Stochastic Calculus of Variations: For Jump Processes [3rd ed.] 9783110675290, 9783110675283

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Yasushi Ishikawa Stochastic Calculus of Variations

De Gruyter Studies in Mathematics

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Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu, Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L. Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany

Volume 54

Yasushi Ishikawa

Stochastic Calculus of Variations �

For Jump Processes 3rd edition

Mathematics Subject Classification 2020 60J25, 60J35, 60G51, 60H07 Author Prof. Dr. Yasushi Ishikawa Ehime University Graduate School of Science and Engineering Mathematics, Physics, and Earth Sciences Bunkyo-cho 2-chome 790-8577 Matsuyama Japan [email protected]

ISBN 978-3-11-067528-3 e-ISBN (PDF) 978-3-11-067529-0 e-ISBN (EPUB) 978-3-11-067532-0 ISSN 0179-0986 Library of Congress Control Number: 2023933917 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface to the first edition There are times in life when you discover what was really going on but too late. F. Forsyth, The outsider: My life in intrigue, Chapter: A day with the arrows

This book is a concise introduction to the stochastic calculus of variations (also known as Malliavin calculus) for processes with jumps. It is written for researchers and graduate students who are interested in Malliavin calculus for jump processes. In this book, ‘processes with jumps’ include both pure jump processes and jump-diffusions. The author has tried to provide many results on this topic in a self-contained way; this also applies to stochastic differential equations (SDEs) ‘with jumps’. This book also contains some applications of the stochastic calculus for processes with jumps to control theory and mathematical finance. The field of jump processes is quite wide-ranging nowadays, from the Lévy measure (jump measure) to SDEs with jumps. Recent developments in stochastic analysis, especially Malliavin calculus with jumps in the 1990s and 2000s, have enabled us to express various results in a compact form. Until now, these topics have been rarely discussed in a monograph. Among the few books on this topic, we would like to mention Bichteler– Gravereaux–Jacod (1987) and Bichteler (2002). One objective of Malliavin calculus (of jump type) is to prove the existence of the density function pt (x, y) of the transition probability of a jump Markov process Xt probabilistically, especially the very important case where Xt is given by an (Itô, Marcus, Stratonovich, …) SDE, cf. Léandre (1988). Furthermore, granting the existence of the density, one may apply various methods to obtain the asymptotic behavior of pt (x, y) as t → 0 where x and y are fixed. The results are known to be different, according to whether x ≠ y or x = y. We also describe this topic. The starting point for this book was July 2009, when Prof. R. Schilling invited me to the Technische Universität Dresden, Germany, to teach a short course on Malliavin’s calculus for jump processes. He suggested that I expand the manuscript, thus creating a book. Prof. H. Kunita kindly read and commented on earlier drafts of the manuscript. The author is deeply indebted to Professors R. Schilling, M. Kanda, H. Kunita, J. Picard, R. Léandre, C. Geiss, F. Baumgartner, N. Privault, and K. Taira. This book is dedicated to the memory of the late Professor Paul Malliavin. Matsuyama, December 2012

https://doi.org/10.1515/9783110675290-201

Yasushi Ishikawa

Preface to the second edition The present edition is an expanded and revised version of the first edition. Several changes have been added. These changes are based on a set of lectures given at Osaka City University and the University of the Ryukyus, and on seminars in several universities. These lectures were addressed to graduate and undergraduate students with interests in Probability and Analysis. In the 1970s Professor P. Malliavin has begun analytic studies of the Wiener space; he gave a probabilistic proof of the hypoellipticity result in PDE theory due to L. Hörmander. The method he used on the Wiener space is called stochastic calculus of variations, now called Malliavin calculus. The spirit of Malliavin’s work was clarified by J.-M. Bismut, S. Watanabe, I. Shigekawa, D. Nualart, and others. In the 1980s and 1990s, the movement of stochastic calculus of variations on the Wiener space has been extended to the Poisson space by J.-M. Bismut, J. Jacod, K. Bichteler–J. M. Gravereaux–J. Jacod, and others. Due to their development of the theory, problems concerning integro-differential operators in the potential theory have come to be resolved. The author has encountered with these techniques and the atmosphere in Strassburg, Clermont-Ferrand, La Rochelle, and Kyoto. The main purpose of this monograph is to summarize and explain analytic and probabilistic problems concerning jump processes and jump-diffusion processes in perspective. Our approach to those problems relies largely on the recent developments in the stochastic analysis on the Poisson space and that of SDEs on it. Several perturbation methods on the Poisson space are proposed, each resulting in an integration-by-parts formula of its own type. The analysis of jump processes has its proper value, since processes with discontinuous trajectories are as natural as processes with continuous trajectories. Professor K. Itô has been interested, and has had a sharp sense, in jump processes (especially in Lévy processes) from the period of his inquiry for the theory of stochastic integration. The theory of stochastic calculus of variations for jump processes is still developing. Its application will cover from economics to mathematical biology, although materials for the latter are not contained in this volume. It has been three years since the first edition had appeared. There have been intense activities focused on stochastic calculus for jump and jump-diffusion processes. The present monograph is an expanded and revised version of the first edition. Changes to the present edition are twofold: on the one hand, there was the necessity to correct typos and small errors, and a necessity for a clearer treatment of many topics to improve the expressions or to give sharp estimates. On the other hand, I have included new material. In Chapter 3, I have added Section 3.6.5 which treats the analysis of the transition density. In particular, it includes recent development on the Hörmander-type hypoellipticity problem for integro-differential operators related to jump-diffusion processes. The notes at the end of the volume are also extended. https://doi.org/10.1515/9783110675290-202

VIII � Preface to the second edition For the help during the preparation of the second edition, I am indebted to many professors and colleagues regarding the contents and motivation. Especially, I am indebted to Professors H. Kunita, M. Tsuchiya, E. Hausenblas, A. Kohatsu-Higa, A. Takeuchi, N. Privault, and R. Schilling. A major part of this work was done at Ehime university (1999–) with the aid of Grantin-Aid for General Scientific Research, Ministry of Education, Culture Sports, Science and Technology, Japan (No. 24540176). I take this opportunity to express my sincere gratitude to all those who are related. The year 2015 is the centennial of the birth of late Professor K. Itô. The development of Itô’s theory has been a good example of international cooperation among people in Japan, Europe, and the U.S. I hope, in this way, we would contribute to peace and development of the world in the future. Matsuyama, October 2015

Yasushi Ishikawa

Preface to the third edition There has been lots of books on stochastic analysis, application to mathematical finance, numerical analysis on Malliavin calculus, and so on. Even after the 1960s, many books appeared. Actually, for an ordinary reader, there is no time to read and understand what the authors assert among all books. Life is finite, and the most important value in life will be the time. Indeed, lots of millionaires do not have the time to spend their affluence, as their work would make them too busy. To answer the above problem that there are too many books, it is necessary (and sufficient) to decide which book one should not read. The world has changed from March 2020 into the epidemic days (with a possible massive financial collapse), and we have changed to lead a docile Zoom life. The 14th century in the West was in the days of pandemic after the crusaders, and now we have contemporary pandemic days after the global economy initiated from Asia. We hope that our present petit Decameron work derived from the Zoom life would influence the future as the Canterbury tales, Heptaméron, … . In the contents of the book, in its third version, some changes have been made in Chapters 1, 2, so that the descriptions are precise and up-to-date. We have changed Chapter 3 largely, mainly due to the fact that detailed stochastic analysis with small jumps has been put among several sections. We need a new framework of analysis for this setting. Namely, we need an accumulation of small jumps of the driving Lévy process to have the existence of density around an area. We have shifted our theory to such events here (and might have not enough treatment of the properties of trajectories of big movements driven by only finite jumps of the Lévy process). As we are interested in the existence of the density function using Malliavin calculus, inquiry into the small jumps is necessary. The basis of the changes in Chapter 3 are [106, 134], and the discussions among Y. Ishikawa and H. Kunita, M. Tsuchiya. Also in the note at the end of the book, an application to neuron cell model has been added at the end of Chapter 4. Applications of jump processes could proceed in computational biology, not only in control and the mathematical finance. The target readership now includes mathematicians, physicists and students whose research are related to measures on infinite-dimensional spaces, distributions of random processes (diffusion and jumps), and pseudodifferential equations. In this version, I cited two Japanese epigrams in classic; one is from Oku-nohosomichi (Chapter 2) and the other is from Hojyoki (Chapter 4). Oku-no-hosomichi by Basho is known worldwide. Hojyoki by Kamo-no-Chomei is well known as a typical medieval literature in Japan, one of »must« items among gymnasium students. Readers might be able to feel an atmosphere which is common with Miyazaki works, as in Chihiro, Mononoke, or with Inu-oh.

https://doi.org/10.1515/9783110675290-203

X � Preface to the third edition For the help during the preparation of the third edition, I am indebted to many professors and colleagues regarding the contents and motivation. Especially, I am indebted to professors H. Kunita, M. Kanda, M. Tsuchiya, A. Takeuchi, N. Privault, R. Schilling, Ch. Geiss, and mes amis S. B., Y. T. I am also grateful to Ehime University for providing the release time that made this project possible. Matsuyama, March 2023

Yasushi Ishikawa

Contents Preface to the first edition � V Preface to the second edition � VII Preface to the third edition � IX 0

Introduction � 1

1 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.3.1

Lévy processes and Itô calculus � 5 Poisson random measure and Lévy processes � 5 Lévy processes � 5 Examples of Lévy processes � 9 The stochastic integral for a finite variation process and related topics � 14 Basic material for SDEs with jumps � 16 Martingales and semimartingales � 17 Stochastic integral with respect to semimartingales � 19 Kunita–Watanabe inequalities � 28 Doléans-Dade exponential and Girsanov transformation � 31 Itô processes with jumps � 35 Pure jump type SDE � 40

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1

Perturbations and properties of the probability law � 47 Integration-by-parts on Poisson space � 47 Bismut’s method � 50 Picard’s method � 61 Some previous methods � 67 Methods of finding the asymptotic bounds (I) � 77 Markov chain approximation � 78 Proof of Theorem 2.3 � 82 Proof of lemmas � 89 Methods of finding the asymptotic bounds (II) � 101 Polygonal geometry � 101 Proof of Theorem 2.4 � 102 Example of Theorem 2.4 – easy cases � 112 Summary of short-time asymptotic bounds � 120 Case that μ(dz) is absolutely continuous with respect to the m-dimensional Lebesgue measure dz � 120 Case that μ(dz) is singular with respect to dz � 121 Auxiliary topics � 123 Marcus’ canonical processes � 124

2.4.2 2.5 2.5.1

XII � Contents 2.5.2 2.5.3 2.5.4

Absolute continuity of the infinitely divisible laws � 126 Chain movement approximation � 132 Support theorem for canonical processes � 142

3 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.2.3 3.3 3.3.1 3.3.2 3.3.3 3.4 3.5 3.5.1 3.5.2 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5 3.7

Analysis of Wiener–Poisson functionals � 146 Calculus of functionals on the Wiener space � 146 Definition of the Malliavin–Shigekawa derivative Dt � 148 Adjoint operator δ = D∗ � 153 Calculus of functionals on the Poisson space � 155 One-dimensional case � 155 Multidimensional case � 159 Characterization of the Poisson space � 162 Sobolev space for functionals on the Wiener–Poisson space � 166 The Wiener space � 167 The Poisson space � 169 The Wiener–Poisson space � 191 Relation with the Malliavin operator � 204 Composition on the Wiener–Poisson space (I) – general theory � 206 Composition with an element in 𝒮 ′ � 206 Sufficient condition for the composition � 216 Smoothness of the density for Itô processes � 223 Preliminaries � 223 Big perturbations � 231 Concatenation (I) � 235 Concatenation (II) – the case that (D) may fail � 238 More on the density � 244 Composition on the Wiener–Poisson space (II) – Itô processes � 261

4 4.1 4.1.1 4.1.2 4.1.3 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5

Applications � 265 Asymptotic expansion of the SDE � 265 Analysis on the stochastic model � 268 Asymptotic expansion of the density � 291 Examples of asymptotic expansions � 295 Optimal consumption problem � 301 Setting of the optimal consumption � 302 Viscosity solutions � 305 Regularity of solutions � 324 Optimal consumption � 328 Historical sketch � 331

A A.1

Appendix � 335 Notes and trivial matters � 335

Contents �

A.1.1 A.1.2 A.1.3

Notes � 335 Ramsey theory � 339 A small application to neural cell theory � 340

Bibliography � 347 List of symbols � 357 Index � 359

XIII

0 Introduction If you only read the books that everyone else is reading, you can only think what everyone else is thinking. Haruki Murakami, Norwegian Wood Vol. 1, p. 67 (in Japanese edition)

A theme of this book is to describe a close interplay between analysis and probability via Malliavin calculus. Compared to other books on this subject, our focus is mainly on jump processes, especially Lévy processes. The concept of (abstract) Wiener space has been well known since the 1970s. Since then, despite a genuine difficulty with respect to the definition of the (abstract) Wiener space, many textbooks have been published on the stochastic calculus on the Wiener space. It should be noted that it is directly connected to Itô’s theory of stochastic differential equations, which Itô invented while inspired by the work of A. N. Kolmogorov. Already at this stage, a close relation between stochastic calculus and PDE theory has been recognized through the transition probability pt (x, dy), whose density pt (x, y) is the fundamental solution to Kolmogorov’s backward equation. Malliavin calculus started with the paper [153] by P. Malliavin (cf. [154]). One of the motivations of his paper was the problem of hypoellipticity for operators associated with stochastic differential equations of diffusion type. At the beginning, Malliavin’s calculus was not very popular (except for his students, and a few researchers such as Bismut, Jacod, Shigekawa, Watanabe, and Stroock) due to its technical difficulties. Malliavin’s paper was presented at the international symposium in Kyoto organized by Prof. K. Itô. At that time, a close relation began between P. Malliavin and the Kyoto school of probability in Japan. The outcome was a series of works by Watanabe [213], Ikeda–Watanabe [87], Shigekawa [196], and others. The relation between Malliavin calculus for diffusion processes and PDEs has been deeply developed by Kusuoka and Stroock [138–140], and others. On the other hand, Paul Lévy began his study on additive stochastic processes before (cf. [147]). The trajectories of his processes are continuous or discontinuous. The additive processes he studied are now called Lévy processes. The discontinuous Lévy processes have an infinitesimal generator of integro-differential type in the semigroup theory in the sense of Hille–Yosida. Such integro-differential operators have been studied in potential theory by, e. g., Ph. Courrège [51] and Bony–Courrège–Priouret [32]. The latter paper is related to the boundary value problem associated with integro-differential operators. The theory developed following that of Fourier integral operators and pseudodifferential operators (cf. [33]). The author’s first encounter with Malliavin calculus for jump processes was the paper by Léandre [142], where he proves p(t, x, dy) P(Xt ∈ dy|X0 = x) = ∼ n(x, dy) t t https://doi.org/10.1515/9783110675290-001

(t → 0),

2 � 0 Introduction if the jump process Xt can reach y by one single jump (y ≠ x). Here, n(x, dy) denotes the Lévy measure. This result has been generalized to the case of n jumps, n = 1, 2, . . . in [93], independently of the work by Picard. As we started the research in this field, we were inspired by the close relation between the theories of integro-differential operators and jump-type (Markov) processes. Consequently, the research on the short time asymptotics of the transition density plays some role in this monograph. Later, Malliavin calculus has found new applications in the theory of finance. The presentation of the contents in this book follows the historical development of the theory. The technical requirements of this book are usual undergraduate calculus, probability, and abstract measure theory. Historically, the theory was started by J.-M. Bismut. The approach by Bismut is based on the Girsanov transform of the underlying probability measure. From an analytic point of view, the main idea is to replace the Radon–Nikodým density function for the perturbed law of the continuous trajectories by that for discontinuous trajectories. Subsequently, the theory was extended to cover singular Lévy measures using perturbation methods (Picard). Most textbooks on Malliavin calculus on the Wiener space (e. g., [154, 196]) adopt a functional-analytic approach, where the abstract Wiener space and the Malliavin operator appear. We do not use such a setting in this book. This is partly because such a setting does not seem very intuitive, and partly because the setting cannot directly be transferred to the Poisson space from the Wiener space. This point is also discussed in Section 3.4. In the spirit of potential theory and (nonlocal) integro-differential operators, we have adopted the method of perturbations of trajectories on the Wiener–Poisson space. This perspective fits well to the Markov chain approximation method used in Sections 2.2, 2.3, and to the technique of path-decomposition used in Section 3.6. Hence, it constitutes one of the main themes of this book. In our approach, both in the Wiener space and in the Poisson space, the main characters are the derivative operator Dt and the finite difference operator D̃ u , and their adjoints δ and δ,̃ respectively. The derivative/finite difference operator is defined to act on the random variable F(ω) defined on a given probability space (Ω, ℱ , P). In the Wiener space, Ω = C0 (T) is the space of continuous functions defined on the interval T equipped with the topology given by the sup-norm. The Malliavin derivative Dt F(ω) of F(ω) is then given in two ways, either as a functional derivative or in terms of a chaos expansion, see Section 3.1.1 for details. The definition via chaos expansion is quite appealing since it gives an elementary proof of the Clark–Ocone formula, and since the definition can be carried over to the Poisson space in a natural way; the details are stated in Section 3.2. Here is a short outline of all chapters.

0 Introduction

� 3

Chapter 1 Here, we briefly prepare basic materials which are needed for the theory. Namely, we introduce Lévy processes, Poisson random measures, stochastic integrals, stochastic differential equations (SDEs) with jumps driven by Lévy processes, Itô processes, canonical processes, and so on. Some technical issues in the stochastic analysis such as Girsanov transforms of measures, quadratic variation, and the Doléans-Dade stochastic exponential are also discussed. The SDEs introduced in Section 1.3 are time independent, i. e., of autonomous (or ‘Markovian’) type. In this chapter, technical details on the material concerning SDEs are often referred to citations, as our focus is to expose basic elements for stochastic analysis briefly. Especially, for material and explanations of diffusion processes, Wiener processes, stochastic integrals with respect to Wiener process, the readers can refer to [118]. Chapter 2 The main subject in Chapter 2 is to relate the integration-by-parts procedure in the Poisson space with the (classical) analytic object, that is, the transition density function. We present several methods and perturbations that lead to the integration-by-parts formula on the Poisson space. We are particularly interested in the Poisson space since such techniques on the Wiener space are already introduced in the textbooks on Malliavin calculus. The integration-by-parts formula induces the existence of the smooth density of the probability law or the functional. Analytically, ‘existence’ and ‘smoothness’ are two different subjects to attack. However, they are obtained simultaneously in the probabilistic method in many cases. Also, we present several upper and lower bounds of the transition densities associated with jump processes. Then, we explain the methods to find those bounds. The results stated here are adopted from several papers written by R. Léandre, J. Picard, and myself in the 1990s. One motivation for Sections 2.2, 2.3, and 2.4 is to provide the short-time asymptotic estimates for jump processes from the view point of analysis (PDE theory). Readers will find sharp estimates of the densities which are closely related to the jumping behavior of the process. Here, the geometric perspectives such as polygonal geometry and chain movement approximation will come into play. Compared to the estimation of the heat kernels associated with Dirichlet forms of jump type, such piecewise methods for the solution to SDEs will give some precise upper and lower bounds for the transition densities. In Section 2.5, we provide some auxiliary material. Chapter 3 Here, we study the Wiener, Poisson, and Wiener–Poisson space. Also, we use stochastic analysis on the path space. In Section 3.1, we briefly review stochastic calculus on the Wiener space using the Malliavin–Shigekawa’s perturbation. We introduce the derivative operator D and its adjoint δ. In Section 3.2, we discuss stochastic calculus on the Poisson space using Picard’s perturbation D.̃ In Section 3.3, we introduce perturbations

4 � 0 Introduction on the Wiener–Poisson space, and define Sobolev spaces on the Wiener–Poisson space based on the norms using these perturbations. This will make the main part of this book in theoretical aspect. A Meyer-type inequality for the adjoint operators on this space is explained in detail. In Sections 3.5 (General theory) and 3.7 (Itô processes), we define the composition Φ ∘ F of a random variable F on the Wiener–Poisson space with a generalized function Φ in the space 𝒮 ′ of tempered distributions, such as Φ(x) = (x − K)+ or Φ(x) = δ(x). These results are mostly new. In Section 3.6, we investigate the smoothness of the density of the processes defined on the Wiener–Poisson space as functionals of Itô processes, and inquire into the existence of the density. Chapter 4 This chapter is devoted to applications of the material from the previous chapters to problems in mathematical finance and optimal control. In Section 4.1, we explain applications to asymptotic expansions using the composition of a Wiener–Poisson functional with tempered distributions. In Section 4.1.1, we briefly repeat the material on compositions of the type Φ ∘ F given in Sections 3.5 and 3.7. Section 4.1.2 treats the asymptotic expansion of the density, which closely relates to the short-time asymptotics stated in Sections 2.2 and 2.3. In Section 4.2, we give an application to the optimal consumption problem associated to a jump-diffusion process. Some historical sketch is given at the end. We tried to make the content as self-contained as possible and to provide proofs to all major statements (formulae, lemmas, propositions, …). Nevertheless, in some instances, we decided to refer the reader to papers and textbooks, mostly if the arguments are lengthy or very technical. Sometimes, the proofs are postponed to the end of the current section due to their length. For the readers’ convenience, we tried to provide several examples. Also, we have repeated some central definitions and notations. How to use this book? Your choices are: – If you are interested in the relation of Malliavin calculus with analysis, read Chapters 1 and 2. – If you are interested in the basic theory of Malliavin calculus on the Wiener–Poisson space, read Chapters 1 and 3. – If you are interested in the application of Malliavin calculus with analysis, read Chapter 4 in reference with Chapter 1. We hope this book will be useful as a textbook and as a resource for researchers in probability and analysis.

1 Lévy processes and Itô calculus Happy families are all alike; every unhappy family is unhappy in its own way. Lev Tolstoy, Anna Karenina

In this chapter, we briefly prepare the basic concepts and mathematical tools which are necessary for stochastic calculus with jumps throughout this book. We consider Poisson processes, Lévy processes, and the Itô calculus associated with these processes. Especially, we consider SDEs of Itô and canonical type. We first introduce Lévy processes in Section 1.1. We provide basic material to SDEs with jumps in Section 1.2. Then, we introduce SDEs for Itô processes (Section 1.3) in the subsequent section. Since the main objective of this book is to inquire into analytic properties of the functionals on the Wiener–Poisson space, not all of the basic results stated in this chapter are provided with full proofs. Throughout this book, we shall denote the Itô process on the Poisson space by xt or xt (x), and the canonical process by Yt . The expressions Xt , X(t) are used for both cases of the above, or just in the sense of a general Itô process on the Wiener–Poisson space. In the text, if we cite formula (l, m, n), we mean the formula (m, n) in Chapter l.

1.1 Poisson random measure and Lévy processes In this section, we recall the basic material related to Poisson random measure, Lévy processes, and variation norms. 1.1.1 Lévy processes We denote by (Ω, ℱ , P) a probability space where Ω is a set of trajectories defined on T = [0, T]. Here, T ≤ ∞ and we mean T = [0, +∞) in the case T = +∞. In most cases, T is chosen finite. However, the infinite interval T = [0, +∞) may appear in some cases. Also ℱ = (ℱt )t∈T is a family of σ-fields on Ω, where ℱt denotes the minimal σ-field, right continuous in t, for which each trajectory ω(s) is measurable up to time t; P is a probability measure on Ω satisfying the consistency conditions. Definition 1.1. A Lévy process (z(t))t∈T on T is an m-dimensional stochastic process defined on Ω such that1 (1) z(0) = 0 a. s. (2) z(t) has independent increments (i. e., for 0 ≤ t0 < t1 < ⋅ ⋅ ⋅ < tn , ti ∈ T, the random variables z(ti ) − z(ti−1 ) are independent) 1 Here and in what follows, a. s. denotes the abbreviation for ‘almost surely’. Similarly, a. e. stands for ‘almost every’ or ‘almost everywhere’. https://doi.org/10.1515/9783110675290-002

6 � 1 Lévy processes and Itô calculus (3) z(t) has stationary increments (i. e., the distribution of z(t + h) − z(t) depends on h, but not on t) (4) z(t) is stochastically continuous (i. e., for all t ∈ T\{0} and all ϵ > 0 P(|z(t +h)−z(t)| > ϵ) → 0 as h → 0) (5) z(t) has càdlàg (right continuous on T with left limits on T \ {0}) paths. Here, m ≥ 1. In case m = 1, we also call z(t) a real-valued process. By the property (5), a trajectory z(t) has at most countable discontinuity a. s. (discontinuity of the first kind). We denote Δz(t) = z(t) − z(t−). The same notation for Δ will be applied for processes Xt , Mt , xt , . . . which will appear later. We can associate the counting measure N to z(t) in the following way: for A ∈ ℬ(Rm \ {0}), we put N(t, A) = ∑ 1A (Δz(s)), 0 0.

