯ ∞ ⊂ L¯ ∞− ⊂ D ¯∞ ¯∞ D ∗ . Elements of D∗ m,n∈N,p>1 D∗ are called generalized Wiener–Poisson functionals. The bilinear form F, G is extended naturally to a generalized functional F and a smooth functional G. ¯ ∞ )d . Let f be a tempered distribution and let F be a nondegenerate element of (D We will define the composition of f and F , following the idea of Watanabe [117] and Hayashi–Ishikawa [37]. The definition is somewhat different from theirs. We prepare a lemma. Lemma 5.12.1 Let F be a δ-nondegenerate Wiener–Poisson functional belonging ¯ ∞ )d . Then for any N ∈ N there exist m, n ∈ N, p > 2 and a positive constant to (D CN such that f (F )∗m,n,p ≤ CN |f |−N

(5.172)

holds for all f ∈ S. Proof For the proof of the lemma, we will apply the Fourier transform. For a ¯ ∞ , consider the (signed) measure μG (A) = E[1A (F )G]. Its Fourier given G ∈ D d transform is equal to (2π )− 2 ψG (v), where ψG (v) is the characteristic function of the measure μG . It is rapidly decreasing with respect to v. Therefore the measure μG has a rapidly decreasing C ∞ -density function, which we denote by fG (y). Let f ∈ S. Then using Parseval equality, we have  E[f (F )G] =

Rd

f (y)fG (y) dy =

 1 d  2 fˆ(v)ψG (v) dv. 2π Rd

(5.173)

242

5 Malliavin Calculus

We shall compute the semi-norm |ψG |N . For a given N ∈ N, take N  ∈ N satisfying γ0 N  > d + 1 + N , where γ0 = 1 − αδ/2. Then by Theorem 5.11.1, there exist m, n, p and positive constants c1 and c2 such that for any |k| ≤ N , the inequality 

|∂vk ψG (v)| = |E[ei(v,F ) F k G]| ≤ c1 |v|−γ0 N Gm,n,p ≤ c2 (1 + |v|2 )−γ0

N 2

Gm,n,p

holds if |v| ≥ 2. Then for any j + |k| ≤ N , there is a positive constant cN such that 

(1 + |v| )

2 j

||∂vk ψG (v)|2 dv

1 2

≤ cN



(1 + |v|2 )−

d+1 2

1 dv

2

Gm,n,p

 ≤ cN Gm,n,p .  G Therefore we have |ψG |N ≤ cN m,n,p . We will prove (5.172). It holds that 

f (F )∗m,n,p = ≤

≤

sup ¯ ∞ ,Gm,n,p ≤1 G∈D

|E[f (F )G]|

(5.174)



 1 d 2 2π  1 d 2 2π

sup 

G;Gm,n,p ≤1

sup  G;Gm,n,p ≤1



Rd

ψG (v)fˆ(v) dv

|ψG |N |fˆ|−N .

Further, ˆ = sup |f, g| = |f |−N . |fˆ|−N = sup |f, g| = sup |fˆ, g| |g| ˆ N ≤1

|g| ˆ N ≤1

|g|N ≤1

Here, we used the fact that F; S → S is one to one and onto, and the Perceval equality. Therefore we get the inequality (5.172).   By the above the composition f (F ) can be extended to any f ∈ S −n .

lemma,  −n Since S = n>0 S , it is extended to any tempered distribution f . We set conventionally E[f (F ) · G] := f (F ), G. We summarize the composition as a theorem.

(5.175)

5.12 Compositions with Generalized Functions

243

¯ ∞ )d . Then for Theorem 5.12.1 Let F be a regular and nondegenerate element of (D any tempered distribution f , the composition f (F ) can be defined as an element of ¯∞ D ∗ . Further, for any N ∈ N, there exist m, n ∈ N, p > 2 and a positive constant CN such that f (F )∗m,n,p ≤ CN |f |−N ,

∀f ∈ S −N .

(5.176)

Further, it satisfies E[f (F ) · G] =

 1 d 2 (Ff, ψG ) 2π

(5.177)

¯ ∞ , where Ff is the Fourier transform of f . for any f ∈ S −∞ and G ∈ D So far we considered the composite of a tempered distribution Φ and a nondegenerate Wiener–Poisson functional F . However, if F is a Wiener functional, m,p the statement of Theorem 5.12.1 should be changed slightly. Let D∗ be the dual of the space of Wiener functionals Dm,p and let  ∗m,p be its norm. Let

m,p ∞ D∞ ∗ = m,p D∗ . Elements of D∗ is called generalized Wiener functionals. If f is a tempered distribution on Rd and F is a d-dimensional nondegenerate Wiener functional, the composition f (F ) is well defined as an element of D∞ ∗ . Further, the assertion of Theorem 5.12.1 is valid if we replace the inequality (5.176) by f (F )∗m,p ≤ CN |f |−N . We are particularly interested in the case where f is the delta function δx of the point x ∈ Rd . Corollary 5.12.1 Let F be a d-dimensional smooth nondegenerate Wiener func¯ ∞ )d . Let G be tional or a regular nondegenerate Wiener–Poisson functional of (D a smooth Wiener functional or smooth Wiener–Poisson functional, respectively. Let ψG (v) be the characteristic function of the law of F weighted by G. Then the density function fG (x) of the law of F weighted by G is represented by fG (x) = E[δx (F ) · G] =

 1 d  e−i(v,x) ψG (v) dv, d 2π R

(5.178)

where δx is the delta function at the point x. Finally, we remark that if F is nondegenerate, the conditional law P (G|F = x) is well defined for all x without ambiguity of measure 0. Indeed, it is given by P (G|F = x) = provided that E[δx (F )] = 0.

E[δx (F ) · G] , E[δx (F )]

(5.179)

244

5 Malliavin Calculus

Note The composite of Schwartz distribution and nondegenerate Wiener functional was first discussed by Watanabe [116, 117], using Sobolev type norm  m,p defined by Orenstein–Uhlenbeck generator (Note’ at the end of Sect. 5.2). The composite of Schwartz distribution and nondegenerate Wiener–Poisson functional is studied in Hayashi-Ishikawa [37] slightly different context.

Chapter 6

Smooth Densities and Heat Kernels

Abstract We discuss the existence of the smooth density of ‘nondegenerate’ diffusions and jump-diffusions determined by SDEs. We will use the Malliavin calculus studied in the previous chapter. In Sects. 6.1, 6.2, and 6.3, we consider diffusion processes. It will be shown that any solution of a continuous SDE defined in Chap. 3 is infinitely H -differentiable. Further, if its generator A(t) is elliptic (or hypo-elliptic), its transition probability Ps,t (x, E) has a density ps,t (x, y), which is a rapidly decreasing C ∞ -function of x and y. We will show further that the weighted transition function has also a rapidly decreasing C ∞ -function of x and y, and it is the heat kernel of the backward heat equation associated with the operator Ac (t), which were defined in Chap. 4. For the proof of these facts, we will apply the Malliavin calculus on the Wiener space discussed in Sects. 5.1, 5.2, and 5.3. In Sects. 6.4, 6.5, and 6.6, we will study jump-diffusions. We will show that if the generator of the jump-diffusion is ‘pseudo-elliptic’, then its weighted transition function has a smooth density. For the proof, we will apply the Malliavin calculus on the Wiener-Poisson space discussed in Sects. 5.9, 5.10, and 5.11. In Sects. 6.7 and 6.8, we discuss short-time estimates of heat kernels, applying the Malliavin calculus again. These facts will be applied to two problems. In Sect. 6.9, we consider the solution of SDEs with big jumps, for which the Malliavin calculus cannot be applied. Instead, we take a method of perturbation and show that the perturbation preserves the smooth density. In Sect. 6.10, we show the existence of the smooth density of the laws of the killed elliptic diffusion or the killed pseudoelliptic jump-diffusions. The density should be the heat kernel of the backward heat equation on the domain with the Dirichlet boundary condition.

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_6

245

246

6 Smooth Densities and Heat Kernels

6.1 H -Derivatives of Solutions of Continuous SDE Let us consider a continuous symmetric SDE on Rd defined on the Wiener space: d   

Xt = X0 +

t

k=0 t0

Vk (Xr , r) ◦ dWrk .

(6.1) 

Here Wr0 = r and V0 (Xr , r) ◦ dWr0 = V0 (Xr , r) dr. Further, (Wt1 , . . . , Wtd ), t ∈ [0, ∞) is a d  -dimensional Wiener process and ◦dW k (t), k = 1, . . . , d  denote symmetric integrals. Coefficients Vk (x, t) = (Vk1 (x, t), . . . , Vkd (x, t)) are Cb∞,1 functions on Rd ×[0, ∞). The solution defines a diffusion process with the generator A(t) defined by (4.4). The operator is rewritten as (4.20). If the matrix A(x, t) := (α ij (x, t)) = V (x, t)V (x, t)T is positive definite for any x, t, the operator A(t) is called elliptic. Further, if the matrix A(x, t) is uniformly positive definite, the operator A(t) is called uniformly elliptic. The associated SDE (6.1) will be called an elliptic SDE and a uniformly elliptic SDE, respectively. Let {Φs,t } be the stochastic flow generated by the above equation. For a while we take s, t from the time interval T = [0, T ]. Then we can note that for any s < t and x ∈ Rd , Φs,t (x) is a Wiener functional discussed in Sect. 5.1. It is an element of (L∞− )d . We want to show (in Theorem 6.1.1) that Φs,t (x) is a d-dimensional Wiener functional belonging to the Sobolev space (D∞ )d . Let H be the Cameron–Martin space defined in Sect. 5.1. For a given h =  (h1 (r), . . . , hd (r)) ∈ H , consider an SDE with one-dimensional parameter λ in Λ = [−1, 1] on the time interval T:  t d  t  λh λh k k Xt = X0 + Vk (Xr , r)(◦dWr + λh (r) dr) + V0 (Xrλh , r) dr. (6.2) k=1 t0

t0

Let Xt be the solution of (6.1) starting from X0 at time t0 ∈ T and let Xtλh be the solution of (6.2). Then it holds that ¯ a.s. (6.3) Xtλh (w) = Xt (w + λh), t ¯ for any λ, where h(t) = 0 h(r) dr. Indeed, we will approximate the solutions Xt and Xtλh as follows. Let N ∈ N. Let XtN and Xtλh,N be solutions of following equations, respectively: d   

XtN = X0 +

k=1 t0

 d

Xtλh,N = X0 +  +

t t0

t

t

k=1 t0

 Vk (XφNN (r) , r) ◦ dWrk +

t

t0

V0 (XφNN (r) , r) dr,

Vk (Xφλh,N , r)(◦dWrk + λhk (r)) dr N (r)

V0 (Xφλh,N , r) dr, N (r)

6.1 H -Derivatives of Solutions of Continuous SDE

where φN (r) =

m 2N

,

if

≤r
s) at which |Xtx,s | ≥ M + 1 occurs. For any N > 2, there is cN > 0 such that the inequality sup P (τ (x, s) < t) ≤ cN (t − s)N

|x|≤M

holds for any s < t.

(6.15)

252

6 Smooth Densities and Heat Kernels

Proof By Chebyschev’s inequality, we have for any p ≥ 2, sup P (τ (x, s) < t) ≤ sup P

|x|≤M

|x|≤M



 sup |Xrx,s − x| ≥ 1

s 4p, there is cN > 0 such that sup|x|≤M P (τ < t) ≤ cN (t − s)N holds for τ = τ (x, s) by Lemma 6.2.1. Then we have  t 1 1   2 (t − s)2p E (t ∧ τ − s)−4p 2≤ (t − s)4p (r − s)−4p P (τ ∈ dr) + P (τ ≥ t) s

 t  1 2 ≤ c (t − s)4p (r − s)−4p+N −1 dr + 1 s

for any 0 ≤ s < t < T , |x| ≤ M. The above is bounded with respect to these s, t, x since we chose N > 4p.

6.3 Density and Fundamental Solution for Nondegenerate Diffusion

253

We can show sup|x|≤M,s,r,t E[|ϕr∧τ |−4p ] < ∞ similarly to the proof of Proposition 6.2.1. Therefore (6.17) is bounded for 0 ≤ s < t < T and |x| ≤ M. Then we get (6.16) in view of Proposition 5.3.1.  

6.3 Density and Fundamental Solution for Nondegenerate Diffusion Let ck (x, t), k = 0, . . . , d  be Cb∞,1 -functions. Let Gs,t (x) be an exponential functional defined by Gs,t (x) =

Gcs,t (x)

= exp

d   k=0 s

t

 ck (Φs,r (x), r) ◦ dWrk .

(6.18)

We saw in Sect. 4.2 that the pair (Φs,t (x), Gs,t (x)) is a solution of an SDE on (d + 1)-dimensional Euclidean space. We will apply Theorem 6.1.1 to the pair process. Then we find that ∂ i Gs,t (x) is a smooth Wiener functional and its Sobolev norm ∂ i Gs,t (x)m,p is bounded with respect to s < t, x for any i and m, p. We define the transition function of Φs,t (x) weighted by c = (c0 , . . . , cd  ) by the formula c (x, E) = E[1E (Φs,t (x))Gcs,t (x)]. Ps,t

Given a differential operator A(t) of (4.4) and the function c, we define an another c (x, E) is the transition function of differential operator Ac (t) by (4.11). Then Ps,t c the semigroup generated by the operator A (t). In this section, we will show the existence of the smooth density of the above transition function, if the system of diffusions with the generator A(t) is nondegenerate. We first define its characteristic function by  c (v) ψs,t,x

=

Rd

c ei(v,y) Ps,t (x, dy) = E[ei(v,Φs,t (x)) Gs,t (x)].

(6.19)

For multi-index i = (i1 , . . . , id ) of nonnegative integers i1 , . . . , id , we set |i| = i1 + · · · id and ∂xi = ∂xi11 · · · ∂xidd . Lemma 6.3.1 Suppose that the system of diffusion processes with the generator c A(t) is nondegenerate. Then for any fixed s < t, ψs,t,x (v) is a C ∞ -function of x ∞ and is a rapidly decreasing C -function of v. Further, for any M > 0 and N ∈ N, there exists a positive constant c such that

(1 + |x|)|j|

j i c

∂v ∂x ψs,t,x (v) ≤ c (1 + |v|)N −|i| holds for all |x| ≤ M and v ∈ Rd .

(6.20)

254

6 Smooth Densities and Heat Kernels

Furthermore, if the system of diffusions is uniformly nondegenerate, the above inequality holds for all x, v ∈ Rd . c Proof We will consider the differentiability of ψs,t,x (v) by x. Differentiating i(v,Φ (x)) s,t e Gs,t (x) with respect to x, we have ∂xi (ei(v,Φs,t (x)) Gs,t (x)) = i(v,Φ (x)) s,t e Gis,t,x,v , where Gis,t,x,v is a finite sum of terms written as

∂ i0 Gs,t (x) · i(v, ∂ i1 Φs,t (x)) · · · i(v, ∂ ik Φs,t (x)), 

(6.21)



where |i0 | + · · · + |ik | = |i|. We know ∂ i Φs,t (x) and ∂ i Gs,t (x) belong to D∞ and   their norms ∂ i Φs,t (x)m,p and ∂ i Gs,t (x)m,p are bounded with respect to s, t, x, in view of Theorem 6.1.1. Then we can change the order of the expectation and the c derivative operator and we find that ψs,t,x (v) is infinitely differentiable with respect to x and we get c (v) = E[ei(v,Φs,t (x)) Gis,t,x,v ] ∂xi ψs,t,x

for any i. Similarly, we can show the differentiability of the above function with respect to v and we get the formula c (v) = i |j| E[ei(v,Φs,t (x)) Φs,t (x)j Gis,t,x,v ] ∂vj ∂xi ψs,t,x

for any j. Further, for any m, p there is a positive constant c such that the inequality Φs,t (x)j Gis,t,x,v m,p ≤ c|v||i| (1 + |x|)|j| holds for any s < t and x, v ∈ Rd . Then in view of Theorem 5.3.1, for any N ∈ N and M > 0, there exist c, m, p such that c (v)| |∂vj ∂xi ψs,t,x  N c ≤ N DΦs,t (x)m,p sup R Φs,t (x) (θ )−1 m,p Φs,t (x)j Gis,t,x,v m,p |v| θ∈Sd−1

≤

c (1 + |x|)|j| |v|N −|i|

holds for all |x| ≤ M and |v| ≥ 1. Further, the inequality holds for any x if Φs,t (x) is uniformly nondegenerate.   Theorem 6.3.1 Assume that the system of diffusion processes with the generator c (x, E) generated by Ac (t) A(t) is nondegenerate. Then the transition function Ps,t ∞ c has a rapidly decreasing C -density ps,t (x, y) for any s < t, x. Further, it satisfies the following properties:

6.3 Density and Fundamental Solution for Nondegenerate Diffusion

255

c (x, y) is a C ∞ -function of x, y and satisfies 1. For any s < t, ps,t c ∂xi ∂yj ps,t (x, y) = (−i)|j|

 1 d  c e−i(v,y) v j ∂xi ψs,t,x (v) dv d 2π R

(6.22)

j

c for any ∂xi and ∂y , where ψs,t,x (v) is defined by (6.19). c 2. For any y, t, ps,t (x, y) is a C ∞,1 -function of x ∈ Rd and s (< t). It satisfies

∂ c c p (x, y) = −Ac (s)x ps,t (x, y). ∂s s,t

(6.23)

c,∗ c . Then for any g ∈ C (Rd ), we have 3. Let Ps,t be the dual operator of Ps,t 0

 c,∗ g(y) = Ps,t

Rd

c ps,t (x, y)g(x) dx,

∀y.

c (x, y) is a C ∞,1 -function of y ∈ Rd and t (> s). It Further, for any x, s, ps,t satisfies

∂ c c p (x, y) = Ac (t)∗y ps,t (x, y), ∂t s,t

(6.24)

where Ac (t)∗ is the differential operator defined by (4.45). Proof In view of Corollary 5.12.1, the transition function has a density c (x, y), y ∈ Rd for any s, t, x and it is given by ps,t  1 d  c e−i(v,y) ψs,t,x (v) dv. 2π Rd (6.25) It is a continuous function of s < t, x, y and further, a rapidly decreasing c (v) is a rapidly decreasing C ∞ -function of v, for C ∞ -function of y, since ψs,t,x any s < t, x. We can change the order of the derivation with respect to x and the integral c (x, y) is infinitely in equation (6.25), because of Lemma 6.3.1, We find that ps,t differentiable with respect to x and satisfies c ps,t (x, y) = E[δy (Φs,t (x)) · Gcs,t (x)] =

 1 d   1 d  c c (v) dv = ∂xi (v) dv e−i(v,y) ∂xi ψs,t,x e−i(v,y) ψs,t,x 2π 2π c = ∂xi ps,t (x, y).

Next, differentiate both sides of the above equation by y. Then we get (6.22). c f −f = t Ac (r)P c f dr for any C ∞ -function We will prove 2. It holds that Ps,t r,t s c (x, y) satisfies f with compact supports. Therefore, the density function ps,t

256

6 Smooth Densities and Heat Kernels

 c ps,t (x, y) − δy

=

t

s

c Ac (r)x pr,t (x, y) dr.

c (x, y) is continuously differentiable with respect to s and the derivative Then ps,t ∂ c c (x, y) for s < t. Consequently, for any t, y, ps,t (x, y) = −Ac (s)x ps,t satisfies ∂s c ∞,1 -function of x and s. ps,t (x, y) is a C c c,∗ g(y) = ps,t (x, y)g(x) dx a.e. y for any We will next prove 3. We have Ps,t c,∗ d c g ∈ C0 (R ), since Ps,t is the dual of Ps,t . The equality should holds for all y, since c,∗ } both terms are continuous with respect to y. Further, the dual semigroup {Ps,t t c ∗ c,∗ c,∗ c satisfies Ps,t g = g + s A (r) Ps,r g dr for any smooth g. Therefore ps,t (x, y) satisfies  t c c (x, y) − δx = Ac (r)∗y ps,r (x, y) dr. ps,t s

Then both sides are differentiable with respect to t and we get c (x, y). Ac (t)∗y ps,t

∂ c ∂t ps,t (x, y)

=  

We shall consider the final value problem for a backward heat equation associated with the operator Ac (t). Let p(x, s, ; y, t), x, y ∈ Rd , 0 < s < t < ∞ be a continuous function of x, y, s, t and C 2,1 -function of x, s for any y, t. It is called the fundamental solution of the backward heat equation (4.18) or simply the backward heat kernel associated with the operator Ac (s), if it satisfies ∂ p(x, s; y, t) = −Ac (s)x p(x, s; y, t), ∂s

x, y ∈ Rd , 0 < s < t < ∞,

(6.26)

and the function  v(x, s) :=

Rd

p(x, s; y, t1 )f1 (y) dy,

x ∈ Rd , 0 < s < t1

(6.27)

is a C 2,1 -function of (x, s) and is a solution of the final value problem of the backward heat equation (4.18) for any slowly increasing continuous function f1 on Rd . Theorem 6.3.2 Assume that the system of diffusion processes with the generator c (x, y) be the density function of the transition A(t) is nondegenerate. Let ps,t c c (x, y) is the function Ps,t (x, E) generated by Ac (t). Then p(x, s; y, t) := ps,t fundamental solution of the backward heat equation associated with the operator Ac (t). c (x, y) satisfies (6.26) by the previous Proof The function p(x, s; y, t) ≡ ps,t theorem. We show that it is the fundamental solution for the backward heat equation. c (x, y) is a rapidly decreasing C ∞ -function of y for any t, x, we can define Since ps,t v(x, s) by (6.27) for any slowly increasing continuous function f1 .

6.3 Density and Fundamental Solution for Nondegenerate Diffusion

257

We show that v(x, s) is a C ∞ -function of x and is differentiable with respect to s. For any M > 0 and positive integers n, N , there is a positive constant

c c (x, y) ≤ c such that for any |i| ≤ n, sup|x|≤M ∂xi ps,t holds for any 1 (1+|y|)N

y, since ∂xi ps,t1 (x, y) are rapidly decreasing with respect to y. Now let p be a  positive integer such that f1 (y)/(1 + |y|)p is a bounded function. Take N satisfying i c N ≥ p + (d + 1). Then ∂x ps,t1 (x, y)f1 (y)(1 + |y|)d+1 is bounded with respect to c |x| < M and y ∈ Rd . Then, for the function v(x, s) = ps,t (x, y)f1 (y) dy, we 1 can change the order of the derivative ∂ i and the integral by (1 + |v|)−(d+1) dv. It means that v(x, s) is n-times differentiable with respect to x and the equality c (x, y)f (y) dy holds. ∂xi v(x, s) = ∂xi ps,t 1 1 ∂ c c (x, y), for the function v(x, s), we Further, since ∂s ps,t1 (x, y) = −Ac (s)x ps,t 1 ∂ can also change the order of the derivative ∂s and the integral. It means that v(x, s) ∂ ∂ v(x, s) = ∂s ps,t1 (x, y)f1 (y) dy holds. is differentiable with respect to s and ∂s ∞,1 Therefore, v(x, s) is a C -function and satisfies the backward heat equation associated with the operator Ac (t). Next, note the equality v(x, s) = E[f1 (Φs,t1 (x))Gs,t1 (x)]. Given x ∈ Rd , the family of random variables {f1 (Φs,t1 (x))Gs,t1 (x); 0 ≤ s < t1 } are uniformly integrable and converges to f1 (x) as s → t1 . Then the function v(x, s) converges to f1 (x) for any x as t → 0. We have thus shown that v(x, s) of (6.27) is a solution of the final value problem for the backward heat equation.   If the differential operator Ac (t) is nondegenerate, the associated semigroup of linear transformations can be extended to that of linear transformations from the space of tempered distributions to the space of C ∞ -functions. The fact will be proved in Theorem 6.6.3 for the the semigroup associated with the integrodifferential operator AJc,d (t). We will next consider the heat equation (4.21) associated with Ac (t). Let p(x, ˇ t; y, s), x, y ∈ Rd , 0 < s < t < ∞ be a continuous function of x, y, s, t and C 2,1 -function of x, t for any y, s. It is called the fundamental solution of the heat equation or simply heat kernel associated with Ac (t) if it satisfies the equation c } {Ps,t

∂ p(x, ˇ t; y, s) = Ac (t)x p(x, ˇ t; y, s), ∂t

x, y ∈ Rd , 0 < s < t < ∞,

(6.28)

and the function  u(x, t) :=

Rd

p(x, ˇ t; y, t0 )f0 (y) dy,

x ∈ Rd , t0 < t < ∞

(6.29)

is a solution of the initial value problem of the heat equation (4.21) for any slowly increasing continuous function f0 on Rd . In order to construct the fundamental solution for the heat equation associated with the operator Ac (t), let us consider the backward SDE defined by (4.24), where coefficients Vk (x, r), k = 0, . . . , d  are Cb∞,1 -functions. The solution generates a backward flow {Φˇ s,t }. It is a backward diffusion with the generator A(t) defined

258

6 Smooth Densities and Heat Kernels

c (x, E) = E[1 (Φ ˇ cs,t (x)], where G ˇ cs,t (x) is given by (4.4). Define Pˇs,t E ˇ s,t (x))G by (4.27). It is a backward transition function of the backward semigroup generated by the operator Ac (t).

Theorem 6.3.3 Assume that the system of backward diffusion processes with the generator A(t) is nondegenerate. Then the backward transition function generated c (x, y) for any t, s, x. It is a C ∞,1 by Ac (t) has a rapidly decreasing C ∞ -density pˇ s,t b d c (x, y) is function of (x, t) for any y ∈ R and t (> s). Further, p(x, ˇ t; y, s) := pˇ s,t the fundamental solution of the heat equation associated with the operator Ac (t). The proof is similar to that of Theorem 6.3.2. It is omitted. If coefficients Vk , k = 0, . . . , d  are time homogeneous, we can define the fundamental solution of the heat equation associated with Ac , using the forward diffusion. c (x, y) with respect to x has The smoothness of the transition density function ps,t not been studied in the Malliavin calculus. We showed the property by using the smoothness of the stochastic flow Φs,t (x) with respect to x. If the operator A(t) is elliptic, both the forward diffusion and the backward diffusion generated by A(t) are nondegenerate. In this case, density functions c (x, y) and pˇ c (x, y) are rapidly decreasing C ∞ -functions of x and y. Indeed, ps,t s,t  since the operator A(t) = 12 k≥1 Vk (t)2 − V0 (t) is elliptic, the dual transition c c,∗ function Ps,t (y, E) := E ps,t (x, y) dx should have a rapidly decreasing C ∞ c (x, y) is a rapidly decreasing C ∞ density by Theorem 6.3.1. This shows that ps,t function of x. Note We can relax the elliptic condition. For two vector fields V and W , we define its Lie bracket by [V , W ]f = V Wf − W Vf. Then [V , W ] is again a vector field. Now let V0 , V1 , . . . , Vd  be the vector fields defining the SDE. Set Σ0 = {Vk (t); k = 1, . . . , d  }, ΣM = {[Vk (t), V (t)]; k = 0, . . . , d  , V (t) ∈ ΣM−1 }.

0 d If there exists N0 ∈ N such that the family of vector fields N M=0 ΣM span R for all x, t, then the operator A(t) is called hypo-elliptic or is said to satisfy the Hörmander condition. There are extensive studies for the existence of the smooth density for timehomogeneous hypo-elliptic operator A(t). It was shown by Malliavin [77], Kusuoka–Stroock [69] that if the operator A(t) satisfies the Hörmander condition, then the diffusion is nondegenerate. The proof is not simple. We refer it to Kusuoka–Stroock [69], Norris [87], Nualart [88] and Komatsu–Takeuchi [58]. Their discussions can be applied for time dependent hypo-elliptic operator.

6.4 Solutions of SDE on Wiener–Poisson Space

259

We can apply Theorems 6.3.1, 6.3.2, and 6.3.3 to hypo-elliptic diffusion. Note that the operator A(t) is hypo-elliptic if A(t) is hypo-elliptic. Then its transition function has a density ps,t (x, y), which is a rapidly decreasing C ∞ -function of x and y. Further, the heat equation associated with the operator Ac (t) satisfying the Hörmander condition has the fundamental solution. Note The existence of the fundamental solution for a backward heat equation is known in analysis. Let A(t) be an elliptic differential operator given by (4.20). We assume that coefficients α ij , β i are bounded and are Hölder continuous. Then there exists a fundamental solution p(x, s; y, t) for the operator A(t), which is continuous in s, t, x, y and is C 2 -class with respect to x. Results may be found in I’lin-Karashnikov-Oleinik [42], Friedman [28] and Dynkin [24]. The fundamental solution p(x, s; y, t) for the operator A(t) is used for constructing a diffusion process with the generator A(t), without using SDEs. Define Ps,t f (x) = p(x, s; y, t)f (y) dy. Then {Ps,t } is a conservative semigroup. There exists a Markov process Xt with transition probability Ps,t (x, E) = E p(x, s; y, t) dy. We can show that Xt is a continuous process and has the strong Markov property. See Dynkin [24], Chap. 5. In this monograph, we started with SDE and proved the existence of the fundamental solution, using the Malliavin calculus. Our result is slightly stronger than those, since p(x, s; y, t) is a C ∞ -function of x and y, although we assumed that coefficients of the operator A(t) are of Cb∞ -class.

6.4 Solutions of SDE on Wiener–Poisson Space ¯ B(Ω), ¯ P ) be the Wiener–Poisson space associated with the Lévy measure Let (Ω, ν discussed in Sect. 5.9. For the Lévy measure ν we assume that at the center it is nondegenerate and satisfies the order condition of exponent 0 < α < 2 with respect to a given family of star-shaped neighborhoods {A0 (ρ)}. Further, we assume that ν has a weak drift. Let us consider a symmetric SDE with jumps defined on Rd : d   

Xt = X0 +

t

k=0 t0

Vk (Xr , r) ◦ dWrk +

 t t0 |z|>0+

{φr,z (Xr− ) − Xr− }N(dr dz).

(6.30) Its diffusion part is the same as equation (6.1). For the jump part, we assume that g(x, t, z) := φt,z (x) − x satisfies Conditions (J.1) and (J.2) in Sect. 3.2 in Chap. 3. Let {Φs,t } be the stochastic flow generated by the above SDE. Let T = [0, T ] be the time parameter of the Wiener–Poisson space. We will show that for any 0 ≤ s < t ≤ T and x ∈ Rd , Φs,t (x) is a regular Wiener–Poisson functional belonging to the ¯ ∞ )d defined in Sect. 5.10. Let u = (r, z) ∈ U and εu+ be the transformation space (D on the Poisson space defined in Sect. 5.4. Then the equality εu+ ω(E) = ω

260

6 Smooth Densities and Heat Kernels

(E) + δu (E) holds for almost all ω. Therefore, for almost all u = (r, z) with respect + to the measure n, Xt := Φs,t (x) ◦ ε(r,z) satisfies the equation d   

Xt = x +

t

k=0 s

Vk (Xr  , r  ) ◦ dWrk +

 t s

+ {φr,z (Xr− ) − Xr− }1(s,t] (r),

{φr  ,z (Xr  − ) − Xr  − }N (dr  dz )

|z|>0+

a.s.

If 0 ≤ r < s or t < r ≤ T , then we have Xt = Φs,t (x). If s ≤ r ≤ t, the solution Xt is equal to Φr,t ◦ φr,z ◦ Φs,r− (x). Since Φs,r (x) = Φs,r− (x) holds for all x, almost surely, we have + = Φs,t (x) ◦ ε(r,z)



Φr,t ◦ φr,z ◦ Φs,r (x) if s ≤ r ≤ t, if r < s or r > t. Φs,t (x),

(6.31)

Next, let j ≥ 2 and let u = {(ri , zi ), i = 1, . . . , j } ∈ Uj . Let u be the subset of u such that s ≤ ri ≤ t. It is written as {(r1 , z1 ), . . . , (rj  , zj  )}, where s < r1 < · · · < rj  < t and j  ≤ j . Then we have Φs,t (x) ◦ εu+ = Φrj  ,t ◦ φrj  ,zj  ◦ · · · ◦ Φr1 ,r2 ◦ φr1 ,z1 ◦ Φs,r1 (x),

a.s.

(6.32)

It is well defined and belongs to (L¯ ∞− )d for any s < t, x and u a.s. We will study j the H -differentiability of D˜ u Φs,t (x). j Lemma 6.4.1 For any x ∈ Rd , 0 ≤ s < t ≤ T and u ∈ Uj , D˜ u Φs,t (x) is H j differentiable. Further, H -derivative Dr D˜ u Φs,t (x) is given a.e. dr dP by



j D˜ u ∇Φr,t (Φs,r (x))Vk (Φs,r (x), r) , 0,

k = 1, . . . , d  , s < r < t, otherwise.



¯ Proof Given h = (h1 (r), . . . , hd (r)) ∈ H , we set h(t) = λh ¯ Φs,t (w) := Φs,t (x)(w + λh). Then,

t 0

(6.33)

h(r) dr. We consider

λh Φs,t ◦ εu+ d   

=x+

t

k=0 s

 t + s

d   

λh Vk (Φs,r

◦ εu+ , r) ◦ dWrk

+λ

k=1 s

λh g(Φs,r− ◦ εu+ , r, z)N(dr dz) +

|z|>0+

t



λh Vk (Φs,r ◦ εu+ , r)hk (r) dr

λh g(Φs,r ◦ εu+ , ri , zi ), i−

i;s≤ri ≤t

6.4 Solutions of SDE on Wiener–Poisson Space

261

λh ◦ ε + is if u = ((r1 , z1 ), . . . , (rj , zj )) and r1 < · · · < rj . Then the solution Φs,t u continuously differentiable with respect to λ in the F -space L¯ ∞− . See Remark after h . Then similarly to Theorem 3.3.2. Let us denote its derivative at λ = 0 by Ξs,t Lemma 6.1.2, it satisfies an inhomogeneous linear SDE; d   

h Ξs,t

=

t

k=0 s

+ +

h ∇Vk (Φs,r ◦ εu+ , r)Ξs,r ◦ dWrk

  t

h ∇g(Φs,r− ◦ εu+ , r, z)Ξs,r− N(dr dz)

s |z|>0+

d   

 i;s≤ri ≤t

h ∇g(Φs,ri − ◦ εu+ , ri , zi )Ξs,r + i−

k=1 s

t

Vk (Φs,r ◦ εu+ , r)hk (r) dr.

On the other hand, the Jacobian matrix ∇Φs,t (x)◦εu+ satisfies a homogeneous linear SDE: d   

∇Φs,t ◦ εu+

=I+

t

k=0 s

+

 t s

+

∇Vk (Φs,r ◦ εu+ , r)∇Φs,r ◦ εu+ ◦ dWrk

|z|>0+



∇g(Φs,r− ◦ εu+ , r, z)∇Φs,r− ◦ εu+ N(dr dz)

∇g(Φs,rj − ◦ εu+ , ri , zi )∇Φs,ri ◦ εu+ .

i;s≤ri ≤t h of the inhomogeneous equation is written, using the solution Then the solution Ξs,t ∇Φs,t ◦ εu+ of the homogeneous equations as

 h Ξs,t

= s

t

∇Φr,t (Φs,r ) ◦ εu+

d 

 Vk (Φs,r ◦ εu+ , r)hk (r) dr.

k=1

This proves that Φs,t (x) ◦ εu+ is H -differentiable and Dr Φs,t (x) ◦ εu+ is given by  ∇Φr,t (Φs,r (x))Vk (Φs,r (x), r) ◦ εu+ , k = 1, . . . , d  , s < r < t, 0, otherwise. j j Since D˜ u Φs,t (x) is written as a linear sum of Φs,t (x) ◦ εv+ with v ⊂ u, D˜ u Φs,t (x) j j is also H -differentiable. Further, we have Dr D˜ u Φs,t (x) = D˜ u Dr Φs,t (x), since D and D˜ are commutative. Therefore we get (6.33) in view of Proposition 6.1.1.  

262

6 Smooth Densities and Heat Kernels

For u = (r, z), we set γ (u) = |z| ∧ 1 and for u = ((r1 , z1 ), . . . , (rj , zj )), we set γ (u) = (|z1 | ∧ 1) · · · (|zj | ∧ 1). We set A(1) = {u = (r, z) ∈ U; |z| ≤ 1}. Lemma 6.4.2 For any 0 ≤ s < t ≤ T , j ∈ N, i and p ≥ 2, we have

p   j E Dr D˜ u ∂ i Φs,t (x) sup sup < ∞, γ (u)p s0+

∇g(r, z)Jr N(dr dz),

where ∇Vk (r) := ∇Vk (Φs,r , r) and ∇g(r, z) := ∇g(Φs,r− , r, z). Now consider an SDE for an unknown d × d-matrix-valued process Vt = Vs,t : d   

Vt = I −

k=0 s

t

Vr ∇Vk (r) ◦ dWrk

−

 t 0

Vr− ∇h(r, z)N(dr |z|>0+

dz),

(6.45)

where ∇h(r, z) := ∇h(φr,z (Φs,r− ), r, z) and h(x, r, z) is given by h(x, r, z) := −1 (x). We shall prove that V is the inverse matrix of J . Apply the rule of x − φr,z t t

6.5 Nondegenerate Jump-Diffusions

267

the differential calculus for jump processes (Theorem 3.10.1) to the product Vt Jt . Using symmetric integrals, it holds that Vt Jt = Vs Js − +

  t



t

j

Vr ∇Vk (r)Jr ◦ dWr +

s

k



 k

t

j

Vr ∇Vk (r)Jr ◦ dWr

0

 Vr− (I −∇h(r, z))(I +∇g(r, z))Jr− −Vr− Jr− dN.

s |z|>0+

We have −1 (I − ∇h(r, z))(I + ∇g(r, z)) = ∇φr,z (φr,z (Φs,r− ))∇φr,z (Φs,r− ) = I.

Consequently, we get Vt Jt = I , proving that Vs,t is the inverse matrix of Js,t . Now, equation (6.45) is a linear equation for Vt , where coefficients ∇Vj (· · · ) and ∇h(· · · ) are bounded uniformly with respect to parameter x. Rewrite the equation using Itô integrals. Then we can apply Lemma 3.3.2 and show the inequality sups 2. Further, the above discussion is valid if we transform Js,t and Vs,t by Js,t ◦ εu+ and Vs,t ◦εu+ , respectively; Vs,t ◦εu+ is the inverse of Js,t ◦εu+ and the former satisfies the inequality of the lemma.   Proof of Theorem 6.5.1 Our discussion is close to that of Proposition 6.2.1. Set ϕr = ϕrs,t,x = θ ∇Φr,t (Φs,r (x)). Then K¯ Φs,t (x) (θ ) is written as K¯ Φs,t (x) (θ ) =



t s

ϕr A(Φs,r (x), r)ϕrT dr.

(6.46)

(s,t,x)

= 0 a.s., since the Jacobian matrix For any s, t, x, it holds that ϕr = ϕr ∇Φr,t (x) is invertible for any x, r, t. Let λr be the minimal eigen-value of the matrix A(Φs,r (x), r). Then we have K¯ Φs,t (x) (θ )−1 ≤

1 (t − s)2

 s

t

1 dr λr |ϕr |2

similarly to the proof of Proposition 6.2.1. The inequality is valid for the transformation εu+ . Since λr ≥ c > 0 holds for any r and ω, we get the inequality E[(K¯ Φs,t (x) (θ ) ◦ εu+ )−p ] ≤ (c(t − s))−p sup E[|ϕr ◦ εu+ |−2p ]. r

We will show that the last term of the above is finite for any p. Let Vr,t (x) be the inverse matrix of ∇Φr,t (x). Since ∇Φr,t (Φs,r (x))Vr,t (Φs,r (x)) = I , we have |ϕr ◦ εu+ |−1 ≤ |Vr,t (Φs,r (x)) ◦ εu+ |.

(6.47)

268

6 Smooth Densities and Heat Kernels

The above is L2p -bounded for any 2p > 2 in view of Lemma 6.5.1. Then we get sups,t,x,r,u E[|ϕr ◦ εu+ |−2p ] < ∞. Consequently, the inequality (6.43) holds. Then the jump-diffusion is uniformly nondegenerate at the center.   We shall next consider the pseudo-elliptic case. We want to show that the jumpdiffusion is nondegenerate at the center. In the case of elliptic diffusions, we proved this by introducing localizing stopping times τ (x, s) of Lemma 6.2.1. Loosely, the lemma shows that the process Φs,t (x) moves slowly, since P (τ (x, s) − s < δ) = O(δ N ) holds for any N > 2. However, the fact does not hold for jump-diffusions. Jump-diffusions could move rapidly. Even so, we will show that a hitting time τ (x, s) of Φs,t (x) to a set V satisfies P (τ (x, s) − s < δ) = O(δ N ), if it jumps at least N times before hitting the set V . We will discuss it precisely. Let {φt,z } be jump-maps of the jump-diffusion. For a subset U of Rd , we set ¯ )= φ(U



φt,z (U ),

¯ φ¯ i−1 (U )), φ¯ i (U ) = φ(

i = 2, 3, . . . ,

where ∪ is taken for all t ∈ T,z ∈ Supp(ν). It holds that U ⊂ φ¯ i (U ) ⊂ φ¯ i+1 (U ) for any set U . Similarly, we define φ¯ −1 (V ) =



−1 (V ), φt,z

φ¯ −i (V ) = φ¯ −1 (φ¯ −i−1 (V )),

i = 2, 3, . . .

For two subsets U, V of Rd , we set d(U, V ) = infx∈U,y∈V d(x, y), where d(x, y) is the metric of two points x, y. In the following argument, we often set Φs,∞ = ∞ conventionally and we use the notation ∞ − ∞ = 0. Lemma 6.5.2 Let N be a positive integer satisfying d(φ¯ N (U ), V ) > 0. For given x, s, let τ = τ (x, s) be the hitting time of the jump-diffusion process Xtx,s = Φs,t (x), t > s to the set V . Then there exists c > 0 such that sup P (τ (x, s) < t) ≤ c(t − s)N

(6.48)

x∈U

holds for any s < t. Proof For > 0 we define the -neighborhood of the set U ⊂ Rd by U = {x; d(x, U ) ≤ }. We set ¯ ), ψ (U ) = φ(U

ψ n (U ) = ψ (ψ n−1 (U )),

n = 2, . . . , N

inductively. Then if d(φ¯ N (U ), V ) > 0 holds, d(ψ N (U ), V¯ ) > 0 holds for sufficiently small > 0. We have U ⊂ ψ (U ) ⊂ · · · ⊂ ψ N (U ) ⊂ V .

6.5 Nondegenerate Jump-Diffusions

269

Let x ∈ U . Let σ1 = σ1 (x, s) be the first leaving time of Xt = Xtx,s = Φs,t (x) from the ball B (x) = {y ∈ Rd ; |y − x| < }. Then Xσ1 ∈ ψ 1 (U ), if σ1 < T a.s. We define stopping times σ2 = σ2 (x, s), . . . , σN = σN (x, s) inductively as σn = σn (x, s) = inf{r > σn−1 ; |Xr − Xσn−1 | ≥ }, = ∞,

if {· · · } is empty.

Then we have Xσn ∈ ψ n (U ) if σn < ∞ a.s. for any n = 2, . . . , N . Consequently, we have XσN ∈ / V if σN < T a.s. This means τ ≥ σN a.s.  Now, we have τ ≥ σN ≥ N n=1 (σn − σn−1 ), where σ0 = s. Therefore, {τ < t} ⊂

N )

{|Xσn − Xσn−1 |1(σn −σn−1 ) MN }, we have d(ψ N (U ), V ) > 0. Therefore (6.49) holds by Lemma 6.5.2.   Theorem 6.5.2 If the operator AJ (t) of (4.52) is pseudo-elliptic, the system of jump-diffusions with the generator AJ (t) is nondegenerate at the center. Further, the inequality  sup (t − s)|K¯ Φs,t (x) (θ )−1 |0,n,p < ∞

sup

(6.50)

s,t∈T,s 2 and M > 0. Proof Consider again the formula (6.46). Let λr be the minimal eigen-value of A(Φs,r (x), r). It may not be uniformly positive with respect to r, ω. So we need a localization. For a given p > 2, we will fix N ∈ N such that N > 2p. Let τ = τ (x, s) be the stopping time satisfying (6.49). Since A(x, t) is positive definite, we can take c > 0 such that θ A(x, t)θ T ≥ c holds for all θ ∈ Sd−1 , t∧τ |x| ≤ M + N (K + ). Then we have K¯ Φs,t (x) (θ ) ≥ s λr |ϕr |2 dr for |x| ≤ M. Then K¯ Φs,t (x) (θ ) are invertible a.s. and inverses satisfy K¯ Φs,t (x) (θ )−1 ≤ c(t ∧ τ − s)−2

 s

t∧τ

1 dr. λr |ϕr |2

Note λr ≥ c for r < τ . Then we get (t − s)p E[K¯ Φs,t (x) (θ )−p ] ≤ c−p ((t − s)4p E[(t ∧ τ − s)−4p ]) 2 sup E[|ϕr∧τ |−4p ] 2 . 1

1

r

We can verify sups s). It satisfies ∂ c,d c,d p (x, y) = AJc,d (t)∗y ps,t (x, y), ∂t s,t

(6.57)

where AJc,d (t)∗ is the operator given by (4.77). The above theorem can be verified similarly to Theorem 6.3.1 (diffusion case), making use of Lemma 6.6.1. We consider the final value problem of the backward heat equation associated with the operator AJc,d (t). Its fundamental solution is defined in the same way as the fundamental solution of the backward heat equation associated with Ac (t) in Sect. 6.3. Theorem 6.6.2 Assume that the system of jump-diffusions with generator AJ (t) is c,d nondegenerate at the center. Let ps,t (x, y) be the smooth density of the transition c,d c,d c,d function Ps,t (x, E) generated by AJ (t). Then p(x, s; y, t) := ps,t (x, y) is the fundamental solution of the backward heat equation associated with the operator AJc,d (t). We can extend the function f1 (x) of the final condition for the backward heat equation to any tempered distribution f1 . Schwartz’s distribution is discussed briefly in Sect. 5.12. We may assume that f1 belongs to S −N for sufficiently large N . Assume that the operator AJc,d (t) is uniformly nondegenerate at the center. Then Theorem 5.12.1 and its proof tells us that for any N ∈ N and fixed s < t, there are m, n, p and a positive constant CN such that

6.6 Density and Fundamental Solution for Nondegenerate Jump-Diffusion

f (Φs,t (x))∗m,n,p ≤ CN |f |−N ,

f ∈ S −N

275

(6.58)

holds for any x, where the norm | |−N is defined by (5.170). Indeed, in Theox | , where rem 5.12.1, the constant CN in (5.176) is given by supG ≤1 |ψG N m,n,p

x (v) = E[ei(v,Φs,t (x)) G]. The above function is a rapidly decreasing C ∞ -function ψG x (v) is of v. Further, for any positive integer m and multi-index i, (1 + |v|2 )m ∂vi ψG  x uniformly bounded with respect to x, v and G with Gm,n,p ≤ 1, since ψG satisfies the inequality (6.54) and

DΦs,t (x)m,n,p ,

˜ s,t (x)m,n,p , DΦ

¯ ρΦs,t (x) (θ )−1 m,n,p sup Q

(ρ,θ)

x | is bounded with respect are bounded with respect to x. Then supG ≤1 |ψG N m,n,p to x.

Theorem 6.6.3 Assume that the system of jump-diffusions with the generator AJ (t) is uniformly nondegenerate at the center. Then the semigroup of linear operators c,d Ps,t is extended to that of linear operators from the space of tempered distributions to the space of C ∞ -functions. c,d For a tempered distribution f1 , set v(x, s) = E[f1 (Φs,t1 (x)) · Gs,t (x)]. Then it 1 is a C ∞,1 -function of (x, s). Further, it is differentiable with respect to s (< t1 ) and satisfies the backward integro-differential equation ∂ v(x, s) = −AJc,d (s)v(x, s) ∂s

(6.59)

and the final condition lims→t1 v(x, s) = f1 . Proof We will approximate f1 by a sequence of smooth functions fm , m = 2, . . . in S with respect to the norm | |−N . Set vm (x, s) = E[fm (Φs,t (x))Gs,t (x)], where c,d Then vm (x, t) converges to v(x, s) uniformly in x, in view of the Gs,t = Gs,t inequality (6.58). Therefore v(x, s) is a continuous function. Further, we have ∂ i vm (x, t) = ∂ i E[fm (Φs,t (x))Gs,t (x)]    = E ∂ i1 fm (Φs,t (x)) · ∂ i2 Φs,t (x), ∂ i3 Gs,t (x) , where the sum is taken for i1 , i2 and i3 such that |i1 |, |i2 |, |i3 | ≤ |i|. The right-hand side of the above converges again uniformly in x in view of (6.58). Since this is valid for any derivative operator ∂ i , the limit function v(x, t) is a C ∞ -function of x. ∂ Now each function vm (x, s) satisfies ∂s vm (x, s) = −AJc,d (s)vm (x, s) for s < t1 . Then the limit function v(x, s) should be differentiable with respect to s < t1 and satisfy the same equation.