Note that this is a counting measure of jumps of z(⋅) in A up to the time t. As the path is càdlàg, for A ∈ ℬ(Rm \ {0}) such that Ā ⊂ Rm \ {0}, we have N(t, A) < +∞ a. s. A random measure on T × (Rm \ {0}) defined by N((a, b] × A) = N(b, A) − N(a, A), where a ≤ b and T = [0, T], is called a Poisson random measure if it follows the Poisson distribution with mean measure E[N((a, b] × A)], and if for disjoint (a1 , b1 ] × A1 , . . . , (ar , br ] × Ar ∈ ℬ(T × (Rm \ {0})), N((a1 , b1 ] × A1 ), . . . , N((ar , br ] × Ar ) are independent. Proposition 1.1 (Lévy–Itô decomposition theorem, [189, Theorem I.42]). Let z(t) be a Lévy process. Then, z(t) admits the following representation: t

t

̃ z(t) = tc + σW (t) + ∫ ∫ zN(dsdz) + ∫ ∫ zN(dsdz), 0 |z| 0 and D ⊂ R is a domain then Af = f = 0

in D implies f = 0 in R.

Here ‘f = 0 in R’ is equivalent to ‘f = 0 in D + R’ (notation in vector sum). Further, if c− = 0, c+ > 0 then f = 0 in D + [0, ∞), and if c− > 0, c+ = 0 then f = 0 in D + (−∞, 0]. Namely, the operator A transmits the analyticity according to the direction of jumps of the stable process. The method applies also to the directional antilocality. Now, we proceed by presenting Itô’s formula. Proposition 1.3 (Itô’s formula (I), [172, Theorems 9.4, 9.5]). (1) Let X(t) be a real-valued process given by t

̃ X(t) = x + tc + σW (t) + ∫ ∫ γ(z)N(dsdz),

t ≥ 0,

0 R\{0}

where γ(z) is such that ∫R\{0} γ(z)2 μ(dz) < ∞. Let f : R → R be a function in C 2 (R), and let Y (t) = f (X(t)). Then, the process Y (t), t ≥ 0 is a real-valued stochastic process which satisfies dY (t) =

df df 1 d2 f (X(t))c dt + (X(t))σ dW (t) + (X(t))σ 2 dt dx dx 2 dx 2 + ∫ [f (X(t) + γ(z)) − f (X(t)) − R\{0}

df (X(t))γ(z)]μ(dz) dt dx

̃ + ∫ [f (X(t−) + γ(z)) − f (X(t−))]N(dt dz). R\{0}

(2) Let X(t) = (X 1 (t), . . . , X d (t)) be a d-dimensional process given by t

̃ X(t) = x + tc + σW (t) + ∫ ∫ γ(z)N(dsdz), 0

t ≥ 0.

12 � 1 Lévy processes and Itô calculus Here, c ∈ Rd , σ is a d ×m-matrix, γ(z) = [γij (z)] is a d ×m-matrix-valued function such that the integral exists, W (t) = (W 1 (t), . . . , W m (t))T is an m-dimensional standard Wiener process, and ̃ N(dtdz) = (N1 (dtdz1 ) − 1{|z1 | 0, then Zt is a positive martingale.

z ∈ supp μ

(2.13)

34 � 1 Lévy processes and Itô calculus It is a crucial problem in mathematical finance, assuming X(t) is a diffusion process, to know whether the exponential local martingale t

t

Zt = exp(∫ h(s)dWs −

1 ∫ h(s)2 ds) 2

(2.14)

0

0

becomes a martingale or not. However, it is not easy to decide just by looking at h(t)(ω). A well-known sufficient condition is that there exists a constant C > 0 such that |h(t)(ω)| ≤ C for all (t, ω). A more general condition is the Novikov condition t

1 E[exp( ∫ h(s)2 ds)] < ∞ 2

for all t.

0

However, it is not easy to apply it in practice except for when h(t) is bounded. (2) In Section 2.1.1 below, we introduce a perturbation method using the Girsanov transform of measures. Bismut [29] used the expression for Zt in terms of the SDE, whereas Bass et al. [22] used the expression ℰ (M)t . (3) Under the assumption that (ℱt ) contains full negligible sets, W1 defined in (2) of the theorem is not necessarily a standard Brownian motion with respect to Q such that the Girsanov’s theorem holds. We need an antienlargement of the filtration in such a case (cf. [25, Warning 3.9.20]). (4) (Exponential expression of continuous positive martingales.) Let (Zt ) be a continuous positive martingale. Then there exists an adapted and measurable process (h(t)) satisfying T

∫ h(t)2 dt < ∞

a. s.

0

such that (Zt ) is represented by (2.14). This is seen as follows. The process Zt is represented by t

Zt = Z0 + ∫ ϕs dWs 0

using Theorem 1.2. We put F(x) = log x and use Itô’s formula to see

(2.15)

1.3 Itô processes with jumps t

t

0

0

� 35

2 ϕ 1 ϕ log Zt = ∫ s dWs − ∫ 2s ds. Zs 2 Zs

Putting h(t) =

ϕt , Zt

we get t

t

0

0

1 log Zt = ∫ h(s)dWs − ∫ h(s)2 ds. 2 This is the needed expression. (5) Let z(t) be a one-dimensional Lévy process without diffusion part and with the Lévy t ̃ measure μ(dz), with z(t) = ∫0 ∫ zN(dsdz) being a martingale. We put θ(t, z) = 1 − eα(t)⋅z where α(t) is fixed below. Then,

log(1 − θ(t, z)) + θ(t, z) = α(t) ⋅ z + 1 − eα(t)⋅z

= −(eα(t)⋅z − 1 − α(t) ⋅ z).

Furthermore, we choose u(t) ≡ 0. Then, t

t

0

0

̃ Zt = exp{∫ ∫ α(s) ⋅ zN(dsdz) − ∫ ∫(eα(s)⋅z − 1 − α(s) ⋅ z)dsμ(dz)} on ℱt (in the sense that it is a ℱt -measurable variable), where ℱ is the σ-field generated by z(t). We see that θ(t, z) satisfies the condition (1) above for the uniformity for a bounded α(t). Hence, putting a new measure dQ by dQ = Zt dP on ℱt , the process t z1 (t) = ∫0 ∫ zÑ 1 (dsdz) is a Lévy process which is a martingale under Q. For application, we can choose α(t) to be some deterministic function. In particular, α(t) =

dL ̇ (h(t)), dq

where L(q) denotes the Legendre transform of the Hamiltonian H(p) associated with the process z(t), H(p) = log E[ep⋅z(1) ], and h(t) is an element in the Sobolev space W 1,p (T), p > 2. This setting is used in the Bismut perturbation in Section 2.1.1, and in [94] to get a lower bound of the density.

1.3 Itô processes with jumps In this section, we introduce an Itô SDE driven by a Lévy process, and processes defined by it. A solution to an Itô SDE is called an Itô process.

36 � 1 Lévy processes and Itô calculus Let z(t) be a Lévy process, Rm -valued, with Lévy measure μ(dz) such that the characteristic function ψt is given by 1 ψt (ξ) = E[ei(ξ,z(t)) ] = exp t(− (ξ, σσ T ξ) + ∫(ei(ξ,z) − 1 − i(ξ, z)1{|z|≤1} )μ(dz)). 2 We may write t

z(t) = (z1 (t), . . . , zm (t)) = σW (t) + ∫ ∫ z(N(dsdz) − 1{|z|≤1} ⋅ μ(dz)ds), 0 Rm \{0}

where N(dsdz) is a Poisson random measure on T × (Rm \ {0}) with mean ds × μ(dz). We denote the index of the Lévy measure μ by β, that is, β = inf{α > 0; ∫ |z|α μ(dz) < +∞}. 0 0, we have |b(x)| ≤ k,

x ∈ Rd ,

|σ(x)|2 + ∫ |g(x, z)|2 μ(dz) ≤ k(1 + |x|2 ),

x ∈ Rd .

(3.6)

Assume further that, for each n = 1, 2, . . . , there exists a nonnegative deterministic function c(n) > 0 such that if |x1 | ≤ n and |x2 | ≤ n then |σ(x1 ) − σ(x2 )|2 + ∫ |g(x1 , z) − g(x2 , z)|2 μ(dz) ≤ c(n) |x1 − x2 |2 ,

(3.7)

and that g −1 (x, z)

exists and is bounded, i. e., |g −1 (x, z)| ≤ k,

(3.8)

for all x ∈ Rd , z ∈ Rm \ {0}. Then (∗) has a weak unique solution Xt , t ∈ [0, T]. Proposition 1.7 (Pathwise uniqueness theorem). Assume |b(x)| + |σ(x)| ≤ kr

for |x| ≤ r,

(3.9)

where kr is a constant depending on r > 0. Assume also that, for each n = 1, 2, . . . , there exist c(n) > 0 and functions ρ(n) (⋅) such that for |x1 | ≤ n, |x2 | ≤ n, 2(x1 − x2 ) ⋅ (b(x1 ) − b(x2 )) + |σ(x1 ) − σ(x2 )|2 + ∫ |g(x1 , z) − g(x2 , z)|2 μ(dz) ≤ c(n) ρ(n) (|x1 − x2 |2 ),

(3.10)

where ρ(n) (⋅) satisfies ∫ 0+

1

ρ(n) (u)

du = ∞,

n = 1, 2, . . .

(3.11)

Then Xt , t ∈ [0, T] is a pathwise unique solution. For the proof of the these results, see Example 134 and Lemma 115 of [200]. See also [6, Theorem 3.1]. A similar condition for the pathwise unique solution for a diffusion SDE is known; cf. [59, (5.3.1)]. Proposition 1.8 (Yamada–Watanabe-type theorem). (1) If the equation (∗) has a weak solution and the pathwise uniqueness holds, then it has a strong solution.

1.3 Itô processes with jumps

� 39

(2) If the equation (∗) has two solutions X 1 , X 2 on two distinct probability spaces (Ωi , ℱ i , Pi ), i = 1, 2, then there exist a probability space (Ω, ℱ , P) and two adapted processes X̃ t1 , X̃ t2 defined on (Ω, ℱ , P) such that the law of X̃ i coincides with that of X i , i = 1, 2 and X̃ i , i = 1, 2 satisfy (∗) on (Ω, ℱ , P). For the proof of this result, see [200, Theorem 137]. The proof in [200] is rather complex. In case that the SDE is driven only by the Wiener process, the result is called ‘Barlow’s theorem’; see [189, Exercise IV.40] (p. 246 in the second edition). An argument on the measurability for strong solutions and the uniqueness is referred to in [114], where its Theorem 1 is slightly different from Proposition 1.8. Based on Propositions 1.6, 1.7, we can use the Yamada–Watanabe-type theorem (Proposition 1.8) to have a strong solution of (∗). Example 1.4. We consider SDE (∗) with b(x) = −

x 1 ̸ . |x| {x =0}

(3.12)

Assume (3.6), (3.7), (3.8), and also (3.10). Then (∗) has a pathwise unique strong solution. Indeed, due to the assumption (3.12), ⟨x1 − x2 , −

x1 x − (− 2 )⟩ + |σ(x1 ) − σ(x2 )|2 + ∫ |g(x1 , z) − g(x2 , z)|2 μ(dz) |x1 | |x2 |

≤ |σ(x1 ) − σ(x2 )|2 + ∫ |g(x1 , z) − g(x2 , z)|2 μ(dz) ≤ cn |x1 − x2 |2 ,

x1 , x2 ∈ Rd ,

(3.13)

by (3.7). Hence Proposition 1.7 applies. Due to Proposition 1.6, we have weak uniqueness, hence a pathwise unique strong solution by Proposition 1.8. In the above example, the effect of the subprincipal part of the main terms is annulled by the form of the drift, which leads to the pathwise unique strong solution. In general, we cannot expect weak (and strong) pathwise uniqueness for the solution. Let (Pt )t≥0 denote the semigroup generated by the transition function (Pt φ)(x) = E[φ(Xt (x))]. It is a strongly continuous semigroup on the Banach space on C(Rd ). Further, it is a Feller semigroup. An invariant probability measure for (Pt ) associated with the SDE (∗) is a Borel probability measure μ on Rd such that ⟨Pt φ, μ⟩ = ⟨φ, μ⟩, for all t ≥ 0. Under conditions (3.6), (3.7), that g(x, z) ≡ 0, and the condition that the right-hand side of (3.10) is changed to −cρ(|x1 − x2 |2 ) + M on Rd (for some M > 0), the SDE (∗) has at least one invariant measure; see [6, Theorem 4.5].

40 � 1 Lévy processes and Itô calculus 1.3.1 Pure jump type SDE In what follows, we consider, in particular, the following SDE on the Poisson space with values in Rd of Itô type:3 t

c

xt (x) = x + ∫ b(xs (x))ds + ∑ γ(xs− (x), Δz(s)), s≤t

0

x0 (x) = x.

(3.14)

Here, ∑c denotes the compensated sum, that is, t

c

∑ γ(x, Δz(s)) = lim{ ϵ→0

s≤t

∑

s≤t,|Δz(s)|≥ϵ

γ(x, Δz(s)) − ∫ ds ∫ γ(x, z)μ(dz)} 0

|z|≥ϵ

(cf. [111, Chapter II]). The functions γ(x, z) : Rd × Rm → Rd and b(x) : Rd → Rd are C ∞ -functions whose derivatives of all orders are bounded, satisfying γ(x, 0) = 0. Equivalently, we may write xt (x) as t

t

t

̃ xt (x) = x + ∫ b (xs (x))ds + ∫ ∫ γ(xs− (x), z)N(ds dz) + ∫ ∫ γ(xs− (x), z)N(ds dz), ′

0

0 |z|≤1

0 |z|>1

(3.15)

̃ where Ñ denotes the compensated Poisson random measure, N(dsdz) = N(dsdz) − ′ dsμ(dz), b (x) = b(x) − ∫|z|≥1 γ(x, z)μ(dz), where the integrability of γ(x, z) with respect to 1{|z|≥1} dμ(z) is assumed. We abandon the restriction (3.1) hereafter, and resume using condition (1.2). We assume that the function γ is of the following form: γ(x, z) =

𝜕γ ̃ z) (x, 0)z + γ(x, 𝜕z

̃ z) = o(|z|) as z → 0. We assume further that for some γ(x, (A.0) There exists some 0 < β < 2 and positive C1 , C2 such that as ρ → 0, C1 ρ2−β I ≤ ∫ zzT μ(dz) ≤ C2 ρ2−β I. |z|≤ρ

(A.1) (a) For any p ≥ 2 and any k ∈ Nd \ {0}, ∫ |γ(x, z)|p μ(dz) ≤ C(1 + |x|)p ,

󵄨󵄨 𝜕k γ 󵄨󵄨󵄨p 󵄨 sup ∫ 󵄨󵄨󵄨 k (x, z)󵄨󵄨󵄨 μ(dz) < +∞. 󵄨󵄨 𝜕x 󵄨󵄨 x

3 The jump function g(x, z) is written as γ(x, z) temporary in this and some later sections.

(3.16)

1.3 Itô processes with jumps

� 41

(b) There exists δ > 0 such that inf{zT (

T

𝜕γ 𝜕γ (x, 0))( (x, 0)) z; x ∈ Rd } ≥ δ|z|2 𝜕z 𝜕z

on Rm . (A.2) For some C > 0, x∈Rd ,

󵄨󵄨 󵄨󵄨 𝜕γ 󵄨󵄨 󵄨 󵄨󵄨det(I + (x, z))󵄨󵄨󵄨 > C. 󵄨󵄨 𝜕x z∈supp μ󵄨󵄨

inf

Due to the previous result (Theorem 1.5), the SDE (3.15) has a unique solution xt (x). Furthermore, the condition (A.2) guarantees the existence of the flow ϕst (x)(ω) : Rd → Rd , xs (x) 󳨃→ xt (x) of diffeomorphisms for all 0 < s ≤ t. Here, we say that ϕst (x)(ω) : Rd → Rd , xs (x) 󳨃→ xt (x) is a flow of diffeomorphisms if it is a bijection a. s. for which ϕst (x) and its inverse are smooth diffeomorphisms a. s. for each s < t. We write φt (x) of xt (x) if s = 0. We remark that at the jump time t = τ of xt (x), xτ (x) = xτ− (x) + Δxτ (x) = xτ− (x) + γ(xτ− (x), Δz(τ)). Hence, 𝜕 𝜕 𝜕 φ (x) = (I + γ(xτ− (x), Δz(τ)))( φτ− (x)), 𝜕x τ 𝜕x 𝜕x and this implies −1

(

𝜕 φ (x)) 𝜕x τ

−1

=(

−1

𝜕 𝜕 φτ− (x)) (I + γ(xτ− (x), Δz(τ))) . 𝜕x 𝜕x

This describes the movements of the coordinate system induced by x 󳨃→ xτ (x). We observe that in order for the map x 󳨃→ φt (x) to be a diffeomorphism a. s., it is necessary that x 󳨃→ x + γ(x, z) is a diffeomorphism for each z ∈ supp μ. Otherwise, we cannot always adequately connect the tangent space at xτ− (x) to that at xτ (x) = xτ (x) + γ(xτ− (x), Δz(τ)). To this end, we assume (A.2) as a sufficient condition. Lp -estimates (moment estimates) The Lp -estimate for the solution of (3.14) is not easy, in general. Here, we provide a simple one. Suppose xt (x) is given by t

t

̃ xt (x) = x + ∫ b(s)ds + ∫ ∫ γ(r, z)N(drdz). 0

0

42 � 1 Lévy processes and Itô calculus Here b(⋅) and γ(⋅, ⋅) are deterministic functions. Under the integrability of γ(r, z) with respect to 1{|z|≥1} ⋅ dμ(z), we have the following Lp -estimate for xt (x). For the proof, we need long discussions. Proposition 1.9 (cf. [130, Theorem 2.11]). For p ≥ 2, we have p

p

t

p

E[ sup |xs (x)| ] ≤ Cp {|x| + E[(∫ |b(r)|dr) ]t 0≤s≤t

0

t

2

p/2

+ E[(∫ ∫ |γ(r, z)| drμ(dz))

t

] + E[∫ ∫ |γ(r, z)|p drμ(dz)]},

0

0

for some Cp > 0. We next consider the case when xt (x) is given generally by t

t

t

̃ xt (x) = x + ∫ b(xs )ds + ∫ ∫ γ(xr− , z)N(drdz) + ∫ ∫ γ(xr− , z)N(drdz), 0

0 |z|≤1

0 |z|>1

where the coefficients satisfy the conditions (3.6)–(3.8). In this case we can only have the following inequality: E[ sup |xt |p ] < ∞ 0≤t≤T

for p ≥ 2, in general, due to the turbulence of jump effect; see Section 7.3 in [119]. We sum up the basic results which are useful in treating processes defined by SDEs in a geometric perspective. Jacobian 𝜕x In order to inquire the flow property of xt (x), we need the derivative ∇xt (x) = 𝜕xt (x) of xt (x). It is known that under the conditions (3.2)–(3.5) and (A.0)–(A.2), we have the following result for the derivative. Proposition 1.10. Under the assumptions (A.0)–(A.2), the derivative ∇xt (x) satisfies the following SDE: t

∇xt (x) = I + ∫ ∇b′ (xs− (x))ds∇xs− (x) 0 t

̃ + ∫ ∫ ∇γ(xs− (x), z)N(dsdz)∇x s− (x) 0 |z|≤1 t

+ ∫ ∫ ∇γ(xs− (x), z)N(dsdz)∇xs− (x). 0 |z|>1

(3.17)

1.3 Itô processes with jumps

� 43

Proof. We skip the proof of the differentiability since it is considerably long (see [130, Theorem 3.3]). The SDE which ∇xt (x) should satisfy can be given as follows. Let e be a unit vector in Rd , and let X ′ (t) =

1 (x (x + λe) − xt (x)), λ t

0 < |λ| < 1.

Then, X ′ (t) satisfies the following SDE: t

t

t

̃ + ∫ ∫ γλ′ (s, z)N(dsdz), X ′ (t) = Ie + ∫ b′λ (s)ds + ∫ ∫ γλ′ (s, z)N(dsdz) 0

0 |z|≤1

0 |z|>1

where 1 ′ (b (xs− (x + λe)) − b′ (xs− (x))), λ 1 γλ′ (s, z) = (γ(xs− (x + λe), z) − γ(xs− (x), z)). λ b′λ (s) =

We let λ → 0. Then, by the chain rule, lim b′λ (s) = ∇b′ (xs− (x)) ⋅ ∇xs− (x)

λ→0

a. s.

and lim γλ′ (s, z) = ∇γ(xs− (x), z) ⋅ ∇xs− (x)

λ→0

a. s.

for each s. Hence, we have t

∇xt (x) = I + ∫ ∇b′ (xs− (x))ds∇xs− (x) 0

t

̃ + ∫ ∫ ∇γ(xs− (x), z)N(dsdz)∇x s− (x) 0 |z|≤1 t

+ ∫ ∫ ∇γ(xs− (x), z)N(dsdz)∇xs− (x). 0 |z|>1

Equation (3.17) is fundamental in considering the stochastic quadratic form (an analogue of the Malliavin matrix for the jump process); see Chapter 2. The derivative ∇xt (x) depends on the choice of e ∈ Rd , and is indeed a directional derivative. We denote the Jacobian matrix by the same symbol, namely, ∇xt (x) = ∇x xt (x). Let m = d. Here is an expression of det ∇xt (x) which will be useful for the analytic (volume) estimates.

44 � 1 Lévy processes and Itô calculus Let t d

At = ∫ ∑ 0 i=1

𝜕 ′i b (xs (x))ds 𝜕xi

t

d

+ ∫ ∫ [det(I + ∇γ(xs− (x), z)) − 1 − ∑ i=1

0 |z|≤1

𝜕 i γ (xs− (x), z)]dsμ(dz) 𝜕xi

t

+ ∫ ∫ [det(I + ∇γ(xs− (x), z)) − 1]N(ds, dz), t

0 |z|>1

̃ Mt = ∫ ∫ [det(I + ∇γ(xs− (x), z)) − 1]N(dsdz). 0 |z|≤1

We then have Lemma 1.3. det(∇xt (x)) = exp At ⋅ ℰ (M)t . Proof. By Itô’s formula applied to det ∘∇xt (x), t d

det(∇xt (x)) = 1 + ∫ ∑ 0 i=1

𝜕 ′i b (xs (x)) det(∇xs (x))ds 𝜕xi

t

+ ∫ ∫ [det((I + ∇γ(xs− (x), z))∇xs− (x)) 0 |z|≤1

̃ − det(∇xs− (x))]N(dsdz) t

+ ∫ ∫ [det((I + ∇γ(xs− (x), z))∇xs− (x)) 0 |z|>1

− det(∇xs− (x))]N(dsdz) t

+ ∫ ∫ [det((I + ∇γ(xs− (x), z))∇xs− (x)) − det(∇xs− (x)) 0 |z|≤1 d

−∑ i=1

𝜕 i γ (xs− (x), z) det(∇xs− (x))]dsμ(dz) 𝜕xi

t

= 1 + ∫ det(∇xs− (x))d(As + Ms ). 0

Here we used the formula

(3.18)

1.3 Itô processes with jumps

(∇ det)(A)(BA) ≡

� 45

d det(A + tBA)|t=0 = trB det A dt

on the right-hand side. Hence, we apply the Doléans-Dade exponential formula and Girsanov’s theorem (Theorem 1.4) from Section 1.2.4. In view of [A, M]s = 0, [A + M] = [M], we have the assertion. Using (3.18), it is possible to show the boundedness of E[supt∈T det(∇xt (x))−p ] by a constant which depends on p > 1 and the coefficients of the SDE (3.14) (cf. [221, Lemmas 3.1, 3.2]). Inverse stochastic flow Using the Jacobian ∇xt (x), we can show the existence of the inverse flow xt−1 (x) as a representation of the flow property. By the inverse mapping theorem, this is due to the (local) invertibility of the Jacobian ∇xt (x). Associated to the above-mentioned process ∇xt (x) = U(t), t

t

̃ U(t) = I + ∫ ∇b (xs− (x))U(s)ds + ∫ ∫ ∇γ(xs− (x), z)U(s)N(dsdz) ′

0

0 |z|≤1

t

+ ∫ ∫ ∇γ(xs− (x), z)U(s)N(dsdz), 0 |z|>1

we introduce 2

V (t) = −U(t) + [U, U]ct + ∑ (I + ΔU(s)) (ΔU(s)) . −1

01

Here, ϕz (x) = x + γ(x, z); cf. [119, Lemma 6.5.1].