276

6 Smooth Densities and Heat Kernels

Let ϕ ∈ S∞ . By the change of variables, we have 

 v(x, s)ϕ(x)dx =

E[f1 (Φs,t1 (x))Gs,t1 (x)ϕ(x)] dx 

=

E[f1 (y)Gs,t1 (Ψˇ s,t1 (y))ϕ(Ψˇ s,t1 (y))| det ∇ Ψˇ s,t1 (y)|] dy.

Let s tend to t1 . Then the last term converges to f1 , ϕ. Therefore v(x, s) converges to f1 as s → t1 .   Finally, we will discuss the fundamental solution for the heat equation associated with the nondgenerate operator AJc,d (t). Consider the backward SDE defined with characteristics (Vj (x, t), j = 0, . . . , d  , g(x, t, z), ν). Let Φˇ s,t (x) be the backward c,d ˇ s,t flow of diffeomorphisms generated by the backward SDE and let G be the expoc,d c,d ˇ ˇ s,t ˇ nential function defined in Sect. 4.5. Then Ps,t (x, E) = E[1E (Φs,t (x))G (x)] is the backward transition function with the generator AJc,d (t). Theorem 6.6.4 Assume that the system of backward jump-diffusions generated by AJ (t) is nondegenerate at the center. Then its backward transition function c,d c,d Pˇs,t (x, E) has a rapidly decreasing C ∞ -density pˇ s,t (x, y) for any 0 ≤ s < t < ∞ d ∞,1 and x ∈ R . Further, it is a C -function of (x, t) ∈ Rd × (s, ∞) for any y, s c,d and p(x, ˇ t; y, s) := pˇ s,t (x, y) is the fundamental solution of the heat equation associated with the operator AJc,d (t). Remark We shall consider a jump-diffusion where its Lévy measure may not satisfy the order condition. In this case Theorem 6.6.1 may not hold. We will consider an elliptic jump-diffusion. Suppose that the generator is elliptic and jump-map φt,z satisfies Condition (J.2). Then the Malliavin covariance R Φs,t (x) of Φs,t (x) defined by (6.11) is invertible and the inverse satisfies sup

1

(t − s)E[R Φs,t (x) (θ )−p ] p < ∞,

∀p ≥ 2

(6.60)

θ∈Sd−1 ,|x|≤M

for any 0 < M < ∞. Indeed, discussions of Sect. 6.3 are also valid for jumpdiffusions, if we use Lemma 6.5.2 instead of Lemma 6.11. Then the transition c,d function Ps,t (x, E) has a rapidly decreasing C ∞ -density. Further, the density c,d function ps,t (x, y) is a rapidly decreasing C ∞ -function of x. See Remark in Sect. 5.11. If jump-maps φt,z do not satisfy Condition (J.2) (φt,z are not diffeomorphic), the smoothness of the density is not known even if AJ (t) is elliptic. Indeed, the flow Φs,t may not be diffeomorphic and hence its Jacobian matrix ∇Φs,t may not invertible. Then the invertibility of the Malliavin covariance is not clear.

6.7 Short-Time Estimates of Densities

277

6.7 Short-Time Estimates of Densities In the remainder of this chapter (Sects. 6.7, 6.8, 6.9, and 6.10) we will study elliptic diffusion and pseudo-elliptic jump-diffusions. We will study short-time asymptotics c (x, y) studied in Sect. 6.3 and p c,d (x, y) studied in of the smooth densities ps,t s,t Sect. 6.6 as t → s. We will be able to obtain the same short-time estimates for the fundamental solutions for heat equations associated with the elliptic operator Ac (t) and pseudo-elliptic operator AJc,d (t). However, we will not repeat these arguments. We will first state the assertion for a diffusion process. Theorem 6.7.1 Assume that the partial differential operator Ac (t) is elliptic. Let c (x, y) be the smooth density of the transition function P c (x, E) generated ps,t s,t j

by Ac (t). Let ∂xi and ∂y be any given differential operators with indexes i and j, respectively. 1. For any T > 0 and M > 0, there exists a positive constant c such that

i j c

∂x ∂y ps,t (x, y) ≤

c (t − s)

|i|+|j|+d 2

,

∀0 ≤ s < t ≤ T

(6.61)

holds for all |x| ≤ M, y ∈ Rd . 2. If Ac (t) is uniformly elliptic, for any T > 0, there exists c > 0 such that the above inequality holds for all x, y ∈ Rd . Proof We will first consider the case where the diffusion is elliptic. For a given T > 0, we consider the Wiener space of time parameter T = [0, T ] and apply the c Malliavin calculus on the Wiener space. Let 0 ≤ s < t ≤ T and set ψs,t,x (v) = c,d i(v,Φ (x)) s,t E[e Gs,t (x)], where Gs,t (x) = Gs,t (x). Then we have from (6.22) c ∂xi ∂yj ps,t (x, y) = (−i)|j|

 1 d  c e−i(v,y) v j ∂xi ψs,t,x (v) dv. d 2π R

It holds that   c ∂xi ψs,t,x (v) = E ei(v,Φs,t (x)) Gis,t,x,v , where Gis,t,x,v is written as a linear sum of terms (6.21). We can rewrite it as Gis,t,x,v =



i ,...,i

0 j |v||i|−|i0 | Gs,t,x,θ ,

where i ,...,i

0 j Gs,t,x,θ = i |i|−|i0 | ∂ i0 Gs,t (x)(θ, ∂ i1 Φs,t (x)) · · · (θ, ∂ ij Φs,t (x)),

(6.62)

278

6 Smooth Densities and Heat Kernels

and the summation is taken for multi-indexes il such that il ≤ i and |i0 | + · · · + |ij | = i ,...,i

i ,...,i

0 j 0 j |i|. Setting ψs,t,x,θ (v) = E[ei(v,Φs,t (x)) Gs,t,x,θ ], we have

 i0 ,...,ij

j i c

|v||i|+|j|−|i0 | |ψs,t,x,θ (v)|

v ∂x ψs,t,x (v) ≤  i0 ,...,ij ≤ (1 ∨ |v|)|i|+|j| |ψs,t,x,θ (v)|.

(6.63)

Therefore we get

 1 d   i0 ,...,ij

i j c

sup ∂x ∂y ps,t (x, y) ≤ (1 ∨ |v|)|i|+|j| |ψs,t,x,θ (v)| dv. 2π Rd y∈Rd Set w = dv = (t

√ t − sv and change the variable. Since 1 ∧ |v| ≤

− s)−d/2 dw,

√ √ T (1 t−s

+ |w|) and

the above is dominated by 

c (t − s)

|i|+|j|+d 2

Rd

˜ (1 ∨ |w|)|i|+|j| |ψ(w)| dw,

where i0 ,...,ij ˜ ψ(w) = E[ei(w,Hs,t,x ) Gs,t,x,θ ],

Φs,t (x) Hs,t,x = √ . t −s

Take any M > 0 and consider DHs,t,x m,p ,

i ,...,i

0 j sup Gs,t,x,θ m,p ,

θ∈Sd−1

sup R Hs,t,x (θ )−1 m,p .

(6.64)

θ∈Sd−1

The first and the second term are bounded with respect to s < t, |x| ≤ M by Theorem 6.1.1. Further, the last term is bounded at the same region by Proposition 6.2.2. Then, for any N there exists a positive constant cN such that cN ˜ inequalities |ψ(w)| ≤ cN (if |w| ≤ 1) and ≤ |w| N (if |w| ≥ 1) hold for any s < t and |x| ≤ M in view of Theorem 5.3.1. Consequently, we obtain from (6.62)



c (x, y) ≤ sup ∂xi ∂yj ps,t

y∈Rd

≤

cN (t − s)

|i|+|j|+d 2

 cN

(t − s)

|i|+|j|+d 2

  1+

|w|≥1

|w||i|+|j|−N dw



(6.65)

for all s < t, |x| ≤ M, if N > d + |i| + |j|. Therefore we get the inequality (6.61) for all |x| ≤ M, y ∈ Rd , proving the first assertion of the theorem.

6.7 Short-Time Estimates of Densities

279

Finally, if the process is uniformly elliptic, all terms in (6.64) are bounded for all s < t, x in view of Theorem 6.1.1 and Proposition 6.2.1. Then (6.65) holds for all s < t, x. This proves the second assertion of the theorem.   Remark The short-time estimate for fundamental solution for the elliptic operator (4.20) is known in analysis. Assuming its coefficients are bounded and Holder continuous, inequality (6.61) is shown in the case |i| ≤ 2 and j = 0 (Il’in–Kalashnikov–Oleinik [42], Dynkin [24]). We got a stronger result using the Malliavin calculus, assuming that coefficients of the operator A(t) is of Cb∞ -class. The short-time estimate of the fundamental solution of a non-elliptic but hypoelliptic operator seems to be open. We will next consider jump-diffusions. Suppose first that their generators are elliptic or uniformly elliptic. Then discussions in the proof of Theorem 6.7.1 are valid for these jump-diffusions. Therefore the assertions of Theorem 6.7.1 are valid for elliptic and uniformly elliptic jump-diffusions. We will consider the density for pseudo-elliptic jump-diffusions. c,d Theorem 6.7.2 Assume that the operator AJ (t) is pseudo-elliptic. Let ps,t (x, y) c,d be the smooth density of transition function Ps,t (x, E) generated by AJc,d (t). Let α0 be an arbitrary constant such that α < α0 < 2, where α is the exponent of the Lévy j measure ν. Let ∂xi and ∂y be any given differential operators with indexes i and j.

1. For any T > 0 and M > 0, there exists a positive constant c such that

c

i j c,d

, ∀0 ≤ s < t ≤ T

∂x ∂y ps,t (x, y) ≤ |i|+|j|+d (t − s) 2−α0

(6.66)

holds for all |x| ≤ M, y ∈ Rd . 2. If the operator AJ (t) is uniformly pseudo-elliptic, for any T > 0 there exists c > 0 such that the above inequality holds for all x, y ∈ Rd . A similar result is shown in Ishikawa–Kunita–Tsuchiya [46] in the case of jump  processes. An interesting point in the inequality (6.66) is that the term t −β , where β  = |i|+|j|+d 2−α0 , is common for all jump-diffusions and all Lévy measures with the same exponent 0 < α < 2 of the order condition, but the constant c should depend on coefficients Vk (x, t), k = 1, . . . , d  , V˜k (x, t), k = 1, . . . , d  . It seems that α0 = α should be the critical value for the estimate (6.66), but we do not know whether the estimate is valid for α0 = α. It may be an interesting problem to find the short-time asymptotics for nonelliptic or non-pseudo-elliptic, but nondegenerate SDEs such as hypoelliptic SDEs.  In such a case the exponent β  in t −β in the inequality (6.66) should be changed. It seems that systematic results are not known. The proof of Theorem 6.7.2 is not so simple as the proof of Theorem 6.7.1. We need four lemmas. Given α < α0 < 2, let us choose 1 < δ < αα0 . For a given T > 0, we set T = [0, T ] and consider the Wiener–Poisson space of time parameter T. ¯ F (ρ) With this constant δ, we consider δ-nondegenerate condition with respect to Q defined by (5.146).

280

6 Smooth Densities and Heat Kernels

Lemma 6.7.1 For any positive integer N0 > 2, there exist m, n ∈ N, p ≥ 2 and a positive constant c such that for |v| ≥ 1 the inequality

 

E ei(v,F ) G ≤

c |v|



(1− αδ 2 )κ

˜  DF m,n,p + DF m,n,2(n+1)p 

 (6.67)

n κ  αδ ˜  ¯ Fρ (θ )−1 m,n,p Gm,n,p sup  Q × 1 + |v|− 2 DF m,n,2(n+1)p (ρ,θ)

¯ ∞. holds for all κ ∈ [1, N0 ] and G ∈ D Proof We modify the proof of Theorem 5.11.1. Let us show that for any positive integer N , there exist m, n, p such that the above inequality holds with κ = N . Instead of (5.153), we use the inequality ˜

e−i(v,DF ) − 1m,n,p;A(ρ)

 n 1 − αδ ˜  ˜  2 DF 1 + |v| ≤ c|v|ϕ(ρ) 2 DF m,n,(n+1)p m,n,(n+1)p 1

which follows from (5.129), replacing the term |v|ρϕ(ρ) 2 by the bigger term αδ |v|− 2 . Then the discussion in the proof of Theorem 5.11.1 leads to (6.67) with κ = N.  Since norms  m,n,p etc. are nondecreasing with respect to m, n, p, we can choose m, n, p, c such that the inequality of the lemma holds for any N = 2, 3, . . . , N0 . Then by the interpolation, there exists another constant c > 0 such that the inequality of the lemma holds for all κ ∈ [0, N0 ].   Let 0 ≤ s < t ≤ T and Φs,t (x) be the solution of a nondegenerate SDE (6.30) starting from x at time s. Let γ be a positive constant. We consider a random variable Hs,t,x =

1 Φs,t (x). (t − s)γ

We are interested in the decay property of characteristic functions of Hs,t,x , as t → s. In the following three lemmas, the operator AJ (t) is assumed to be pseudoelliptic. Lemma 6.7.2 For any M > 0 and κ0 > 2, there exist m, n, p and a positive constant c such that for any |x| ≤ M and 1 < κ < κ0 the inequality

  (t − s)γ −1 κ 

Gm,n,p

E ei(v,Hs,t,x ) G ≤ c (1− αδ ) 2 |v| 2γ

holds for all 0 ≤ s < t ≤ T and |v| ≥ (t − s)− αδ .

(6.68)

6.7 Short-Time Estimates of Densities

281

Further, If AJ (t) is uniformly pseudo-elliptic, for any κ0 > 2 there exist c, m, n, p such that for any x ∈ Rd and 1 < κ < κ0 , the inequality (6.68) holds for 2γ all 0 ≤ s < t ≤ T and |v| ≥ (t − s)− αδ . Proof We will apply Lemma 6.7.1, setting F = Hs,t,x . Consider the right-hand side of (6.67). Since   ˜ s,t (x) c1 := sup DΦs,t (x)m,n,p + DΦ (6.69) m,n,2(n+1)p s 2, there exist m, n, p and a positive constant c such that |E[ei(v,Φs,t (x)) G]| ⎧ ⎨ Gm,n,p if v ∈ Rd , c  ≤ Gm,n,p , if |v| ≥ (t − s)−c0 γ ⎩ αδ (1− αδγ )κ (1− )κ 2 2 (t − s) |v| holds for all 1 < κ < N0 , 0 ≤ s < t ≤ T and |x| ≤ M, where c0 =

2 αδ

+ 1.

(6.74)

282

6 Smooth Densities and Heat Kernels

Further, if AJ (t) is uniformly pseudo-elliptic, for any N0 > 2, there exist m, n, p and a positive constant c such that (6.74) holds for all 1 < κ < N0 , 0 ≤ s < t ≤ T and x ∈ Rd . Proof Since the equality E[ei(v,Φs,t (x)) G] = E[ei((t−s) from Lemma 6.7.2 the inequality |E[ei(v,Φs,t (x)) G]| ≤ c ≤



γ v,H

κ

(t − s)γ −1 αδ

((t − s)γ |v|)(1− 2 ) c

(t − s)(1−

αδγ 2

)κ

αδ

s,t,x )

G] holds, we get

Gm,n,p

|v|(1− 2 )κ

Gm,n,p ,

(6.75)

2

if |v| ≥ (t − s)−( αδ +1)γ , for any |x| ≤ M, s < t and 1 ≤ κ ≤ N0 . Further, if the SDE is uniformly nondegenerate at the center, the above holds for all x ∈ Rd , s < t and 1 ≤ κ ≤ N0 .   c,d c,d Lemma 6.7.4 Let ψs,t,x (v) be the characteristic function of Ps,t (x, E). For any M > 0 and N0 > 2, there exist positive constants c3 , c4 such that

⎧ |i| if v ∈ Rd ,

⎨ c3 (1 ∨ |v|) ,

i c,d

c4 αδ

∂x ψs,t,x (v) ≤ |v||i|−(1− 2 )κ , if |v| ≥ (t − s)−c0 γ ⎩ (1− αδγ )κ 2 (t − s)

(6.76)

holds for all 1 ≤ κ ≤ N0 , 0 ≤ s < t ≤ T and |x| ≤ M. If the SDE is uniformly nondegenerate at the center, there exist k, p, c > 0 such that the above holds for all 0 ≤ s < t ≤ T , x ∈ Rd and 1 ≤ κ ≤ N0 . c,d Proof The function ψs,t,x (v) is infinitely differentiable with respect to x. It holds that   c,d (6.77) (v) = E ei(v,Φs,t (x)) Gis,t,x,v , ∂xi ψs,t,x

where Gis,t,x,v is written as a linear sum of terms written as (6.21). Then the i ,...,i

0 j c,d . Apply (6.75) for each G = Gs,t,x,θ and sum inequality (6.63) holds for ψs,t,x up for i0 , . . . , ij . Then we get (6.76) for any |x| ≤ M, s < t and 1 ≤ κ ≤ N0 . The latter assertion will be obvious.  

Proof of Theorem 6.7.2 It is sufficient to prove (6.66) in the case 0 < t − s < 1. We want to prove the inequality by applying the following formula: c,d ∂xi ∂yj ps,t (x, y) = (−i)|j|

 1 d  c,d e−i(v,y) v j ∂xi ψs,t,x (v) dv. 2π Rd

Note that (t − s)−c0 γ > 1. From Lemma 6.7.4, for any M > 0 there exist positive constants c1 , c2 such that the inequality

6.7 Short-Time Estimates of Densities

283

⎧ if |v| ≤ 1, ⎪ c0 , ⎨

⎪

|i|+j| , if 1 < |v| ≤ (t − s)−c0 γ , c1 |v|

j i c,d

v ∂x ψs,t,x (v) ≤ c αδ 2 ⎪ ⎪ |v||i|+|j|−(1− 2 )κ , if |v| > (t − s)−c0 γ ⎩ (1− αδγ )κ 2 (t − s) (6.78) holds for all s < t, |x| < M. Note that the right-hand side is an integrable function |i|+|j|+d of v, if and only if κ satisfies |i| + |j| − (1 − αδ αδ 2 )κ < −d or equivalently κ > 1−

holds. We will take γ = γ0 satisfying γ0 ≥ (2 − αδ) ∨

1 αδ .

Next, set κ0 =

2

|i|+|j|+d . α 1− 20

Then the right-hand side of (6.78) is an integrable function of v. In fact, |i| + |j| − (1 − αδ 2 )κ0 < −d holds, because αδ < α0 . Further, if |x| < M, we have from (6.78)





c,d e−i(v,y) v j ∂xi ψs,t,x (v) dv

Rd



+ c1

|v||i|+|j| dv +

|v|≤(t−s)−c0 γ0

≤ c0 +

since 1 c0 γ0

have

c1 (t − s)

+

|i|+|j|+d c0 γ0

 ≤ c0

(t − s)

(t − s)

(t − s)

≤

1 (t − s)

|i|+|j|+d 2−αδ

(1−

(6.79)



αδγ0 2 )κ0

αδ

|v||i|+|j|−(1− 2 )κ0 dv

|v|≥1

c2 (2−αδγ0 ) |i|+|j|+d 2−α

|i|+|j|−(1− αδ 2 )κ0 dv is finite. |v|≥1|v| 1 ≤ 2−α . Furthermore, 2 − αδγ0 ≤ 0

|i|+|j|+d γ0

1 dv

c2



1

|v|≤1

,

0

We have

|i|+|j|+d c0 γ0

1 holds because γ0 ≥ 1

,

≤

(t − s)

(2−αδγ0 ) |i|+|j|+d 2−αδ

≤

|i|+|j|+d because 2−α0 1 αδ . Therefore we

1 (t − s)

|i|+|j|+d 2−αδ

.

Consequently, we get the inequality c,d sup |∂xi ∂yj ps,t (x, y)| =

y∈Rd

≤

 1 d  c,d |v j ∂xi ψs,t,x (v)| dv 2π Rd c3 (t − s)

|i|+|j|+d 2−αδ

,

(6.80)

for all 0 ≤ s < t ≤ T , if |x| < M. If AJ (t) is uniformly pseudo-elliptic, we can take constants c1 , c2 in (6.78) such that the inequality holds for all 0 ≤ s < t ≤ T and x ∈ Rd . Then the inequality (6.80) holds for all x, y ∈ Rd and 0 ≤ s < t ≤ T . Then the second assertion of the theorem follows.  

284

6 Smooth Densities and Heat Kernels

Remark If the generator of a jump-diffusion satisfying Condition (J.2) is elliptic, the same short-time estimate of (6.61) is valid for its transition function. Indeed, the proof of Theorem 6.7.1 can be applied to elliptic jump-diffusions satisfying Condition (J.2).

6.8 Off-Diagonal Short-Time Estimates of Density Functions We shall next consider the off-diagonal short-time asymptotics of density functions for elliptic diffusions and pseudo-elliptic jump-diffusions. These properties are quite different between diffusions and jump-diffusions. We first consider a diffusion process. c (x, y) be the smooth density of transition function Theorem 6.8.1 Let ps,t j

c (x, E), where its generator Ac (t) is elliptic. Let ∂ be any given differential Ps,t y operator of index j. Suppose that U, V are bounded disjoint open subsets of Rd such that d(U¯ , V¯ ) > 0. Then for any T > 0 and N ∈ N, there exists a positive constant c = cT ,N,j,U,V such that

sup x∈U,y∈V



j c

∂y ps,t (x, y) ≤ c(t − s)N ,

∀0 ≤ s < t ≤ T .

(6.81)

For the proof of the theorem, we will consider a diffusion Xtx,s = Φs,t (x) pinned at the point y ∈ Rd at time t, or a diffusion conditioned by {Xtx,s = y}. Later we will consider a similar problem for a jump-diffusion. So we will define the pinned process for a jump-diffusion. We defined in Sect. 5.12 the composite of a tempered distribution and a nondegenerate smooth Wiener–Poisson functional. Let Φs,t (x) be the jump-diffusion −1 (y). Then determined by a pseudo-elliptic SDE (6.30) and let Ψˇ s,t (y) = Φs,t both Φs,t (x) and Ψˇ s,t (y) are nondegenerate Wiener–Poisson functionals for any s < t, x, y. Let δx and δy be the delta functions of points x and y, respectively. Then the composite δy (Φs,t (x)) and δx (Ψˇ s,t (y)) are well defined as generalized c,d Wiener–Poisson functionals. It holds that ps,t (x, y) = E[δy (Φs,t (x))Gs,t (x)], where Gs,t (x) is given by (6.53) (see Corollary 5.12.1). Lemma 6.8.1 We have E

n  i=1

n    fi (Φs,ti (x))δy (Φs,t (x)) = E fi (Ψˇ ti ,t (y))δx (Ψˇ s,t (y))| det ∇ Ψˇ s,t (y)| i=1

for any s < t1 < · · · tn < t, x, y ∈ Rd and C0∞ -functions fi , i = 1, . . . , n.

(6.82)

6.8 Off-Diagonal Short-Time Estimates of Density Functions

285

Proof Applying the change-of-variable formula and then taking expectations (Sect. 4.6), we get the dual formula  Rd

=

ψ(x)E



 fi (Φs,ti (x))ϕ(Φs,t (x)) dx

i

 Rd

E



 fi (Ψˇ ti ,t (y))ψ(Ψˇ s,t (y))| det ∇ Ψˇ s,t (y)| ϕ(y) dy

i

for C0∞ -functions fi , i = 1, . . . , n and ϕ, ψ. Let ϕ and ψ tend to δy and δx , respectively. Then we get the formula of the lemma.   We need an another lemma. Lemma 6.2.1 is modified as follows, whose proof can be done similarly to the proof of Lemma 6.2.1. Lemma 6.8.2 Let U, V be bounded open subsets of Rd such that d(U¯ , V ) > 0. Let τ (x, s) be the hitting time of the diffusion process Xt = Xtx,s := Φs,t (x) to the set V . Then for any N ≥ 2, there exists a positive constant c such that sup P (τ (x, s) < t) ≤ c(t − s)N ,

∀s < t.

(6.83)

x∈U

Proof of Theorem 6.8.1 We will consider the case Gs,t (x) = 1 only, since discussions for the case Gs,t (x) > 0 are similar. We first consider the case j = 0. Let U, V be given sets in the assertion of Theorem 6.8.1. Take a bounded open set W which satisfies U¯ ⊂ W ⊂ V c , d(U¯ , W c ) > 0 and d(W¯ , V¯ ) > 0. Let x ∈ U and y ∈ V . Let τ = τ (x, s) be the first leaving time of the diffusion process Xt = Xtx,s = Φs,t (x) from the set W . Then it holds that P (τ ≤ t) ≤ c(t − s)N by Lemma 6.8.2. Since d(W¯ , y) > 0, we have   ps,t (x, y) = E δy (Φs,t (x)); s < τ ≤ t (6.84)     = E δy (Φs,t (x)); s < τ ≤ (t + s)/2 + E δy (Φs,t (x)); (t + s)/2 < τ ≤ t . Using the strong Markov property, the first term of the right-hand side is estimated as     E δy (Φs,t (x)); s < τ ≤ (t + s)/2 = E Pτ,t δy (Φs,τ (x)); s < τ ≤ (t + s)/2   = E pτ,t (Φs,τ (x), y); s < τ ≤ (t + s)/2 . Note that Φs,τ ∈ W¯ . Then we have pτ,t (Φs,τ (x), y) ≤ c(t − τ )−d0 ≤ c

 t − s −d0 , 2

(6.85)

286

6 Smooth Densities and Heat Kernels

where d0 = d2 , in view of Theorem 6.7.1. Therefore, we get  t − s −d0   P (s < τ ≤ (t + s)/2) E δy (Φs,t (x)); s < τ ≤ (t + s)/2 ≤ c1 2 ≤ c2 (t − s)N −d0 .

(6.86)

We shall next consider the second term of (6.84). It holds by Lemma 6.8.1 that E[δy (Φs,t (x)); (t + s)/2 < τ ≤ t]

(6.87)

= E[| det ∇ Ψˇ s,t (y)|δx (Ψˇ s,t (y)); (t + s)/2 < τ ≤ t]. If ps,t (x, y) > 0, the conditional probability measure given {Φs,t (x) = y} is defined by P (A|Ψˇ s,t (y) = x) =

E[1A δx (Ψˇ s,t (y))] . ps,t (x, y)

Under this measure, it holds that Φs,r (x) = Ψˇ r,t (Φs,t (x)) = Ψˇ r,t (y) a.s. Then τ may be regarded as the last exit time of the backward process Yˇr := Ψˇ r,t (y) (t, y are fixed) from W¯ . Let σ∗ be the first exit time of the same process Yˇr from W¯ c . It is defined by σ∗ = sup{r; Yˇr ∈ W¯ c }. Then it holds that t − τ ≥ t − σ∗ . We will apply Lemma 6.8.2 to the backward process Yˇr . Then, similarly to (6.86), we have E[δy (Φs,t (x)); (t + s)/2 < τ ≤ t]

(6.88)

≤ E[| det ∇ Ψˇ s,t (y)|δx (Ψˇ s,t (y)); (t + s)/2 < σ∗ ≤ t] ≤ c3 (t − s)N −d0 . Substituting (6.86) and (6.88) to (6.84), we get the inequality (6.81) of the theorem if we rewrite N − d0 as N . Finally, if j = 0, we consider the composite of the generalized function j   (−1)j ∂x δy and Φs,t (x). Then we get the assertion (6.81) similarly. In the case of a jump-diffusion, the problem is more complicated, since Lemma 6.8.2 does not hold. In order that the inequality (6.83) is valid, the distance of two disjoint sets U and V should be big so that the point x ∈ U reaches the point y ∈ V after sufficiently many jumps. It is explained below. Let {φr,z } be jump-maps of the jump-diffusion. Let U be an open subset of Rd . In Sect. 6.5, we defined the ¯ ) = ∪φr,z (U ) etc. set φ(U The following is a consequence of Lemma 6.5.2. Lemma 6.8.3 Let N be a positive integer satisfying d(φ¯ N (U ), φ −N (V )) > 0. Let τ = τ (x, s) be the hitting time of the process Xtx,s := Φs,t (x), t ∈ [s, ∞) to the set V . Then there exists c > 0 such that sup P (τ (x, s) < t) ≤ c(t − s)N , x∈U

∀s < t.

(6.89)

6.8 Off-Diagonal Short-Time Estimates of Density Functions

287

c,d Theorem 6.8.2 Let ps,t (x, y) be the smooth density of transition function c,d (x, E), where its generator AJc,d (t) is pseudo-elliptic. Let α0 be an arbitrary Ps,t positive constant such that α < α0 < 2, where α is the exponent of the Lévy measure. Suppose that U, V are bounded open sets of Rd such that d(φ¯ N (U ), φ¯ −N (V )) > 0 holds for some positive integer N . Then for any T > 0 there exists a positive constant c = cT ,j,U,V such that

sup x∈U,y∈V

N − |j|+d

j c,d

∂y ps,t (x, y) ≤ c(t − s) 2−α0 ,

∀0 ≤ s < t ≤ T .

(6.90)

Proof We will consider the case Gs,t (x) = 1, i.e., c = 0, d = 1, since discussions c,d (x, y) by ps,t (x, y). We first for the case Gs,t (x) = 1 are similar. We denote ps,t consider the case j = 0. Let x ∈ U and y ∈ V . Let τ be the first leaving time of the process Xt = Φs,t (x) from φ¯ N (U ). Then it holds that P (τ ≤ r) ≤ c(r − s)N by Lemma 6.8.3. Further, we have   (6.91) ps,t (x, y) = E δy (Φs,t (x)); s < τ ≤ t   = E pτ,t (Φs,τ (x), y) : s < τ ≤ (t + s)/2 + E[| det ∇ Ψˇ s,t (y)|δx (Ψˇ s,t (y)); (t + s)/2 < τ < t], as in the proof of the previous theorem. Note that Φs,τ (x) ∈ φ¯ N (U ) and φ¯ N (U ) is a bounded set, because g(x, t, z) is a bounded function. Then we have  t − s −d0 pτ,t (Φs,τ (x), y) ≤ c(t − τ )−d0 ≤ c , 2 where d0 =

d 2−α0 ,

in view of Theorem 6.7.2. Therefore, we get

 t − s −d0   P (s < τ ≤ (t + s)/2) E pτ,t (Φs,τ (x), y) : s < τ ≤ (t + s)/2 ≤ c1 2 ≤ c2 (t − s)N −d0 . We shall next consider the last term of (6.91). Under the conditional measure P (A|Ψˇ s,t (y) = x), it holds that Φs,r (x) = Ψˇ r,t (Φs,t (x)) = Ψˇ s,t (y) a.s. Then τ may be regarded as the last exit time of the left continuous backward process Yˇs = Ψˇ s−,t (y) (t is fixed) from φ¯ −N (U c ). Let σ∗ be the first exit time of the same process Yˇs from φ¯ −N (U c ). It is defined by σ∗ = sup{s; Yˇs ∈ φ¯ −N (V )}. Then it holds that t − τ ≥ t − σ∗ , since d(φ¯ N (U ), φ¯ −N (V )) > 0. We will apply Lemma 6.8.1 to the backward process Ψˇ s,t (y) (t being fixed). Then, similarly to (6.86), we have E[| det ∇ Ψˇ s,t (y)|δx (Ψˇ s,t (y)); (t + s)/2 < σ∗ < t] ≤ c3 (t − s)N −d0 . Substituting these two inequalities into (6.91), we get the inequality (6.90) of the theorem in the case |j| = 0.

288

6 Smooth Densities and Heat Kernels j

Finally, if j = 0, we consider the composite (−1)j ∂x δy and Φs,t (x). Then we get the assertion (6.90) similarly.   Note The short-time estimate and the off-diagonal short-time estimate for diffusion processes seems to be known more or less. These estimates for jump processes are studied by Picard [93] and Ishikawa [43], not using the Malliavin calculus. Ishikawa et al. [46] obtained the short-time estimates for jump processes using the Malliavin calculus. Their results are close to our Theorem 6.7.1. The present off-diagonal short-time estimate using the Malliavin calculus might be new.

6.9 Densities for Processes with Big Jumps Let us return to SDE (6.30). For drift and diffusion coefficients V0 (x, t), V1 (x, t), . . ., Vd  (x, t), we assumed that they are of Cb∞,1 -class; for jump coefficients φt,z , we assumed that they satisfy Conditions (J.1) and (J.2) stated in Sect. 3.2. In this section we will assume the same condition for drift and diffusion coefficients. But for jump coefficients, we will assume Conditions (J.1)K and (J.2)K stated in Sect. 3.9. For the Lévy measure ν, we assume again that it has a weak drift and that at the center it satisfies the order condition of exponent 0 < α < 2 with respect to a family of star-shaped neighborhoods. We assume further that its generator is pseudoelliptic. We should remark that the pseudo-elliptic property is concerned with the diffusion part and small jumps only, since vector fields V˜k (t), k = 1, . . . , d  are determined by jump-maps φr,z ; |z| < γ for arbitrary small γ > 0. We are again interested in the existence of the smooth density of the law of the solution Φs,t (x). However, since equation (6.30) may admit big jumps, the solution may not belong to L∞− . So we cannot apply the Malliavin calculus to the solution directly. To avoid this difficulty, we will first consider an SDE where big jumps are truncated, for which the smooth density of the law exists. Then we will adjoin big jumps as a perturbation and show that such a perturbation should preserve the smoothness of the density. In this section, we will restrict our attention to the law of the solution Φs,t (x) and will avoid some complicated discussions and notations related the weighted laws. Our discussions will be extended to the weighted law of Φs,t (x) as in Sects. 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, and 6.8. Suppose we are given an SDE (6.30), where coefficients of the equation satisfy Conditions (J.1)K and (J.2)K . We will consider an SDE truncating jumps which are bigger than δ > 0. The truncated equation is given by d   

Xt = X0 +

t

k=0 t0

+

Vk (Xr , r) ◦ dWrk

 t t0

δ≥|z|>0+

{φr,z (Xr− ) − Xr− }N(dr dz).

(6.92)

6.9 Densities for Processes with Big Jumps

289

The solutions of the above equation define a stochastic flow of diffeomorphisms {Φ¯ s,t } by Theorem 3.9.1, since the truncated function g(x, t, z)1|z|≤δ satisfies Conditions (J.1) and (J.2). For a given T > 0, we set T = [0, T ]. We will fix 0 ≤ s < t ≤ T and a point x. We shall construct the solution of equation (6.30) starting from x at time s, by  adding big jumps to Φ¯ s,t (x). Let U = T × Rd0 and u = ((t1 , z1 ), . . . , (tn , zn )) be an element of Un (n ≥ 1) such that |zi | > δ, i = 1, . . . , n and s < t1 < · · · < tn ≤ t. u (x) := Φ ¯ s,t (x) ◦ εu+ . It is represented by We consider Φ¯ s,t u (x) = Φ¯ tn ,t ◦ φtn ,zn ◦ · · · ◦ Φ¯ t1 ,t2 ◦ φt1 ,z1 ◦ Φ¯ s,t1 (x), Φ¯ s,t

(6.93)

φ as in (6.32). We set Φ¯ s,t (x) = Φ¯ s,t (x) if u = ∅ ∈ U0 (empty set). Now, let q(ω) be a  Poisson point process on U = [s, T ] × Rd0 with intensity measure dtνδ (dz), where νδ (dz) = 1(δ,∞) (|z|)ν(dz). Let N(dr dz) be the associated Poisson random measure (Sect. 1.4). Let Dq be the domain of the point process q. Since λδ ≡ ν({|z| > δ}) is finite, the set Dq ∩ (s, t] can be written, if it is non-empty, as {s = τ0 < τ1 < · · · < τn }, where τm are stopping times defined inductively by τ0 = s and by τm = inf{t > τm−1 ; Nt − Nτm−1 ≥ 1} (= ∞ if the set {· · · } is empty). Here Nt := N ((s, t] × {|z| > δ}) is a Poisson process with intensity λδ . We set u(q) = ∅ if τ1 > t and u(q) = ((τ1 , q(τ1 )), . . . , (τn , q(τn ))) if τn ≤ t < τn+1 . Then we u(q) define Φs,t (x) := Φ¯ s,t (x). It is written as



Φ¯ s,t (x), if t < τ1 , Φ¯ τn ,t ◦ φτn ,q(τn ) ◦ · · · ◦ Φ¯ τ1 ,τ2 ◦ φτ1 ,q(τ1 ) ◦ Φ¯ s,τ1 (x), if τn ≤ t < τn+1 .

Therefore Φs,t (x) is a solution of the original equation (6.30) starting from x at time s (see Sect. 3.2 in Chap. 3). Now, we assume that the generator of the process Φs,t (x) is pseudo-elliptic. Then the law of Φ¯ s,t (x); P¯s,t (x, E) = P (Φ¯ s,t (x) ∈ E) has a rapidly decreasing C ∞ density by Theorem 6.6.1. We denote it by p¯ s,t (x, y). Further, the stochastic process u u (x) is a Markov process with fixed discontinuity, jumping from Φ ¯ s,t Xt = Φ¯ s,t to i− u ∞ ¯ φti ,zi (Φs,ti − ) at fixed times ti ; i = 1, . . . , n. Its transition probability has a C density given by  u (x, y) p¯ s,t

=

 ···

(Rd )n

p¯ s,t1 (x, x1 )p¯ t1 ,t2 (φt1 ,z1 (x1 ), x2 ) × · · ·

× p¯ tn ,t (φtn ,zn (xn ), y) dx1 · · · dxn . φ

Further, setting p¯ s,t (x, y) = p¯ s,t (x, y), we have P (Φs,t (x) ∈ E) = P (Φ¯ s,t

u(q)



u(q)

∈ E) = E

E[p¯ s,t (x, y)] dy.

(6.94)

290

6 Smooth Densities and Heat Kernels

Therefore, the function defined by u(q)

ps,t (x, y) ≡ E[p¯ s,t (x, y)]

(6.95)

is the density functions of transition probability Ps,t (x, E) of Φs,t (x). We want to prove the smoothness of the above ps,t (x, y) under additional conditions. Theorem 6.9.1 Consider a system of uniformly pseudo-elliptic jump-diffusions, which satisfies Conditions (J.1)K and (J.2)K . 1. Assume that {φt,z } satisfies  sup t,x

|z|>δ



−1 (x) ν(dz) < ∞ for some δ > 0,

det ∇φt,z

(6.96)

−1 −1 where det ∇φt,z (x) is the Jacobian determinant of the inverse map φt,z . Then the function ps,t (x, y) given by (6.95) is a continuous function of x, y ∈ Rd for any 0 ≤ s < t ≤ T . Further, for any α < α0 < 2, there exists c0 > 0 such that the following holds for all 0 ≤ s < t ≤ T and x, y ∈ Rd :

ps,t (x, y) ≤

c0 d

(6.97)

.

(t − s) 2−α0

2. Let m0 be a positive integer. Assume further that  sup t,x

|z|>δ

p

i

∂x φt,z (x) ν(dz) < ∞ for some δ > 0

(6.98)

holds for ∂xi with 1 ≤ |i| ≤ m0 and p > 1. Then ps,t (x, y) is a Cbm0 -function of x. Further, there exists cm0 > 0 such that the following holds for all 0 ≤ s < t ≤ T and x, y ∈ Rd if |i| ≤ m0 :

i

∂x ps,t (x, y) ≤

cm0 |i|+d

.

(6.99)

(t − s) 2−α0

−1 and φ Conditions (6.96) and (6.98) indicate that derivatives of jump-maps φr,z r,z should not be big but moderate, though φr,z can be big. Loosely, the image of a ball by these maps should not be too distorted. We discuss two lemmas. Let u = {(t1 , z1 ), . . . , (tn , zn )}, where s ≤ t1 < · · · < tn ≤ t. We write it as u = (t, z), where t = (t1 , . . . , tn ) and   z = (z1 , . . . , zn ) ∈ (Rd0 )n . We define a probability measure on (Rd0 )n by (n)

1

νδ (dz) = νδ (dz1 ) · · · νδ (dzn ), where νδ (dz) = |z|>δ λδ ν(dz) and λδ = ν({|z| > δ}). We fix δ > 0 stated in the theorem and we set for t ∈ T and x, y ∈ Rd

6.9 Densities for Processes with Big Jumps

291

 t p¯ s,t (x, y) :=

(t,z)

 (Rd0 )n

(n)

p¯ s,t (x, y)νδ (dz),

(t,z)

where p¯ s,t (x, y) is defined by (6.94). Then we get the formula ps,t (x, y) =

∞    u(q) E p¯ s,t (x, y)1τn ≤t 0 such that if |j| ≤ n0 , the following inequality holds for all 0 ≤ s < t ≤ T and x, y ∈ Rd :

j

∂y ps,t (x, y) ≤

cn0 |j|+d

.

(6.107)

(t − s) 2−α0

Proof We need the dual property of the jump-diffusion with respect to L2 (dx) on t f by Rd . We will define the dual of P¯s,t t,∗ g(y) = P¯s,t

 Rd

t p¯ s,t (x, y)g(x) dx.

(6.108)

t,∗ Then it holds that |P¯s,t g|∞ ≤ c∗ |g|∞ . Further, similarly to Lemma 6.9.1, there exists a positive constant Cˇ such that



t,∗ sup ∂ j P¯s,t g

t∈(s,t]n+

∞

≤ Cˇ n+1

 

∂ j g , ∞

n = 1, 2, . . .

(6.109)

j ⊂j

Then we can verify that ps,t (x, y) is a Cbn0 -function of y and satisfies the short-time estimate (6.107) by a similar argument.   In Theorem 6.6.1, we showed that the density function ps,t (x, y) is a rapidly decreasing C ∞ -function of x and y, under Conditions (J.1) and (J.2). Hence jumps are assumed to be bounded. However, in Theorem 6.9.2, the density function ps,t (x, y) may not be rapidly decreasing, because of big jumps. For example, any

6.10 Density and Fundamental Solution on Subdomain

295

stable process satisfies conditions of the above theorem, and its law has a C ∞ density function, but the density function is not rapidly decreasing if it is not Gaussian. c,d Assertions of Theorem 6.9.1 can be extended to the smooth density ps,t (x, y) of c,d a weighted law Ps,t (x, E) = E[1E (Φs,t (x))Gs,t (x)], if Gs,t (x) is an exponential functional given by (6.53), where dr,z (x) is assumed to be a bounded positive function. For details, see the proof of Theorem 7.5.1 in Sect. 7.5, Note The existence of the measurable density can be shown without Condition (J.2)K . It means that maps {φt,z , |z| > δ} may not be diffeomorphic for some > 0. In fact, ps,t (x, y) of (6.95) is well defined and is the density function of the transition probability, without condition (J.2)K . However, it seems to be a hard question whether we can remove Condition (J.2)K for the existence of the smooth density. The continuity and the differentiability are not clear without the diffeomorphic property of maps φr,z . In Theorem 3.2 in Kunita [61], the existence of the smooth density is asserted without (J.2)K , but his discussion contains gaps.

6.10 Density and Fundamental Solution on Subdomain In Sect. 4.8, we considered the weighted transition function of a jump-diffusion c,d killed outside of a subdomain D of Rd . The weighted transition function Qs,t (x, E) is equal to c,d c,d Qs,t (x, E) = E[1E (Φs,t (x))Gs,t (x)1τ (x,s)>t ],

where τ (x, s) is the first leaving time of the process Xt = Xtx,s := Φs,t (x), t in [s, ∞) from the set D. We saw in Sects. 6.3 and 6.6 that, if Xt is nondegenerate, the c,d c,d weighted transition function Ps,t (x, E) = E[1E (Φs,t (x))Gs,t (x)] has a smooth c,d c,d c,d density ps,t (x, y). Since Qs,t (x, E) ≤ Ps,t (x, E) holds for any s, x, t, the measure c,d c,d c,d c,d Qs,t (x, E) has always a density qs,t (x, y) such that qs,t (x, y) ≤ ps,t (x, y) holds a.e. y. We want to show its smoothness. Suppose that the jump-diffusion is pseudoelliptic. If there are no jumps entering D c from D a.s., it should have a C ∞ -density. However, if the process has jumps from D c into D, the result is not simple. The smoothness of the density is related to the off-diagonal short-time estimate studied in Sect. 6.8. c,d We first consider a diffusion process. Since d = 1, we denote Qs,t (x, E) and c,d c c qs,t (x, y) by Qs,t (x, E) and qs,t (x, y), respectively. Lemma 6.10.1 Suppose that Xtx,s := Φs,t (x) is an elliptic diffusion process of initial state (x, s). Then for any x ∈ D and s < t, Qcs,t (x, E); E ⊂ D has a C ∞ c (x, y). It is continuously differentiable with respect to t and satisfies density qs,t

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6 Smooth Densities and Heat Kernels

∂ c c (x, y). q (x, y) = Ac (t)∗y qs,t ∂t s,t

(6.110)

Further, the density has the following two properties: 1. Let U be an open subset of Rd such that U¯ ⊂ D. Then for any multi-index j and T > 0, there exists a positive constant c such that the inequality

j c

∂y qs,t (x, y) ≤

c (t − s)

|j|+d 2

,

∀0 ≤ s < t ≤ T

(6.111)

holds for any x ∈ D,y ∈ U . 2. Let V be an open subset of D satisfying V¯ ⊂ U ⊂ U¯ ⊂ D. Then for any j and T > 0 there exists a positive constant c such that the inequality c (x, y)| ≤ c , |∂yj qs,t

∀0 ≤ s < t ≤ T

(6.112)

holds for all x ∈ U c ∩ D, y ∈ V . Proof For a given T > 0, we take 0 ≤ s < t ≤ T . Let τ = τ (x, s) be the first leaving time of the process Xt = Xtx,s = Φs,t (x) from the set D. We have by the strong Markov property   c c f (x) − E Pτ,t f (Φs,τ (x))Gcs,τ (x)1s 0 such that the inequality

j c,d

∂y qs,t (x, y) ≤ holds for all x ∈ D, y ∈ U .

c |j|+d

(t − s) 2−α0

,

∀0 ≤ s < t ≤ T

(6.123)

6.10 Density and Fundamental Solution on Subdomain

301

2. Let V be an open subset of U such that d(φ¯ N (V ), φ¯ −N (U c )) > 0. Then for any j and T > 0 there exists c > 0 such that the inequality c,d (x, y)| ≤ c , |∂yj qs,t

∀0 ≤ s < t ≤ T

(6.124)

holds for all x ∈ U c ∩ D, y ∈ V . Proof We will repeat an argument similar to the proof of Lemma 6.10.1. Let U be an open subset of D satisfying properties stated in the lemma. Define the function c,d c (x, y) by p c,d (x, y). It is the density function qs,t (x, y) by (6.114) replacing ps,t s,t j

c,d (x, E) for any x ∈ Rd and y ∈ U . We can define the function qs,t (x, y) of Qs,t c (x, y) by p c,d (x, y). Indeed, since Φ (x) ∈ D c if by (6.115) replacing ps,t s,τ s,t τ = τ (x, s) < t, we have by Theorem 6.8.2, c,d |∂yj pτ,t (Φs,τ (x), y)| ≤ c(t − τ )

|j|+d N − 2−α

0

≤ c < ∞.

j

Then qs,t (x, y) is well defined and is continuous with respect to y ∈ U for any c,d j with |j| ≤ n0 . Therefore, qs,t (x, y) is n0 -times continuously differentiable with j c,d j (x, y) = qs,t (x, y) holds for |j| ≤ n0 . respect to y ∈ V and the equality ∂y qs,t j c,d (x, y) has the same short-time Since the last term of (6.115) is bounded, ∂y qs,t j c,d c,d estimate as ∂y ps,t (x, y), which proves (6.123) for qs,t (x, y). j c,d (x, y)| is We show the second assertion of the lemma. We know that |∂y ps,t c bounded with respect to x ∈ U ∩ D, y ∈ V and s < t, in view of Theorem 6.8.2. Further, the last term of equation (6.115) is also bounded with respect to these j   x, y, t. Consequently, qs,t (x, y) is also bounded with respect to these x, y, t. −1 Now if jump-maps φt,z map D onto D, diffeomorphically, both φt,z and φt,z should map D c into itself. Therefore we have d(φ¯ N (U ), φ¯ −N (D c )) > 0 for any N c,d if U¯ is a compact subset of D. Therefore Qs,t (x, E) has a C ∞ -density if x ∈ U¯ . Next, we will apply the same argument to the dual process. Then we find that c,d qs,t (x, y) is a C ∞ -function of x, similarly to the case of diffusion. Therefore, we have the following theorem.

Theorem 6.10.4 Consider a system of pseudo-elliptic jump-diffusions on Rd with the generator AJ (t) of (4.52). Assume that jump-coefficients φt,z map a bounded domain D ⊂ Rd onto itself for all t, z. Then the process killed outside of D is c,d a jump-diffusion. Its weighted transition function Qs,t (x, E) has a C ∞ -density c,d ∞,1 qs,t (x, y). The density function is a C -function of x ∈ D, s (< t) for any y, t. It satisfies the short-time estimates (6.123), (6.124). Further, p(x, s; y, t) = c,d qs,t (x, y) is the fundamental solution of the backward heat equation with the stochastic Dirichlet boundary condition on D, associated with the operator AJc,d (t).