2 Perturbations and properties of the probability law Days and months are the travellers of eternity. So are the years that pass by. Those who steer a boat across the sea, or drive a horse over the earth till they succumb to the weight of years, spend every minute of their lives travelling. Matsuo Basho (Haiku Poet, 1644–1694), Oku-no-hosomichi

In this chapter, we briefly sketch the perturbation methods in the Poisson space and their applications. Namely, we relate Lévy processes (and SDEs) to perturbations for functionals on the Poisson space. The perturbation induces the integration-by-parts formula which is an essential tool leading to the existence of the smooth density of the probability law. In Section 2.1, we recall several types of perturbations on the Poisson space. In Sections 2.2, 2.3, granting the existence of the density, we present the short-time asymptotics from both above and below the density function pt (x, y) obtained as a result of the integration-by-parts calculation, which uses a perturbation method given in Section 2.1. We can observe distinct differences in the short-time bounds of densities from those obtained for diffusion processes with analytic methods. The existence of a smooth density is closely related to the behavior of the Fourier ̂ as |v| → ∞ of the density function y 󳨃→ pt (x, y). We will inquire into the transform p(v) existence and behavior of pt (x, y) more precisely in Sections 3.3, 3.6, relating to the SDE with jumps. In Section 2.4, we give a summary on the short-time asymptotics of the density functions of various types. Section 2.5 will be devoted to the auxiliary topics concerning the absolute continuity of the infinitely divisible laws and the characterization of the “support” of the probability laws in the path space.

2.1 Integration-by-parts on Poisson space We are going to inquire into the details on the absolute continuity and on the smoothness of the transition density for the functionals given as solutions to the SDE of jump and diffusion type, by using Malliavin calculus for processes with jumps. In Malliavin calculus, we make perturbations of the trajectories of processes with respect to the jump times or to the jump size and directions. Our general aim is to show the integration-by-parts formula for a certain d-dimensional Poisson functional F. The formula is described as follows. For any smooth random variable G, there exists a certain Lp -random variable determined by F and G, denoted by ℋ(F, G), such that the equality E[𝜕ϕ(F)G] = E[ϕ(F)ℋ(F, G)]

https://doi.org/10.1515/9783110675290-003

(1.1)

48 � 2 Perturbations and properties of the probability law holds for any smooth function ϕ. Formula (1.1) is called an integration-by-parts formula. We will show in this section that the formula holds for ‘nondegenerate’ Poisson functionals in various cases. The formula can be applied to proving the existence of the smooth density of the law of F. Indeed, let pF (dx) denote the law of F. Its Fourier transform p̂ F (v) is written by E[ϕv (F)], where ϕv (x) = exp(−i(x, v)). We apply the formula above k times by setting ϕ = ϕv and G = 1, and thus E[𝜕k ϕv (F)] = E[𝜕k−1 ϕv (F)ℋ(F, 1)]

= ⋅ ⋅ ⋅ = E[ϕv (F)ℋ(F, ℋ(F, . . . , ℋ(F, 1)))].

Since |ϕv (F)| ≤ 1 holds for all v ∈ Rd , the above formula implies that there exists a positive constant Ck such that 󵄨󵄨 k 󵄨 󵄨󵄨E[𝜕 ϕv (F)]󵄨󵄨󵄨 ≤ Ck ,

∀v

(1.2)

for each multiindex k. Since 𝜕k ϕv = (−i)|k| vk ϕv , we have 󵄨󵄨 k 󵄨󵄨󵄨󵄨 ̂ 󵄨 󵄨󵄨v 󵄨󵄨󵄨󵄨pF (v)󵄨󵄨󵄨 ≤ Ck ,

∀v

for each k, where p̂ F (v) = E[ϕv (F)]. Hence, d

(

1 ) ∫ ei(x,v) i|l| vl p̂ F (v) dv 2π

is well defined for l such that |l| < |k| − d. Then, the theory of the Fourier transform tells us that pF (dx) has a density pF (x) and the above integral is equal to 𝜕xl pF (x). This result is closely related to analysis and the classical potential theory. Lemma 2.1 ([191, Lemma (38.52)]). Suppose l ∈ N and that a random variable F with values in Rd is such that, for some constant Kl and any f ∈ Cbl+d+1 (Rd ), 󵄨󵄨 󵄨 α 󵄨󵄨E[(D f )(F)]󵄨󵄨󵄨 ≤ Kl ‖f ‖∞ for |α| ≤ l + d + 1, where ‖ ⋅ ‖∞ denotes the sup-norm. Then F has a density function p(⋅) belonging to Cbl (Rd ). Proof. Let μ be the law of F and let φ denote its characteristic function, φ(θ) = E[exp(i(θ, F))], We assume f (x) = exp(i(θ, x)), and find that

θ ∈ Rd .

2.1 Integration-by-parts on Poisson space

󵄨 󵄨 |θ|α 󵄨󵄨󵄨φ(θ)󵄨󵄨󵄨 ≤ Cl

� 49

for |α| ≤ l + d + 1,

where α denotes a multiindex. Hence if β is a multiindex with |β| ≤ l then d

󵄨 󵄨 |θ|β 󵄨󵄨󵄨φ(θ)󵄨󵄨󵄨 ≤ Cl ∏ |θk |−1−1/d , k=1

and the law μ has a density p with (Dβ p)(y) = (2π)−d (−i)|β| ∫ exp(−i(θ, y))θβ φ(θ)dθ. Here the integral converges absolutely and uniformly. The virtue of this lemma is that we can express the order of regularity for p(⋅) concretely. The original Malliavin calculus for diffusion processes by Malliavin [154] began with this observation, and the aim was to derive the integration-by-parts formula on a probability space. The aim of early works by Bismut [28, 29] for jump processes was also the same. In this perspective, one of the most important subjects was to decide which kind of perturbation on the trajectories should be chosen in order to derive the integrationby-parts formula. Malliavin’s approach uses the Malliavin operator L and makes a formalism for the integration-by-parts. It has succeeded in the analysis on the Wiener space, and signified the importance of the integration-by-parts formula. Bismut’s approach, on the other hand, uses the perturbation of trajectories by making use of the derivative operator D, and then makes use of the Girsanov transform of the underlying probability measures to show the integration-by-parts formula. We can adopt Bismut’s approach in this section. Bismut’s approach was originally developed on the Poisson space under the condition that the Lévy measure μ of the base Poisson random measure has a smooth density function. Accordingly, along the idea of the Bismut’s approach, the condition has been relaxed by Picard to accept singular Lévy measures where we use the difference operator. See Section 3.3 for Picard’s approach. Malliavin’s approach can adopt general functionals on the Wiener (and on the Poisson) space. On the other hand, Bismut’s approach, in the formulation treated in this book, can only adopt those Poisson functionals which appear as solutions to SDEs with jumps. This aspect has been relaxed in Picard’s approach. For an interpretation of the use of Malliavin’s approach to the solutions of SDEs with jumps on the Poisson space, see [24, Section 10]. Remark 2.1. Bismut also considered the approach regarding the Poisson process as a boundary process of a diffusion process in a higher-dimensional space, that is, the method of subordination (cf. [30] and [144]). The calculation looks a bit messy. We will not adopt this approach.

50 � 2 Perturbations and properties of the probability law 2.1.1 Bismut’s method In this subsection, we assume the Lévy measure μ has a density with respect to the Lebesgue measure. The Bismut method is the way to make a variation of the intensity, i. e., the density of the Lévy measure. We can use the Girsanov transform of the underlying probability measures of the driving Lévy processes to make perturbations on the jump size and the direction, leading to the integration by parts formula. This method has been initiated by Bismut [29], and has been developed by Norris [163], Bichteler et al. [24, 26], and Bass [22]. Although the idea is simple, the calculation using this method is a bit complicated. We denote by Ω1 the set of continuous functions from T to Rm , by ℱ1 the smallest σ-field associated to W ’s in Ω1 , and by P1 the probability of W for which W (0) = 0 and W (t) := ω(t) behaves as a standard Wiener process (Brownian motion). The triplet (Ω1 , ℱ1 , P1 ) is called a Wiener space. Let Ω2 be the set of all nonnegative integer-valued measures on 𝕌 = T × (Rm \ {0}) ̂ such that ω(T × {0}) = 0, ω({u}) ≤ 1 for all u, and that ω(A) < +∞ if N(A) < +∞, A ∈ 𝒰 . Here, u = (t, z) denotes a generic element of 𝕌, 𝒰 denotes the Borel σ-field on 𝕌, and ̂ N(dtdz) = dtμ(dz), where μ is a Lévy measure. Let ℱ2 be the smallest σ-field of Ω2 with respect to which ω(A) ∈ ℱ2 , A ∈ 𝒰 . Let P2 be a probability measure on (Ω2 , ℱ2 ) such that ̂ N(dtdz) := ω(dtdz) is a Poisson random measure with intensity measure N(dtdz). The space (Ω2 , ℱ2 , P2 ) is called the Poisson space. Given α ∈ (0, 2), let zj (t) be the independent, symmetric jump processes on R such that zj (0) ≡ 0, j = 1, . . . , m. We assume they have the same law, and that they have the infinitesimal generator Lφ(x) = ∫ [φ(x + z) − φ(x)]|z|−1−α dz,

x ∈ R, φ ∈ C0∞ (R).

R\{0}

We put g(z)dz ≡ |z|−1−α dz, that is, μ(dz) = g(z)dz is the Lévy measure of zj (t) having the law of a symmetric α-stable process. The assumption for the Lévy measure μ to be of α-stable type plays an essential role in this method. See the proof of Lemma 2.3 below. [A] Integration-by-parts of order 1 (Step 1) The symmetric Lévy process zj (s) has a Poisson random measure representation s

zj (s) = ∫ ∫ zÑ j (dsdz), 0

2.1 Integration-by-parts on Poisson space

� 51

where Ñ j (dsdz) is the compensated Poisson random measure on R associated to zj (s), with the common mean measure ds × g(z)dz.

(1.3)

Here, g(z)dz = |z|−1−α dz,

α ∈ (0, 2).

We denote by P the law of zj (s), j = 1, . . . , m. Let the process xs (x) be given by the following linear SDE, that is, dxs (x) = ∑m j=1 Xj (xs− (x)) dzj (s) + X0 (xs (x))ds,

{

x0 (x) ≡ x.

Here, the Xj (x)’s denote d-dimensional smooth vector fields on Rd whose derivatives of all orders are bounded, j = 0, 1, . . . , m, satisfying Span(X1 , . . . , Xm ) = Tx (Rd ) at each point x ∈ Rd . We assume (A.0) to (A.2) in Section 1.3 for xt (x), with b(x) = X0 (x) and γ(x, z) = ∑m j=1 Xj (x)zj . In particular, (A.2) takes the following form: for some C > 0, 󵄨󵄨 𝜕Xj (x) 󵄨󵄨󵄨 󵄨󵄨 z )󵄨󵄨 > C, 󵄨󵄨det(I + 𝜕x j 󵄨󵄨󵄨 x∈Rd ,z∈supp μ󵄨󵄨 inf

j = 1, . . . , m.

We denote by φs the mapping x 󳨃→ xs (x). We use the Girsanov transform of measures, see Section 1.2.4 above (cf. [24, Section 6]). Let v = (v1 , . . . , vm ) be a bounded predictable process on [0, +∞) to Rm . We consider the perturbation θjλ : zj 󳨃→ zj + λν(zj )vj ,

λ ∈ R, j = 1, . . . , m.

Here, ν(⋅) is a C 2 -function such that ν(zj ) ∼ z2j for zj small, j = 1, . . . , m. We consider the above transformation for |λ| sufficiently small. Let Njλ (dsdz) be the Poisson random measure defined by t

∫ ∫ ϕ(z)Njλ (dsdz) 0

t

= ∫ ∫ ϕ(θjλ (z))Nj (dsdz), 0

ϕ ∈ C0∞ (R), λ ∈ R.

(1.4)

52 � 2 Perturbations and properties of the probability law It is a Poisson random measure with respect to P. We set Λλj (z) = {1 + λν′ (z)vj } m

t

j=1

0

g(θjλ (z)) , g(z)

and t

Ztλ = exp[∑{∫ ∫ log Λλj (zj )Nj (dsdzj ) − ∫ ds ∫(Λλj (zj ) − 1)g(zj )dzj }].

(1.5)

0

We will apply the m-dimensional version of Theorem 1.4 in Section 1.2. It implies that Ztλ is a martingale with mean 1. We define a probability measure Pλ by Pλ (A) = E[Ztλ 1A ], A ∈ ℱt . Then, due to (1.5), the compensator of Nj (dsdzj ) with respect to Pλ is dsΛλj (zj )g(zj )dzj .

Furthermore, we can compute the compensator of Njλ (dsdzj ) with respect to Pλ as follows. We have E

Pλ

t

[∫ ∫ ϕ(zj )Njλ (dsdzj )] 0

Pλ

t

= E [∫ ∫ ϕ(θj (zj ))Nj (dsdzj )] 0

= t ∫ ϕ(θj (zj ))Λλj (zj )g(zj )dzj .

(1.6)

Setting yj = θjλ (zj ), we find by the definition that the above integral is equal to

t ∫ ϕ(yj )g(yj )dyj . Consequently, the compensator of Njλ (dsdzj ) with respect to Pλ is equal

to dsg(zj )dzj . Hence, as Poisson random measures, the law of Njλ (dsdzj ) with respect to Pλ coincides with that of Nj (dsdzj ) with respect to P. Consider the perturbed process xsλ (x) of xs (x) defined by λ λ λ dx λ (x) = ∑m j=1 Xj (xs− (x))dzj (s) + X0 (xs (x))ds, { λs x0 (x) ≡ x. t

(1.7)

λ

Here, zλj (t) = ∫0 ∫ zNjλ (dsdz). Then, E P [f (xt (x))] = E P [f (xtλ (x))] = E P [f (xtλ (x))Ztλ ], and

𝜕 P E [f (xtλ (x))Ztλ ], f ∈ C0∞ (Rd ). A result of the differentiability of ODE can we have 0 = 𝜕λ be applied jump by jump to show that, for |λ| small, we have

𝜕x λ (x) 𝜕 (f ∘ xtλ (x)) = Dx f (xtλ (x)) ⋅ t , 𝜕λ 𝜕λ

f ∈ C0∞ (Rd ).

We would like to differentiate E[f ∘ xtλ (x)] at λ = 0. To this end, we introduce the following notion. Definition 2.1. A family (Gλ )λ∈Λ of processes is called F-differentiable at 0 with derivative 𝜕G if (i) supt∈T |Gλ | ∈ ⋂p 0.

(1.10)

This lemma is proved at the end of this subsection. With the choice of vj below (given in the proof of Lemma 2.3 below), we can regard Ht as a linear mapping from Rd to Rd which is nondegenerate a. s.; see (1.24) below. By the inverse mapping theorem, we can guarantee the existence and the differentiability of the inverse of Htλ for |λ| small, which we denote by Htλ,−1 , Htλ,−1 = [Htλ ]−1 . (Step 3) We carry out the integration-by-parts procedure for Ftλ (x) = f (xtλ (x))Htλ,−1 as follows. 𝜕 |λ=0 on both sides Recall that we have E[Ft0 (x)] = E[Ftλ (x) ⋅ Ztλ ]. Taking the F-derivative 𝜕λ yields 0 = E[Dx f (xt (x))Ht−1 Ht ] + E[f (xt (x)) Here,

𝜕 λ,−1 H 𝜕λ t

⟨

𝜕 λ,−1 H | ] + E[f (xt (x))Ht−1 ⋅ Rt ]. 𝜕λ t λ=0

(1.11)

is defined by the formula

𝜕 𝜕 λ,−1 H , e⟩ = trace[e′ 󳨃→ ⟨−Htλ,−1 ( Htλ ⋅ e′ )Htλ,−1 , e⟩], 𝜕λ t 𝜕λ

e ∈ Rd ,

𝜕 λ where 𝜕λ Ht is the second F-derivative of xtλ (x) defined as in [24, Theorem 6-44]. As H λ 𝜕 λ 𝜕 λ,−1 is F-differentiable at λ = 0, we put DHt = 𝜕λ Ht |λ=0 . Then 𝜕λ Ht |λ=0 = −Ht−1 DHt Ht−1 . This yields the integration-by-parts formula

E[Dx f (xt (x))] = E[f (xt (x))ℋt(1) ],

(1.12)

ℋt = Ht DHt Ht − Ht Rt .

(1.13)

where (1)

−1

−1

−1

2.1 Integration-by-parts on Poisson space

� 55

Lastly, we compute Ht−1 DHt Ht−1 . By the argument similar to that in [28, p. 477], we have m

′ DHt = ∑ φ∗t {∑(φ∗−1 s Xj )(x)ν (Δzj (u))vj (s)ν(Δzj (u))vj (s)} s≤t

j=1 m

= ∑ φ∗t {∑ Mj(1) (t, s)ν(Δzj (s))} s≤t

j=1

(say).

(1.14)

Here, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 (1) p 󵄩󵄩∑ ν(Δzj (s))Mj (t, s)󵄩󵄩󵄩 ∈ L , 󵄩󵄩s≤t 󵄩󵄩

p ≥ 1, j = 1, . . . , m.

(1.15)

(This type of ‘pushforward formula’ appears to be important in many places.) Combining (1.10), (1.15) (t = 1) gives E[Dx f (x1 (x))] = E[f (x1 (x))ℋ1(1) ].

(1.16)

This leads our assertion of order 1. [B] Higher-order formulae and smoothness of the density We can repeat the above argument to obtain the integration-by-parts formulae for higher orders by a direct application of the above method. Instead of describing the detail, we will repeat the argument in [A] in a general setting, and paraphrase the argument mentioned in the introduction to Section 2.1. Let g(ω2 ) be a functional on (Ω2 , ℱ2 , P2 ), and put G0 = g(N(dudz)), Gλ = g(N λ (dudz)) for λ ≠ 0.

(1.17)

Theorem 2.1. Suppose that the family (Gλ )λ∈(−1,1) is F-differentiable at λ = 0. Then, we have an integration-by-parts formula E[Dx f (F)G0 ] = E[f (F)ℋ(F, G0 )]

(1.18)

for any smooth function f . Here, 0

0

ℋ(F, G ) = (Ht DHt Ht − Ht Rt )G − Ht 𝜕G,

where 𝜕G =

−1

−1

−1

−1

(1.19)

𝜕Gλ | . 𝜕λ λ=0

Proof. We put Φλ = f (F λ )Htλ,−1 with F λ = xtλ (x). Using Girsanov’s theorem (Theorem 1.4), we have E[Φ0 G0 ] = E[Φλ Zt Gλ ],

(1.20)

56 � 2 Perturbations and properties of the probability law where Ztλ is given by (1.5). We take λ = 0 after differentiating the two sides, and obtain 𝜕 λ,−1 H | G0 ] 𝜕λ t λ=0 𝜕 + E[f (F)Ht−1 Rt G0 ] + E[f (F)Ht−1 Gλ |λ=0 ] 𝜕λ

0 = E[Dx f (F)Ht−1 Ht G0 ] + E[f (F)

using (1.8), (1.9). This leads to the integration-by-parts formula (1.18) with 0

ℋ(F, G ) = −(

𝜕 λ,−1 𝜕 H | + Ht−1 Rt )G0 − Ht−1 Gλ |λ=0 . 𝜕λ t λ=0 𝜕λ

The formula (1.16) is obtained from this theorem by putting G = 1. If we apply the above result with G0 = ℋ(F, 1) and Gλ = ℋ(F λ , 1) for λ ≠ 0, then the family (Gλ )λ∈(−1,1) is F-differentiable at λ = 0 by (1.10). This leads to the second-order formula E[D2x f (F)] = E[Dx f (F)ℋ(F, 1)] = E[f (F)ℋ(F, ℋ(F, 1))]. Repeating this argument, we obtain the integration-by-parts formulae of higher orders. We then can prove the smoothness of the density pF (x) as explained in the introduction to Section 2.1. Proof of Lemma 2.3 Proof. Part (i). We begin by computing Htλ ≡ We recall the SDE (1.7), namely

𝜕xtλ 𝜕λ

at λ = 0.

m

dGλ = ∑ Xj (Gλ (x))dzλj (t) + X0 (Gλ )dt,

(1.7)′

j=1

t

where zλj (t) = ∫0 ∫ zNjλ (dsdz). Recalling the definition of θjλ (zj ) = zj + λν(zj )vj , we differentiate with respect to λ both sides of (1.7), and put λ = 0. Then, Ht = Ht0 = 𝜕G is obtained as the solution of the following equation, that is, m

c

j=1

s≤t

Ht = ∑{∑(

𝜕Xj 𝜕x

t

(xs− ))Hs− Δzj (s) + ∑ Xj (xs− )vj (s)ν(Δzj (s))} + ∫ s≤t

0

𝜕X0 (x )H ds (1.21) 𝜕x s− s−

(cf. [24, Theorem 6-24]). Then, Ht is given according to the method of variation of constants, similar to the Duhamel’s formula, by

2.1 Integration-by-parts on Poisson space

e

t(A+B)

� 57

t

tA

= e (1 + ∫ e−sA Bes(A+B) ds), 0

with B corresponding to the second term on the right-hand side of (1.21), by m

φ∗t { ∑ ∑( j′ =1 s≤t

m 𝜕 𝜕 φs− (x)) {∑(I + Xj (xs− (x))Δzj (s)) }ν(Δzj′ (s))Xj′ (xs− (x))vj′ (s)} 𝜕x 𝜕x j=1 −1

−1

(1.22)

(cf. [24, (6-37), (7-16)]). In the above equation, ∑cs≤t denotes the compensated sum, and φ∗t denotes φ∗t Y (x) = 𝜕φt 𝜕φ ( 𝜕x (x))Y (x) by φt , that is, φ∗t = ( 𝜕xt (x)). Below we will also use the following notation: 𝜕 −1 φ∗−1 denotes the pushforward φ∗−1 t t Y (x) = ( 𝜕x φt (x)) Y (xt− (x)) by φt , where −1

(

𝜕 φ (x)) 𝜕x t

m 𝜕 𝜕 φt− (x)) {∑(I + Xj (xt− (x))Δzj (t)) }. 𝜕x 𝜕x j=1 −1

=(

−1

We remark Ht is the F-derivative of xtλ (x) to the direction of vj . Notice also that we can regard Ht as a linear functional: Rd → R by (using the above notation) m

⟨Ht , p⟩ = ⟨φ∗t ∑ ∑ ν(Δzj (s))(φ∗−1 s Xj )(x)vj (s), p⟩, j=1 s≤t

p ∈ Rd .