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6 Smooth Densities and Heat Kernels

By the similar argument, we can obtain the fundamental solution for the heat equation on D associated with the integro-differential operator Ac,d (t). Details are omitted. c,d Note The smoothness of the density function qs,t (x, y) with respect to y in the case c = 0, d = 1, was shown by Picard–Savona [95] for jump processes. The existence of fundamental solutions for various type of heat equations are known in analysis and mathematical physics. A well known case is that the operator A(t) is a Laplacian and domain D of the heat equation is an interval, a squares or a ball. Results for arbitrary operators Ac (t), AJc,d (t) and arbitrary domains D might be new. The stochastic Dirichlet boundary condition should be transformed to the usual Dirichlet boundary condition if the boundary ∂D is regular, say if it is of C 1 -class. For the proof, further discussions will be needed. c (x, y) for elliptic So far, we showed the existence of the smooth density qs,t diffusions (or pseudo-elliptic jump-diffusions) on a subdomain of an Euclidean space. We can extend the result for a nondegenerate diffusion (or a nondegenerate c (x, y) satisfies the short time estimate jump-diffusion) if its transition density ps,t like (6.81) (or (6.90)). It is expected that a hypo-elliptic diffusion (and pseudohypo-elliptic jump diffusions) admits such an estimate, but for the proof further discussions will be needed.

Chapter 7

Stochastic Flows and Their Densities on Manifolds

Abstract In this chapter, we will study stochastic flows and jump-diffusions on manifolds determined by SDEs. If the manifold is not compact, SDEs may not be complete; solutions may explode in finite time. Then solutions could not generate stochastic flow of diffeomorphisms; instead they should define a stochastic flow of local diffeomorphisms. These facts will be discussed in Sect. 7.1. In Sect. 7.2, it will be shown that the stochastic flow defines a jump-diffusion on the manifold. Then, the dual process with respect to a volume element will be discussed. Further, in Sect. 7.3, the Lévy process on a Lie group and its dual with respect to the Haar measure will be discussed. In Sect. 7.4, we consider an elliptic diffusion process on a connected manifold. We show the existence of the smooth density with respect to a volume element, by piecing together smooth densities on local charts which were obtained in Sect. 6.10. The result of the section can be applied to diffusion processes on Euclidean space with unbounded coefficients, where the explosion may occur. For a pseudo-elliptic jump-diffusion on a connected manifold, we need additional arguments, since sample paths may jump from a local chart to other local charts. It will be discussed in Sect. 7.5. Finally, in Appendix, we collect some basic facts on manifolds and Lie groups that are used in this chapter.

7.1 SDE and Stochastic Flow on Manifold Let M be a C ∞ -manifold of dimension d. Related notations and terminologies for manifolds are collected in the Appendix at the end of the chapter. If M is not compact, we consider its one point compactification M ∪ {∞}. If M is compact, ∞ is adjoined to M as an isolated point. Given a function f on M, we set f (∞) = 0, conventionally. A vector field V (t) with parameter t ∈ T = [0, ∞) is called a C ∞,1 -vector field, if V (t)f (x) is a C ∞,1 -function of (x, t) for any C ∞ function f on M. With a local coordinate (x1 , . . . , xd ), V (t)f (x) is represented by  ∂f i 1 d ∞,1 -functions. i V (x, t) ∂xi (x), where V (x, t), . . . , V (x, t) are C

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4_7

303

304

7 Stochastic Flows and Their Densities on Manifolds 

Let V0 (t), . . . , Vd  (t) be C ∞,1 -vector fields on M. Let {φt,z , (t, z) ∈ T × Rd } be a family of C ∞ -maps from M into itself, satisfying: 

Condition (J.1)’. (x, t, z) → φt,z (x) is a C ∞,1,2 -map from M × T × Rd to M. Further, it satisfies φt,0 (x) = x for any x, t. −1 be Condition (J.2)’. φt,z ; M → M are diffeomorphic maps for all t, z. Let φt,z −1 the inverse map. Then (x, t, z) → φt,z (x) is a C ∞,1,2 -map. 



Let z = (z1 , . . . , zd ) ∈ Rd . We set

V˜k (t)f (x) = ∂zk f (φt,z (x))

z=0

,

k = 1, . . . , d  .

(7.1)

Then V˜k (t), k = 1, . . . , d  are C ∞,1 -vector fields, called the tangent vector fields of maps {φt,z } at z = 0. We shall define a symmetric stochastic differential equation on manifold M associated with the above Vk (t) and φt,z . Let {Fs,t , 0 ≤ s < t < ∞} be the two-sided filtration generated by a d  -dimensional Wiener process Wt =   (Wt1 , . . . , Wtd ), t ∈ T and a Poisson random measure N(dr dz) on T×Rd0 with the Lévy measure ν, which is independent of Wt . We set Ft = F0,t . Let 0 ≤ t0 < ∞. Let τ∞ be a stopping time with values in [t0 , ∞) ∪ {∞}. Let Xt , t0 ≤ t < τ∞ be a cadlag M-valued {Ft }-adapted process such that limt↑τ∞ Xt = ∞ whenever τ∞ < ∞. The stopping time τ∞ is called the terminal time or the explosion time. Let C0∞ (M) be the set of all C ∞ -functions of compact supports. Xt is called a solution of a symmetric SDE with characteristics (Vk (t), k = 0, 1, . . . , d  , φt,z , ν), starting from X0 at time t0 , if f (Xt ), t < τ∞ is a local semi-martingale for any f ∈ C0∞ (M) and satisfies d   

f (Xt ) = f (X0 +

t

k=0 t0

+ lim

→0

 t

Vk (r)f (Xr ) ◦ dWrk

(7.2)

 t d     f (φr,z (Xr− ))−f (Xr− ) N(drdz)− b k V˜k (r)f (Xr− )dr

t0 |z|≥

k=1

t0

for t0 < t < τ∞ . Here ◦dWrk , k = 1, . . . , d  mean symmetric integrals by Wiener k  0 processes Wr , kk = 1, . . . , d and ◦dWr means the usual integral dr. Further, k b = ≤|z|≤1 z ν(dz). The terminal time τ∞ depends on the initial condition of the process Xt . We denote it by τ∞ (X0 , t0 ). Let us consider the relation with SDE on Euclidean space Rd defined in Chap. 3. If Xt is a solution of equation (3.10) on Rd , it satisfies (7.2) for any smooth function f by setting φt,z (z) = g(x, t, z) + x, as it was shown in (3.69). Conversely, if a process Xt on Rd satisfies (7.2) for any smooth function f , Xt satisfies (3.10). However, in this chapter, we do not assume the boundedness of coefficients Vk (x, t), g(x, t, z) := φt,z (x) − x and their derivatives in equation (7.2). Hence the

7.1 SDE and Stochastic Flow on Manifold

305

explosion may occur for the solution in the Euclidean space. Hence our SDE (7.2) may be regarded as a generalization of SDE (3.10). Theorem 7.1.1 For the symmetric SDE (7.2) on the manifold M, we assume that Vk (t), k = 0, . . . , d  are C ∞,1 -vector fields and {φt,z } satisfies Condition (J.1)’. Then, given x ∈ M and 0 ≤ s < ∞, the equation (7.2) has a unique solution Xtx,s (ω), t < τ∞ (x, s, ω), starting from x at time s. Further, for any s < t, the solution has a modification Φs,t (x, ω), t < τ∞ (x, s, ω) such that there is a measurable subset Ω˜ s,t of Ω with P (Ω˜ s,t ) = 1 and for any ω ∈ Ω˜ s,t the following properties hold. The terminal time τ∞ (x, s, ω) is lower semi-continuous with respect to x for any s, so that the set Ds,t (ω) := {x; τ∞ (x, s, ω) > t} are open subsets of M. Further, maps Φs,t (ω); Ds,t (ω) → M are C ∞ . The family of pairs {Φs,t , Ds,t } satisfying properties of the above theorem is called a stochastic flow of C ∞ -maps on the manifold M. Proof We will construct the solution of equation (7.2) by piecing together solutions defined on local charts of the manifold M. Let us first choose a family of local charts Uα ⊂ Dα , α = 1, 2, . . . satisfying the following:

(i) α Uα = M. (ii) The number of α such that x ∈ Uα is at most finite for any x. (iii) For any α, D¯ α is compact and satisfies U¯ α ⊂ Dα . We will consider SDE on Dα . Let ψα be a diffeomorphism from Dα to an open set in Rd (local coordinate). Then the SDE is transformed to an SDE on ψα (Dα ) (⊂ Rd ). Set V´k (t) = dψα (Vk (t)), V˘k (t) = dψα (V˜k (t)) and g( ´ x, ´ t, z) = ψα (φt,z (x)) − x´ if ´ φt,z (x) ∈ Dα , where x´ = ψα (x). The functions {Vk (x, ´ t), k = 0, . . . , d  , g( ´ x, ´ t, z)} are defined on the set ψα (Dα ). Since the set is relatively compact in Rd , these functions are extended to Rd as Cb∞,1 -functions and Cb∞,1,2 -functions, respectively. Consider an SDE on Rd : d   

X´ t = x´ +

k=0 s

+ lim

→0

t

V´k (X´ r , r) ◦ dWrk

  t s |z|≥

(7.3) 



g( ´ X´ r− , r, z)N(dr dz) −

d  k=1

b k

t

 V˘k (X´ r− , r) dr .

s

It generates a stochastic flow {Φ´ s,t } of C ∞ -maps on Rd (Theorem 3.4.4). Given x ∈ M and 0 ≤ s < ∞, we will define an M-valued process as follows. If x ∈ Uα , we set (α) (x) = ψα−1 (Φ´ s,t (x)), ´ Φs,t

if t < τDα (x, s) := inf{t > s; Φ´ s,t (x) ´ ∈ / ψα (Dα )}.

306

7 Stochastic Flows and Their Densities on Manifolds (α)

(β)

Then, by the pathwise uniqueness of the solution, it holds that Φs,t (x) = Φs,t (x) if x ∈ Uα ∩ Uβ and t < τDα (x, s) ∧ τDβ (x, s). Now set τ1 (x, s) :=

min τDα (x, s).

{α;x∈Uα }

(7.4)

Then τ1 (x, s) − s is strictly positive for any x, s a.s. We can define the solution of (7.2) up to time τ1 (x, s) as follows. For t < τ1 (x, s), take α such that τUα (x, s) > (α) t. Then we define Φs,t (x) by Φs,t (x). Then the process Φs,t (x), t < τ1 (x, s) is indeed a solution of SDE (7.2) on the manifold M. d We will prolong the solution for t ≥ τ1 = τ1 (x, s). Define an R0 -valued random variable by S1 = {(τ ,z);z∈Rd  } zN(dr dz). Set Φs,τ1 = φτ1 ,S1 ◦ Φs,τ1 − and define 1

0

Φs,t (x) = Φτ1 ,t (Φs,τ1 (x)),

if τ1 (x, s) ≤ t < τ2 (x, s) := τ1 (Φs,τ1 (x), τ1 ).

Then it satisfies SDE (7.2) for s < t < τ2 (x, s). We will continue this procedure. Then we can define Φs,t (x) for s < t < τ∞ (x, s) := limk→∞ τk (x, s), where τk (x, s) = τ1 (Φs,τk−1 (x), τk−1 ). It satisfies SDE (7.2) for s < t < τ∞ (x, s). The above τ∞ (x, s) is a terminal time. Indeed, suppose τ∞ (x, s) < ∞. Then the sequence of M-valued random variables {Φs,τn − (x), n = 1, 2, . . .} should converges to a point y ∈ M or diverges to ∞. But the former case cannot occur, because of the definition of stopping times τn (x, s). Therefore, the latter case occurs only, proving that τ∞ (x, s) is the terminal time. The functional Φs,t (x), s < t < τ∞ (x, s) constructed above has the flow property: There exists Ω˜ 1 ⊂ Ω with P (Ω˜ 1 ) = 1 such that for any fixed ω ∈ Ω˜ 1 , Φs,t (x, ω) satisfies Φr,t (Φs,r (x, ω), ω) = Φs,t (x, ω),

s < ∀r < t < τ∞ (x, s, ω).

Set Ds,t (ω) = {x ∈ M; τ∞ (x, s, ω) > t}. Then Φs,t (ω) is a map from Ds,t (ω) to M. We will prove that it is a C ∞ -map for almost all ω. Take any x0 ∈ Ds,t and consider the sample path Φs,t (x0 , ω). There is Ω˜ 2 ⊂ Ω with P (Ω˜ 2 ) = 1 such that for fixed ω ∈ Ω˜ 2 , we may choose s = t0 < t1 < · · · < tn = t and local charts Uαi , i = 1, . . . , n from the family {Uα } such that the trajectory {Φs,r (x0 , ω); r ∈ [ti−1 , ti )} is included in Uαi for any i = 1, . . . , n. Then there is an open neighborhood U0 of x0 (depending on ω) such that 

{Φs,r (x, ω); r ∈ [ti−1 , ti )} ⊂ Dαi ,

i = 1, . . . , n.

x∈U0

If ω ∈ Ω˜ 1 ∩ Ω˜ 2 , Φs,t (ω) is a C ∞ -map, since the flow property Φs,t (ω) = Φtn−1 ,t (ω) ◦ Φtn−1 ,tn−2 (ω) ◦ · · · ◦ Φs,t1 (ω)

7.1 SDE and Stochastic Flow on Manifold

307

holds and the right hand side is composites of C ∞ -maps. Therefore Φs,t (ω) is a C ∞ -map on U0 if ω ∈ Ω˜ 1 ∩ Ω˜ 2 . Further, U0 ⊂ Ds,t (ω) and hence Ds,t (ω) is an open set. We have thus shown the assertion of the theorem.   Now, for the symmetric SDE (7.2), we will assume further that jump-maps φt,z satisfy Condition (J.2)’. We shall study the diffeomorphic property of the flow. We need the backward symmetric SDE on the manifold M. Let 0 < t1 < ∞. An {Fs,t1 }adapted M-valued backward caglad process Xˇ s , t1 ≥ s > τˇ∞ is called a solution −1 of a backward symmetric SDE with characteristics (−Vk (t), k = 0, . . . , d  , φt,z , ν) ˇ starting from x at time t1 if f (Xs ), s ∈ (τˇ∞ , t1 ] is a backward local semi-martingale for any f ∈ C0∞ (M) and satisfies d   

f (Xˇ s ) = f (x) −

k=0 s

+ lim

→0

 t1 s

t1

Vk (r)f (Xˇ r ) ◦ dWrk

−1 ˇ {f (φr,z (Xr+ ))−f (Xˇ r+ )}N(dr |z|≥

(7.5) d   t1   dz)+ b k V˜k (r)f (Xˇ r ) dr , k=1

s

for t1 > s > τˇ∞ . Here ◦dWrk , k = 1, . . . , d  mean backward symmetric integrals by  processes Wˇ tk = Wtk − WTk , t ∈ T, where Wt = (Wt1 , . . . , Wtd ), t ∈ T is a Wiener process and ◦dWr0 means the usual integral dr. Further, V˜k (t), k = 1, . . . , d  are tangent vector fields of {φt,z } at z = 0. τˇ∞ is the (backward) terminal time of the backward process Xˇ s ; it means that it is a backward stopping time with values in [0, t1 ] ∪ {−∞} such that limt↓τˇ∞ Xˇ t = ∞ holds if τˇ∞ > 0. The equation has a unique backward solution, which we denote by Xˇ sx,t1 (ω), s > τˇ∞ . The backward terminal time is denoted by τˇ∞ (x, t1 , ω). Then for any 0 < s < t, the backward solution has a modification Ψˇ s,t (x, ω), s > τˇ∞ (x, t, ω) such that for any ω ∈ Ω˜ s,t ⊂ Ω with P (Ω˜ s,t ) = 1, the set Dˇ s,t (ω) = {x ∈ M; τˇ∞ (x, t, ω) < s}

(7.6)

is open and maps Ψˇ s,t (ω); Dˇ s,t (ω) → M are C ∞ -maps. {Ψˇ s,t (x, ω), Dˇ s,t (ω)} is called the backward stochastic flow of C ∞ -maps generated by the backward symmetric SDE (7.5). We first consider a continuous SDE. Lemma 7.1.1 Let {Φs,t , Ds,t } be the continuous stochastic flow of C ∞ -maps on the manifold M defined by the continuous symmetric SDE with coefficients Vk (x, t), k = 0, . . . , d  . Let {Ψˇ s,t , Dˇ s,t } be the continuous backward stochastic flow of C ∞ -maps defined by a continuous symmetric backward SDE with coefficients −Vk (x, t), k = 0, . . . , d  . Then we have the following:

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7 Stochastic Flows and Their Densities on Manifolds

1. For any 0 < s < t, Φs,t are C ∞ -maps from Ds,t to Dˇ s,t a.s. Further, these are one to one and onto and inverse maps are C ∞ a.s. 2. For any 0 < s < t < T , it holds a.s. that Ψˇ s,t (Φs,t (x)) = x if x ∈ Ds,t and Φs,t (Ψˇ s,t (x)) = x if x ∈ Dˇ s,t . ´ t) be vector fields on the Euclidean space Rd defined in the proof Proof Let V´k (x, of the previous theorem. Let {Φ´ s,t } be the forward flow on Rd generated by SDE with coefficients V´k (x, ´ t), k = 0, . . . , d  and let {Ψ¯ s,t } be the backward flow on Rd generated by the backward SDE with coefficients −V´k (x, ´ t), k = 0, . . . , d  . Since ∞,1 ´ is ∞ for any these coefficients are of Cb -class, the explosion time of Φ´ s,t (x) x, ´ s a.s. and the explosion time of the backward flow Ψ¯ s,t (x) ´ is −∞ for any t, x´ ´ = a.s. as we have seen in Chap. 3. Further, we saw in Sect. 3.7 that Φ´ s,t (Ψ¯ s,t (x)) Ψ¯ s,t (Φ´ s,t (x)) ´ = x´ holds for all 0 < s < t and x´ ∈ Rd , a.s. Let Uα be a local chart of M. We transform Φ´ s,t and Ψ¯ s,t by the inverse map ψα−1 . Set Φs,t = ψα−1 (Φ´ s,t ) and Ψˇ s,t = ψα−1 (Ψ¯ s,t ). Then if x ∈ Uα and τUα (x, s) > t, we have τˇ∞ (Φs,t (x), t) < s and Ψˇ s,t (Φs,t (x)) = x holds. The same fact holds if x ∈ M satisfies τ∞ (x, s) > t. Repeating this argument inductively, we find that τ∞ (x, s) > t &⇒ τˇ∞ (Φs,t (x), t) < s and Ψˇ s,t (Φs,t (x)) = x. Consequently, Φs,t maps Ds,t into Dˇ s,t , a.s. Next, apply a similar argument to Ψˇ s,t instead of Φs,t . Then we find that Ψˇ s,t maps Dˇ s,t into Ds,t and the equality Φs,t (Ψˇ s,t (x)) = x holds for x ∈ Dˇ s,t . Consequently, the map Φs,t : Ds,t → Dˇ s,t is one to one and onto, and further, the inverse is a C ∞ -map. See the discussion at the beginning of Sect. 3.7. Therefore we get the assertion of the lemma.   −1 We next consider the case with jumps. Let Vˆk (t) be tangent vector fields of {φt,z }  ˜ ˜ ˆ at z = 0. Then it holds that Vk (t)f = −Vk (t)f, k = 1, . . . , d , where Vk (t)f are tangent vector fields of {φt,z } at z = 0.

Lemma 7.1.2 Let {Φs,t , Ds,t } be the stochastic flow of C ∞ -maps on the manifold M defined by symmetric SDE (7.2). Let {Ψˇ s,t , Dˇ s,t } be the backward stochastic flow of C ∞ -maps on the manifold M defined by the backward symmetric SDE (7.5). Then assertions 1 and 2 of Lemma 7.1.1 are valid. Before the proof, we prepare another lemma. Lemma 7.1.3 Let (Uα , ψα ) be a local chart of M. For x ∈ Uα , set x´ = ψα (x), which is an element of Rd . For maps φt,z (x) satisfying Condition (J.1)’, there exist ´ x, ´ t, z) such that φ´ t,z (x) ´ := g( ´ x, ´ t, z) + x are diffeomorphisms on Rd δ0 > 0 and g( for |z| < δ0 , t ∈ T and ψα (φt,z (x)) = φ´ t,z (x) ´ holds for x ∈ Uα .

7.1 SDE and Stochastic Flow on Manifold

309

Proof Choose a sufficiently small δ0 > 0 and a function g( ´ x, ´ t, z), x´ ∈ Rd , |z| < δ0 such that for any z the equality g( ´ x, ´ t, z) = ψα (φt,z (x)) − x´ holds if x´ ∈ ψα (Uα ) ´ x, ´ t, z)| < 1/2 holds for any x´ ∈ Rd . Then the map φ´ t,z ; Rd → Rd is and |∂x´ g( one to one for any t, z, since |∂x´ g( ´ x, ´ t, z)| ≤ 1/2 for any x, ´ t, z. We will show that ¯ d be the one point compactification of Rd . It maps φ´ t,z are onto for any z. Let R is homeomorphic to the d-dimensional unit sphere. We set φ´ t,z (∞) = ∞. Since ¯d → R ¯ d is φ´ t,z (x) ´ = ∞ holds, the extended map φ´ t,z (x); ´ {|z| ≤ δ0 } × R limx→∞ ´ continuous. Further, φ´ t,0 is the identity. Then the map φ´ t,z is an onto map for any t, z by the homotopy theory. This indicates that the map φ´ t,z ; Rd → Rd is also an onto map for any z. We have thus seen that φ´ t,z are diffeomorphisms for any t and |z| < δ0 .   Proof of Lemma 7.1.2 Let δ0 be the positive number of Lemma 7.1.3 and let 0 < δ < δ0 . We consider a forward SDE with jumps on the manifold M with characteristics (Vk (x, t), k = 0, . . . , d  , φt,z (x), νδ ), where the associated Poisson random measure is N δ (dr dz) = 1(0,δ] (z)N (dr dz) and its Lévy measure νδ is given by νδ (dz) = 1(0,δ] (|z|)ν(dz). The forward stochastic flow generated by the SDE is δ , D δ }. Further, we consider a backward SDE with jumps with denoted by {Φs,t s,t −1 characteristics (−Vk (x, t), k = 0, . . . , d  , φt,z , νδ ). The backward stochastic flow δ ,D δ }. ˇ s,t generated by the backward SDE is denoted by {Ψˇ s,t The forward equation on M is transformed to a forward equation on Rd with characteristics (V´k (x, ´ t), k = 0, . . . , d  , g( ´ x, ´ t, z), νδ ), where g(x, ´ t, z) is the function defined in Lemma 7.1.3. The backward SDE on M is transformed to a backward ´ x, equation on Rd with characteristics (−V´k (x, ´ t), k = 0, . . . , d  , −h( ´ t, z), νδ ), −1 δ d ´ ´ ´ where h(x, ´ t, z) = x − φt,z (x). Let Φs,t be the flow on R generated by the forward δ be the backward flow on Rd generated by the backward SDE. SDE and let Ψ´ s,t −1 ´ x, ´ = g( ´ x, ´ t, z) + x´ are diffeomorphic and φ´ t,z (x) ´ = h( ´ t, z) + x, ´ these Since φ´ t,z (x) δ δ δ δ flows satisfy Φ´ s,t (Ψ´ s,t (x)) ´ = Ψ´ s,t (Φ´ s,t (x)) ´ = x´ on the Euclidean space. Further, δ (x) = ψ −1 (Φ δ (x)) ´ s,t it holds that Φs,t ´ for t < τDδ α (x, s) (the first leaving time of α x,s,δ δ (x) from D ) and Ψ δ (x) = ψ −1 (Ψ δ (x)) ˇ s,t ´ s,t = Φs,t ´ for s < σ δ (x, t) Xt = Xt α α Dα

δ (x) from D ). Therefore we have (the backward first leaving time of Xˇ sx,t,δ = Ψ´ s,t α δ δ δ δ ˇ ˇ Φs,t (Ψs,t (x)) = Ψs,t (Φs,t (x)) = x for x ∈ Dα if t < τDδ α (x, s). Then we can δ (Ψ δ (x)) = Ψ δ (Φ δ (x)) = x for x ∈ M and t < τ (x, s). ˇ s,t ˇ s,t prolong the equality Φs,t 1 s,t Now the stochastic flow Φs,t (x) on the manifold M is decomposed as