(1.23)

Furthermore, the process vj may be replaced by the process ṽj with values in Tx (Rd ), which can be identified with the former by the expression vj = ⟨ṽj , q⟩, q ∈ Rd . That is, vj = ṽj (q). We put ṽj ≡ (

−1

𝜕 φ (x)) Xj (xs− (x)) 𝜕x s

in what follows. Using this expression, Ht defines a linear mapping from Tx∗ (Rd ) to Tx (Rd ) defined by m

q 󳨃→ Ht (q) = φ∗t ∑ ∑ ν(Δzj (s))(φ∗−1 s Xj )(x)ṽ j (q). j=1 s≤t

We shall identify Ht with this linear mapping. Let Kt = Kt (x) be the stochastic quadratic form on Rd × Rd , m

∗−1 Kt (p, q) = ∑ ∑ ν(Δzj (s))⟨(φ∗−1 s Xj )(x), p⟩⟨q, (φs Xj )(x)⟩. j=1 s≤t

(1.24)

58 � 2 Perturbations and properties of the probability law Here, ν(z) is a function such that ν(z) ∼ |z|2 for |z| small. For 0 < s < t, Ks,t = Ks,t (⋅, ⋅) is the bilinear form m

∗−1 Ks,t (p, q) = ∑ ∑ ν(Δzj (u))⟨(φ∗−1 u Xj )(x), p⟩⟨q, (φu Xj )(x)⟩. j=1 s η−1 } = P{K1 < η} = ov,k (η∞ )

(1.26)

as η → 0. Here, and in what follows, for η > 0 small, we write fi (η) = oi (1) if limη→0 fi (η) = 0 uniformly in i, and fi (η) = oi (η∞ ) whenever (fi (η)/ηp ) = oi (1) for all p > 1. Furthermore, since P(supk≤ 1 |(

sufficient to show, for some r > 0,

η

𝜕φkη −1 ) | 𝜕x

>

1 ) η

= ok (η∞ ) (cf. [143, (1.11)]), it is

P{K(k+1)η (v)̄ − Kkη (v)̄ < ηr | ℱkη } = ov,k (1)

(1.27)

for (k + 1)η ≤ 1. Here v̄ = |( 𝜕xkη )−1 |−1 v. To this end, it is enough to show 𝜕φ

(k+1)η

P{[ ∫ ds1(η−r1 ,∞) ( ∫ dνs (k, η)(u))] ≤ ηr2 | ℱkη } = ov,k (1)

(1.28)

|u|>ηr

kη

for some r1 > 0, r2 > 0, r1 > r2 suitably chosen. Here, dνs (u) = dνs (k, η)(u) is the Lévy measure of ΔKs given ℱkη for s > kη (i. e., ds × dνs (u) is the compensator of ΔKs with respect to dP). Indeed, since t

Yt = exp[ ∑ 1(ηr ,∞) (ΔKs ) − ∫ ds ∫ (e−1 − 1)dνs (u)] kη≤s≤t

kη

|u|>ηr

2.1 Integration-by-parts on Poisson space

� 59

is a martingale (given ℱkη ), we have P{

∑

kη≤s≤(k+1)η

1(ηr ,∞) (ΔKs ) = 0 | ℱkη } (k+1)η

≤ E[exp[ ∫ ds ∫ (e−1 − 1)dνs (u)] |u|>ηr

kη (k+1)η

: {ω; [ ∫ ds1(η−r1 ,∞) ( ∫ dνs (u))] > ηr2 } | ℱkη ] |u|>ηr

kη

(k+1)η

+ E[exp[ ∫ ds ∫ (e−1 − 1)dνs (u)] |u|>ηr

kη (k+1)η

: {ω; [ ∫ ds1(η−r1 ,∞) ( ∫ dνs (u))] ≤ ηr2 } | ℱkη ]. |u|>ηr

kη

By (1.28), the right-hand side is inferior to (k+1)η

ok (1) + E[exp[ ∫ ds ∫ (e−1 − 1)dνs (u)] kη

|u|>ηr

(k+1)η

: {ω; ∫ ds1(η−r1 ,∞) ( ∫ dνs (u)) > ηr2 } | ℱkη ] |u|>ηr

kη

≤ ov,k (1) + exp[(e−1 − 1)(η−r1 × ηr2 )] = ov,k (1) since ηr2 −r1 → ∞ as η → 0. Hence, we have (1.27). The proof of (1.28). For each vector field X, we put the criterion processes Cr(s, X, v, k, η) ≡ ⟨(

−1

𝜕φs ̄ ) X(xs− (x)), v⟩, 𝜕x

d

󵄨 󵄨 Cr(s, v, k, η) ≡ ∑󵄨󵄨󵄨Cr(s, Xj , v, k, η)󵄨󵄨󵄨 j=1

for s ∈ [kη, (k + 1)η]. By dνs′ (X, v, k, η) we denote the Lévy measure of Cr(s, X, v, k, η). To get (1.28), it is sufficient to show that for the given η > 0, there exist integers n = n(η), n1 = n1 (η) such that

60 � 2 Perturbations and properties of the probability law (k+1)η

P{ ∫ ds1(ηn ,∞) (Cr(s, v, k, η)) < ηn1 | ℱkη } = ov,k (1).

(1.29)

kη

Indeed, consider the event {Cr(s, v, k, η) ≥ c > 0, s ∈ [kη, (k + 1)η]}. Then, we can show on this event ∫ dνs (k, η)(u) ≥ Cη−αr/2

(1.30)

|u|>ηr

for η > 0 small. Here, α > 0 is what appeared in the definition of g(z). This is because dνt (k, η)(u) is the sum of transformed measures of gj (z) by the mapping 2

̄ . Fj,t : z 󳨃→ ν(z)⟨φ∗−1 t Xj (x), v⟩ Since ν(z) is equal to |z|2 in a neighborhood of 0, ∞

∫ ∫ dνt (k, η)(u) ≥ C1 ∫ gj (z)dz |z|>ηr

ηr/2

for each j. By the definition of g(z), the right-hand side is equal to ∞

C1 ∫ |z|−1−α dz > Cη−αr/2 . ηr/2

Hence, we can choose r > 0 such that Cη−αr/2 ≥ η−r1 for η small, and thus (k+1)η

∫ ds1(η−r1 ,∞) ( ∫ dνs (k, η)(u)) ≥ ηr2

kη

(1.31)

|u|>ηr

for some r1 > 0, r2 > r1 on this event. Following (1.29), the probability of the complement of this event is small (= oe,k (1)). Hence, (1.28) follows. To show (1.29), note that it is equivalent to (k+1)η

P{ ∫ ds1(ηn ,∞) (Cr(s, v, k, η)) ≥ ηn1 | ℱkη } = 1 − ov,k (1). kη

This is proved similarly to [94, Section 3], and we omit the details.

(1.32)

2.1 Integration-by-parts on Poisson space

� 61

Remark 2.2. According to [143, Section 3], the result (1.28) follows under a weaker condition (URH) (cf. Section 2.5.1). If Yt = xt (x) is a canonical process in Section 2.5.1, we do not need the condition (A.2). 2.1.2 Picard’s method Picard’s method is also a way to make a variation of the jump intensity, however, it is more general. That is, in the previous method for the jump size perturbation, the key assumption is that the Lévy measure μ has a density g which is differentiable. In case that this assumption fails (e. g., the cases of the discrete Lévy measures), we have to use a completely different method for the perturbation. Picard [178] has introduced a method of perturbation using the difference operator based on the previous work by Nualart–Vives [170]. We put φ(ρ) = ∫|z|≤ρ |z|2 μ(dz), ρ > 0, which is supposed to satisfy the order condition φ(ρ) ≥ cρα

as ρ → 0

(∗)

for some c > 0 and α ∈ (0, 2). Here, we remark, contrary to the Bismut’s case, the Lévy measure can be singular with respect to the Lebesgue measure. Remark 2.3. The order condition is a nondegeneracy condition for the Lévy measure. It is a basic requirement in the stochastic calculus for jump processes by Picard’s method. The quantity 2 − α is called the characteristic exponent. The power α describes the order of concentration of masses or ‘weight’ of μ around a small neighborhood of the origin. Indeed, suppose we extend μ to μ̄ = aδ{0} + μ on Rm , a ≥ 0, and we interpret a > 0 ̄ ̄ as though the continuous (Wiener) component is nonempty. Let φ(ρ) = ∫|z|≤ρ |z|2 μ(dz). ̄ Then, the condition (∗) holds for φ(ρ) with α = 0. On the other hand, in case the order condition (∗) holds for α less than 2, we remark that the condition √φ(ρ) →∞ ρ

as ρ → 0

holds, which means the mass of μ is so concentrated around small neighborhoods of {0} (origin) that φ(ρ) > cρ2 as ρ → 0 holds. We write t

̃ zρ (t) = ∫ ∫ zN(dsdz) 0 |z|ρ

In the sequel, we will study a stochastic calculus of variations (Malliavin calculus) on the Poisson space (Ω2 , ℱ2 , P2 ). We work on Ω2 for a moment, and will denote by ω elements of Ω2 in place of ω2 . ̂ ̂ ̃ ̂ In the sequel, we set u = (t, z) and N(du) = N(dtdz), N(du) = N(du) − N(du). Here, μ is m 2 a Lévy measure on R such that μ({0}) = 0 and ∫Rm (|z| ∧ 1)μ(dz) < +∞. On the measurable space (𝕌 × Ω2 , 𝒰 ⊗ ℱ2 ), we introduce the measures μ+ , μ− by μ+ (dudω) = N(ω, du)P(dω), ̂ μ− (dudω) = N(ω, du)P(dω). Let |μ| = μ+ + μ− . Next, we shall introduce the difference operator D̃ u , u ∈ 𝕌, acting on the Poisson space as follows. For each u = (t, z) = (t, z1 , . . . , zm ) ∈ 𝕌, we define a map εu− : Ω2 → Ω2 by εu− ω(E) = ω(E ∩ {u}c ), and εu+ : Ω2 → Ω2 by

2.1 Integration-by-parts on Poisson space

� 63

εu+ ω(E) = ω(E ∩ {u}c ) + 1E (u). Here, ω(E) denotes the value of the random measure on the set E. We write εu± ω = ω ∘ εu± , respectively. These operations are extended to Ω by setting εu± (ω1 , ω) = (ω1 , εu± ω). We observe if

u1 ≠ u2

then εuθ11 ∘ εuθ22 = εuθ22 ∘ εuθ11 ,

θ1 , θ2 ∈ {+, −}

and εuθ1 ∘ εuθ2 = εuθ1 ,

θ1 , θ2 ∈ {+, −}.

̂ It holds that εu− ω = ω a. s. P2 for N-almost all u since ω({u}) = 0 holds for almost all + ̂ ω for N-almost all u. Also, we have εu ω = ω a. s. P2 for N-almost all u. In what follows, we denote a random field indexed by u ∈ 𝕌 by Zu . Let ℐ ⊂ 𝒰 × ℱ2 be the sub-σ-field generated by the set A × E, A ∈ 𝒰 , E ∈ ℱAc . Here ℱB for B ∈ 𝕌 denotes the σ-field on Ω2 generated by N(C), C ⊂ B. We remark that for Zu positive, Zu ∘εu+ and Zu ∘εu− are ℐ -measurable by the definition ± of εu . We cite the following property of the operators εu+ , εu− . Lemma 2.4 ([179, Corollary 1]). Let Zu be a positive, μ± -integrable random field. Then, we have ̂ E[∫ Zu N(du)] = E[∫ Zu ∘ εu+ N(du)], ̂ = E[∫ Zu ∘ εu− N(du)]. E[∫ Zu N(du)] The difference operators D̃ u for a ℱ2 -measurable random variable F is defined by D̃ u F = F ∘ εu+ − F.

(1.33)

Since the image of P2 by εu+ is not absolutely continuous with respect to P2 , D̃ u is not well ̂ defined for a fixed u. However, it is defined by N(du)⊗P 2 -almost surely due to Lemma 2.4. Since D̃ u is a difference operator, it enjoys the property D̃ u (FG) = F D̃ u G + GD̃ u F + D̃ u F D̃ u G, assuming that the left-hand side is finite a. s. The next proposition is a key to the integration-by-parts (duality) formula which appears below. Proposition 2.1 ([179, Theorem 2]). Let Zu1 , Zu2 be processes such that Zu1 D̃ u Zu2 and Zu2 D̃ u Zu1 are |μ|-integrable. Then, we have

64 � 2 Perturbations and properties of the probability law ̃ ̃ ̂ E[∫ Zu1 D̃ u Zu2 N(du)] = E[∫ Zu2 D̃ u Zu1 N(du)] = E[∫ D̃ u Zu1 D̃ u Zu2 N(du)]. Proof. We see ̃ ̂ E[∫ Zu1 D̃ u Zu2 N(du)] = E[∫ Zu1 D̃ u Zu2 N(du)] − E[∫ Zu1 D̃ u Zu2 N(du)] ̂ ̂ = E[∫(Zu1 ∘ εu+ )D̃ u Zu2 N(du)] − E[∫ Zu1 D̃ u Zu2 N(du)] ̂ = E[∫ D̃ u Zu1 D̃ u Zu2 N(du)], due to Lemma 2.4. If we interchange Zu1 and Zu2 , we have the other equality. This leads to the following proposition. Proposition 2.2 ([179, Corollary 5]). Let F be a bounded variable, and let Zu be a random field which is μ± -integrable. Then, we have (1)

̃ E[∫ Zu D̃ u FN(du)] = E[F ∫(Zu ∘ εu+ )N(du)],

(2)

̂ ̃ = E[F ∫(Zu ∘ εu− )N(du)]. E[∫ Zu D̃ u F N(du)]

In particular, if Zu is ℐ -measurable, ̂ ̃ E[∫ Zu D̃ u F N(du)] = E[F ∫ Zu N(du)]. Proof. Assertion (2) is obtained from the previous proposition by putting Zu1 = F and Zu2 = Zu N({u}). Assertion (1) is obtained similarly by putting Zu1 = Zu , Zu2 = F and from assertion (2). The adjoint δ̃ of the operator D̃ = (D̃ u )u∈𝕌 is defined as follows. We denote by 𝒮 the set of the random fields Zu which are ℐ -measurable, bounded, and of compact support.1 Let 2

𝒮 ̄ = {Z ∈ L (𝕌 × Ω2 , ℐ , μ ); there exist Zn ∈ 𝒮 , Zn → Z in ‖ ⋅ ‖}. −

Here, the norm ‖ ⋅ ‖ is given temporarily by 2

̂ ̃ ‖Z‖2 = E[∫ |Zu |2 N(du)] + E[(∫ Zu N(du)) ].

1 Meaning it is zero outside of a set which is of finite μ− measure.

2.1 Integration-by-parts on Poisson space

� 65

For Z ∈ 𝒮 , we define δ0 (Z) by ̂ E[Fδ0 (Z)] = E[∫ Zu D̃ u F N(du)] as an adjoint operator of D̃ u , where F is bounded and of compact support. The last as̃ sertion of Proposition 2.1 implies that the mapping Z 󳨃→ ∫ Zu N(du) can be extended to ̄ ̄ the element in 𝒮 . Hence, we define δ0 (Z) as above for Z ∈ 𝒮 . Let Zu ∈ L2 (𝕌 × Ω2 , 𝒰 ⊗ ℱ2 , μ− ) be such that ‖Z‖ < +∞. We denote by 𝒵 the set of all such random fields Zu . We observe Zu ∘ εu− ∈ 𝒮 .̄ Hence, we put ̃ δ(Z) = δ0 (Zu ∘ εu− ) to extend δ0 to δ.̃ In view of the last assertion of Proposition 2.2, we are ready to define the adjoint operator. We set for Z ∈ 𝒵 , ̃ ̃ δ(Z) = ∫ Zu ∘ εu− N(du).

(1.34)

𝕌

By Proposition 2.2 and the definition of δ,̃ the operator satisfies the adjointness property ̃ ̂ E[F δ(Z)] = E[∫ D̃ u FZu N(du)]

(1.35)

𝕌

for any bounded ℱ2 -measurable random variable F [178, (1.12)]. We also have the commutation relation ̃ D̃ u δ(Z) = δ(̃ D̃ u Z) + Zu ,

a. e. n(du)dP2 .

Remark 2.4. The origin of the operators seems to be what studied in connection with quantum mechanics by Mecke [156], and Meyer [158, 159]. Operators εu+ , εu− are originally εt+ , εt− , respectively, used in Carlen-Pardoux [47], Nualart–Vives [170], and J. Picard [179], + − where εt+ = ε(t,1) and εt− = ε(t,1) . A main point in those days was to show the duality formula using the operator Dt given by Dt F = F ∘ εt+ − F ∘ εt− . Nualart–Vives [171] showed the duality formula (1.35) for Dt by using the chaos decomposition on the Poisson space. See also Lemma 1.4 of Picard [178]. N. Privault has repeatedly treated the basic problems concerning the operators ε+ , ε− in [183, 185]. We will discuss this topic in Section 3.2.

66 � 2 Perturbations and properties of the probability law We have the Meyer-type inequality (Theorem 3.4 in Section 3.3), and the operator δ̃ is well defined on the Sobolev spaces D0,k,p , D0,k,p;A , D∗0,k,p for k ≥ 0 and p ≥ 2. This implies D̃ u is closable for p ≥ 2. (For the Sobolev space, Dl,k,p , Dl,k,p;A , D∗l,k,p over the Wiener– Poisson space, see Section 3.3.) In the remaining part of this subsection, we shall briefly sketch how to derive the integration-by-parts setting using these operators. We introduce a linear map Q̃ ρ by Q̃ ρ Y =

1 ̂ ∫ (D̃ u F)D̃ u Y N(du). φ(ρ)

(1.36)

A(ρ)

Lemma 2.5. The adjoint of Q̃ ρ exists and is equal to ̃ T X), Q̃ ρ∗ X = δ̃ ρ ((DF)

(1.37)

where δ̃ ρ (Z) =

1 ̃ 1 ̃ δ(Z1A(ρ) ) = ∫ Zu ∘ εu− N(du). φ(ρ) φ(ρ)

(1.38)

A(ρ)

̃ T denotes the transpose of DF. ̃ Here A(ρ) = {u = (t, z) ∈ 𝕌; |z| ≤ ρ}, and (DF) Let f (x) be a C 2 -function with bounded derivatives. We claim a modified formula of integration-by-parts. Concerning the difference operator D̃ u , we have, by the meanvalue theorem, 1

D̃ u (f (G)) = (D̃ u G)T ∫ 𝜕f (G + θD̃ u G)dθ

(1.39)

0

for a random variable G on the Poisson space. This implies 1

Q̃ ρ f (F) = R̃ ρ 𝜕f (F) +

1 ̂ ∫ D̃ u F(D̃ u F)T (∫{𝜕f (F + θD̃ u F) − 𝜕f (F)}dθ)N(du). φ(ρ)

(1.40)

0

A(ρ)

Here, R̃ ρ =

1 ̂ ∫ D̃ u F(D̃ u F)T N(du). φ(ρ)

(1.41)

A(ρ)

Use (1.40) and then take the inner product of this with Sρ X. Its expectation yields the following.

2.1 Integration-by-parts on Poisson space

� 67

Proposition 2.3 (Analogue of the formula of integration by parts, [100]). For any X, we have E[(X, 𝜕f (F))] = E[Q̃ ρ∗ (Sρ X)f (F)] 1

1 ̂ E[(X, Sρ ∫ D̃ u F(D̃ u F)T (∫{𝜕f (F + θD̃ u F) − 𝜕f (F)}dθ)N(du))]. − φ(ρ) A(ρ)

0

(1.42)

Here, Sρ = R̃ −1 ρ . Remark 2.5. If F is just a Wiener functional, then the formula is written shortly as E[(X, 𝜕f (F))] = E[Q∗ (R−1 X)f (F)] = E[δ((R−1 X, DF))f (F)].

(1.43)

See [100] for details. A similar operator to Q̃ ρ appears (but in simpler form) in Section 3.5 as QF (v) on the Wiener–Poisson space (Proposition 3.6). On the other hand, if R̃ ρ is not zero, or equivalently Q̃ ρ is not zero, we have a remaining term (the last term of (1.42)). We have this term even if Zt is a simple Poisson process Nt or its sums. However, if we take f (x) = ei(w,x) , w ∈ Rd \ {0}, then 𝜕f (x) = iei(w,x) w and ̃ ei(w,F+θDu F) − ei(w,F) = ei(1−θ)(w,F) D̃ u (ei(w,θF) ).

Hence, we have an expression of the integration-by-parts for the functional E[(X, w)𝜕x (ei(w,F) )] = E[(Q̃ ρ∗ + R∗ρ,w )Sρ X ⋅ ei(w,F) ],

∀w.

(1.44)

Here, 1

R∗ρ,w Y

i ̃ i(1−θ)(w,F) χ DF( ̃ DF) ̃ T Y ), ei(θ−1)(w,F) w)dθ. =− ∫(δ(e ρ φ(ρ)

(1.45)

0

We will continue the analysis using this method in Section 3.3.

2.1.3 Some previous methods Apart from perturbation methods by Bismut and Picard (stated in Sections 2.1.1, 2.1.2) on the Poisson space, there are previously examined perturbation methods which may seem intuitively more comprehensive. We will briefly recall two of them in this section.

68 � 2 Perturbations and properties of the probability law Bismut–Graczyk method Apart from the Bismut method in Section 2.1.1, a very naive integration-by-parts formula based on Bismut’s idea (perturbation on the jump size and the direction) can be carried out. This is described by P. Graczyk [42, 74] in details. In effect, Graczyk has applied this method on a nilpotent group G. Here, we adapt his method to the case G = Rd for simplicity. A key point of this method is that we can explicitly describe the probability law (and the existence of the density) of the jumps caused by the Lévy process by using the integration-by-parts formula, where it does not use the Girsanov formula. Let m = d. We consider a jump process given by the stochastic differential equation (SDE) d

dxt (x) = ∑ Xj dzj (t),

x0 (x) = x.

j=1

Here, Xj′ s denote d-dimensional vectors in Rd which constitute a linear basis, and zj (t)’s are scalar Lévy processes. That is, xt (x) = Az(t) + x,

x0 (x) = x.

Here, A is a d × d-matrix A = (X1 , . . . , Xd ), and z(t) = (z1 (t), . . . , zd (t)). By taking a linear transformation, we may assume A is a unitary matrix without loss of generality. We choose a positive function ρ ∈ C ∞ , 0 ≤ ρ ≤ 1 such that ρ(z) = 1 if |z| ≥ 1, ρ(z) = 0 if |z| ≤ 21 . We put ρη (z) ≡ ρ( ηz ), η > 0. We would like to have the integration-by-parts formula E[f (α) (xT (x))] = E[f (xT (x)) ⋅ MT(α) ],

α = 1, 2, . . . , f ∈ C ∞ .

This can be done by elementary calculus as below. We put T = 1 for simplicity. In fact, Graczyk [74] has studied a kind of stable process on a homogeneous group. Viewing Rd as a special case of a nilpotent group, we mimic his calculation to illustrate the idea in the simple case. (Step 1) Let ν(z) be a C 2 -function such that ν(z) ∼ z2 for |z| small. Choose ϵ > 0 and fix it. Let Njϵ (dsdz) be the Poisson counting measure with mean measure ds × g ϵ (z)dz ≡ ds × ρ2ϵ (z)g(z)dz.

(1.46)

2.1 Integration-by-parts on Poisson space

� 69

s

We write zϵj (s) = ∫0 ∫ zNjϵ (dsdz), j = 1, . . . , d. We denote by N = N ϵ ≡ N1ϵ + ⋅ ⋅ ⋅ + Ndϵ where 1

Njϵ = ∫0 ∫ Njϵ (dsdz), and by S1 , S2 , . . . , the jump times of N ϵ . Let ⟨e1 , . . . , ed ⟩ be a standard basis of Rd , and let ⟨ē1 , . . . , ēd ⟩ = ⟨Ae1 , . . . , Aed ⟩. Let xsϵ be the solution of the SDE d

dxsϵ = ∑ Xj dzϵj (s), j=1

x0ϵ ≡ x.

(1.47)

First, we integrate by parts with respect to the process {xsϵ }. We shall calculate the expectation of d

⟨Df (x1ϵ ), e⟩ = ∑ Dr f (x1ϵ )er r=1

𝜕f )i=1,...,d below. Here and in the following calculation, we use Df and Dr f in the sense ( 𝜕x

and

i

𝜕f , 𝜕xr

respectively. Since |Δzϵj (s)| > ϵ uniformly, we have d

d

d

E[ρη ( ∑ ∑ ∑ ν(Δzϵj (s))⟨Xj , er′ ⟩2 ) × ∑ ⟨Df (x1ϵ ), er ⟩] r ′ =1 s≤1 j=1

r=1

d

∞

n

d

d

= ∑ P(N = n)E[ρη ( ∑ ∑ ∑ ν(Δzϵj (Sk ))⟨Xj , er′ ⟩2 ) ∑ ⟨Df (x1ϵ ), er ⟩]. n=1

r=1

r ′ =1 k=1 j=1

We introduce the following notations to simplify the argument: d

d

(X ∗ ν)(1) = ∑ ∑ ∑ ν(Δzϵj (s))⟨Xj , er′ ⟩2 , r ′ =1 s≤1 j=1 d

n

d

X ∗ ν = ∑ ∑ ∑ ν(Δzϵj (Si ))⟨Xj , er′ ⟩2 , r ′ =1 i=1 j=1 d

d

(X ∗ ν)k = ∑ ∑ ν(Δzϵj (Sk ))⟨Xj , er′ ⟩2 . r ′ =1 j=1

Then, X ∗ ν = ∑nk=1 (X ∗ ν)k , and d

d

∑ ∑ ∑ ν(Δzϵj (Si ))⟨Xj , er′ ⟩2 = ∑ (X ∗ ν)i .