 Φs,t (x) =

δ (x), if t < τ1 Φs,t δ (x), if τ ≤ t < τ Φτδn ,t ◦ φτn ,Sn · · · φτ1 ,S1 ◦ Φs,τ n n+1 , 1

(7.7)

where τn are jumping times of the Poisson process N tδ = N ((s, t] × {|z| > δ}) and S1 , . . . , Sn are random variables given by Si = {(τi ,z);|z|>δ} zN(dr dz), i = 1, . . . , n. Further the backward flow Ψˇ s,t (x) is represented by

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7 Stochastic Flows and Their Densities on Manifolds

 Ψˇ s,t (x) =

δ (x), Ψˇ s,t if t < τ1 , δ ◦ φ −1 · · · φ −1 ◦ Ψ ˇ τδ ,t (x), if τn ≤ t < τn+1 . Ψˇ s,τ τ1 ,S1 τn ,Sn n 1

(7.8)

Therefore we get Φs,t (Ψˇ s,t (x)) = Ψˇ s,t (Φs,t (x)) = x if t < τDα (x, s). We can prolong the equality Φs,t (Ψˇ s,t (x)) = Ψˇ s,t (Φs,t (x)) = x for any x and t < τ1 (x, s). Repeating this argument, the equality holds for any x ∈ M and t < τ∞ (x, s). Finally, the equality implies that maps Φs,t ; Ds,t → Dˇ s,t are one to one and onto a.s. and further, the inverse maps are C ∞ . Hence we established the lemma.   From the above lemmas, we get the following assertion. Theorem 7.1.2 Assume that Vk (t), k = 0, . . . , d  are C ∞,1 -vector fields and {φt,z } satisfy Conditions (J.1)’ and (J.2)’ for the symmetric SDE (7.2) on the manifold M. Then maps Φs,t ; Ds,t → M of Theorem 7.1.1 are diffeomorphic a.s. for any s < t. Let {Ψˇ s,t , Dˇ s,t } be the backward flow defined by SDE (7.5) on the manifold M −1 with characteristics (−Vk (t), k = 0, . . . , d  , φt,z , ν). Then Φs,t are maps from Ds,t −1 onto Dˇ s,t . Further, it holds that Ψˇ s,t (x) = Φs,t (x) if x ∈ Dˇ s,t . The family of pairs {Φs,t , Ds,t } satisfying properties of the above theorem is called a stochastic flow of local diffeomorphisms on the manifold M. SDE (7.2) is called complete if τ∞ (x, s) = ∞ holds a.s. for any x, s. Further, it is called strongly complete if infy∈M τ∞ (x, s) = ∞ holds a.s for any s. It is equivalent to that Ds,t = M holds a.s. for any s < t. Corollary 7.1.1 Assume that SDE (7.2) on the manifold M is strongly complete. Then Ds,t = M holds a.s. and {Φs,t } satisfies the same properties stated in Theorem 3.9.1, replacing the state space Rd by the manifold M. Proposition 7.1.1 If the manifold M is compact and SDE (7.2) is time homogeneous, the SDE is strongly complete. Proof Let us consider the stopping time τ1 (x, s) defined by (7.4). We define a stopping time by σ1 (s) = infx∈M τ1 (x, s). Then σ1 (s) > s a.s. since M is compact. Further, set σn (s) = σ1 (σn−1 (s)) for n ≥ 2. Then it holds that σn (s) ≤ τn (x, s) for all x. Set σ∞ (s) = limn→∞ σn (s). We want to prove σ∞ (s) = ∞ for any s a.s. This should imply that the SDE is strongly complete. Consider a sequence of random variables σn+1 (s) − σn (s), n = 1, 2, . . .. Note that σn+1 (s) − σn (s) = σ1 (σn (s)) − σn (s). Since σ1 (t) − t is Ft,∞ -measurable and 1σn (s)≤t is F0,t -measurable, these two random variables are independent. Therefore, for any bounded continuous function f we have E[f (σn+1 (s) − σn (s))|Fσn ] = E[f (σ1 (t) − t)]|t=σn (s) = E[f (σ1 (0))], because laws of σ1 (t) − t are common for all t. This proves that σn+1 (s) − σn (s) and F0,σn are independent.

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold

311

The above discussion shows that σn+1 (s) − σn (s), n = 1, 2, . . . are independent and identically distributed. Since σN (s) − s =

N −1 

{σn+1 (s) − σn (s)},

(σ0 (s) = s)

n=0

(σN (s) − s)/N converges to a positive constant, by law of the large numbers. This proves σ∞ (s) = ∞ a.s for all s > 0. Finally, σN (s) is right continuous in s for any N . Then σ∞ (s) = ∞ holds for all s a.s.   Remark Let N be a connected component of the manifold M. Then jump-maps φt,z maps N onto itself diffeomorphically. Indeed, since φt,0 are identity maps and φt,z (x) are continuous with respect to t, z, x, we have φt,z (N ) ⊂ N . Then the stochastic flow Φs,t should also map N onto itself diffeomorphically a.s. Hence it is sufficient that we consider SDE and stochastic flow on each connected components. We will relax Conditions (J.1)’ and (J.2)’ to the following: Condition (J.1)K . (i) The map (x, t, z) → φt,z (x) is of C ∞,1,2 -class on the space M ×T×{|z| ≤ c} for some c > 0. (ii) It is of C ∞ -class on M for any t ∈ T, |z| > c and is piecewise continuous in (t, z) ∈ T × {|z| > c}. Condition (J.2)K . For any t, z, the map φr,z ; M → M is a diffeomorphism. −1 Further, the inverse maps φt,z satisfy Condition (J.1)K . Then the assertion of Theorem 7.1.2 is valid. In this case, the stochastic flow Φs,t may jump from a connected component N to other connected components. Note The continuous SDE on manifolds was studied by Ikeda–Watanabe [41]. Our definition of the SDE on manifold is close to their definition. We saw in Chap. 3 that the SDE on a Euclidean space is strongly complete, if coefficients Vk (x, t) are of Cb∞,1 -class and g(x, t, z) are Cb∞,2,1 -class. It is expected to get criteria for vector fields Vk (t) and jump-maps φt,z under which an SDE is strongly complete. For continuous SDEs or diffusion processes, we refer to XueMei Li [76]. The SDE with jumps on manifolds was studied by Fujiwara [29] and Kunita [60]. The present definition of the SDE is taken from [60]. For the construction of stochastic flows on a compact manifold, Fujiwara [29] applied the Whitney embedding theorem of a manifold into a higher-dimensional Euclidean space. Assumptions required in the paper are different from ours.

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold In Sect. 7.1, we studied a symmetric SDE on a C ∞ -manifold M of dimension d. In the sequel, we assume that the Lévy measure has a weak drift. Then the last term (jump part) of (7.2) is split into the difference of two terms:

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7 Stochastic Flows and Their Densities on Manifolds

 t

 t d   f (φr,z (Xr− ))−f (Xr− ) N(dr dz)− b0k V˜k (r)f (Xr ) dr,

t0 |z|>0+

t0 k=1

where the first term is the improper integralby the Poisson random measure N and b0k = lim →0 b k . Then rewriting V0 (t) − k b0k V˜k (t) as V0 (t), SDE (7.2) can be rewritten simply as d   

f (Xt ) = f (X0 )+ +

t

k=0 t0

 t



Vk (r)f (Xr ) ◦ dWrk

 f (φr,z (Xr− ))−f (Xr− ) N(dr dz).

t0 |z|>0+

(7.9)

We define an integro-differential operator AJ (t) by d

1 AJ (t)f = Vk (t)2 f + V0 (t)f + 2 k=1

 {f ◦ φt,z −f }ν(dz).

|z|>0+

(7.10)

Then, using the Itô integral, Eq. (7.9) is rewritten as  f (Xt ) = f (X0 )+

t



AJ (r)f (Xr ) dr +

t0

+

 t



d  

t

k=1 t0

Vk (r)f (Xr ) dWrk

 ˜ f (φr,z (Xr− ))−f (Xr− ) N(dr dz).

t0 |z|>0+

(7.11)

It is called a symmetric SDE with characteristics (Vk (t), k = 0, . . . , d  , φt,z , ν). We will define stochastic flow generated by the above SDE slightly different from that defined in Sect. 7.1. Let M  = M ∪ {∞}, where ∞ is a cemetery adjoined to the manifold M as a one-point compactification if M is noncompact, and as an isolated point if M is compact. For a given function f on M, we extend it to a function on M ∪ {∞} by setting f (∞) = 0. Let Xtx,s , t < τ∞ (x, s) be the solution starting from x at time s, where τ∞ (x, s) is the terminal time. For t ≥ τ∞ (x, s), we set Xtx,s = ∞. Then Xtx,s , t ∈ [s, ∞) is an M ∪ {∞}-valued process and satisfies the above equation (7.11) for all t > s. Theorems 7.1.1 and 7.1.2 tell us that Xtx,s , t > s has a modification {Φs,t (x), t > s} satisfying the following properties: 1. The terminal time τ∞ (x, s) is lower semi-continuous with respect to x for any s. 2. Set Ds,t = {x; τ∞ (x, s) > t}. These are open subsets of M a.s., for any s < t. Further, maps Φs,t ; Ds,t → M are C ∞ a.s. 3. If jump-maps φt,z satisfy Condition (J.2)’ or identity maps, then maps Φs,t : D → M are into diffeomorphisms a.s.

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold

313

We set Φs,t (∞) = ∞ for any t ≥ s. Then Φs,t are maps from M ∪ {∞} into itself and satisfy Φt,u (Φs,t (x)) = Φs,u (x) for any x ∈ M ∪ {∞} and s < t < u. The family of maps {Φs,t , s < t} is called again the stochastic flow of local diffeomorphisms generated by a forward SDE with characteristics (Vk (t), k = 0, . . . , d  , φt,z , ν). Now, since Φs,t (x) and Φt,u (x) are independent for any s < t < u, the stochastic process Xtx,s = Φs,t (x) is a Markov process of initial state (x, s) with transition function Ps,t (x, E) = P (Φs,t (x) ∈ E) = P (Φs,t (x) ∈ E, τ∞ (x, s) > t) for x ∈ M and E ⊂ M. The Markov process Xtx,s is called a jump-diffusion process on the manifold M with characteristics (Vk (t), k = 0, . . . , d  , φt,z , ν). For f ∈ Cb (M), we set Ps,t f (x) := M Ps,t (x, dy)f (y). It is written as Ps,t f (x) = E[f (Φs,t (x))] = E[f (Φs,t (x))1τ∞ (x,s)>t ]. Then it is a bounded continuous function of x. Hence {Ps,t } is a semigroup of linear transformations of Cb (M). The above Markov process Xtx,s = Φs,t (x), t ∈ [s, ∞) satisfies equation (7.11) with the initial condition X0 = x and t0 = s. Take expectations for each term of (7.11). Expectations of the third and the fourth terms of the right-hand side are 0, since these are martingales. Consequently, we get E[f (Φs,t (x))] = f (x) + E



t

 AJ (r)f (Φs,r (x)) dr .

s

t

Therefore we have Ps,t f (x) = f (x) + s Ps,r AJ (r)f (x) dr for any f ∈ C0∞ (M). This shows that the semigroup {Ps,t } satisfies Kolmogorov’s forward equation with respect to the operator AJ (t). If φt,z are identity maps for any t, z in equation (7.9), the Markov process Xt is called a diffusion process on the manifold M with coefficients (Vk (t), k = 0, . . . , d  ). In this case, we define a differential operator A(t) by d

1 A(t)f (x) = Vk (t)2 f (x) + V0 (t)f (x). 2

(7.12)

k=1

t Then the semigroup satisfies Ps,t f (x) = f (x) + s Ps,r A(r)f (x) dr for any f ∈ C0∞ (M). Hence the semigroup {Ps,t } satisfies Kolmogorov’s forward equation with respect to the operator A(t). It is expected that if f is in C0∞ (M), Ps,t f (x) should be smooth with respect to x and should satisfy Kolmogorov’s backward equation. If the associated stochastic flow is strongly complete, these fact is true, but if it is not strongly complete, these facts are not obvious, since it is not evident whether Ps,t f (x) is smooth or not with respect to x. Then it is not clear whether the (backward) heat equation associated

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7 Stochastic Flows and Their Densities on Manifolds

the operator A(t) or AJ (t) has a solution. In Sect. 7.4, we will show these facts for elliptic diffusion, by constructing the fundamental solution for the differential operator A(t). Let us consider the dual processes, assuming that the manifold M is orientable. We will restrict our attention to jump-diffusions. A similar result for diffusions will be obtained with the obvious change. From now, we assume further that jump-maps φt,z satisfy Condition (J.2)’ in Sect. 7.1. Hence these maps are diffeomorphic. Let dx be a given volume element of the manifold. We will study the dual of the above jump-diffusion with respect to dx. An operator AJ (t)∗ ; C0∞ (M) → C ∞ (M) is called a dual of the operator AJ (t) with respect to dx if it satisfies 



f (x) · AJ (t)∗ g(x) dx

AJ (t)f (x) · g(x) dx = M

M

for any f, g ∈ C0∞ (M). It is computed directly, using formulas of the integration by parts and the change of variables. Indeed, similarly to the dual in the Euclidean space (Sect. 4.6), it is written for any g ∈ C0∞ (M) as d

1 (Vk (t) + div Vk (t))2 g − (V0 (t) + div V0 (t))g AJ (t) g = 2 k=1    −1 |Jφ −1 |g ◦ φt,z + − g ν(dz). ∗

|z|>0+

(7.13)

t,z

Here, div V is the divergence of the vector field V with respect to the volume element dx (Appendix). Further, for a diffeomorphic map φ; M → M, Jφ is defined as a C ∞ -function such that the pullback φ ∗ dx coincides with Jφ dx. It coincides with the Jacobian determinant of φ if M is a Euclidean space and dx is the Lebesgue measure. We call Jφ the Jacobian of φ. Let Ψˇ s,t (x), s > τˇ∞ (x, t) be the backward flow of local diffeomorphisms generated by the backward SDE on the manifold M. f (Xˇ s ) = f (x)−

d  

t

Vk (r)f (Xˇ r )◦dWrk +

 t

k=0 s

−1 ˇ {f (φr,z (Xr ))−f (Xˇ r )}N(dr dz)

s |z|>0+

−1 for t > s > τˇ∞ for any f ∈ C0∞ (M). Let Φs,t be the inverse map of the flow −1 ˇ Φs,t . We saw in Sect. 7.1 that Φs,t (x) = Ψs,t (x) holds for any t < τ∞ (x, s). For s < τˇ∞ (t, x), we set Ψˇ s,t (x) = ∞ and Ψˇ s,t (∞) = ∞ for any 0 < s < t. Then Xˇ s = Ψˇ s,t (x) is a backward Markov process. Its generator is given by d

1 Vk (t)2 g − V0 (t)g + AJ (t)g = 2 k=1

 |z|>0+

−1 {g ◦ φt,z − g}ν(dz).

(7.14)

7.2 Diffusion, Jump-Diffusion and Their Duals on Manifold

315

Then the dual operator AJ (t)∗ is obtained by the transformation of the operator AJ (t) as AJ (t)∗ g = AJ (t)

−div V,|Jφ −1 |

(7.15)

g.

For the construction of the dual process, we need an additional condition. Condition (D) (Dual condition) (i) div Vk (t), Vk (t)(div Vk (t)), k = 0, . . . , d  are bounded C ∞,1 -functions on the space M × T.   (ii) The improper integral |z|>0+ |Jφ −1 | − 1 ν(dz) is a bounded function of x, t t,z and is smooth with respect to x. The above condition is always satisfied if the manifold M is compact. In the case where the manifold M is an Euclidean space and dx is the Lebesgue measure, Condition (D) tells us that coefficients Vk (x, t), φt,z (x) − x = g(x, t, z) of the SDE might be unbounded, but div Vk (x, t), Vk (t)(div Vk (t))(x) and |z|>0 (| det ∇φt,z (x)| − 1)ν(dz) should be bounded. The function c∗ (t) = c∗ (x, t) = AJ (t)∗ 1(x) is written as c∗ (t) = −div V0 (t) +

d

1 {Vk (t)(div Vk (t)) + (div Vk (t))2 } 2 k=1

 +

|z|>0+

(|Jφ −1 | − 1)ν(dz).

(7.16)

t,z

Then the dual condition makes sure that the function c∗ (x, t) is a bounded function on M × T. Then we can construct the dual jump-diffusion with generator (7.13). We apply the change-of-variable formula (A.4) in the Appendix for open submanifolds Ds,t ≡ {x; τ∞ (x, s) > t}. Replace f (x) by f (Φs,t (x))g(x) and φ by Ψˇ s,t . Note that Ψˇ s,t are diffeomorphic maps from Dˇ s,t ≡ {x; τˇ∞ (x, t) < s} to Ds,t by Theorem 7.1.2, and the equality Φs,t (Ψˇ s,t (x)) = x holds. Then we get the formula of the change of variables: 

 Ds,t

f (Φs,t (x))g(x) dx =

Dˇ s,t

f (x)g(Ψˇ s,t (x))|JΨˇ s,t (x)| dx,

a.s. P ,

c , we have f (Φ (x)) = f (∞) = 0, for any functions f, g ∈ C0 (M). On the set Ds,t s,t c ˇ ˇ and on the set Ds,t we have g(Ψs,t (x)) = g(∞) = 0. If τˇ∞ (x, t) > s, we set JΨˇ s,t (x) = 0. Then we get the equality



 f (Φs,t (x))g(x) dx = M

M

f (x)g(Ψˇ s,t (x))|JΨˇ s,t (x)| dx,

a.s. P .

(7.17)

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7 Stochastic Flows and Their Densities on Manifolds

Lemma 7.2.1 |JΨˇ s,t |, τˇ∞ (x, t) < s < t is written as a backward exponential functional with coefficients −div V = (−div V0 (x, t), . . . , −div Vd  (x, t)) and |Jφ −1 | = (|Jφ −1 (x)|): t,z

exp

d   k=0 s

t

−div Vk (t)(Ψˇ r,t ) ◦ dWrk +

 t s

|z|>0+

 log |Jφ −1 (Ψˇ r,t )| dN . r,z

(7.18)

Further, under condition (D), E[|JΨˇ s,t (x)|] is bounded with respect to s < t and x. Proof We showed the above lemma for the stochastic flow on a Euclidean space (Proposition 4.6.2). What we need in the proof are the following three facts: 1. Differential rule for the flow {Φs,t } with respect to the backward variable s. 2. Change-of-variable formula. 3. Formula of integration by parts. We know that the first one holds for stochastic flows on manifold. The second and the third are known as we have mentioned above. Therefore, discussions in the proof of Lemma 4.6.1 are valid to the present case and we get the expression (7.18). Further, (7.18) is written as the product of three backward exponential functionals (0) (1) (2) Gs,t , Gs,t and Gs,t as in Sect. 4.6. Since c∗ (x, t) is a bounded function, we find that E[|JΨˇ s,t (x)|1τˇ∞ (x,t) σn−1 ; Xσ−1 X ∈ / U }. σ1 = inf{t > 0; Xt ∈ n−1 t

(7.21)

Lemma 7.3.1 The sequence of random variables σ1 , σ2 − σ1 , . . . , σn − σn−1 is independent and identically distributed. Proof Set G = G ∪ {∞} and g · ∞ = ∞ for any g ∈ G. We set Xt = ∞ if t > τ∞ . Then for a function f on G such that f (∞) = 0 and is C ∞ on G, we have d   

f (Xt ) = f (Xs∧σ1 ) +

t

k=0 s∧σ1

Vk f (Xr ) ◦ dWrk .

318

7 Stochastic Flows and Their Densities on Manifolds

Since Vk are left-invariant vector fields, we have d   

f (xXt ) = f (xXs∧σ1 ) +

t

k=0 s∧σ1

Vk f (xXr ) ◦ dWrk

−1 for any x ∈ G. Setting x = Xs∧σ , we get 1 d   

−1 f (Xs∧σ X ) = f (e) + 1 t

t

k=0 s∧σ1

−1 Vk f (Xs∧σ X ) ◦ dWrk . 1 r

−1 X coincides with the solution Xte,s∧σ1 of equation (7.20), in Consequently, Xs∧σ 1 t view of the pathwise uniqueness of the solution. This means that, for any s and σ1 , −1 stochastic processes {Xt , t ≤ s ∧ σ1 } and {Xs∧σ X , s ∧ σ1 < t < 2s ∧ σ2 } 1 t−s∧σ1 are independent and have the same law. Then random variables σ1 and σ2 − σ1 are independent and identically distributed. Repeating this argument, we find that σ1 , . . . , σn − σn−1 are independent and identically distributed.  

Now, the expectation of σ1 is positive. Then by the law of large numbers, limn→∞ σn = ∞ holds a.s. Therefore we have τ∞ = ∞ a.s. Further, since the solution Xt is defined for all t, we have the formula d   

f (Xs−1 Xt ) = f (e) +

k=0 s

t

Vk f (Xs−1 Xr ) ◦ dWrk .

(7.22)

Then Xs−1 Xt is independent of Fs for any s < t. It shows that Xt has left independent increments. Therefore we get the following theorem. Theorem 7.3.1 Any continuous SDE on a Lie group G with left-invariant vector fields Vk , k = 0, . . . , d  is strongly complete. The solution is a time homogeneous   diffusion on G with the generator Af = 12 dk=1 Vk2 f + V0 f. Let Xt be the solution starting from the unit element e ∈ G at time 0. Then it is a Brownian motion on the Lie group G. Further, Φs,t (x) := xXs−1 Xt is the stochastic flow of diffeomorphisms. We will consider the dual of Xt with respect to a left Haar measure m. Let exp sVk , s ∈ (−∞, ∞) be the one-parameter subgroup of G generated by the leftinvariant vector field Vk . Then the modular function Δ(exp sVk ) is a one-parameter d subgroup of the multiplicative group (0, ∞). We set ck = ds Δ(exp sVk )|s=0 for  k = 0, . . . , d . These constants satisfy 

 Vk f (x)g(x) dm = −

 f (x)Vk g(x) dm − ck

f (x)g(x) dm

for functions f, g ∈ C0∞ (M). Therefore, the constant ck coincides with div Vk . Then the dual of A is represented by

7.3 Brownian Motion, Lévy Process and Their Duals on Lie Group

319

d

1 A g(x) = (Vk + ck )2 g(x) − (V0 + c0 )g(x). 2 ∗

k=1

In particular, if the Lie group G is unimodular, then ck = 0 holds for all k. Therefore  the dual is represented by A∗ f = 12 k≥1 Vk2 f −V0 f. Hence the dual is a backward Brownian motion. We shall next consider an SDE with jumps. We assume that the Lévy measure of the Poisson random measure has a weak drift. Let Vk , k = 0, . . . , d  be left invariant vector fields on the Lie group G. Let φ(z) be a smooth map Rd → G such that φ(0) = e. We consider an SDE on G with characteristics (Vk , k = 0, . . . , d  , Rφ(z) , ν), where Rφ(z) is the right translation. Let Xt , t < τ∞ be the solution starting from e at time 0, where τ∞ is the terminal time. It satisfies, for any smooth function f and 0 ≤ s < t < τ∞ , d   

f (Xt ) = f (Xs ) + +

  t

t

k=0 s

Vk f (Xr ) ◦ dWrk

(7.23)

{f (Xr− φ(z)) − f (Xr− )}N(dr dz).

s |z|>0+

For the process Xt , we define a sequence of stopping times σ1 , σ2 , . . . by (7.21). Then the assertion of Lemma 7.3.1 is valid. Therefore τ∞ = ∞ holds a.s. We will show that Xt has independent increments. Since Vk are left-invariant vector fields, we have d   

f (xXt ) = f (xXs ) + +

  t

k=0 s

t

Vk f (xXr ) ◦ dWrk

{f (xXr− φ(z))−f (xXr− )}N(dr dz).

s |z|>0+

The above equality holds for x = Xs−1 . Then we find that Xs−1 Xt is a solution of the SDE with characteristics (Vk , k = 0, . . . , d  , Rφ(z) , ν) starting from e at time s. Then Xs and Xs−1 Xt are independent. This proves that Xt is a Lévy process on the Lie group G. Theorem 7.3.2 SDE (7.23) on a Lie group G with left-invariant vector fields Vk , k = 0, . . . , d  and right translation Rφ(z) is strongly complete. Let Xt be the solution starting from the unit element e ∈ G at time 0. Then it is a Lévy process on the Lie group G. Further, Φs,t (x) := xXs−1 Xt is the stochastic flow of diffeomorphisms generated by the SDE (7.23). Its generator is given by

320

7 Stochastic Flows and Their Densities on Manifolds d

1 2 AJ f (x) = Vk f (x) + V0 f (x) + 2 k=1



  f (xφ(z)) − f (x) ν(dz).

|z|>0+

(7.24)

We will consider the dual of the operator AJ with respect to the left Haar measure m. Consider the transformation Rφ(z) . Regarding m as a positive d-form, we defined ∗ the function JRφ(z) by the relation Rφ(z) dm = JRφ(z) dm in Sect. 4.8. Since m is a ∗ Haar measure, we have Rφ(z) dm = Δ(φ(z)) dm. Therefore JRφ(z) (x) is a constant function of x and we have JRφ(z) (x) = Δ(φ(z)). If  |z|>δ

Δ(φ(z))ν(dz) < ∞

(7.25)

holds for some δ > 0, the dual of AJ is well defined. It is written as A∗J g(x)

d

1 = (Vk + ck )2 g(x) − (V0 + c0 )g(x) 2 k=1    Δ(φ(z))g(xφ(z)−1 ) − g(x) ν(dz). + |z|>0+

For a unimodular Lie group, it holds that ck = 0 and Δ(φ(z)) = 1. Then the operator A∗J is written simply as A∗J g(x)

d

1 2 = Vk g(x) − V0 g(x) + 2 k=1



 |z|>0+

 g(xφ(z)−1 ) − g(x) ν(dz).