̸ j=1 r ′ =1 i=k

i=k ̸

In the right-hand side of the above formula, we freeze the variable Δzϵj (sk ) and view it as zℓ ∈ supp g ϵ , showing that

70 � 2 Perturbations and properties of the probability law n

d

E[⋅ ⋅ ⋅] = ∑ ∑ E[( k=1 r=1 n

d

d

ρη (X ∗ ν) X ∗ν

= ∑ ∑ ∑ E[∫( k=1 r=1 ℓ=1 d

)(X ∗ ν)k ⟨Df (x1ϵ ), er ⟩]

ρη (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 ) ∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2

× { ∑ ν(zℓ )⟨Xℓ , er′ ⟩2 }⟨Df (x1ϵ ), er ⟩ × r ′ =1

)

g ϵ (zℓ ) dzℓ ]. ∫ g ϵ (zℓ )dzℓ

Here, we integrate by parts on Rd with respect to zℓ , viewing x1ϵ as a functional of z = (z1 , . . . , zd ). To this end, we compare Df (x1ϵ (z)) = 𝜕x1ϵ 𝜕z

𝜕x1ϵ

(z). Here 𝜕z (z) = A if |zi | > ϵ, i = 1, . . . , d. Then, we have ∫ ⋅ ⋅ ⋅ dzℓ = − ∫[ −

×

𝜕f 𝜕 (x ϵ (z)) with 𝜕z (f 𝜕x 1

∘ x1ϵ )(z) = Df (x1ϵ (z)) ⋅

ρ′η (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 ) (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 )

ρη (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 ) (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 )2

d 𝜕 { ∑ ν(zℓ )⟨Xℓ , ēr′ ⟩2 } 𝜕zℓ r′ =1 d

× { ∑ ν(zℓ )⟨Xℓ , er′ ⟩2 }f (x1ϵ ) r ′ =1

− ∫( d

×∑

]

g ϵ (zℓ ) dzℓ ∫ g ϵ (zℓ )dzℓ

ρη (∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2 ) ∑i=k̸ (X ∗ ν)i + ∑dr′ =1 ν(zℓ )⟨Xℓ , er′ ⟩2

∇zℓ {ν(zℓ )⟨Xℓ , ēr′ ⟩2 ⋅ g ϵ (zℓ )} g ϵ (z

r ′ =1

ℓ)

f (x1ϵ )

)

g ϵ (zℓ ) dzℓ . ∫ g ϵ (zℓ )dzℓ

(1.48)

We suppress below the superscript ϵ for simplicity. Note that we have ∇zℓ {ν(zℓ )⟨Xℓ , ēr′ ⟩2 ⋅ g (zℓ )} = ∇zℓ {ν(zℓ ) ⋅ g (zℓ )} × ⟨Xℓ , ēr′ ⟩2 .

(1.49)

We put Φ0 =

ρη ((X ∗ ν) (1))

Φ1 = [

, (X ∗ ν) (1) ρ′η ((X ∗ ν) (1)) ((X ∗ ν) (1))

Then, we have the following expression:

(1.50) −

ρη ((X ∗ ν) (1)) ((X ∗ ν) (1))2

].

(1.51)

2.1 Integration-by-parts on Poisson space

� 71

d

E[ρη ((X ∗ ν) (1)) × ∑ ⟨Df (x1ϵ ), er ⟩] r=1

N

d

d

𝜕 {ν(zℓ )⟨Xℓ , ēr ⟩2 }(Δzℓ (Sk )) 𝜕z ℓ k=1 ℓ=1 r=1

= −E[Φ1 { ∑ ∑ ∑ d

× { ∑ ν(Δzℓ (Sk ))⟨Xℓ , er′ ⟩2 }f (x1ϵ )}] r ′ =1

N

d

d

− E[Φ0 { ∑ ∑ ∑ {(

∇zℓ {ν(zℓ ) ⋅ g(zℓ )} g(zℓ )

k=1 ℓ=1 r=1

)(Δzℓ (Sk )) × ⟨Xℓ , ēr ⟩2 }f (x1ϵ )}].

We put (cf. (1.48), (1.49)) d

𝜕 {ν(z)⟨Xℓ , ēr ⟩2 }(z), 𝜕z r=1

ψ1,ℓ (z) = ∑ ψ2 (z) =

∇z {ν(z) ⋅ g(z)} . g(z)

(1.52)

Then, 󵄨󵄨 󵄨󵄨 d 󵄨󵄨 󵄨 󵄨󵄨E[ρη ((X ∗ ν) (1)) ∑ ⟨Df (x ϵ ), er ⟩]󵄨󵄨󵄨 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 r=1 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 2 󵄨󵄨󵄨 󵄨 ≤ ‖f ‖∞ {E[󵄨󵄨Φ1 ∑ ∑ ψ1,ℓ (Δzℓ (s)){ ∑ ν(Δzℓ (s))⟨Xℓ , er′ ⟩ }󵄨󵄨] 󵄨󵄨 s≤1 ℓ=1 󵄨󵄨 󵄨 󵄨 r ′ =1 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 󵄨󵄨 + E[󵄨󵄨󵄨Φ0 ∑ ∑ ∑ {ψ2 (Δzℓ (s))⟨Xℓ , ēr ⟩2 }󵄨󵄨󵄨]}. 󵄨󵄨 s≤1 ℓ=1 r=1 󵄨󵄨 󵄨 󵄨 Hence, we can let ϵ → 0 to obtain d

E[ρη ((X ∗ ν) (1)) ∑ ⟨Df (x1 ), er ⟩] r=1

d

d

= −E[Φ1 (∑ ∑ ψ1,ℓ (Δzℓ (s)) × ( ∑ ν(Δzℓ (s))⟨Xℓ , er′ ⟩2 ))f (x1 )] s≤1 ℓ=1

d

r ′ =1

d

− E[ ∑ Φ0 (∑ ∑ ψ2 (Δzℓ (s)) × ⟨Xℓ , ēr ⟩2 )f (x1 )]. r=1

s≤1 ℓ=1

(1.53)

In the left-hand side of (1.53), we can show [(X ∗ ν) (1)]

−1

∈ Lp ,

p ≥ 1.

(1.54)

The proof proceeds similarly to that of Lemma 2.3, and we omit the details. Indeed, we consider the form

72 � 2 Perturbations and properties of the probability law d

d

t 󳨃→ ∑ ∑ ∑ ν(Δzϵj (s))⟨Xj , er′ ⟩2 r ′ =1 s≤t j=1

instead of Kt (p), and the idea of the proof is the same as in Section 2.1.1. (Step 2, η → 0) By the latter expression, we have 󵄨󵄨 󵄨󵄨 d 󵄨󵄨 󵄨 󵄨󵄨E[ρη ((X ∗ ν) (1)) ∑ ⟨Df (x1 ), er ⟩]󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 r=1 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 󵄨󵄨 ≤ ‖f ‖∞ {E[󵄨󵄨󵄨Φ1 ∑ ∑ ψ1,ℓ (Δzℓ (s)){ ∑ ν(Δzℓ (s))⟨Xℓ , er′ ⟩2 }󵄨󵄨󵄨] 󵄨󵄨 s≤1 ℓ=1 󵄨󵄨 󵄨 󵄨 r ′ =1 󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 󵄨 󵄨 + E[󵄨󵄨󵄨Φ0 {∑ ∑ ∑ {ψ2 (Δzℓ (s))⟨Xℓ , ēr ⟩2 }}󵄨󵄨󵄨]}, 󵄨󵄨󵄨 󵄨󵄨󵄨 s≤1 ℓ=1 r=1 and the right-hand side is independent of η > 0. By (1.54), d

d

{ ∑ ∑ ∑ ν(Δzj (s))⟨⋅ ⋅ ⋅⟩2 } > 0 r ′ =1 s≤1 j=1

a. e.

Hence, by letting η → 0, we have d

d

d

d

r=1

r=1

E[ρη ( ∑ ∑ ∑ ν(Δzj (s))⟨Xj , er′ ⟩2 ) ∑ ⟨Df (x1 ), er ⟩] → E[ ∑ ⟨Df (x1 ), er ⟩] in Lp . r ′ =1 s≤1 j=1

(1.55)

On the other hand, d

∑ ∑ ψ1,ℓ (Δzℓ (s))⟨Xℓ , er′ ⟩2 ∈ Lp ,

s≤1 ℓ=1

d

∑ ∑ ψ2 (Δzℓ (s))⟨Xℓ , ēr ⟩2 ∈ Lp ,

s≤1 ℓ=1

p > 1,

and

p>1

since ν(⋅) and ν′ (⋅) are C ∞ and have compact support. Hence, we can let η → 0 in the above as before, and we have d

E[ ∑ ⟨Df (x1 ), er ⟩] = E[f (x1 )ℋ1 ], r=1

where

(1.56)

2.1 Integration-by-parts on Poisson space d

� 73

d

× ∑ ∑ ψ1,ℓ (Δzℓ (s)) ∑ ⟨Xℓ , er′ ⟩2

ℋ1 = [(X ∗ ν) (1)]

−2

s≤1 ℓ=1

− [(X ∗ ν) (1)]

−1

d

d

r ′ =1

× ∑ ∑ ∑ ψ2 (Δzℓ (s))⟨Xℓ , ēr ⟩2 . s≤1 ℓ=1 r=1

We have thus an integration-by-parts formula of order 1. Calculation for the integration-by-parts of order 2 is similar, and we omit the details. We can proceed to the higher-order calculation in a similar way, and have integrationby-parts formula of arbitrary order. Time shifts Consider a one-dimensional standard Poisson process Nt and denote one of its jump times (jump moment) by T. We consider the shift of the jump times of Nt . This works in principle since the inter-jump times of the Poisson process have an exponential law which has smooth density. We can compare this method with the previous ones by Bismut which make shifts with respect to the jump size (and direction) of the driving process. The method was first studied by E. Carlen and E. Pardoux [47] in 1990, and subsequently (and presumably, independently) by R. J. Elliot and A. H. Tsoi [60] in 1993. Elliot and Tsoi have obtained the following integration-by-parts formula: T

E[

𝜕G (T) ∫ xs ds] = −E[(∑ xs ΔNs )G(T)], 𝜕t s≥0 0

where G(t) denotes some suitable function and T denotes a jump time. Note that the right-hand side = −E[δ(u)G(T)] in the contemporary notation. Subsequently, V. Bally et al. [15] have proceeded along this idea. Namely, let T1 , T2 , . . . be the successive jump times of Nt and let F = f (T1 , . . . , Tn ) be a functional of N. The gradient operator D = Dt is given by n

Dt F = ∑ i=1

𝜕 f (T1 , . . . , Tn ) ⋅ 1{Ti >t} . 𝜕xi

(1.57)

This gradient operator is a differential operator and enjoys the chain rule. This is a main point of the operator, compared to the difference operator D̃ u which appeared in Section 2.1.2. However, by applying D again to both sides of (1.57), D2 F would contain the term D1{Ti >t} , which is singular. This is the price one pays for enjoying the chain rule. As a result, there are some problems with respect to showing the smoothness of the density. Precisely, since we cannot expect the iterated use of the operator, we do not apply the higher-order derivatives at the same point of time within this theory.

74 � 2 Perturbations and properties of the probability law In Bally et al. [15, 16], to avoid the iterated use of the operation by D, a more moderate expression of the gradient operator using the approximation procedure is adopted. We introduce it below. We might have to endure some intricate expressions instead. Let us introduce dumping functions (πi ) which cancel the effect of singularities coming from D2 F, D3 F, . . . in the manner of (1.57). Namely, let ai , bi , i = 1, 2, . . . , m be random variables on Ω2 such that −∞ ≤ ai < bi ≤ +∞, and consider weight functions (πi ) such that 0 ≤ πi ≤ 1, πi is smooth on (ai , bi ), πi (ai ) = πi (bi ) = 0, and that πi > 0 only on (ai , bi ), i = 1, 2, . . . , m. We remark that the intervals (ai , bi ) are not necessarily distinct. For F = f (T1 , . . . , Tm ), we give the dumped derivative by Dπi F = πi ⋅ 𝜕i f (T1 , . . . , Tm ).

(1.58)

Let Dπ F = (Dπi F)i=1,...,m , and we call them π derivatives. Then, we can use Dπ F in place of DF in (1.57) (viewed at t = Ti ). One advantage of this notation for Dπ is that it is a differential operator, and thus it still satisfies the chain rule Dπi φ(F) = φ′ (F) ⋅ Dπi F

(1.59)

for F = f (T1 , . . . , Tm ). The other advantage is that, due to the dumping effect of πi at each edge of (ai , bi ), we can obtain the integration-by-parts formula: for Fm = f (T1 , . . . , Tm ), E[𝜕β g(Fm )] = E[g(Fm )Hβm ],

β = 1, 2, . . . ,

(1.60)

where Hβm is a random variable defined by using Dπ . More precisely, consider a functional F = f ((Ti )) on Ω2 of the form ∞

f ((Ti )) = ∑ f j (T1 , . . . , Tj )1{J(ω)=j} . j=1

(1.61)

Here, J(ω) is an integer-valued random variable on Ω2 . A functional F of such form with some f ∈ Cπ∞ is called a simple functional. The set of simple functionals is denoted by S. Here, we put Cπk = {f ; f admits partial π derivatives up to order k on (ai , bi ) with respect to each variable ti }

k and Cπ∞ = ⋂∞ k=1 Cπ . A sequence U = (Ui ) of simple functionals Ui ∈ S is called a simple process. The set of simple processes is denoted by 𝒫 in this subsection. The adjoint operator δ of Dπ is defined for U ∈ 𝒫 by

δi (U) = −(𝜕i (πi Ui ) + Ui 1{pJ >0} 𝜕iπ log pJ )

(1.62)

2.1 Integration-by-parts on Poisson space

� 75

and J

δ(U) = ∑ δi (U). i=1

Here, pJ denotes the density of the law of (T1 , . . . , Tj ) with respect to the Lebesgue measure on {J = j}. We assume {pJ > 0} is open in RJ and pJ ∈ Cπ∞ on {pJ > 0}. We remark that πi is a localization factor (weight), and thus it is expected that j

⋂ supp πi ⊂ supp pj i=1

on {J = j}.

Then, we have Proposition 2.4. Under the above assumption, E[(Dπ F, U)] = E[Fδ(U)]

(1.63)

for F ∈ S, U ∈ 𝒫 . See [16, Proposition 1]. Repeating this calculus and using the chain rule (1.59), we can reach the formula (1.60) under the assumption that 󵄨 󵄨p E[󵄨󵄨󵄨γ(F)󵄨󵄨󵄨 ] < +∞

for all p ≥ 1.

Here, γ(F) = σ −1 (F) J

with σ(F) = ∑j=1 (Dπj F)2 . In particular, H1m = δ(γ(F)Dπ F) for β = 1. The formula (1.60) then gives 󵄨󵄨 ̂ 󵄨 −β 󵄨 m 󵄨 󵄨󵄨pFm (v)󵄨󵄨󵄨 ≤ |v| E[󵄨󵄨󵄨Hβ 󵄨󵄨󵄨],

β = 1, 2, . . . ,

by choosing g(x) = ei(x,v) . Instead of showing directly by letting Fm → F in L2 (Ω2 ) as m → +∞ in the above, we would be able to show 󵄨 󵄨 sup E[󵄨󵄨󵄨Hβm 󵄨󵄨󵄨] < +∞ m

with a suitable choice of (πi ). This then implies 󵄨󵄨 ̂ 󵄨 2 −β/2 , 󵄨󵄨pF (v)󵄨󵄨󵄨 ≤ C(1 + |v| ) as desired.

β = 1, 2, . . . ,

(1.64)

76 � 2 Perturbations and properties of the probability law The time stratification method by A. Kulik [122, 123] can also be regarded as a variation of the time shift method. See also Fournier [65]. One benefit of the time shift method would be that one can avoid the strict nondegeneracy condition of the Lévy measure such as the order condition (∗) given in Section 2.1.2. There exist cases where the Lévy measure is scarce around the center (origin) where perturbation described above can not be applied. In such a case a specially designed method may be necessary. For example, we have a case where the density exists but is irregular. Let d = 1 and we consider the case Xt is a solution of SDE t

Xt = X0 + ∫ AXs ds + Z(t), 0

where A is a nonzero constant and Z(t) is a Lévy process. 1 k k Example 2.1 ([31, Example 2)]). Let Xt = ∑∞ k=1 k! Nt where (Nt ) are independent copies of a Poisson process. Then the density pt (x, y) of Xt exists but for every t and x, y 󳨃→ pt (x, y) is not in Lp , p > 1.

We remark that Z(t) = f ((T1 , T2 , . . .)) is a Lévy process with Lévy measure ∑∞ k=1 δ{ k!1 } , which has singular distribution. This may reflect a limitation of the method in case of scarce Lévy measure (low intensity for small jumps). In such a case, in order to guarantee the existence of the density, one can use the perturbation with respect to time, and also invoke the drift term to compensate the low intensity for small jumps for the Lévy measure [31, Proposition 2]. For full generality, we introduce an SDE in Rd , ̃ t− )dz(t), dXt = b(Xt )dt + g(X

X0 = x ∈ Rd .

̃ are Lipschitz-continuous vector- and matrix-valued functions, respecHere b(x), g(x) tively, and z(t) is a Lévy process given by t

t

̃ z(t) = ∫ ∫ zN(dsdz) + ∫ ∫ zN(dsdz), 0 |z|≤1

0 |z|>1

where ∫|z|≤1 |z|μ(dz) < ∞. We introduce a matrix ̃ ̃ B(x) = ∇b(x)g(x) − ∇g(x)b(x), which is assumed to be nondegenerate.

x ∈ Rd ,

(1.65)

2.2 Methods of finding the asymptotic bounds (I)

� 77

For l ∈ Rm \ {0} and ϵ > 0, we put Vl,ρ = {z : ⟨z, l⟩ ≥ ρ|z||l|} for every ρ ∈ (0, 1). Then we have Proposition 2.5 ([125, Proposition 3.4.7]). We assume the matrices B(x) are nondegenerate, and μ(Vl,ρ ∩ {|z| ≤ ϵ}) > 0,

ρ ∈ (0, 1)

(1.66)

for any l ∈ Rm \ {0} and ϵ > 0. Then Xt satisfies the local Dobrushin condition. The local Dobrushin condition (a mixing condition) is closely related to the existence of the density (cf. [124, Section 1, Theorem 1.3]). We remark that the matrices B(x) play a similar role as the series of Lie brackets in Section 3.6.5 spanned by jump and drift terms. Historical sketch In a process of inquiring into the Fock space representation of the Poisson space, Nualart and Vives [170] have introduced the translation operator Ψt (ω) = ω + δ{t} ,

(1.67)

where ω is a generic element in the Poisson space. This operation can be viewed as adding a new jump time at t to ω associated to N. . This operator induces a translation of the functionals by Ψt (F)(ω) = F(ω + δ{t} ) − F(ω),

(1.68)

which has led to the difference operator D̃ u by Picard [178]; refer to Section 2.1.2. Background of considering the time perturbation (time shift) method would be then in physical models. Nowadays we feel no difficulty in considering a perturbation in time-space (t, z), but the ideas have evolved gradually as time went by.

2.2 Methods of finding the asymptotic bounds (I) In this section and the next, we recall some results stated in Sections 2.1.1, 2.1.2 and, assuming the existence of a transition density pt (x, y) for the process Xt , we explain various ideas of finding the asymptotic bounds of the density function pt (x, y) as t → 0 based on methods employed in the proof. Here, Xt denotes a solution of some SDE. The analysis to obtain the existence and smoothness of the density function is carried out throughout Chapter 3.

78 � 2 Perturbations and properties of the probability law The estimate is closely related, as will be seen in proofs, with fine properties of the trajectories in the short time, and thus with the perturbation methods stated in the previous section. Also, from an analytical point of view, it provides fine properties of the fundamental solution in the short time associated with the integro-differential equation. In this section, we construct Markov chains which approximate the trajectory of Xt . Throughout this and the next sections, we confine ourselves to the case of the Itô process Xt = xt (x) given in Section 1.3. In the proof of Theorem 2.3(b) below, we introduce a supplementary process xs (r, St , x) which is obtained from xs (x) by directing the times (moments) of big jumps of xs (x) during [0, t]. We will encounter this again in Section 3.6.4. Our method stated in this and in the next section is not effective to yield the largetime (t > 1) estimate of the density, and we cannot obtain sharp results in such a case. We do not mention the large-time case in this book.

2.2.1 Markov chain approximation Let z(s) be an Rm -valued Lévy process with Lévy measure μ(dz), such that ∫Rm (|z|2 ∧ 1)μ(dz) < +∞. That is, the characteristic function ψt is given by ψt (v) = E[ei(v,z(t)) ] = exp(it(v, c) + t ∫(ei(v,z) − 1 − i(v, z)1{|z|≤1} )μ(dz)). s

We may write z(s) = cs + ∫0 ∫Rm \{0} z(N(dudz) − du1{|z|≤1} ⋅ μ(dz)), where N is a Poisson

random measure with mean du × μ(dz). Let γ(x, z) : Rd × Rm → Rd and b(x) : Rd → Rd be C ∞ -functions, whose derivatives of all orders are bounded, satisfying γ(x, 0) = 0. We carry out our study in the probability space (Ω, (ℱt )t≥0 , P), where Ω = D([0, T]) (Skorohood space, space D in Section 1.1.3), (ℱt )t≥0 is the filtration generated by z(s), and P is a probability measure on ω of z(s). That is, μ(dz) = P(z(s + ds) − z(s) ∈ dz|z(s))/ds. Consider the following SDE studied in Section 1.3: t

c

xt (x) = x + ∫ b(xs (x))ds + ∑ γ(xs− (x), Δz(s)). s≤t

0

(2.1)

Equivalently, we may write xt (x) as t

t

̃ xt (x) = x + ∫ b′ (xs (x))ds + ∫ ∫ γ(xs− (x), z)N(dsdz) 0 t

0 |z|≤1

+ ∫ ∫ γ(xs− (x), z)N(dsdz), 0 |z|>1

(2.2)

2.2 Methods of finding the asymptotic bounds (I)

� 79

̃ where Ñ denotes the compensated Poisson random measure: N(dsdz) = N(dsdz) − ′ dsμ(dz), b (x) = b(x) − ∫|z|≥1 γ(x, z)μ(dz), where the integrability of γ(x, z) with respect to 1{|z|≥1} ⋅ dμ(z) is assumed. We remark that 𝜕γ ̃ z) (x, 0)z + γ(x, 𝜕z

γ(x, z) =

(2.3)

̃ z) such that for some γ(x, 󵄨󵄨 󵄨󵄨 𝜕 k 󵄨󵄨 ̃ z)󵄨󵄨󵄨󵄨 ≤ Ck |z|α 󵄨󵄨( ) γ(x, 󵄨󵄨 𝜕x 󵄨󵄨 for some α > β ∨ 1 and for k ∈ Nd on {|z| ≤ 1}. Here β > 0 is what appeared in (A.0). Throughout this section, we assume the following four assumptions: (A.0)–(A.2) in Section 1.3, and (A.3), namely (3-a)If 0 < β < 1, we assume that c = ∫|z|≤1 zμ(dz), b = 0 and for all u ∈ S d−1 , ∫ ⟨z, u⟩2 1{⟨z,u⟩>0} (z)μ(dz)≍ρ2−β

as ρ → 0.

(2.4)

{|z|≤ρ}

(3-b)If β = 1, then 󵄨󵄨 󵄨 lim sup󵄨󵄨󵄨 ϵ→0 󵄨󵄨

∫

{ϵ1

and that equation (2.6) has a unique solution due to [127, Theorem 3.1]. Furthermore, due to the invertibility assumption (A.2), there exists a stochastic flow of diffeomorphisms denoted by ϕs,t (s < t) : Rd → Rd such that xt (x) = ϕs,t (xs (x)), which is invertible ([142, Section 1], [24]; see also [189, Theorem V.65] for the simple case γ(x, z) = X(x)z). We cite the following basic result due to Picard [181] using the perturbation stated in Section 2.1.2.

80 � 2 Perturbations and properties of the probability law Proposition 2.6 ([181, Theorem 1]). Under the conditions (A.0)–(A.3), xt (x) has a Cb∞ density for each t > 0, which we denote by y 󳨃→ pt (x, y). Methodologies leading to the existence of the smooth density under various conditions have much developed following this result. We shall go a bit into detail in Sections 3.5, 3.6. The following result is an extension of Theorem 1 in [181] to the above mentioned process. Theorem 2.2 (General upper bound). The density pt (x, y) satisfies the following estimate: (a)

sup pt (x, y) ≤ C0 t x,y

pt (x, x) ≍ t (b)

− βd

− dβ

as t → 0, for some C0 > 0,

(2.7)

as t → 0 uniformly in x.