Denote the inverse Xs−1 of the Lévy process Xs by Xˇ s . Then Xˇ s is a backward jump-diffusion with the generator AJ given by (7.14). It coincides with the above A∗J . Thus if the Lie group G is unimodular, the inverse process Xˇ s coincides with the dual process with respect to the left Haar measure m. Note It was shown by Itô [49] that a stationary Brownian motion ona Lie group is a stationary diffusion with the generator A represented by Af = k a k Xk f + 1 ik ik i,k a Xi Xk f, where {Xi } is a basis of L(G) and (a ) is a nonnegative definite 2 symmetric matrix. By the method of the diagonalization, the above operator is  rewritten as Af = V0 f + 12 k≥1 Vk2 f , where V0 , . . . , Vd  are elements of L(G). Therefore all Brownian motions on a Lie group can be obtained by solving SDEs of the form (7.20). A Lévy process on a Lie group is studied in Hunt [39]. He obtained its generator explicitly, using left-invariant vector fields and a Lévy measure on the Lie group.

7.4 Smooth Density for Diffusion on Manifold

321

7.4 Smooth Density for Diffusion on Manifold We will consider a diffusion on the orientable manifold M determined by SDE d   

f (Xt ) = f (X0 ) +

t

k=0 t0

Vk (r)f (Xr ) ◦ dWrk ,

(7.26)

where f are C ∞ -functions on M. Let {Φs,t (x), t < τ∞ (x, s)} be the stochastic flow generated by SDE (7.26), where τ∞ (x, s) is the explosion time of the process Xtx,s := Φs,t (x). We set Xtx,s = Φs,t (x) = ∞ if t > τ∞ (x, s). Then Xtx,s is a (not necessarily conservative) diffusion process on M. If the tangent vectors Vk (x, t), k = 1, . . . , d  span the tangent space Tx (M) for any x ∈ M and t, the SDE is called elliptic and the above Xtx,s is called an elliptic diffusion. Let ck (x, t), k = 0, . . . , d  be C ∞,1 -functions on M ×T. We assume that ck (x, t) and Vk (t)ck (x, t), k = 0, . . . , d  are bounded functions. Define the differential operator Ac (t) by d

1 A (t)f (x) = (Vk (t) + ck (t))2 f (x) + (V0 (t) + c0 (t))f (x). 2 c

(7.27)

k=1

Its potential part c(t) = Ac (t)1 is a bounded function, since it is equal to d

d

k=1

k=1

  1  c(x, t) = c0 (x, t) + Vk (t)ck (x, t) + ck (x, t)2 . 2 We set ck (∞, t) = 0 for k = 0, . . . , d  and define the exponential functional Gs,t (x) by Gs,t (x) = Gcs,t (x) = exp

d   k=0 s

 ck (Φs,r (x), r) ◦ dWrk .

t

(7.28)

It satisfies sups 1 (see the proof of Lemma 4.2.1). Let f be a bounded continuous function on M. We set f (∞) = 0. Then c f (x) = E[f (Φ (x))Gc (x)] satisfies Kolmogorov’s forward the semigroup Ps,t s,t s,t equation associated with the differential operator Ac (t) defined by (7.27). We will fix a volume element of M and will denote it by dx, dy etc. We will study the smooth density of the weighted transition function of an elliptic diffusion c (x, E) = E[1 (Φ (x))Gc (x)], with respect to dy. defined by Ps,t E s,t s,t For a positive integer i, d i φ is the i-th differential of φ and for i = 0, d 0 φ ≡ φ. When differentials are operated to the variable x of the C ∞ (M × M)-function

322

7 Stochastic Flows and Their Densities on Manifolds

φ(x, y), these are denoted by dxi φ and their localization at the point (x, y) are denoted by dxi φ(x, y). An objective of this section is to prove the following. Theorem 7.4.1 Consider an elliptic diffusion on the orientable manifold M determined by SDE (7.26). Assume that ck (x, t), Vk (t)ck (x, t) are bounded C ∞,1 c (x, E) functions. Then, for any 0 ≤ s < t and x ∈ M, the transition function Ps,t c ∞ c weighted by Gs,t (x) of (7.28) has a C -density ps,t (x, y), y ∈ M with respect to the volume element. Further, for any compact subset K of M, nonnegative integer j and T > 0, there exists a positive constant c such that

j c

sup dy ps,t (x, y) ≤

x,y∈K

c (t − s)

j +d 2

,

0 ≤ s < t ≤ T.

(7.29)

Assume further Condition (D) (i). Then the density function is a C ∞,1 -function of x, s (< t) for any y, t. Further, for any compact subset K of M, nonnegative integer i and T > 0, there exists a positive constant c such that



c sup dxi ps,t (x, y) ≤

x,y∈K

c (t − s)

i+d 2

,

0 ≤ s < t ≤ T.

(7.30)

We will first show the existence of the smooth density on each local chart of M. Let V , U, D be a triple of local charts such that V¯ ⊂ U ⊂ U¯ ⊂ D and D¯ is compact. Then D is diffeomorphic to a bounded open set D´ of Rd by a diffeomorphism ´ SDE (7.26) on D is transformed to an SDE on D´ by the diffeomorphic ψ : D → D. map ψ. Indeed, vector fields Vk (t) on the local chart D are transformed to vector fields dψ(Vk (t)) on D´ by the map ψ. Since D´ is relatively compact in Rd , we can ´ t) on the whole space Rd such that these extend them to Cb∞,1 -vector fields V´k (x, are uniformly elliptic. We consider an equation on Rd : 

d X´ t =

d 

V´k (X´ t , t) ◦ dWtk .

(7.31)

k=0

We denote the solution starting from x´ at time s by Φ´ s,t (x). ´ Let c´k (x, ´ t) be Cb∞,1 functions on Rd such that ψ(ck (x, t)) = c´k (ψ(x), t) holds on the local chart D. We ´ s,t (x) define a positive functional G ´ similarly to Gs,t (x), making use of functions  d ´ c´k , k = 0, . . . , d on R and Rd -valued process Φ´ s,t (x). ´ ´ We Let τ´ (x, ´ s) be the first leaving time of the process X´ tx,s = Φ´ s,t (x) ´ from D. shall consider the law of the killed process given by   ´ cs,t (x, ´ s,t (x)1 Q . ´ E) = E 1E (Φ´ s,t (x)) ´ G ´ τ´ (x,s)>t ´

(7.32)

We can apply Lemma 6.10.1 for the killed process. Let U´ = ψ(U ). Then the closure ´ Then the measure Q ´ cs,t (x, of U´ in Rd is included in the open set D. ´ ·) restricted to

7.4 Smooth Density for Diffusion on Manifold

323

c (x, ´ y), ´ y´ ∈ U´ with respect to the Lebesgue measure by U´ has a C ∞ -density q´s,t Lemma 6.10.1. It satisfies (6.111) and (6.112). For a given T > 0, take 0 ≤ s < t ≤ T . Let Φs,t (x) be the solution on the manifold M. We shall consider its killed process at D c . Let τ (x, s) be the first leaving time of the process Xtx,s = Φs,t (x) from D. Then the weighted law

  Qcs,t (x, E) = E 1E (Φs,t (x))Gcs,t (x)1τ (x,s)>t c (x, y), y ∈ U with respect to the volume element dy, has also a C ∞ -density qs,t −1 c ´ since Φs,t (x) = ψ (Φs,t (ψ(x))) holds for t < τ (x, s) a.s. P. Further, since q´s,t  satisfies (6.111) and (6.112), for any j, i there are positive constants c, c such that c satisfies qs,t

sup x∈D,y∈U

sup

x∈U c ∩D,y∈V



j c

dy qs,t (x, y) ≤

c (t − s)



i c

dy qs,t (x, y) ≤ c

j +d 2

,

(7.33) (7.34)

for all 0 ≤ s < t ≤ T . c (x, E) We want to show that for any x ∈ M the weighted transition function Ps,t ∞ has a C -density with respect to the volume element dy. This will be done by c (x, y) of the killed process. For piecing together the above density functions qs,t this purpose, we will fix a pair (U, D) of the above local charts. For the process Xt = Xtx,s := Φs,t (x) we define stopping times τm = τm (x, s), m = 0, 1, 2, . . . and σm = σm (x, s), m = 1, 2, . . . by induction as τ0 = s and σm = inf{t ≥ τm−1 ; Xt ∈ D c } (= ∞ if {· · · } is empty), τm = inf{t ≥ σm ; Xt ∈ U } (= ∞ if {· · · } is empty).

(7.35)

Then we have for E ⊂ U , E[1E (Φs,t (x))Gcs,t (x)] =

∞ 

  E 1E (Φs,t (x))Gcs,t (x)1τm a − c2 t for some t < c1 T ). x,s ´

Since Bt is a Wiener process, the last probability denoted by c is less than 1. This proves the assertion of the lemma.   Lemma 7.4.2 Let U, D be a pair of local charts satisfying U¯ ⊂ D. Let τm = τm (x, s) be stopping times defined by (7.35). Then there exists a positive constant 0 < c < 1 such that sup P (τm (x, s) < t) ≤ cm

(7.39)

s 0. Then, τ1 (x, s) ≥ σ (x, s) a.s. Therefore, we have by Lemma 7.4.1 ´ s) < t) ≤ c. sup P (τ1 (x, s) < t) ≤ sup P (σ (x, x∈ ´ U´

x∈U

m Then we have P (τl − τ l−1 < t) ≤ c < 1. Further, since τm = l=1 (τl − τl−1 ), we have {τm < t} ⊂ m {τ − τ < t}. Therefore, using the strong Markov l l−1 l=1 property, we have P (τm < t) ≤ P (

m ) {τl − τl−1 < t}) l=1

m−1 )   ≤ E P (τm − τm−1 < t|Fτm−1 ); {τl − τl−1 < t} l=1



≤ E P (τ1 (y) < t) y=Φ ≤ cP (

m−1 )

τm−1

; (x)

m−1 )

{τl − τl−1 < t}



l=1

{τl − τl−1 < t})

l=1

≤ c2 P (

m−2 )

{τl − τl−1 < t})

l=1

≤ cm , proving (7.39).

 

326

7 Stochastic Flows and Their Densities on Manifolds (n)

Lemma 7.4.3 The sequence {ps,t (x, y)} of functions defined by (7.37) converges c (x, y) be its limit. It is a C ∞ uniformly with respect to x ∈ M, y ∈ V . Let ps,t c (x, E ∩ V ), so that it function of y and it is the density of transition function Ps,t does not depend on the choice of local charts U, D. Furthermore, for any j ∈ N and T > 0, there exists a constant cj > 0 such that sup x∈D,y∈V



j c

dy ps,t (x, y) ≤

cj (t − s)

j +d 2

,

∀0 ≤ s < t ≤ T .

(7.40)

Proof Since Φs,τm (x) ∈ ∂U , qτcm ,t (Φs,τm (x), y) is uniformly bounded with respect to y ∈ V by (7.34). Further, E[|Gcs,τm (x)|p ] is bounded uniformly with respect to x, s and m for any p > 1. Consequently, there is a positive constant K such that   E qτcm ,t (Φs,τm (x), y)Gcs,τm (x)1τm 0+



where ck (x, t), k = 0, . . . , d  are C ∞,1 -functions of x, t on M × Rd and dt,z (x) is  a C ∞,1,2 -function of (x, t, z) on M × T × Rd satisfying dt,0 (x) = 1 for any x, t. Suppose we are given a volume element dx on the manifold M. In the following we will fix it. It is also written as dy. We will study the existence of the smooth c,d c,d density of Ps,t (x, E) = E[1E (Φs,t (x))Gs,t (x)] with respect to the volume element

7.5 Density for Jump-Diffusion on Compact Manifold

329

dy. In order to construct the smooth density, we need more careful discussions than those for diffusions, since sample paths may jump to other local charts. Our discussion will be divided into two. First, we show that if the size of jumps is sufficiently small, say smaller than δ, we can construct the density function by a method similar to that of diffusion discussed in Sect. 7.4. Next, we adjoin jumps which are bigger than δ and show the existence of the smooth density by the method of perturbation discussed in Sect. 6.9. Let δ be a positive number. We will truncate jumps which are bigger than δ > 0 from the above equation: The equation is written as d   

f (Xt ) = f (X0 ) + +

  t

t

k=0 t0

Vk (r)f (Xr ) ◦ dWrk

(7.43)

{f (φr,z (Xr− )) − f (Xr− )}N(dr dz).

t0 δ≥|z|>0+

We denote the stochastic flow of diffeomorphisms generated by the above equation δ }. We set by {Φs,t Gδs,t

:= exp

d   k=0 s

t

δ ck (Φs,r , r) ◦ dWrk +

  t

 δ log dr,z (Φs,r− )N (dr dz) ,

s δ≥|z|>0+

δ (x, E)(= P c,d,δ (x, E)) := E[1 (Φ δ (x))Gδ (x)]. For a given n ∈ and define Ps,t E 0 s,t s,t s,t δ (x, E) has a C n0 -density with respect to N, we want to show that the measure Ps,t the volume element dy, if δ (depending on n0 ) is taken sufficiently small. We will discuss how such δ can be taken. Let D be a local chart of M and let τ δ (x, s) be the first leaving time of the process x,s δ (x) from D. We consider the transition function of the killed process Xt = Φs,t defined by δ (x))Gδs,t (x)1τ δ (x,s)>t ]. Qδs,t (x, E) = E[1E (Φs,t

(7.44)

In the first step, we want to show the existence of the smooth density of the above Qδs,t (x, E). Let α0 be any fixed positive number satisfying α < α0 < 2. Lemma 7.5.1 Let n0 be a positive integer. Let V , U, D be a triple of local charts such that V¯ ⊂ U ⊂ U¯ ⊂ D. Then there exists a positive constant δn0 ,V ,U,D (depending on n0 and V , U, D) such that for any 0 < δ < δn0 ,V ,U,D , the killed δ (x, y), y ∈ U with respect transition function Qδs,t (x, E ∩ U ) has a C n0 -density qs,t to the volume element dy for any s < t and x ∈ D. Further, for any integer 0 ≤ j ≤ n0 and T > 0, there are positive constants cT ,δ,j and cT ,δ,j such that

330

7 Stochastic Flows and Their Densities on Manifolds

sup x∈D,y∈U

sup

sup

x∈U c ∩D,y∈V 0≤s 0 such that for any |z| ≤ δ0 and t ∈ T, we can on D. construct a family of diffeomorphic maps φ´ t,z on Rd such that ψ(φt,z (x)) = φ´ t,z (x) ´ holds if x ∈ D (Lemma 7.1.3). 0 +d Let N be a positive integer such that N > n2−α . Let U´ = ψ(U ) and V´ = ψ(V ). 0 There exists a positive constant δN,V ,U,D , less than or equal to δ0 , such that for any 0 < δ < δN,V ,U,D inequalities d(φ´ δN (U´ ), φ´ δ−N (D´ c )) > 0, hold, where φ´ δ (U´ ) =

t∈T,|z| 0

(7.47)

φ´ t,z (U´ ) and φ´ δN (U´ ) are defined by iteration.

Indeed, note that φ´ δN (U´ ) ↓ U´ and φ´ δ−N (D´ c ) ↓ D´ c as δ ↓ 0. Then d(φ´ δN (U´ ), φ´ δ−N (D´ c )) ↑ d(U´ , D´ c ) > 0. Therefore the first inequality of (7.47) is valid. The second inequality is verified in the same way. Now, take 0 < δ < δN,V ,U,D and consider an SDE on Rd ; 

d X´ t =

d  k=0

V´k (X´ t , t) ◦ dWtk +



{φ´ t,z (X´ t− ) − X´ t− }N(dt dz).

(7.48)

δ≥|z|>0+

δ } of diffeomorphisms on Rd . Let τ´ δ (x, It generates a stochastic flow {Φ´ s,t ´ s) be the δ ´ We shall consider the law of the killed ´ from D. first leaving time of X´ t = Φ´ s,r (x) process given by

  δ ´ δs,t (x)1 ´ δs,t (x, . ´ E) = E 1E (Φ´ s,t (x)) ´ G ´ τ´ δ (x,s)>t Q ´

(7.49)

δ (x) ´ δs,t (x, Since the process Φ´ s,t ´ is pseudo-elliptic, the measure Q ´ ·) restricted to U´ = n δ 0 ´ ´ y), ´ y´ ∈ U with respect to the Lebesgue measure by ψ(U ) has a C -density q´s,t (x, c,d Lemma 6.10.3. For a given T > 0, it satisfies (6.123) and (6.124) in place of qs,t .

7.5 Density for Jump-Diffusion on Compact Manifold

331

δ (x) be the solution of equation (7.43) on the manifold M. Let Now let Φs,t δ (x) from D. Then the weighted τ δ (x, s) be the first leaving time of Xtδ,x,s = Φs,t δ δ law Qs,t (x, E) of the killed process Φs,t (x), t < τ δ (x, s) has also a C n0 -density δ (x, y), y ∈ U with respect to the volume element dy, since Φ δ (x) = qs,t s,t δ (ψ(x))) holds for t < τ δ (x, s) almost surely. Further, since q´ δ satisψ −1 (Φ´ s,t s,t c,d δ satisfies (7.45) and (7.46). fies (6.123) and (6.124) (in place of qs,t ), qs,t   δ (x, E) has a Next, we will show that for a sufficiently small δ > 0, the law Ps,t for any x ∈ M, s < t, by piecing δ (x, y) of the killed process. Our discussion together the above density function qs,t proceeds in parallel with the discussion for diffusion processes in Sect. 7.4, but we need some rectifications due to jumps of the process. Given a pair of local charts V , D such that V¯ ⊂ D, we can choose open sets U and W such that V¯ ⊂ U ⊂ U¯ ⊂ W ⊂ W¯ ⊂ D and a positive constant δN,V ,U,W,D with 0 < δN,V ,U,W,D ≤ δN,V ,U,D such that

C n0 -density with respect to the volume element dy

d(φδN (V ), φδ−N (U c )) > 0,

d(φδ (U ), W c ) > 0,

d(φδN (W ), φδ−N (D c )) > 0

0 +d hold for any 0 < δ < δN,V ,U,W,D , where N ≥ n2−α . We want to prove that 0 δ n 0 Ps,t (x; E ∩ V ) has a C -density for any x ∈ M and 0 ≤ s < t ≤ T . For this purpose, we define stopping times τmδ = τmδ (x, s), m = 0, 1, 2, . . ., and δ (x) by induction σmδ = σmδ (x, s), m = 1, 2, . . . for the process Xtδ = Xtδ,x,s = Φs,t as τ0δ = s and

δ σmδ = inf{t ≥ τm−1 ; Xtδ ∈ D c } (= ∞ if {· · · } is empty),

τmδ = inf{t ≥ σmδ ; Xtδ ∈ W } (= ∞ if {· · · } is empty). Then we have for E ⊂ U , δ (x))Gδs,t (x)] = E[1E (Φs,t

∞ 

  δ E 1E (Φs,t (x))Gδs,t (x)1τ δ 0, there exists a positive constant c = cδ,n0 such that the inequality

j δ

dy ps,t (x, y) ≤

c j +d

(t − s) 2−α0

,

∀0 ≤ s < t ≤ T

(7.52)

holds for all x, y ∈ M. Proof Let {Vα ⊂ Uα ⊂ Wα ⊂ Dα } be a family of quaternions of local charts discussed above, where {Vα } covers the manifold M. Since M is compact, it is covered by a finite number of local charts {Vαi , i = 1, 2, . . . , n}. We set δ1 =

7.5 Density for Jump-Diffusion on Compact Manifold

333

mini δVαi ,Uαi ,Wαi ,Dαi and T1 = mini TUαi ,Vαi ,Wαi ,Dαi . Suppose 0 ≤ s < t ≤ T . δ (x, y) of (7.51) is uniformly convergent for all x, y ∈ M Then, if 0 < δ < δ1 , ps,t δ (x, E) with respect to the and 0 < t − s < T1 . It is the density function of Ps,t volume element dy for any x ∈ M, 0 < t − s < T1 . δ (x, y) is a C n0 -function of y. Similarly to the proof of We will show that ps,t Lemma 7.5.2, we can show that for any j ≤ n0 , the infinite sum j

δ dy qs,t (x, y) +

∞ 

  j δ δ E dy qτδδ ,t (Φs,τ δ 0 there exist positive constants c, c such that inequalities

j c,d

sup dy ps,t (x, y) ≤

x,y∈M



c,d sup dxi ps,t (x, y) ≤

x,y∈M

hold.

c j +d

,

∀0 ≤ s < t ≤ T ,

(7.54)

,

∀0 ≤ s < t ≤ T .

(7.55)

(t − s) 2−α0 c i+d

(t − s) 2−α0

334

7 Stochastic Flows and Their Densities on Manifolds

c,d Further, p(x, s; y, t) := ps,t (x, y) is the fundamental solution of the backward heat equation on the manifold M, associated with the operator AJc,d (t) defined by d

AJc,d (t)f (x)

1 = (Vk (t) + ck (t))2 f (x) + (V0 (t) + c0 (t))f (x) 2 k=1  + {dt,z (x)f (φt,z (x)) − f (x)}ν(dz). (7.56) |z|>0+

Proof We want to apply Proposition 7.5.1 for the proof of the theorem. Given a positive integer n0 , let δ1 > 0 be a number such that for any 0 < δ < δ1 , the δ (x) has a C n0 -density. We weighted transition function of the truncated process Φs,t will show the existence of the smooth density of the weighted law of Φs,t (x) by the method of perturbation studied at Sect. 6.9. For u = ∅ ∈ U0 (empty set), we δ,φ δ (x). Further, for u = ((t , z ), . . . , (t , z )) ∈ Un such that define Φs,t (x) = Φs,t 1 1 n n  s < t1 < · · · < tn ≤ t and zi ∈ Rd0 , we define δ,u δ (x) := Φtδn ,t ◦ φtn ,zn ◦ · · · ◦ Φtδ1 ,t2 ◦ φt1 ,z1 ◦ Φs,t (x). Φs,t 1

(7.57)

In the following arguments, we will let n run on n = 0, 1, . . .. Let q be a Poisson point process with intensity dtνδ (dz), where νδ (dz) = 1|z|≥δ ν(dz). Then Dq ∩ (s, t] is an empty set or it is written as {τ1 < · · · < τn } (stopping times) as in Sect. 6.9. We set u(q) = ∅ or u(q) = ((τ1 , q(τ1 )), . . . , (τn , q(τn ))). Then it holds that Φs,t (x) = δ,u(q) Φs,t (x) a.s. See Sect. 6.9. δ (x, y), x, y ∈ M be the density function of the weighted transition Let ps,t δ (x, E) of Φ δ (x). For u = ∅, we define p δ,φ (x, y) = p δ (x, y) and function Ps,t s,t s,t s,t δ,u for u = ((t1 , z1 ), . . . , (tn , zn )) ∈ Un , we define ps,t (x, y) by  δ,u ps,t (x, y) =

 ···

M

M

δ ps,t (x, x1 ) dt1 ,z1 (x1 )ptδ1 ,t2 (φt1 ,z1 (x1 ), x2 ) × · · · 1

× dtn ,zn (xn )ptδn ,t (φtn ,zn (xn ), y) dx1 · · · dxn .

(7.58)

Set δ,u(q)

c,d (x, y) := E[ps,t ps,t

(x, y)].