(2.8)

For all k ∈ Nd , there exists Ck > 0 such that − |k|+d 󵄨 󵄨󵄨 β sup󵄨󵄨󵄨p(k) t (x, y)󵄨󵄨 ≤ Ck t x,y

as t → 0,

(2.9)

where p(k) denotes the kth derivative with respect to y. Below, we give a refinement of this theorem (Theorem 2.3) and provide the proof (Section 2.2.2). We can give examples that the supremum in (a) is in fact attained on the diagonal {x = y} (diagonal estimate): pt (x, x)≍t

− dβ

as t → 0

uniformly in x. It is well known in the case of a one-dimensional symmetric stable process with index β = α that the density function satisfies pt (0) ∼ Ct −1/α as t → 0 [92, Section 2-4]. The estimate above is more exact than that by Hoh–Jacob [81] using functionalanalytic methods: there exist C > 0, ν ∈ (1, +∞) such that sup pt (x, y) ≤ Ct −ν x,y

as t → 0.

Below, we construct a Markov chain associated to xt (x). Let ν(dz) be the probability measure on Rd given by ν(dz) =

(|z|2 ∧ 1)μ(dz) . ∫(|z|2 ∧ 1)μ(dz)

(2.10)

Then, dν ∼ dμ (μ and ν are mutually absolutely continuous), and the Radon–Nikodým dν derivative dμ is globally bounded from above.

2.2 Methods of finding the asymptotic bounds (I)

� 81

d+m×n Consider a series of functions (An )∞ → Rd defined by A0 (x) = x n=0 , An : R and An+1 (x, x1 , . . . , xn+1 ) = An (x, x1 , . . . , xn ) + γ(An (x, x1 , . . . , xn ), xn+1 ). Fix x ∈ Rd . We put 𝒮n to be the support of the image measure of μ⊗n by the mapping (x1 , . . . , xn ) 󳨃→ An (x, x1 , . . . , xn ), and 𝒮 ≡ ⋃n 𝒮n .

Definition 2.2 (Accessible points). Points in 𝒮 , regarded as points in Rd , are called accessible points. Points in 𝒮 ̄ \ 𝒮 are called asymptotically accessible points. Intuitively, accessible points are those points which can be reached by xt (x) by only using a finite number of jumps of z(s). We remark that 𝒮 is not necessarily closed, although each 𝒮n is. We define for each x the mapping Hx : supp μ → Px ≡ x + {γ(x, z); z ∈ supp μ} by z 󳨃→ x + γ(x, z). Let Px(n) = {y ∈ Pzn−1 ; z1 ∈ Px , zi ∈ Pzi−1 , i = 2, . . . , n − 1}, n = 1, 2, . . . , (z0 = x). Then, Px(1) = Px , and Px(n) can be interpreted as points which can be reached from x by n jumps along the trajectory xt (x). Given x, y ∈ Rd (y ≠ x), let α(x, y) be the minimum number l such that y ∈ Px(l) if such l exists, and put α(x, y) = +∞ otherwise. Or equivalently, α(x, y) = inf{n; y ∈ ⋃k≤n 𝒮k }. To be more precise, we introduce a concrete ‘singular’ Lévy measure of z(s) which has already been described in [207, Example 3.7]. Let μ(dz) = ∑∞ n=0 kn δ{an } (dz) be an m-dimensional Lévy measure such that (an ; n ∈ N) and (kn ; n ∈ N) are sequences of points in Rm and real numbers, respectively, satisfying (i) |an | decreases to 0 as n → +∞, (ii) kn > 0, ∑i≥n ki |ai |2 ≥ cnα (α ∈ (0, 2)), 2 (iii) ∑∞ n=0 kn |an | < +∞. For this Lévy measure, we can show the unique existence of the solution xt (x) of (2.2) (see [207, Theorems 1.1, 2.1]), and the existence of the density under the assumptions (A.0)–(A.3). We further assume that 1 N = N(t) ≡ max{n; |an | > t 1/β }≍ log( ). t

(2.11)

Here, since |an | → 0 as n → ∞, β > 0 is an index which indicates the speed of concentration of masses in supp μ around the origin, |an | > t 1/β . Hence it is related to the order condition (∗) in Section 2.1.2. Part (b) of the next theorem can be viewed as an extension of Proposition 5.1 in [180]. Theorem 2.3 ([95, Theorem 2]). Assume μ is given as above. Let y ≠ x. (a) Assume y ∈ 𝒮 , that is, α(x, y) < +∞. Then, we have pt (x, y)≍t α(x,y)−d/β

as t → 0.

(2.12)

82 � 2 Perturbations and properties of the probability law (b) Assume y ∈ 𝒮 ̄ \ 𝒮 (α(x, y) = +∞). Suppose b(x) ≡ 0 and let β′ > β. Then, log pt (x, y) is bounded from above by the expression of type Γ = Γ(t), N

Γ ≡ − min ∑ (wn log( n=0

1 1 1 ) + log(wn !)) + O(log( ) log log( )) tkn t t

(2.13)

as t → 0. Here, the minimum is taken with respect to all choices of a0 , . . . , aN by ξn for n = 1, 2, . . . , n1 and n1 ∈ N such that 󵄨󵄨 󵄨 1/β′ 󵄨󵄨y − An1 (x, ξ1 , . . . , ξn1 )󵄨󵄨󵄨 ≤ t ,

(2.14)

where wn = # of an in the choice and n1 = ∑Nn=0 wn . We remark that in finding the above minimum, conditions (2.11) and (2.14) work complementarily. That is, as t > 0 gets smaller, (2.14) becomes apparently more strict, whereas we may use more combinations of ai ’s to approximate y due to (2.11). Since β′ > β, condition (2.14) does not prevent An1 (x, ξ1 , . . . , ξn1 ) from attaining y as t → 0, using a0 , . . . , aN under (2.11). We also remark that if x and an are rational points (e. g., m = ̄ [0, 1]) relative to the 1, x = 0, an = 2−n ), then the result (b) holds for almost all y ∈ 𝒮 (= Lebesgue measure, whereas for t > 0, y 󳨃→ pt (x, y) is smooth on 𝒮 ̄ due to Proposition 2.6. 2.2.2 Proof of Theorem 2.3 In this subsection, we prove Theorem 2.3. The main idea is to replace the Lévy measure μ by another probability measure ν which is absolutely continuous with respect to μ, and consider another process x̃. of pure jump (bounded variation) whose Lévy measure is proportional to ν. The Markov chain which approximates the trajectory of x. is constructed on the basis of x̃. . First, we prove two lemmas which are essential for our estimation. These lemmas are inspired by Picard [181]. Lemma 2.6 (Lower bound for accessible points). Let (ξn )n∈N be the Rd -valued independent and identically distributed random variables obeying the probability law ν(dz) given by (2.10), independent of z(s). We define a Markov chain (Un )n∈N by U0 = x and Un+1 = Un + γ(Un , ξn+1 ), n ∈ N. Assume that for y ∈ Rd , there exist some n ≥ 1, γ = γn ≥ 0 and c > 0 such that for all ϵ ∈ (0, 1], P(|Un − y| ≤ ϵ) ≥ cϵγ . Then, we have pt (x, y) ≥ Ct n+(γ−d)/β

as t → 0.

(2.15)

We notice that the lower bound on the right-hand side depends on (n, γn ). Put g(x, dz) = d(Hx∗ ν)(z), z ∈ Px \ {x}, where Hx∗ ν = ν ∘ Hx−1 . Then, we have an expression of the probability above:

2.2 Methods of finding the asymptotic bounds (I)

P(|Un − y| ≤ ϵ) = ∫ ⋅ ⋅ ⋅ ∫ 1{zn ;|zn −y|≤ϵ} (zn )g(x, dz1 )⋅ ⋅ ⋅g(zn−1 , dzn ). Px

� 83

(2.16)

Pzn−1

Hence, the condition P(|Un − y| ≤ ϵ) ≥ cϵγ implies that y can be attained with the singular Lévy measure (dim supp ν = 0) if γ = 0. In order to obtain the upper bound, we introduce the perturbation of the chain. Let (φn )n∈N be a sequence of smooth functions, Rd → Rd . We define another Markov chain (Vn )n∈N by V0 = φ0 (x) and Vn+1 = Vn + (φn+1 ∘ γ)(Vn , ξn+1 ). Furthermore, we define a sequence of real numbers (Φn )n∈N by Φn ≡

󵄨 󵄨 󵄨 󵄨 sup (󵄨󵄨󵄨φk (y) − y󵄨󵄨󵄨 + 󵄨󵄨󵄨φ′k (y) − I 󵄨󵄨󵄨). d

k≤n,y∈R

(2.17)

Under this setup, we have: Lemma 2.7 (Upper bound). Choose y ≠ x. Assume there exist a sequence (γn )n∈N , γn ∈ [0, +∞], and a nondecreasing sequence (Kn )n∈N , Kn > 0, such that the following condition holds true for each n and for any (φk )nk=0 satisfying Φn ≤ Kn : Vn defined as above satisfies with some Cn > 0 that if

γn < +∞,

then

P(|Vn − y| ≤ ϵ) ≤ Cn ϵγn

if

γn = +∞,

then

P(|Vn − y| ≤ ϵ) = 0

for all ϵ > 0,

(2.18)

for ϵ > 0 small.

(2.19)

and

Furthermore, we put Γ ≡ minn (n + (γn − d)/β). Then we have the following assertions: (i) If Γ < +∞, then pt (x, y) = O(t Γ ) as t → 0. (ii) If Γ = +∞, then for any n ∈ N pt (x, y) = o(t n ) as t → 0. Note that Γ depends implicitly on the choice of (Kn ). Whereas, for each n, the bigger the γn , the better the upper bound. Given the perturbations (φn ), we define Qx(0) = {φ(x)} and the sequence (Qx(n) ) successively by Qx(n) ≡ {zn−1 + (φn ∘ γ)(zn−1 , z); z ∈ supp ν, zn−1 ∈ Qx(n−1) } for n = 1, 2, . . . Hence, the set Qx(n) can be interpreted as the points which can be reached from x by Vn , and we have P(|Vn − y| ≤ ϵ) = ∫ ⋅ ⋅ ⋅ Pφ(x)

∫ Pφn−1 (zn−1 )

1{zn ;|φn (zn )−y|≤ϵ} (zn )g(φ(x), dz1 )⋅ ⋅ ⋅g(φn−1 (zn−1 ), dzn ) (2.20)

84 � 2 Perturbations and properties of the probability law for n = 0, 1, 2, . . . We remark that if y ∉ Px(n) (n ≥ 1), then by choosing φn such that the size Φn of perturbations is small enough, we can let y ∉ Qx(n) . Therefore, by choosing Kn > 0 small and ϵ > 0 small, γn may be +∞ in the above. The proofs of Lemmas 2.6, 2.7, and that of Lemma 2.8 (below), will be provided in the next subsection. Proof of statement (a) of Theorem 2.3 For n ≥ α(x, y), we have P(|Un − y| ≤ ϵ) ≥ c for ϵ ∈ (0, 1] since μ has point masses. Hence, pt (x, y) ≥ Ct α(x,y)−d/β by Lemma 2.6. For the upper bound, if n < α(x, y), then by choosing Kn small, we have y ∉ Qx(n) . That is, P(|Vn − y| ≤ ϵ) = 0

for ϵ > 0 small,

and we may choose γn = +∞ in (2.18) and (2.19). On the other hand, if n ≥ α(x, y), then we must choose γn = 0 in (2.18) (φn = id must satisfy it). Hence, we may choose Γ = α(x, y) − d/β in the conclusion. These imply pt (x, y)≍t α(x,y)−d/β .

(2.21)

Proof of the statement (b) of Theorem 2.3 We set 𝒮t,k = ([0, t]k / ∼) and 𝒮t = ∐k≥0 𝒮t,k , where ∐k≥0 denotes the disjoint sum and ∼ means the identification of the coordinates on the product space [0, t]k by permutation. Let r = r(t) = t 1/β . We denote by z̃r (s) the Lévy process having the Lévy measure μ ⋅ 1{|z|>r} (z). That is, z̃r (s) = ∑u≤s Δz(u)1{|Δz(u)|>r} . The distribution P̃ t,r of the moments (instants) of jumps related to z̃r (s) during [0, t] is given by k

∫ {#St =k}

1 f (St )d P̃ t,r (St ) = {(t ∫ 1{|z|>r} (z)μ(dz)) ( ) exp (−t ∫ 1{|z|>r} (z)μ(dz))} k! t

t

0

0

1 × k ∫ ⋅ ⋅ ⋅ ∫ f (s1 , . . . , sk )ds1 ⋅ ⋅ ⋅ dsk , t

(2.22)

where f is a function on 𝒮t,k (a symmetric function on [0, t]k ). Given St ∈ 𝒮t , we introduce the process xs (r, St , x) as the solution of the following SDE:

2.2 Methods of finding the asymptotic bounds (I)

� 85

s

xs (r, St , x) = x − ∫ du ∫ 1{|z|>r} (z)γ(xu (r, St , x), z)μr (dz) 0 c

+ ∑ γ(xu− (r, St , x), Δzr (u)) + u≤s

∑

si ∈St ,si ≤s

γ(xsi − (r, St , x), ξir ),

(2.23)

where (ξnr )n∈N denotes a sequence of independent and identically distributed random variables obeying the probability law μr (dz) =

1{|z|>r} (z) ⋅ μ(dz)

∫ 1{|z|>r} (z) ⋅ μ(dz)

.

We remark that xs (r, St , x) is a martingale for each 0 < r < 1 due to the assumpr tion b(x) ≡ 0. We define a new Markov chain (Unr )n∈N by U0r = x and Un+1 = Unr + r γ(Unr , ξn+1 ), n ∈ N. We can prove, due to Proposition 2.6, that under (A.0)–(A.3), the law of xs (r, St , x) for (ds-a. e.) s > 0 has a Cb∞ -density denoted by ps (r, St , x, y). Indeed, let 0 ≤ s ≤ t. In the case St = 0, we can choose 1{|z|≤r} ⋅ μ(dz) for the measure μ(dz) in Proposition 2.6. Hence, we have the existence of the density for the law ps (r, 0, x, dy) of xs (r, 0, x) (= xs (r, St , x)|St =0 ), ps (r, 0, x, dy) = ps (r, 0, x, y)dy. Next, we consider the general case. Since z̃r (s) and zr (s) are independent, we have by the Markov property that the law ps (r, St , x, dy) ≡ P(xs (r, St , x) ∈ dy) of xs (r, St , x) is represented by ps (r, St , x, dy) = ∫ dz′0 ∫ ps1 (r, 0, x, z′0 )gr (z′0 , dz1 ) ∫ dz′1 ∫ ps2 −s1 (r, 0, z1 , z′1 )gr (z′1 , dz2 ) Pz′

Pz′

1

0

⋅ ⋅ ⋅ ∫ dz′n1 −1

∫ Pz′

psn −sn −1 (r, 0, zn1 −1 , z′n1 −1 )gr (z′n1 −1 , dzn1 )pt−sn (r, 0, zn1 , dy) 1 1 1

n1 −1

if St ∈ 𝒮t,n1 . Here, gr (x, dz) = P(x + γ(x, ξir ) ∈ dz). On the other hand, once again, we have by the independence of z̃r (s) and zr (s), ps (x, dy) = ∫ ps (r, St , x, dy)d P̃ t,r (St ), 𝒮t

using the factorization of the measure (disintegration of measure) N (cf. [111, p. 71]). By Proposition 2.6, the left-hand side has a density ps (x, y) with respect to dy. Hence, ps (r, St , x, dy) is absolutely continuous with respect to dy (d P̃ t,r -a. s.). Hence, we have, by differentiating under the integral sign,

86 � 2 Perturbations and properties of the probability law ps (x, y) = ∫ (ps (r, St , x, dy)/dy)(y)d P̃ t,r (St ). 𝒮t

We denote by ps (r, St , x, y) the derivative ps (r, St , x, dy)/dy(y) which is defined uniquely d P̃ t,r ⊗ dy-a. e. (Since d P̃ t,r |𝒮 is the uniform distribution on 𝒮t,k , ps (r, St , x, y) is defined t,k

uniquely as ds ⊗ dy-a. e.) Since y 󳨃→ ps (x, y) is smooth, so is y 󳨃→ ps (r, St , x, y) ds-a. s., and hence ps (r, St , x, y) is defined as a smooth density ds-a. e. Thus, by taking s = t, ∞

pt (x, y) = ∫ pt (r, St , x, y)d P̃ t,r (St ) = ∑ pt (k, r, x, y), k=0

𝒮t

(2.24)

where pt (k, r, x, y) = ∫ pt (r, St , x, y)d P̃ t,r (St ). 𝒮t,k

Hence, pt (x, y)dy = E Pt,r [P(xt (r, St , x) ∈ dy)] ̃

r ⊗#St

= E Pt,r E (μ ) ̃

r [P(xt (r, St , x) ∈ dy|St , ξ1r , . . . , ξ#S )]. t

(2.25)

For each St ∈ 𝒮t , we put n1 = k if St ∈ 𝒮t,k . We then have r ⊗#St

P(xt (r, St , x) ∈ dy|St ) = E (μ )

′ 󵄨 󵄨 [P(xt (r, St , x) ∈ dy : 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β |St , ξ1r , . . . , ξnr1 )]

r ⊗#St

+ E (μ )

′ 󵄨 󵄨 [P(xt (r, St , x) ∈ dy : 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 > t 1/β |St , ξ1r , . . . , ξnr1 )]. (2.26)

First, we have to compute ′ r ⊗n1 ′ 󵄨 󵄨 󵄨 󵄨 P(󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β ) ≡ E (μ ) [P(󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β |ξ1r , . . . , ξnr1 )]

for a given y = xt (r, St , x) with St ∈ 𝒮t,n1 . We denote by the random variable Wn the n1 number of an in (ξir )i=1 . Given n1 ∈ N, let (wn )Nn=0 and wn ∈ N ∪ {0} be a sequence of integers such that n1 = ∑Nn=0 wn . We then have N ′ 1 󵄨 󵄨 (tkn )wn e−tkn , P(for all n ≤ N, Wn = wn and 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β ) ≤ ∏ w ! n n=0

since each Wn is a Poisson random variable with mean tkn . Hence,

2.2 Methods of finding the asymptotic bounds (I)

� 87

󵄨 󵄨 log P(for all n ≤ N, Wn = wn and 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β ) ′

N

≤ − ∑ (wn log(1/(tkn )) + log(wn !) + tkn ) n=0 N

= − ∑ (wn log(1/(tkn )) + log(wn !)) + O(tN).

(2.27)

n=0

We introduce the set 𝒲 of all (wn )Nn=0 such that for some n1 ∈ N, Unr1 directed by the

following condition (∗) satisfies |y − Unr1 | ≤ t 1/β , that is, ′

(∗) wn = (# of an ’s which appear in ξ1r , . . . , ξnr1 ) and n1 = ∑Nn=1 wn . Then, from (2.27), it follows that log P (

there exists (wn ) ∈ 𝒲 such that for all n ≤ N, N

Wn = wn and |y − Unr1 | ≤ t 1/β

′

)

≤ − min ∑ (wn log(1/(tkn )) + log(wn !)) + O(log |𝒲 |). 𝒲

n=0

(2.28)

On the other hand, we have 3

󵄨 󵄨 P(Wn ≥ N 3 and 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β ) ≤ Ce−N ≤ Ce−c(log(1/t)) ′

3

(2.29)

since Wn is a Poisson variable with mean tkn . Since this is very small relative to the probability above, we may put the restriction wn ≤ N 3 and n1 ≤ N 4 . Hence, we may write |𝒲 | = O((N 3 )N ), and log |𝒲 | = O(N log N) = O(log(1/t) log log(1/t)). Let xt (r, 0, x) denote the process defined by (2.23) with St = 0, and pt (r, 0, x, y) as its density. By Theorem 2.2(a), pt (r, 0, x, y) ≤ Ct −d/β as t → 0, and this implies, for a given St ∈ 𝒮t,n1 , ξ1r , . . . , ξnr1 , that P(xt (r, St , x) ∈ dy|St , ξ1r , . . . , ξnr1 )/dy = O(t −d/β ) as t → 0. Since zr (u) and (ξir ) are independent, by (2.24) and Fubini’s theorem, we have log(E

P̃ t,r |𝒮t,n

1

r ⊗n

1 󵄨 󵄨 E (μ ) [P(xt (r, St , x) ∈ dy : 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 ≤ t 1/β |St , ξ1r , . . . , ξnr1 )]/dy) ′

N

≤ log(t −d/β exp(− min ∑ (wn log(1/(tkn )) + log(wn !)) + O(log(1/t) log log(1/t)))) 𝒲

N

n=0

≤ − min ∑ (wn log(1/(tkn )) + log(wn !)) + O(log(1/t) log log(1/t)). 𝒲

n=0

(2.30)

For the second term of (2.26), we have the following lemma, whose proof is given later.

88 � 2 Perturbations and properties of the probability law Lemma 2.8. Given y = xt (r, St , x), St ∈ 𝒮t,n1 and Unr1 = An1 (x, ξ1r , . . . , ξnr1 ), there exist k > 0 and C0 > 0 such that for every p > k, E

P̃ t,r |𝒮t,n

1

r ⊗n1 ′ 2 󵄨 󵄨 E (μ ) [P(󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 > t 1/β |St , ξ1r , . . . , ξnr1 )] ≤ n1 C0 exp[−(p − k)(log(1/t)) ]

as t → 0. Given ξ1r , . . . , ξnr1 , we have, as above, E

P̃ t,r |𝒮t,n

1

[P(xt (r, St , x) ∈ dy|St , ξ1r , . . . , ξnr1 )/dy] = O(t −d/β ) as t → 0.

We integrate this with respect to (μr )⊗n1 on {|y − Unr1 |> t 1/β }. Since zr (u) and (ξir ) are independent, by Lemma 2.8, we then have ′

E

P̃ t,r |𝒮t,n

1

r ⊗n1 ′ 󵄨 󵄨 E (μ ) [P(xt (r, St , x) ∈ dy : 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 > t 1/β |St , ξ1r , . . . , ξnr1 )]/dy

2

≤ n1 C0′ t −d/β exp[−(p − k)(log(1/t)) ]

as t → 0.

We get log(E

P̃ t,r |𝒮t,n

1

r ⊗n1 ′ 󵄨 󵄨 E (μ ) [P(xt (r, St , x) ∈ dy : 󵄨󵄨󵄨y − Unr1 󵄨󵄨󵄨 > t 1/β |St , ξ1r , . . . , ξnr1 )]/dy)

2

≤ −(p − k)(log(1/t)) + max log n1 + log C0′ + O(log(1/t)) 2

n1 ≤N 4

≤ −(p − k)(log(1/t)) + O(log(1/t)) since maxn1 ≤N 4 log n1 = log N 4 = O(log(1/t)). Since p > k is arbitrary, this can be neglected in view of the right-hand side of (2.30) and (2.26). After summing up E

log(E

P̃ t,r |⋃

k≤N 4

𝒮t,k

P̃ t,r |𝒮t,n

r ⊗#St

E (μ )

N

1

r ⊗n1

E (μ ) [. . .] with respect to n1 = 0, . . . , N 4 , we get

r [P(xt (r, St , x) ∈ dy|St , ξ1r , . . . , ξ#S )]/dy) t

≤ − min ∑ (wn log(1/(tkn )) + log(wn !)) + O(log(1/t) log log(1/t)) + O(log(1/t)). 𝒲

n=0

(2.31)

In view of (2.25), (2.31), N

log pt (x, y) ≤ − min ∑ (wn log(1/(tkn )) + log(wn !)) + O(log(1/t) log log(1/t)). 𝒲

n=0

(2.32)

Since there is no difference between the trajectories of the deterministic chain An1 (x, ξ1 , . . . , ξn1 ) and Unr1 obtained by using {an ; n = 0, . . . , N} under (2.11), we have the assertion.