(7.59)

c,d (x, E). Then it is a density function of the weighted transition function Ps,t c,d n 0 We want to show that the function ps,t (x, y) is of C -class with respect to y. Our discussion is close to those in Theorem 6.9.1. Condition (6.96) should be modified as

7.5 Density for Jump-Diffusion on Compact Manifold

335

 sup t,x

|z|>δ

|Jφ −1 |ν(dz) < ∞. t,z

However, the above condition is satisfied in the present case, since M is a compact c,d manifold and the support of the Lévy measure ν is compact. Therefore, ps,t (x, y) is a continuous function of y. See the proof of Theorem 6.9.1, (1) The n0 -times differentiability with respect to y can be verified similarly to Theorem 6.9.1, (2) Finally, remark that the above fact is valid for any positive integer n0 . Then the c,d δ (x, y) density ps,t (x, y) is in fact  a C ∞ -function of y. It satisfies (7.54), since qs,t satisfies (7.45) and the term i≥1 · · · in (7.51) is bounded. c,d,∗ } be the dual semigroup Let Ψˇ s,t be the inverse map of Φs,t and let {Ps,t c,d of the semigroup {Ps,t } with respect to a volume element dx. Apply the above discussion to the backward jump-diffusion Ψˇ s,t . Then we find that its transition c,d,∗ c,d,∗ (y, E) has a C ∞ -density ps,t (y, x) for any s < t and y. Further, function Ps,t c,d,∗ c,d we can show as before that ps,t (y, x) = ps,t (x, y) holds for any s < t and c,d ∞ x, y ∈ M. Therefore ps,t (x, y) is a C -function of x and satisfies (7.55). We can c,d ∂ c,d verify that it satisfies ∂s ps,t (x, y) = −AJc,d (s)x ps,t (x, y), making use of the dual c,d (x, y)f1 (y) dy is a C ∞,1 semigroup as in Lemma 6.10.2. Then v(x, s) := ps,t 1 function of x, s and in fact a solution of the final value problem of the backward heat equation associated with the operator AJc,d (t),   The next corollary is an immediate consequence of Theorem 7.5.1. c,d Corollary 7.5.1 Let {Ps,t } be the semigroup of Theorem 7.5.1. Then it maps C(M) ∞ to C (M) for any s < t. Further it satisfies Kolomogorov’s backward equation.

We will study the smooth density of the law of a Lévy process on a compact Lie group. We will consider a Lévy process on a Lie group G with the generator (7.23), where Vk are left-invariant vector fields and xg(z) is the right translation by g(z).

∂R

. Then V˜k , k = 1, . . . , d  are left-invariant vector fields. We set V˜k = ∂zg(z) z=0 k Theorem 7.5.2 Let Xt be a Lévy process on a compact Lie group. We assume that left-invariant vector fields {Vk , k = 1, . . . , d  } ∪ {V˜k ; k = 1, . . . , d  } span the Lie algebra of left-invariant vector fields. Then the law of Xt has a C ∞ -density with respect to the Haar measure for any t > 0. Note It is expected that similar assertions of Theorem 7.5.1 and Theorem 7.5.2 should hold for non-compact manifold and non-compact Lie groups, respectively. It remains open. Picard–Savona [95] studied the smooth density for Markov process of pure jumps on non-compact manifolds or non-compact Lie group with another additional conditions. An analytic approach is taken in Applebaum [2] for the problem on a compact Lie group.

336

7 Stochastic Flows and Their Densities on Manifolds

Appendix: Manifolds and Lie Groups Let M be a Hausdorff topological space with the second countability. It is called a manifold of dimension d if each point of M has an open neighborhood that is homeomorphic to an open set in Rd . If U is an open neighborhood of x ∈ M and ψ is a homeomorphism from M to an open subset of Rd , we call (U, ψ) a chart at x. For y ∈ U , we write ψ(y) = (x1 (y), . . . , xd (y)) and call x1 , . . . , xd local coordinates at U . An atlas is a collection of charts {(Uα , ψα ); α ∈ I } so that (Uα ) covers M. If M has an atlas for which mappings ψα ◦ ψβ−1 ; Rd → Rd are C ∞ for all α, β ∈ I , then M is said to be a C ∞ -manifold. A function f : M → R is called smooth or C ∞ , if f ◦ ψα−1 are C ∞ from ψα (Uα ) to R for any α. Let C ∞ (M) be the collection of smooth functions on M. A tangent vector V (x) at x ∈ M is a linear functional on C ∞ (M) which satisfies V (x)(f g) = f (x)V (x)(g) )g(x) for any f, g ∈ C ∞ (M). In local charts, d + i(V (x)f ∂ we can write V (x) = i=1 V (x) ∂xi . The set of all tangent vectors at x ∈ M forms a d-dimensional vector

space called the tangent space and it is denoted by Tx (M). The set T (M) = x∈M Tx (M) can be regarded as a 2d-dimensional manifold, called the tangent bundle to M. A C ∞ -map V from  M to T (M) is called a vector field. In each local chart (U, ψ), we can write V (x) = di=1 V i (x) ∂x∂ i , where V i (x) are C ∞ -functions from ψ(U ) to Rd . The collection of all vector fields is denoted by L(M). For V , W ∈ L(M), its Lie bracket [V , W ] is defined by [V , W ]f = V (Wf ) − W (Vf ). It is an element of L(M). Thus L(M) is an Lie algebra. Let M1 and M2 be C ∞ -manifolds of dimension d1 and d2 , respectively and let {(Uα , ψα ); α ∈ I } be an atlas of M1 and let {(Uβ , ψβ ); β ∈ J } be an atlas of M2 . A mapping f ; M1 → M2 is said to be C ∞ if ψβ ◦ f ◦ ψα−1 is C ∞ from Rd1 to Rd2 . A C ∞ -mapping f ; M1 → M2 is said to be a diffeomorphism if it is a bijection and the inverse f −1 is also a C ∞ -mapping. Let ϕt , t ∈ R be a family of diffeomorphisms of M such that ϕt ◦ ϕs = ϕt+s and (t, x) → ϕt (x) is a C ∞ -map. {ϕt } is called a one–parameter group of transformations. For a given {ϕt }, we set for f ∈ C ∞ (M) Vf (x) =

d

f ◦ ϕt (x) . t=0 dt

(A.1)

Then V is a vector field. Conversely, for a given vector field V , if there exists a one–parameter group of transformations ϕt satisfying (A.1), the vector field V is called complete. The associated ϕt (x) is denoted by exp(tV )(x). If the manifold M is compact, it is known that any vector field is complete, but if M is not compact, non-complete vector fields can exist. Let M be a connected manifold of dimension d and let L(M) be the linear space of vector fields on M. For f ∈ C ∞ (M), we define a linear map df : L(M) → C ∞ (M) by df (V ) = Vf . We call df a one-form or the differential of f . Localizing df at x ∈ M, dfx is a linear mapping from the tangent space Tx (M) to R, i.e., it is

Appendix: Manifolds and Lie Groups

337

the dual of Tx (M). It is denoted by Tx (M)∗ and is called the cotangent space. Let r Tx∗ (M)⊗ be the r-times tensor product to Tx∗ (M). Its element is called an r-tensor. The alternating r-tensor of v1 , . . . , vr ∈ Tx∗ (M) is defined by Alt(v1 , . . . , vr ) =

1  sign(σ )vσ (1) ⊗ · · · ⊗ vσ (r) , r!

(A.2)

σ ∈Σ(r)

where v1 , . . . , vr ∈ Tx∗ (M), vσ (1) ⊗ · · · ⊗ vσ (r) is an r-tensor, and Σ(r) is the permutation group of r letters. If an r-tensor v satisfies v = Alt(v1 , . . . , vr ), v is called an alternating tensor. r Let Λr (Tx∗ (M)) be the collection of alternating r-tensors in Tx∗ (M)⊗ . It is a vector &' d space of dimension if r ≤ d and it is equal to {0} if r > d. We set Λr (M) = r

r ∗ ∞ r x∈M Λ (Tx (M)). A smooth r-form ω is a C -mapping from x ∈ M to Λ (M). r s If v ∈ Λ (M) and w ∈ Λ (M), the exterior product (wedge product) v ∧ w is defined by v∧w =

(r + s)! Alt(v1 , . . . , vr , w1 , . . . , ws ). r!s!

At a local chart U , let x1 , . . . , xd be a local coordinate and let dxi , i = 1, . . . , d be 1-forms. Then a smooth r-form ω can be written at x ∈ U as  hωi1 ,...,ir (x) dxi1 ∧ · · · ∧ dxir , ω(x) = {i1 ,...,ir }⊂{1,...,d}

where hωi1 ,...,ir are smooth maps from U to R. In the case r = d, ω(x) is written as ω(x) = hω (x) dx1 ∧ · · · ∧ dxd .

(A.3)

If there exists a continuous d-form ω such that ω(x) is not 0 everywhere, then the manifold is called orientable. If hω (x) > 0, the d-form ω is called a positive d-form or a volume element. Let Ω d (M) be the collection of smooth d-form on M. Let φ; M → M be a ∞ C -map. The pullback φ ∗ is the linear map from Ω d (M) into itself defined by φ ∗ (ω)(V1 , . . . , Vd )(x) = ωφ(x) (dφx (V1 (x)), . . . , dφx (Vd (x))), for all V1 , . . . , Vd ∈ L(M) and x ∈ M. We will define the integral by a positive d-form (volume element) ω. Let C0∞ (M) be the collection of C ∞ -functions on M with compact supports. For f ∈ C0∞ , we will define the integral M f ω by the formula 

 fω = M

Rd

f ◦ φ −1 (x1 , . . . , xd )hω (x1 , . . . , xd ) dx1 · · · dxd ,

338

7 Stochastic Flows and Their Densities on Manifolds

if the support of f is included in a chart U . If the support of f is not included in a chart, we need a partition of unity. It is a collection of functions (ψi , i ∈ I ) in C ∞ (M) for which: (1) At each x ∈ M, only finite number of ψi are nonzero. (2) For each i, supp(ψi ) is compact.  (3) For each x ∈ M, i ∈ I, ψi (x) ≥ 0 and i∈I ψi (x) = 1. We can define the integral of a continuous function f with compact support by  fω =



M

i

f φi ω.

M

We will fix a positive d-form (volume element) ω and denote it by dx. The integral by dx of a continuous function f on M is denoted by M f (x) dx if it exists. It is usually denoted by M f (y) dy etc. Let φ : M → M be a locally diffeomorphic C ∞ -map. Let φ ∗ dx be the pullback of dx by φ. Then there exists a positive or negative C ∞ -function Jφ on M such that φ ∗ dx = Jφ dx. We have a formula of the change of variables: 

 f (x) dx =

f (φ(x))|Jφ (x)| dx

M

(A.4)

M

for any continuous functions f of compact supports. Let V be a complete vector field on M and let {ϕt } be the one–parameter group of diffeomorphisms generated by V . Then the Lie derivative of dx denoted by LV dx is defined by limt→0 1t (ϕt∗ dx −dx). Since LV dx is a d-form, it is written as LV dx = h(x) dx with a smooth scalar function h, which we denote by div V (x). We have for any f, g ∈ C0∞ (M), 

 f (x)(g(ϕt (x)) − g(x)) dx =

(f ◦ ϕt−1 (x) − f (x))g(x) dx 

+

f ◦ ϕt−1 (x)g(x)((ϕt−1 )∗ dx − dx).

Since the inverse map ϕt−1 is generated by the vector field −V , we have   1 lim f (x)(g(ϕt (x)) − g(x)) dx = f (x)V g(x) dx, t→0 t   1 −1 lim (f ◦ ϕt (x) − f (x))g(x) dx = − Vf (x)g(x) dx, t→0 t   1 lim f ◦ ϕt−1 (x)g(x)((ϕt−1 )∗ dx − dx) = − f (x)g(x)div V (x) dx. t→0 t

Appendix: Manifolds and Lie Groups

339

Therefore we have a formula of the integration by parts: 





f (x)·V g(x) dx = − M

Vf (x)·g(x) dx − M

div V (x)f (x)g(x) dx

(A.5)

M

for any f, g ∈ C0∞ (M). The above formula holds for any (not necessarily complete) C ∞ -vector field on M We should remark that values of scalar functions Jφ and div V depend on the choice of the volume element dx. Remark If M is a Euclidean space Rd , we can take dx as the Lebesgue measure. Then the C ∞ -function Jφ coincides with the Jacobian determinant det ∇φ of the map φ. Further, equality (A.4) for M =  Rd is a well known formula for change of variables. Further, div V coincides with i ∂Vi /∂xi . Formula (A.5) for M = Rd is well known as a formula for the integration by parts. As a volume element, however, we can take any measure m(dx) = m(x) dx on the Euclidean space, where m is a positive C ∞ -function. Then Jφ is no longer a  i Jacobian determinant and div V is non longer equal to i ∂V . ∂x i A set G is called a topological group if it satisfies the following three properties. (1) G is a group and, at the same time, it is a topological space. (2) The map (x, y) → xy from the product space G × G into G is continuous. (3) The map x → x −1 from G into itself is continuous, where x −1 is the inverse element of x. Let G be a topological group and let g ∈ G. Two maps Lg , Rg ; G → G are defined by Lg (x) = gx,

Rg (x) = xg,

(x ∈ G).

These two maps are homeomorphisms of G. Lg is called the left translation of G by g, and Rg is called the right translation of G by g. A topological group G is called a Lie group if G is a C ∞ -manifold with a countable basis and maps (x, y) → xy and x → x −1 are differentiable. Let V be a C ∞ -vector field on G. It is called left-invariant if for any f ∈ C0 (G), it satisfies V (f ◦ Lg )(x) = Vf (Lg x) for any g ∈ G. It is called right invariant if V (f ◦ Rg )(x) = Vf (Rg x) holds for any g ∈ G. We denote by L(G) the set of all left-invariant vector fields on the Lie group G.

340

7 Stochastic Flows and Their Densities on Manifolds

Let G be a connected Lie group of dimension d. A nontrivial regular Borel measure m on G is called a left Haar measure if m(A) = m(gA) holds for any Borel subset A of G and g ∈ G. A right Haar measure is defined similarly. It is known that the left Haar measure exists and it is unique up to a positive multiplicative constant. Let m be a left Haar measure on G. For g ∈ G, we define mg (A) = m(Ag), for each Borel set A. Then mg is another left Haar measure on G. Then there exists a positive constant Δ(g) such that mg (A) = Δ(g)m(A) holds for all Borel subsets A. We call the mapping g → Δ(g) from G to (0, ∞) the modular function of G. If the modular function is identically equal to 1, the Lie group is called unimodular. Examples of unimodular Lie groups are abelian Lie groups, compact Lie groups, semisimple Lie groups and connected nilpotent Lie groups.

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Symbol Index



| · |0,n,p , norms, 200  · 0,n,p;A , Sobolev norms, 197  · m,n,p;A , Sobolev norms, 226  · m,p , Sobolev norms, 174   · 0,n,p , Sobolev norms, 199 

 · m,n,p , Sobolev norms, 227 ∂ j , differential operator, 3 Cb∞,m -class, function, 77 Cb∞,m,n -class, function, 78 D∞ , D∞ T , Sobolev spaces, 174 ¯ ∞, D ¯ ∞ , Sobolev spaces, 227 ¯ ∞, D D T U ∞ ∞ ˜ , Sobolev spaces, 199 ˜ ,D D U det ∇ Ψˇ s,t , Jacobians of Ψˇ s,t , 141 Φs,t , stochastic flow, 79 Φˇ s,t , backward stochastic flow, 101 K F , Malliavin covariance at the center, 210

K¯ F , Malliavin covarance at the center, 237 LT , space, 51 LU , space of predictable random fields, 64, 67 LT , LT (Λ), spaces of predictable processes, 48, 49 Lip,p n+Lip,p LT (Λ), LT (Λ), spaces of predictable processes, 56 n+Lip,p (Λ), space, 72 LU p LT , space of predictable processes, 52 p LU , space of predictable random fields, 70 ˇ Ψs,t , inverse stochastic flow, 80 R¯ ρF , Malliavin covariance, 238 R˜ ρF , Malliavin covariance, 209 R F , Malliavin covariance, 184 Xt , quadratic variation, 31 X, Y t , quadratic covariation, 35

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4

347

Index

Symbols β-stable, 10 μ-continuous, 4 σ -field, 1 H -derivative, 170, 223 H -differentiable, 170, 223, 249 C ∞ -manifold, 336 Cbn -function, 4 F -space, 168 i-th H -derivative, 170 Lp -martingale, 29 n-th moment, 3 r-form, 337 r-tensor, 337

Backward martingale, 30, 73 Backward predictable processes, 73 Backward process, 15 Backward semi-group, 40 Backward semi-martingale, 31 Backward stochastic integral, 73 Backward stopping time, 30 Backward symmetric integral, 74 Backward symmetric SDE, 80 Backward transition function, 40 Bounded kernel, 37 Brownian motion, 16 Brownian motion on Lie group, 317 Burkholder-Davis-Gundy, 52

A Adapted, 28 Adjoint operator, 172 Admissible, 28 Almost surely, 14 Alternating r-tensor, 337 Alternating tensor, 337 Anticipating stochastic integral, 176 Atlas, 336

C Cadlag process, 15 Cameron Martin space, 168 Center of the intensity n, 201 Center of the Lévy measure, 10 Chapman–Kolmogorov equation, 37 Characteristic function, 3 Characteristics of distribution, 10 Characteristics of SDE, 122 Chart, 336 Commutation relation, 172, 194 Compensated random measure, 23, 64 Complete, 310, 336 Compound Poisson distribution, 9 Compound Poisson process, 22 Conditional expectation, 25 Conditional probability, 2, 26 Condition (D), 315 Condition (J.1), 82, 304

B Backward diffusion, 138 Backward exponential functional, 138 Backward filtration, 30 Backward heat kernel, 256 Backward Itô integral, 74 Backward local martingale, 31 Backward Markov process, 40

© Springer Nature Singapore Pte Ltd. 2019 H. Kunita, Stochastic Flows and Jump-Diffusions, Probability Theory and Stochastic Modelling 92, https://doi.org/10.1007/978-981-13-3801-4

349

350 Condition (J.2), 82, 304 Condition (J.1)K , 118 Condition (J.2)K , 118 Condition (R), 210 Conditions 1-3 for master equation, 88 Conservative, 38 Continuous master equation, 88 Continuous process, 15 Continuous semi-martingale, 30 Continuous stochastic flow, 80 Converge almost surely, 2 Convergence in Lp , 2 Convergence in probability, 2 Convolution, 3 Cotangent space, 337 Covariance matrix, 2

D Deterministic flow, 84 Diffeomorphism, 78, 336 Difference operator, 192 Differential, 336 Diffusion on manifold, 313 Diffusion process, 40 Distribution, 2 Doob’s inequality, 29

E Elliptic, 265 Elliptic operator, 246 Elliptic SDE, 246 Equivalent, 14 Equivalent in law, 15 Event, 1 Expectation, 2 Explosion time, 304 Exponential distribution, 8 Exponential functional, 130 Exterior product, 337

F Feynman–Kac–Girsanov formula, 136 Feynman–Kac transformation, 135 Filtration, 26 Filtration with continuous parameter, 28 Final value problem, 298 Final value problem for backward heat equation, 133, 256 First exit time, 34 Fisk–Stratonovitch’s integral, 59

Index Flow of local diffeomorphisms, 313 Formula of the integration by parts, 189 Forward process, 15 Fourier inversion formula, 240 Fundamental solution, 256, 257

G Gamma distribution, 8 Gaussian distribution, 9 Generalized Itô’s formula, 57 Generator, 133, 148 Girsanov transformation, 55, 136 Gronwall’s inequality, 88

H Haar measure, 340 Heat kernel, 257 Hitting time, 34 Hölder’s inequality, 175 Hörmander condition, 258 Hypo-elliptic, 258

I Improper integral, 119 Increasing process, 30 Independent, 2 Independent copy, 18 Independent increments, 15 Index, 10 Infinitely divisible distribution, 9 Initial value problem, 136, 300 Integrable random variable, 2 Intensity, 18 Inverse stochastic flow, 80 Itô integral, 47 Itô process, 67, 75 Itô SDE, 79 Itô SDE with jumps, 82 Itô’s formula, 49 Itô–Wentzell formula, 57

J Jacobian determinant, 105 Jacobian matrix, 105 Jump-diffusion on manifold, 313 Jump-diffusion process, 147 Jumping times, 20 Jump process, 147

Index K Kernel, 37 Killed process, 164 Kolmogorov’s backward equation, 128, 132 Kolmogorov’s criterion, 41 Kolmogorov’s forward equation, 128, 132 Kolmogorov–Totoki’s theorem, 41

L Law, 2 Law or random field, 15 Law of random variable, 2 Left-invariant, 339 Left translation, 339 Lévy–Itô decomposition, 25 Lévy-Khintchine formula, 10 Lévy measure, 9 Lévy process, 15 Lévy process on Lie group, 317 Lévy’s inversion formula, 4 Lie derivative, 338 Lie group, 339, 340 Local coordinate, 336 Local semi-martingale, 30

M Malliavin covariance, 183, 209 Malliavin covariance at the center, 210 Manifold, 336 Markovian, 38 Markov process, 38 Martingale part, 30 Martingale transform, 26 Martingale with continuous time, 28 Martingale with discrete time, 26 Master equation, 86 Mean vector, 2 Measurable, 2, 14 Measurable space, 1 Measure, 1 Meyer’s equivalence, 182 Modification, 15 Modular function, 340 Morrey’s Sobolev inequality, 114, 264

N Negative parameter martingale, 28 Nondegenerate, 12, 184, 209, 238, 250 Nondegenerate at the center, 237

351 O Off diagonal short-time, 284 One-form, 336 Order condition of exponent α, 12 Order condition of exponent α at the center, 11 Orientable, 337 P Parameter of Poisson distribution, 9 Parseval’s inequality, 240 Point function, 25 Point process, 25 Poisson distribution, 9 Poisson functional, 190 Poisson point process, 25 Poisson process, 18 Poisson random measure, 20 Poisson space with the intensity n, 190 Poisson variable, 222 Polynomial Wiener functional, 171 Predictable, 64 Predictable process, 45 Predictable sequence, 26 Predictable σ -field, 45 Probability measure, 1 Probability space, 1 Process of bounded variation, 30 Pseudo-elliptic jump-diffusion, 328 Q quadratic covariation, 35 quadratic variation, 31 R Rademacher system, 16 Random field, 14 Random variable, 2 Rapidly decreasing, 7, 8 Regular, 236 Regular functional, 210 Right continuous stochastic flow, 86 Right translation, 339 S Sample, 1 Schrodinger operator, 137 Schwartz space, 240 Semigroup of linear transformation, 37 Semi-martingale, 30 Short-time asymptotics, 277

352 Simple functional, 224 Skorohod equation, 177 Skorohod integral, 172, 176, 193, 224 Slowly increasing, 127 Smooth Wiener functional, 174 Sobolev norms, 226 Sobolev norms for Wiener functionals, 174 Stable, 10 Star-shaped neighborhood, 12 Stochastic Dirichlet condition, 299 Stochastic flow of C ∞ -maps, 79 Stochastic flow of diffeomorphisms, 79 Stochastic integral, 46, 47 Stochastic process, 15 Stopping time, 27, 28 Strong drift, 119 Strongly complete, 310 Strongly nondegenerate, 239 Strong Markov property, 39 Sub-martingale, 28 Super-martingale, 28 Symmetric integral, 59 Symmetric SDE, 78 Symmetric stochastic differential equation on manifold, 304 Symmetric stochastic differential equation with jumps, 82 T Tangent bundle, 336 Tangent space, 336 Tangent vector, 336 Tangent vector fields, 304 Tangent vector of maps, 82 Tensor, 337 Terminal time, 304 Time homogeneous, 15, 38 Topological group, 339

Index Transition function, 37 Transition probability, 38 Two-sided filtration, 31, 78, 81

U Uniformly elliptic, 265 Uniformly elliptic operator, 246 Uniformly elliptic SDE, 246 Uniformly integrable, 2 Uniformly Lipschitz continuous, 87 Uniformly Lp -bounded, 87 Uniformly nondegenerate, 250 Uniformly pseudo-elliptic, 265 Unimodular, 340

V Vector field, 336 Volume element, 337 Volume-gaining, 161 Volume-gaining in the mean, 163 Volume-losing, 161 Volume-preserving, 161 Volume-preserving in the mean, 163

W Weak drift, 119 Weighted law, 3, 183 Weighted transition function, 132 Wiener functional, 168 Wiener measure, 168 Wiener–Poisson functional, 222 Wiener-Poisson space, 222 Wiener process, 16 Wiener space, 168 Wiener variable, 222