2.2 Methods of finding the asymptotic bounds (I)

� 89

In Section 2.5.3, we will see the image of (Markov) chain approximations using planar chains. 2.2.3 Proof of lemmas In this subsection, we give proofs of Lemmas 2.6, 2.7, and 2.8 which appeared in the previous subsection. (A) Proof of Lemma 2.6 To show Lemma 2.6, we prepare several sublemmas. We recall that μ satisfies (A.0). As stated above, we introduce a probability measure ν by ν(dz) =

(|z|2 ∧ 1)μ(dz) . ∫(|z|2 ∧ 1)μ(dz)

dν . Then, μ ∼ ν. Let 1/c0 be the upper bound of dμ We decompose z(s) into an independent sum

̃ + Z(s), z(s) = z(s) ̃ is a Lévy process with the Lévy measure c0 ν which is a pure jump process, where z(s) and Z(s) is a Lévy process with the Lévy measure μ − c0 ν. Let N = N(t) the number of jumps of z̃ before time t of size bigger than r, and let r = r(t) = t 1/β . We further decompose Z(s) into an independent sum Z(s) = Z r (s) + Z̃ r (s) as Z̃ r (s) = ∑ ΔZ(u)1{|ΔZ(u)|>r} u≤s

and Z r (s) = Z(s) − Z̃ r (s). Then, z,̃ Z r , Z̃ r are chosen to be independent processes, making z = z̃ + Z r + Z̃ r . We introduce three processes xsr (x), xs(r) (x), x̃s (x) by using (3.14) in Chapter 1, and replacing z(⋅) with Z r , z̃ + Z r , z,̃ respectively. Let Γ = {ω; Z̃sr ≡ 0, s ∈ [0, t]}. By (A.0), we have P(Γ) > ct > 0

90 � 2 Perturbations and properties of the probability law by using the small deviations property ([96], see also Section 2.5.4). We observe xt = xt(r) on Γ. Hence, it is sufficient to estimate the density of xt(r) at y for the lower bound. Here, we give three sublemmas. Sublemma 2.1 ([181, Lemma 3]). Let h > 0 and k ∈ Nd . Denote by pr,t (x, y) the density of xtr (x). Then, |p(k) (x, y)| ≤ Ckh r −(|k|+d) for 0 < r ≤ 1. r,r β h 1 r x (x). Then x̄hr (x) satisfies x̄hr (x) = r rβ h h ∫0 br (x̄hr ′ (x))dh′ +∑ch′ ≤h γr (x̄hr ′ − (x), Δz̄r (h′ ))+ r1 x, where z̄r (t) ≡ r1 Z r (r β t), br (y) ≡ r β−1 b(ry), and γr (x, ζ ) ≡ r1 γ(rx, rζ ). Assumption (A.3)(3-b) is used here to guarantee that the scaled drift parameter cr = r β−1 (c − ∫{r 0 and r > 0. We put x̄hr (x) ≡

Also γr (x, ζ ) satisfies the assumptions (A.0)–(A.3) uniformly in r. Hence, by Proposition 2.6, (x̄hr (x); r > 0) have Cb∞ -densities p̄ h ( xr , y : r) which are uniformly bounded with respect to r > 0. Hence, by the above relation, pr,rβ h ( xr , y : r) = r1d p̄ h ( xr , yr : r). This im-

plies |pr,rβ h ( xr , y : r)| ≤ Ch r −d . Estimates for the derivatives with respect to y follow by differentiating the equality. Sublemma 2.2 ([181, Lemma 8]). There exists a matrix-valued function A such that c

̃ xt(r) (x) = x̃t (x) + ∑ A(s, z(⋅)) s≤t

1 𝜕 γ(x̃s (x), 0)ΔZsr + o(t β ) 𝜕z

(2.33)

in case β ≠ 1, and c

̃ xt(r) (x) = x̃t (x) + ∑ A(s, z(⋅))( s≤t

1 𝜕 γ(x̃ (x), 0)ΔZsr + b(x̃s (x))ds) + o(t β ) 𝜕z s

(2.34)

in case β = 1. The function A(s, z)̃ and its inverse are bounded by some C(n) on {N = n}. Proof. (Idea of proof.) We prove the case β ≠ 1. Let N = N(t) be the number of jumps of z̃ before time t. In order to see the behavior of zr − z̃ on {N = n}, we see the case for small values of n. For n = 0, z̃s = 0, x̃s = x for s ≤ t and xs(r) is reduced to the solution of the equation driven by Z r . We can then have xt(r) = x + γ1 (x)Ztr + o(t 1/β ), and the assertion is obtained with A = I. For n = 1, let T denote the time of jumps of z̃ in [0, t].

(2.35)

2.2 Methods of finding the asymptotic bounds (I)

� 91

On the intervals where z̃ does not jump, we can apply (2.35) to consider the semigroup by xt(r) conditioned on z,̃ which coincides with the semigroup of the Z r -driven process. In particular, on [0, T), we obtain (r) xT− = x + γ1 (x)ZTr + o(t 1/β ) = x + O(t 1/β ).

At time T, we have x̃T = x + γ(x, Δx̃T ) and (r) (r) xT(r) = xT− + γ(xT− , Δx̃T ) = x + γ1 (x)ZTr + γ(x + γ1 (x)ZTr , Δx̃T ) + o(t 1/β ).

Hence xT(r) = x̃T + (I +

𝜕γ (x, Δx̃T ))γ1 (x)ZTr + o(t 1/β ). 𝜕x

In particular, xT(r) − x̃T is of order o(t 1/β ). On [T, t), we have, as on [0, T), xt(r) = xT(r) + γ1 (xT(r) )(Ztr − ZTr ) + o(t 1/β ) = xT(r) + γ1 (x̃T )(Ztr − ZTr ) + o(t 1/β ) and x̃t = x̃T . Hence xt(r) = x̃t + (I +

𝜕γ (x, Δx̃T ))γ1 (x)ZTr + γ1 (xT(r) )(Ztr − ZTr ) + o(t 1/β ). 𝜕x

Hence the assertion holds for A(s, z)̃ = (I +

𝜕γ (x, Δx̃T ))1{s≤T} + I ⋅ 1{s>T} . 𝜕x

Generally, if Ti , i = 1, . . . , N, are the times of jumps of z̃ on [0, t] where T0 = 0, TN+1 = t, we have the assertion for A(s, z)̃ = (I +

𝜕γ 𝜕γ (x̃ , Δx̃TN )) ⋅ ⋅ ⋅ (I + (x̃Ti x, Δx̃Ti+1 )) 𝜕x TN−1 𝜕x

on {Ti < s < Ti+1 }, i = 0, 1, . . . , N − 1, and A(s, z)̃ = I if i = N.

92 � 2 Perturbations and properties of the probability law Sublemma 2.3 ([181, Lemma 6]). Consider a family of Rd -valued infinitely divisible random variables Θit indexed by t > 0 and i ∈ ℐ , and another family of Rd -valued random variables Yti . We assume: (i) Yti has a Cb∞ transition density uniformly in (t, i). (ii) ‖Yti − Θit ‖L1 → 0 as t → 0 uniformly in i. (iii) The drift parameter of Θit is uniformly bounded. (iv) We denote by μit the Lévy measure of Θit . There exists a compact K ⊂ Rm such that supp μit ⊂ K uniformly in i, the measure |x|2 μit (dx) is bounded uniformly in (t, i), and μit satisfies the condition (A.0), or (A.3)(3-a) if β < 1 uniformly, respectively. Then, the density of Yti is bounded away from zero as t → 0 on any compact set uniformly in i. The proof of this Sublemma is somewhat complex, and we refer to [181]. Proof of Lemma 2.6. (continued) ̃ Put An = {N = n} ∩ {Tn ≤ 2t }. Here, T1 , T2 , . . . are jump times of z(⋅). On the set An , z̃ has n jumps on [0, 2t ] and no jump on [ 2t , t]. We have P(An ) ≥ ct n

with c > 0,

since the left-hand side is greater than n!1 (λt )n e−λt with λt = 2t × ν(Rd \ {0}). (r) On the other hand, the conditional law of xs(r) (x) given (z,̃ xt/2 (x)) coincides with the transition function of xs(r) (x) on [ 2t , t]. We apply Sublemma 2.1 to see that the conditional density of process

xt(r) (x) t 1/β

=

xt(r) (x) r

given z̃ is uniformly Cb1 on An . By the definition of x̃t (x), the

Ut =

1 (x (r) (x) − x̃t (x)) t 1/β t

satisfies the same property. By Sublemma 2.2 applied to the process Ut , Ut is equivalent to an infinitely divisible variable 1

t 1/β

t

(∫ A(s, x̃s (x)) 0

𝜕 γ(x̃ (x), 0)dZsr ) + o(1) 𝜕z s

̃ conditionally on z(t). We use Sublemma 2.3 (the index i is for the path z)̃ to obtain that P(Ut ∈ dz|z)̃ ≥ cdz on An , z in a compact set. This implies

2.2 Methods of finding the asymptotic bounds (I)

P(xt(r) (x) ∈ dy|z)̃ ≥ ct −d/β dy

� 93

on {|y − x̃t (x) |≤ t 1/β } ∩ An .

Thus 󵄨 󵄨 P(xt(r) (x) ∈ dy) ≥ ct −d/β P(󵄨󵄨󵄨x̃t (x) − y󵄨󵄨󵄨 ≤ t 1/β , An )dy.

(2.36)

̃ of z(t) ̃ is independent with the common law ν, On the other hand, each jump Δz(t) and the jumps are independent of (Ti ). Hence, conditionally on An , the variables x̃t (x) and Un have the same law. Thus, 󵄨 󵄨 P(󵄨󵄨󵄨x̃t (x) − y󵄨󵄨󵄨 ≤ t 1/β , An ) = P(|Un − y| ≤ t 1/β , An )

= P(|Un − y| ≤ t 1/β ) × P(An ) ≥ ct n+γ/β .

(2.37)

From (2.36) and (2.37), we get the conclusion. (B) Proof of Lemma 2.7 Note that if y is not an accessible point, then y is not in the support of Un . By choosing the size of Kn small enough, it is not in the support of Vn . Hence, we can take γn = +∞ for any n, and hence Γ = +∞. It is sufficient to prove the case γn < +∞. To show Lemma 2.7, we again prepare several sublemmata which are cited from [181], with simple ideas of proofs. We adjust the definition of z̃r (s) and zr (s) to be z̃r (s) = ∑ Δz(u)1{|Δz(u)|>r} u≤s

and zr (s) = z(s) − z̃r (s), and x̃sr (x) (resp. xtr (x)) to be given as in (3.14) of Chapter 1 using z̃r (s) (resp. as in (2.1) using zr (s)). This is for an extrapolation. Sublemma 2.4 ([181, Lemma 9]). Consider a family of d-dimensional variables H admitting uniformly Cb1 densities p(y). Then, for any q ≥ 1, p(y) ≤ Cq (

1/(d+1)

E[|H|q ] + 1 ) 1 + |y|q

.

(2.38)

Proof. The proof is not hard. It is omitted. Sublemma 2.5 ([181, Lemma 10]). Consider the pure jump process z̃r (s) and x̃sr (x) as defined above for r = r(t) = t 1/β . Let N = N(t) be the number of jumps of z̃r before t. Then, for any λ > 0, we have

94 � 2 Perturbations and properties of the probability law 󵄨 󵄨 P(󵄨󵄨󵄨x̃tr (x) − x 󵄨󵄨󵄨 ≤ λt 1/β ) ≤ C(1 + λL + N L )t Γ+β/d

(2.39)

for some L > 0. Proof. (Idea of proof.) Recall that N is the number of jumps of size > t 1/β , which is a Pois∞ son variable with mean ∫t1/β μ(dz). Note that t 1/β < t 1/(2β) < t 1/4 . We divide the argument

into cases N ≤ t −1/4 and N > t −1/4 . Further, an essential part is for the case L ≥ 4(Γ+d/β). First, we assume N ≤ t −1/4 (case of a small number of jumps). Let an integer l be chosen to satisfy 4(Γ + d/β) ≤ l ≤ L. Let A = Al = {∑ 1{|Δzs |>t1/(2β) } < l}, s≤t

and consider the event {l ≤ N ≤ t −1/4 }. Conditionally on N, the law of the number of jumps of size greater than t 1/(2β) is binomial with parameter N, where κ(t) =

μ{z; |z| > t 1/(2β) } = O(t 1/2 ) μ{z; |z| > t 1/β }

as t → 0.

(a) On the set Ac , we have N

N−j

P(Ac |N) = ∑ N Cj κ(t)j (1 − κ(t)) j=l

N

≤ ∑ N j κ(t)j = ON (t l/4 ) = O(t Γ+d/β ) j=l

if we choose l ≥ 4(Γ + d/β). (b) We can work on the set A. We consider z̃r and z̃ρ for r = r(t) and ρ a constant. Let J = J(t) be the number of jumps of z̃ρ before time t. Let (Sj ), j = 1, . . . , J be the times of such jumps, where S0 = 0 and SJ+1 = t. Hence J < l on A. j,y1

We consider, for each j and y1 , the pure jump process Wt = Wt ΔWs = γ(Ws− , Δz̃r (s) − Δz̃ρ (s)),

j,y1

WS

j

= y1

given by

(2.40)

for s ≥ Sj . Let φj (y1 ) be the value of Wt at time Sj+1 , hence x̃Srj+1 − = φj (x̃Srj ).

(2.41)

We construct a chain (Vj ) with this perturbation φj in (2.17), and another chain (V̄ j ) with random variables (ξn̄ ) with the law μ(dz) conditioned on {|z| > ρ}. By definition, x̃tr and V̄ J have the same law, conditionally on (z̃r − z̃ρ , z̃ρ − z̃r , (Sj )). On the other hand, the law of V̄ j is dominated by that of Vj .

2.2 Methods of finding the asymptotic bounds (I)

� 95

Hence for any λ > 0, 󵄨 󵄨 P(󵄨󵄨󵄨x̃tr − y󵄨󵄨󵄨 ≤ λt 1/β |z̃r − z̃ρ , z̃ρ − z̃r , (Sj )) ≤ Cj P(|Vj − y| ≤ λt 1/β |z̃r − z̃ρ , z̃ρ − z̃r ) on {J = j}. As j = J < l on A, the dependence of Cj on j does not make a problem. We can verify that if ρ is chosen small, the perturbations φj satisfy the assumptions of Lemma 2.7. We prepare the following to estimate the difference due to z̃ρ and z̃r . We notice that on A ∩ {N ≤ t −1/4 } one has 󵄨 ̃ r ̃ ρ 󵄨󵄨󵄨󵄨 = ∑󵄨󵄨󵄨󵄨Δz(s)󵄨󵄨󵄨󵄨1{t1/β r

= O(t),

l−1

󵄨 󵄨 P(󵄨󵄨󵄨x̃tr − y󵄨󵄨󵄨 ≤ λt 1/β |N) ≤ Ct Γ+d/β + Cl ∑ N j λγj t j+γj /β ≤ Cl t Γ+d/β (1 + N j λγj ). j=0

This proves the assertion for this case. Next, for the case N > t −1/4 , the result follows as even for N ≤ t −1/4 , so the upper bound is valid. Sublemma 2.6 ([181, Lemma 11]). With the same notation as in Sublemma 2.5, for any ̃ of xt given z̃r satisfies k ∈ Nd , the conditional density p(y) 󵄨󵄨 ̃ (k) 󵄨󵄨 −(d+|k|)/β exp(Ck N) 󵄨󵄨p (y)󵄨󵄨 ≤ Ck t

(2.42)

96 � 2 Perturbations and properties of the probability law for some Ck > 0. Proof. Let T1 , . . . , TN be the times of jumps of z̃r , as T0 = 0, TN+1 = t. We seek for an t ; and we fix one as [TI , TI+1 ]. interval [Ti , Ti+1 ] with its length at least N+1 t Then the conditional density of x. from TI to S = TI + N+1 coincides with that of x.r − dβ

t on {N = n}. In particular, its density is dominated by ( n+1 ) − d+|k| β

, and its k-times density

t is dominated by ( n+1 ) . Integrating with respect to the conditional law of xI , we see that the conditional density p̃ S (⋅) of xS given z̃r satisfies − d+|k| β

t 󵄨󵄨 ̃ (k) 󵄨󵄨 ) 󵄨󵄨pS (y)󵄨󵄨 ≤ Ck ( N +1

.

The density propagates up to time t, in order to get the conditional density of xt = ϕS,t (xS ) given z̃r , where ϕs,t is the stochastic flow of diffeomorphisms generated by (2.1), to have ′ 󵄨󵄨 ̃ −1 r ̃ = E[󵄨󵄨󵄨󵄨det(ϕ−1 p(y) S,t (y)) (y)󵄨󵄨pS (ϕS,t (y))|z̃ ].

The map (diffeomorphism) ϕ−1 S,t (⋅) is decomposed into −1 −1 −1 −1 −1 ϕ−1 S,t = ϕS,TI+1 − ∘ ϕTI+1 −,TI+1 ∘ ⋅ ⋅ ⋅ ∘ ϕTN−1 ,TN − ∘ ϕTN −,TN ∘ ϕTN ,t . r Here the maps ϕ−1 Tj −,Tj+1 − are z̃ -measurable and have bounded derivatives, the maps r ϕ−1 Tj ,Tj+1 − are independent conditionally on z̃ , and they satisfy (l) 󵄨 󵄨󵄨q r E[󵄨󵄨󵄨(ϕ−1 Tj ,Tj+1 − ) (y)󵄨󵄨 |z̃ ] ≤ Cl,q

for l ∈ Nd \ {0}. Viewing the decomposition above, we see (l) 󵄨 󵄨󵄨 r E[󵄨󵄨󵄨(ϕ−1 S,t ) (y)󵄨󵄨|z̃ ] ≤ exp(Cl N).

This then implies − d+|k| β

t 󵄨󵄨 ̃ (k) 󵄨󵄨 ) 󵄨󵄨p (y)󵄨󵄨 ≤ Ck ( N +1

exp(Ck N).

Sublemma 2.7 ([181, Lemma 12]). With the same notation as in Sublemma 2.5, for each q ≥ 1, we have 1/q 󵄨 󵄨q E[󵄨󵄨󵄨xt (x) − x̃tr (x)󵄨󵄨󵄨 |z̃r ] ≤ cq t 1/β exp(cq N).

Proof. On {N = 0}, xt = xtr and x̃t = x0 , so xtr − x0 = O(t 1/β ) in Lq .

(2.43)

2.2 Methods of finding the asymptotic bounds (I)

� 97

From the Doob–Meyer decomposition, the only term which must be studied is Mt . Since Mt is dominated by [M, M]1/2 t by the Burkholder–Davis–Gundy inequalities, it is sufficient to show [M, M]t is of order t 2/β . By the Doob–Meyer decomposition, t

󵄨2 󵄨 [M, M]t = ∫ ∫ 󵄨󵄨󵄨γ(xur , z)󵄨󵄨󵄨 μ(dz)du + M̄ t ≤ Ct 2/β + M̄ t 0 |z|≤r

for the martingale part M̄ t of [M, M]t . As the jumps of Mt are bounded by Ct 1/β , ̄ t = ∑ |ΔMs |4 ≤ Ct 2/β [M, M]t , [M,̄ M] s≤t

and hence 󵄩󵄩 󵄩 2/β 󵄩 ̄ t 󵄩󵄩󵄩 ≤ Ct 2/β + Ct 1/β 󵄩󵄩󵄩[M, M]t 󵄩󵄩󵄩1/2 . 󵄩󵄩[M, M]t 󵄩󵄩󵄩q ≤ Ct + 󵄩󵄩󵄩[M,̄ M] 󵄩q 󵄩 󵄩q/2 We see [M, M]t is O(t 2/β ) in L1 as above, hence [M, M]t is O(t 2/β ) in L2 and in Lq , q ≥ 1, by above. The result then follows for {N = 0}. For a general case, if Ti are the times of jumps of x̃ r with T0 = 0 and TN+1 = t, we can observe 1/q

E[|xTi+1 − − xTi |q |x̃ r ]

≤ Cq t 1/β

as above. One has x̃Tr i+1 − = x̃Tr i so 1/q 1/q 󵄨 󵄨q 󵄨 󵄨q E[󵄨󵄨󵄨xTi+1 − − x̃Tr i+1 − 󵄨󵄨󵄨 |x̃ r ] ≤ E[󵄨󵄨󵄨xTi − x̃Tr i 󵄨󵄨󵄨 |x̃ r ] + Cq t 1/β .

On the other hand, from the Lipschitz continuity of x 󳨃→ x + γ(x, z), we have 󵄨 󵄨 󵄨󵄨 r 󵄨 r 󵄨󵄨xTi+1 − x̃Ti+1 󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨xTi+1 − − x̃Ti+1 − 󵄨󵄨󵄨. From these two formulas and using induction we get n−1

1/q 󵄨 󵄨q E[󵄨󵄨󵄨xTn − x̃Tr n 󵄨󵄨󵄨 |x̃ r ] ≤ Cq t 1/β ∑ C i . i=1

Then we choose n = N + 1. Proof of Lemma 2.7. (continued). Let H = exp(CN)(xt − x̃tr )/t 1/β .

98 � 2 Perturbations and properties of the probability law Here, the constant C is chosen large enough so that the conditional density of H given z̃r is uniformly Cb1 , which is possible due to Sublemma 2.6. By Sublemma 2.7, the conditional moments of H given z̃r have at most of exponential growth with respect to N. Hence, we apply Sublemma 2.4 to obtain that for each q ≥ 1, P(H ∈ dh|z̃r )/dh ≤ cq (1 + |h|)

−q

exp(cq N)

̃ of xt (x) satisfies as a conditional density. Hence, the conditional density p(y) −q ̃ ≤ cq t −d/β {exp(cq N)(1 + t −1/β 󵄨󵄨󵄨󵄨y − x̃tr (x)󵄨󵄨󵄨󵄨 exp(CN)) } p(y) 󵄨 󵄨 −1 q ≤ cq t −d/β {exp(cq N)(1 + t −1/β 󵄨󵄨󵄨y − x̃tr (x)󵄨󵄨󵄨) }

due to Sublemma 2.6. ̃ The density pt (x, y) is the expectation of p(y). To this end, we apply the distribution identity ∞

E[Z q ] = q ∫ uq−1 P(Z ≥ u)du,

Z ≥ 0.

0

Then, we have ∞

󵄨 󵄨 pt (x, y) ≤ qcq t −d/β ∫ uq−1 P(1 + t −1/β 󵄨󵄨󵄨y − x̃tr (x)󵄨󵄨󵄨 ≤ exp(cq N)/u)du 0

∞

≤ cq t Γ ∫ uq−1 E[(1 + N L + u−L exp(Lcq N)) × 1{N≥(log u)/cq } ]du 0

by Sublemma 2.5. The variable N = N(t) is a Poisson variable with bounded exponential mean. Hence, the expectation above is uniformly o(u−k ) for any k as u → +∞, and uniformly O(u−L ) as u → 0. Hence, the integral is bounded for q > L. We need fine estimates of the above integral for further development. (C) Proof of Lemma 2.8 We fix 0 < r < 1 and put ξ(r, ⋅, x) ≡ x + γ(x, ⋅) when ‘⋅’ is occupied by the random variables ξir . Choose St = 0 and consider the process s 󳨃→ xs (r, 0, x) as the solution of (2.23). Given (s1 , y), the solution for s2 ≥ s1 with the initial value z at s = s1 is given by a smooth stochastic semiflow ϕs1 ,s2 (z). This is proved in [68]. Assuming y = xt (r, St , x), St ∈ 𝒮t,n1 , we shall estimate |Unr1 − xt (r, St , x)|, and show that for p > k and t ≤ 1,

2.2 Methods of finding the asymptotic bounds (I)

E

P̃ t,r |𝒮t,n

1

� 99

r ⊗n1 ′ 󵄨 󵄨 E (μ ) [P(󵄨󵄨󵄨Unr1 − xt (r, St , x)󵄨󵄨󵄨 > t 1/β |St , ξ1r , . . . , ξnr1 )]

2

≤ n1 C0 exp[−(p − k)(log(1/t)) ].

(2.44)

We first recall that Unr1 and xt (r, St , x) can be represented by Unr1 = An1 (x, ξ1r , . . . , ξnr1 ) = ξ(r, ξnr1 , ⋅) ∘ ξ(r, ξnr1 −1 , ⋅) ∘ ⋅ ⋅ ⋅ ∘ ξ(r, ξnr1 , x) and xt (r, St , x) = ϕtn

1

,t

∘ ξ(r, ξnr1 , ⋅) ∘ ϕtn −1 ,tn ∘ ⋅ ⋅ ⋅ ∘ ξ(r, ξ2r , ⋅) ∘ ϕt1 ,t2 ∘ ξ(r, ξ1r , ⋅) ∘ ϕ0,t1 (x). 1

1

(1) Case n1 = 0. In this case, St = 0 and Unr1 = x. The process xt (r, 0, x) is a martingale whose quadratic variation [x(r, 0, x), x(r, 0, x)]t is bounded by t r 󵄨󵄨 2 󵄨󵄨 𝜕γ 2 󵄨 󵄨 ∫ du ∫ {γ(xu− (r, 0, x), ζ )} μ(dζ ) ≤ t ⋅ (2 sup󵄨󵄨󵄨 (x, ζ )󵄨󵄨󵄨) ⋅ ∫(ζ 2 ∧ 1)μ(dζ ) ≤ kt 󵄨󵄨 󵄨 𝜕z 󵄨 x,ζ −r 0

for some k > 0, which is uniform in 0 < r < 1 and x. We have that, for given p > k and t ≤ 1, ′ 2 󵄨 󵄨 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 > t 1/β ) ≤ C0 exp[−(p − k)(log(1/t)) ]

0≤s≤t,x

with C0 not depending on x, r, and hence the assertion follows for this case. Indeed, by the upper bound of exponential type (see [145, Theorem 13]), 1 󵄨 󵄨 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 ≥ C) ≤ 2 exp[−λC + (λS)2 kt(1 + exp λS)] 2 0≤s≤t,x for C > 0, λ > 0. We choose C = pS log(1/t) and λ = (log(1/t))/S. Then, 󵄨 󵄨 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 ≥ pS log(1/t)) 0≤s≤t,x

2

2

2

≤ 2 exp[−p(log(1/t)) + (log(1/t)) kt + k(log(1/t)) ] 2

≤ C0 exp[−(p − k)(log(1/t)) ]

as t → 0.

We choose S = (1/p)t 1/β . Then, ′ 󵄨 󵄨 󵄨 󵄨 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 ≥ t 1/β ) ≤ P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 ≥ t 1/β log(1/t))

0≤s≤t,x

0≤s≤t,x

2

≤ C0 exp[−(p − k)(log(1/t)) ], as t → 0. Here, β′ > β and the constant C0 does not depend on x, r.

(2.45)

100 � 2 Perturbations and properties of the probability law (2) Case n1 ≥ 1. Assume St = {t1 , . . . , tn1 } with t1 < ⋅ ⋅ ⋅ < tn1 . Given ξ1r , . . . , ξnr1 , we put In1 ≡ Unr1 − xt (r, St , x) and 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝜕ξ 𝜕γ 󵄩 󵄩 󵄩 󵄩 M ≡ max(sup󵄩󵄩󵄩 (r, ζ , x)󵄩󵄩󵄩, 1) = max(sup󵄩󵄩󵄩I + (x, ζ )󵄩󵄩󵄩, 1), 󵄩󵄩 󵄩󵄩 𝜕x 󵄩 󵄩 𝜕x x,ζ 󵄩 x,ζ 󵄩 which exists by the assumption on γ(x, z). Here, ‖ ⋅ ‖ denotes the norm of the matrix. Rewriting the above, by an elementary calculation, we have 󵄨 󵄨 󵄨 󵄨 |In1 | ≤ sup󵄨󵄨󵄨ϕtn ,t (y) − y󵄨󵄨󵄨 + M sup󵄨󵄨󵄨ϕtn −1 ,tn (y) − y󵄨󵄨󵄨 1 1 1 y y 󵄨 󵄨 󵄨 󵄨 + M 2 sup󵄨󵄨󵄨ϕtn −2 ,tn −1 (y) − y󵄨󵄨󵄨 + ⋅ ⋅ ⋅ + M n1 sup󵄨󵄨󵄨ϕ0,t1 (x) − x 󵄨󵄨󵄨. 1 1 x y We assume |In1 | > t 1/β . Then, for some j ∈ {1, . . . , n1 }, ′

′ 󵄨 󵄨 sup󵄨󵄨󵄨ϕtj−1 ,tj (y) − y󵄨󵄨󵄨 > (1/n1 )M −n1 t 1/β . y

Indeed, if for all j, supy |ϕtj−1 ,tj (y) − y| ≤ (1/n1 )M −n1 t 1/β , then ′

󵄨 󵄨 󵄨 󵄨 |In1 | ≤ sup󵄨󵄨󵄨ϕtn ,t (y) − y󵄨󵄨󵄨 + M sup󵄨󵄨󵄨ϕtn −1 ,tn (y) − y󵄨󵄨󵄨 y

y

1

2

1

1

󵄨 󵄨 󵄨 󵄨 + M sup󵄨󵄨󵄨ϕtn −2 ,tn −1 (y) − y󵄨󵄨󵄨 + . . . + M n1 sup󵄨󵄨󵄨ϕ0,t1 (x) − x 󵄨󵄨󵄨 1 1 x y

≤ (1/n1 )M −n1 t 1/β + (1/n1 )M −n1 +1 t 1/β + ⋅ ⋅ ⋅ + (1/n1 )t 1/β ≤ t 1/β , ′

′

′

′

which is a contradiction. Hence, E

P̃ t,r |𝒮t,n

r ⊗n1 ′ 󵄨 󵄨 E (μ ) [P(|In1 | > t 1/β 󵄨󵄨󵄨St , ξ1r , . . . , ξnr1 󵄨󵄨󵄨St )] ′ P̃ | 󵄨 󵄨 ≤ E t,r 𝒮t,n1 [P(∃j such that sup󵄨󵄨󵄨ϕtj−1 ,tj (y) − y󵄨󵄨󵄨 > (1/n1 )M −n1 t 1/β )] y 1

′ 󵄨 󵄨 ≤ n1 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 > (1/n1 )M −n1 t 1/β ).

0≤s≤t,x

We choose β′′ = β′′ (n1 ) such that β < β′′ < β′ and t 1/β < t 1/β < (1/n1 )M −n1 t 1/β < t 1/β for t > 0 small. Then, by the proof of (2.45), ′′

E

P̃ t,r |𝒮t,n

r ⊗n1

E (μ ) [P(|In1 | > t 1/β |St , ξ1r , . . . , ξnr1 )] ′′ 󵄨 󵄨 ≤ n1 P( sup 󵄨󵄨󵄨xs (r, 0, x) − x 󵄨󵄨󵄨 > t 1/β ) 1

0≤s≤t,x

′

2

≤ n1 C0 exp[−(p − k)(log(1/t)) ] as t → 0. This proves (2.44).

′

′

2.3 Methods of finding the asymptotic bounds (II)

� 101

2.3 Methods of finding the asymptotic bounds (II) Assume xt (x) is an Itô process given by (3.15). In this section, we use an approximation method called the polygonal geometry method. Intuitively, we make polygonal lines along with the trajectories of jump paths connecting jump points in-between, instead of chains described in the previous section. This intuitive view will help us understand the support theorems stated in Section 2.5. In contrast to the Markov chain method, this method is applied (at least here) to the case where the Lévy measure has a density. By using this method, we can obtain similar results on the short-time asymptotic bounds of the density. Statements are simple compared to those obtained by the previous method, although the conditions are more restrictive. 2.3.1 Polygonal geometry We assume, in particular, the following assumptions on the Lévy measure μ(dz): μ(dz) has a C ∞ -density h(z) on Rm \ {0} such that supp h( ⋅ ) ⊂ { z ∈ Rm ; |z| ≤ c},

where 0 < c ≤ +∞,

2

∫ min(|z| , 1) h(z)dz < +∞, Rm \{0}

∫ ( |z|≥ϵ

|𝜕h/𝜕z|2 )dz < +∞ h(z)

for ϵ > 0,

(3.1)

z ) ⋅ |z|−m−α |z|

(3.2)

and h(z) = a(

in a neighborhood of the origin for some α ∈ (0, 2) and some strictly positive function a( ⋅ ) ∈ C ∞ (S m−1 ). Here, we are assuming the smoothness of the Lévy measure. This is due to the fact that we employ Bismut–Graczyk perturbation method. The latter half of the condition in (3.1) is also the requirement from this method. It may seem too restrictive, however, this method will well illustrate the behavior of the jump trajectories, as will be seen below. We repeat several notations that appeared in Section 2.2. Set Px = x + γ(x, supp h) = x + {γ(x, z); z ∈ supp h}. Then, each Px is a compact set in Rd since γ(x, z) is bounded. For each y ∈ Rd (y ≠ x), we put α(x, y) ≡ l0 (x, y) + 1.

102 � 2 Perturbations and properties of the probability law Here, l0 (x, y) denotes the minimum number of distinct points z1 , . . . , zl ∈ Rd such that z1 ∈ Px , zi ∈ Pzi−1 , i = 2, . . . , l, and y ∈ Pzl (z0 = x).

(3.3)

We always have α(x, y) < +∞ for each given x, y ∈ Rd (x ≠ y) by (3.2) and (A.1) in Section 1.3. We set 󵄨 󵄨 g(x, z) = h( Hx−1 (z) )󵄨󵄨󵄨[Jγ]−1 (x, Hx−1 (z))󵄨󵄨󵄨 for z ∈ Px \ {x},

(3.4)

and g(x, z) = 0 otherwise. Here, we put Hx : supp h → Px , z 󳨃→ x + γ(x, z) (= z), and Jγ = (𝜕γ/𝜕z)(x, z) is the Jacobian matrix of γ(x, z). The kernel g(x, z) is well defined and satisfies ∫ f (z)g(x, z)dz = ∫ f (x + γ(x, z)) h(z)dz,

for f ∈ C(Px ).

(3.5)

That is, g(x, dz) = g(x, z)dz is the Lévy measure of xt (x); supp g(x, ⋅) ⊂ Px by definition. Here continuous functions on Px are denoted by C(Px ). Remark 2.6. An intuitive view of the polygonal geometry is that one can cover polygonal lines connecting x and y ∈ Rd by a sequence of ‘small balls’ Pzi ’s with zi ’s at the jump points of xt (x) and connecting those zi ’s by piecewise linear lines. Historically, the polygonal geometry method has been employed earlier than the Markov chain method. In the Markov chain approximation setting, assumptions stated in (3.1) are not necessary. Theorem 2.4 ([93]). Under the assumptions (3.1), (3.2) and (A.1), (A.2) in Section 1.3, we have, for each distinct pair (x, y) ∈ Rd for which κ = α(x, y) (< +∞), lim t→0

pt (x, y) = C, tκ

where {(1/κ!){∫P dz1 ⋅ ⋅ ⋅ ∫P dzκ−1 g(x, z1 ) ⋅ ⋅ ⋅ g(zκ−1 , y)}, κ ≥ 2, x zκ−2 C = C(x, y, κ) = { g(x, y), κ = 1. {

(3.6)

The proof of this theorem will be given below.

2.3.2 Proof of Theorem 2.4 Decomposition of the transition density In this section we give a decomposition of pt (x, y), which plays a crucial role. We partly repeat a similar argument to what appeared in Section 2.2.2.

2.3 Methods of finding the asymptotic bounds (II)

�

103

Given ϵ > 0, let ϕϵ : Rm → R be a nonnegative C ∞ -function such that ϕϵ (z) = 1 if |z| ≥ ϵ and ϕϵ (z) = 0 if |z| ≤ ϵ/2. Let zt (ϵ) and z′t (ϵ) be two independent Lévy processes whose Lévy measures are given by ϕϵ (z) h(z)dz and (1 − ϕϵ (z))h(z)dz, respectively. Then the process z(t) has the same law as that of zt (ϵ) + z′t (ϵ). Since the process zt (ϵ) has finite Lévy measure on Rm \{0}, the corresponding Poisson random measure Ns (ϵ) on [0, +∞)× (Rm \ {0}) jumps only finite times in each finite interval [0, t]. We set 𝒮t,k = ([0, t]k / ∼) and 𝒮t = ∐k≥0 𝒮t,k , where ∐k≥0 denotes the disjoint sum and ∼ means the identification of the coordinates on the product space [0, t]k by permutation. The distribution P̃ t,ϵ of the moments (instants) of jumps related to Ns (ϵ) in [0, t] is given by k

∫

f (St )d P̃ t,ϵ (St ) = {(t ∫ ϕϵ (z)h(z)dz) (

{#St =k} t

t

×

1 ) exp (−t ∫ ϕϵ (z)h(z)dz)} k!

1 ∫ ⋅ ⋅ ⋅ ∫ f (s1 , . . . , sk )ds1 ⋅ ⋅ ⋅ dsk , tk 0

(3.7)

0

where f is a function on 𝒮t,k (a symmetric function on [0, t]k ). Let J(ϵ) be a random variable whose law is ϕϵ (z) h(z)dz/(∫ ϕϵ (z) h(z)dz), and choose a family of independent random variables Ji (ϵ), i = 1, 2, . . ., having the same law as J(ϵ). Choose 0 ≤ s ≤ t < +∞. For a fixed St ∈ 𝒮t , we consider the solution of the following SDE with jumps: c

xs (ϵ, St , x) = x + ∑ γ(xu− (ϵ, St , x), Δz′u (ϵ)) + u≤s

∑

si ∈St ,si ≤s

γ(xsi − (ϵ, St , x), Ji (ϵ)).

(3.8)

(Here we are assuming the drift vector b′ (⋅) to be 0 for simplicity.) Then, the law of xs (ϵ, St , x) has a smooth density denoted by ps (ϵ, St , x, y) (cf. [142, p. 87]). Then, ps (x, y) = ∫ ps (ϵ, St , x, y) d P̃ t,ϵ (St )

(3.9)

𝒮t

because zt (ϵ) and z′t (ϵ) are independent, which is written in a decomposed form (by putting s = t) N−1

pt (x, y) = ∑ pt (i, ϵ, x, y) + p̄ t (N, ϵ, x, y), i=0

where pt (i, ϵ, x, y) = ∫ pt (ϵ, St , x, y) d P̃ t,ϵ (St ) 𝒮t,i

(3.10)

104 � 2 Perturbations and properties of the probability law and p̄ t (N, ϵ, x, y) is the remaining term. Then, it follows from the definition and (3.7) that k

pt (k, ϵ, x, y) = (1/k!) ⋅ exp(−t ∫ ϕϵ (z) h(z)dz) ⋅ (∫ ϕϵ (z) h(z)dz) t

t

× ∫⋅ ⋅ ⋅ ∫ pt (ϵ, {s1 , . . . , sk }, x, y)dsk ⋅ ⋅ ⋅ ds1 . 0

(3.11)

0

We denote by xs (ϵ, 0, x) the process in (3.8) with St ∈ 𝒮t,0 , and its density by ps (ϵ, 0, x, y). For the random variable J(ϵ) introduced above, there exists a density gϵ (x, z) such that P(x + γ(x, J(ϵ)) ∈ dz) = gϵ (x, z)dz. Indeed, gϵ (x, z) is given by gϵ (x, z) =

ϕϵ (Hx−1 (z))g(x, z) ∫ ϕϵ (z) h(z)dz

,

(3.12)

where g(x, z) is the density of the Lévy measure of xt (x) (cf. (3.4)). Note that by definition, supp gϵ (x, ⋅) ⊂ Px , and that gϵ (x, z) is of class C ∞ whose derivatives are uniformly bounded (since g(x, z) only has a singularity at x = z). Now, for each s1 < ⋅ ⋅ ⋅ < sk < t, we have pt (ϵ, {s1 , . . . , sk }, x, y) = ∫ dz′0 ∫ dz1 ⋅ ⋅ ⋅ ∫ dz′k−1 ∫ dzk {ps1 (ϵ, 0, x, z′0 )gϵ (z′0 , z1 ) ps2 −s1 (ϵ, 0, z1 , z′1 )gϵ (z′1 , z2 ) Pz′

0

×

Pz′

k−1

ps3 −s2 (ϵ, 0, z2 , z′2 ) ⋅ ⋅ ⋅ gϵ (z′k−1 , zk ) pt−sk (ϵ, 0, zk , y)}.

(3.13)

Indeed, the increment xsi +u (ϵ, {s1 , . . . , sk }, x)−xsi (ϵ, {s1 , . . . , sk }, x) has the same law as that of xu (ϵ, 0, x)−x on (0, si+1 −si ) for i = 0, . . . , k (s0 = 0, sk+1 = t), and x(si +u)− (ϵ, {s1 , . . . , sk }, x) is going to make a ‘big jump’ (i. e., a jump derived from Ji+1 (ϵ)) at u = si+1 − si according to the law gϵ (x(si+1 )− (ϵ, {s1 , . . . , sk }, x), z)dz. Lower bound Let ϵc ≡ sup{ϵ > 0; {|z| < ϵ} ⊂ supp h} > 0, and choose 0 < ϵ < ϵc . First, we note that for each η > 0 and a compact set K, uniformly in y ∈ K, we have lim

s→0

∫

ps (ϵ, 0, z, y)dz = 1,

(3.14)

{z;|z−y|≤η}

by Proposition I.2 in Léandre [142]. Then, we have: Lemma 2.9. Let 𝒦 be a class of nonnegative, equicontinuous, uniformly bounded functions whose supports are contained in a fixed compact set K. Then, given δ > 0, there exists a t0 > 0 such that

2.3 Methods of finding the asymptotic bounds (II)

�

inf ∫ f (z)pt−s (ϵ, 0, z, y)dz ≥ f (y) − δ

105 (3.15)

s≤t

for every f ∈ 𝒦, y ∈ K and every t ∈ (0, t0 ). Proof. Given δ > 0, choose η > 0 so that |f (z) − f (y)| < δ/2 for each |z − y| < η. Then, ∫ f (z)pt−s (ϵ, 0, z, y)dz ≥

f (z)pt−s (ϵ, 0, z, y)dz

∫ {z;|z−y|≤η}

=

(f (z) − f (y))pt−s (ϵ, 0, z, y)dz

∫ {z;|z−y|≤η}

+ f (y)

pt−s (ϵ, 0, z, y)dz

∫ {z;|z−y|≤η}

≥ (f (y) − δ/2)

pt−s (ϵ, 0, z, y)dz.

∫ {z;|z−y|≤η}

By (3.14), we can choose t0 > 0 so that if t < t0 , then { inf

pt−s (ϵ, 0, z, y)dz}(f (y) − δ/2) ≥ f (y) − δ

∫

y∈K,s≤t

for all y ∈ K.

{z;|z−y|≤η}

̄ Now, we choose an arbitrary compact neighborhood U(x) of x and arbitrary comd pact sets K1 , . . . , Kk−1 of R , and set ̄ 𝒦 = {gϵ (z0 , ⋅), gϵ (z1 , ⋅), . . . , gϵ (zk−1 , ⋅); z0 ∈ U(x), z1 ∈ K1 , . . . , zk−1 ∈ Kk−1 }. ′

′

′

′

′

′

Since 𝒦 has the property in Lemma 2.9 (cf. (3.15)), it follows from (3.13) that for every δ > 0, there exists t0 > 0 such that for every 0 < t < t0 , pt (ϵ, {s1 , . . . , sk }, x, y) ≥ ∫ ps1 (ϵ, 0, x, z′0 )dz′0 ∫ dz′1 ⋅ ⋅ ⋅ ∫ dz′k−1 (gϵ (z′0 , z′1 ) − δ) ⋅ ⋅ ⋅ (gϵ (z′k−1 , y) − δ). K1

̄ U(x)

(3.16)

Kk−1

However, for each fixed η > 0, we have lim

inf

t→0 s1 ≤t,x∈Rd

∫

ps1 (ϵ, 0, x, z′0 )dz′0 = 1,

{|x−z′0 |≤η}

by (2.13) in [142]. Therefore, for every sufficiently small t > 0, it holds that

(3.17)

106 � 2 Perturbations and properties of the probability law pt (ϵ, {s1 , . . . , sk }, x, y) ≥ (1 − δ) ∫ dz′1 ∫ dz′2 ⋅ ⋅ ⋅ ∫ dz′k−1 (gϵ (z′0 , z′1 ) − δ)(gϵ (z′1 , z′2 ) − δ) ⋅ ⋅ ⋅ (gϵ (z′k−1 , y) − δ). K1

K2

Kk−1

(3.18)

Combining (3.11) with (3.18), we have lim inf( t→0

1 )pt (k, ϵ, x, y) tk

k

≥ (1/k!) ⋅ (1 − δ) ⋅ (∫ ϕϵ (z) h(z)dz)

× ∫ dz′1 ∫ dz′2 ⋅ ⋅ ⋅ ∫ dz′k−1 {(gϵ (x, z′1 ) − δ)(gϵ (z′1 , z′2 ) − δ) ⋅ ⋅ ⋅ (gϵ (z′k−1 , y) − δ)}. K1

K2

Kk−1

(3.19)

Since δ > 0 and K1 , . . . , Kk−1 are arbitrary, and supp gϵ (z′i−1 , ⋅) ⊂ Pz′ , i = 1, . . . , k − 1, we i−1 have lim inf( t→0

≥

1 )pt (k, ϵ, x, y) tk

k

1 ⋅ (∫ ϕϵ (z) h(z)dz) ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzk−1 gϵ (x, z1 )gϵ (z1 , z2 ) ⋅ ⋅ ⋅ gϵ (zk−1 , y). k! Px

Pz1

Pzk−2

(3.20)

Hence, it follows from (3.10) that lim inf( t→0

≥

1

t α(x,y)

)pt (x, y) k

1 ⋅ (∫ ϕϵ (z) h(z)dz) ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzk−1 gϵ (x, z1 )gϵ (z1 , z2 ) ⋅ ⋅ ⋅ gϵ (zk−1 , y). α(x, y)! Px

Pz1

Pzk−2

(3.21)

Since ϵ > 0 is arbitrary, in view of (3.12), we have 1 )pt (x, y) t α(x,y) 1 ≥ ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzk−1 {g(x, z1 )g(z1 , z2 ) ⋅ ⋅ ⋅ g(zk−1 , y)}. α(x, y)!

lim inf( t→0

Px

Pz1

Pzk−2

The proof for the lower bound is now complete.

(3.22)

2.3 Methods of finding the asymptotic bounds (II)

� 107

Upper bound The proof of the upper bound lim supt→0 (1/t α(x,y) )pt (x, y) is rather delicate and is carried out in the same way as in [142], but it is a little more tedious in our case. First, we choose and fix N ≥ α(x, y) + 1. Noting that sup{pt (ϵ, St , x, y); #St ≥ 2, x, y ∈ ̃ (cf. [142, (3.23)]), we have: Rd } ≤ C(ϵ) Lemma 2.10. For every ϵ > 0 and t > 0, we have sup p̄ t (N, ϵ, x, y) ≤ C(ϵ) t N .

(3.23)

y∈Rd

Proof. Recall that p̄ t (N, ϵ, x, y) =

∫

pt (ϵ, St , x, y) d P̃ t,ϵ (St ).

𝒮t \(⋃N−1 i=0 𝒮t,i )

Since P̃ t,ϵ (#St ≥ N) ≤ C(ϵ)t N by the Poisson law, we have p̄ t (N, ϵ, x, y) ≤ sup pt (ϵ, St , x, y)P̃ t,ϵ (#St ≥ N) #St ≥N

≤ sup{pt (ϵ, St , x, y); #St ≥ 2, x, y ∈ Rd }C(ϵ)t N N ̃ ≤ C(ϵ)C(ϵ)t .

Lemma 2.11 ([142, Proposition III.2]). For every p > 1 and any η > 0, there exists ϵ > 0 such that for all t ≤ 1 and every ϵ′ ∈ (0, ϵ), sup pt (0, ϵ′ , x, y) ≤ C(ϵ′ , η) t p .

(3.24)

|x−y|≥η

Combining Lemma 2.10 with Lemma 2.11, we see that it is sufficient to study pt (k, ϵ, x, y) for 1 ≤ k ≤ α(x, y). For a given η > 0, make a subdivision of the space A ≡ {(z′0 , z1 , . . . , z′k−1 , zk ) ∈ Rd × ⋅ ⋅ ⋅ × Rd ; z1 ∈ Pz′ , z2 ∈ Pz′1 , . . . , zk ∈ Pz′ } 0

k−1

k+1

2 as A = ⋃i=1 Ai (η), where

A1 (η) = {(z′0 , z1 , . . . , z′k−1 , zk ) ∈ Rd × ⋅ ⋅ ⋅ × Rd ; z1 ∈ Pz′ , z2 ∈ Pz′1 , . . . , zk ∈ Pz′ 0 k−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 and 󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z1 − z′1 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z2 − z′2 󵄨󵄨󵄨 ≤ η, . . . , |zk − y| ≤ η},

108 � 2 Perturbations and properties of the probability law A2 (η) = {(z′0 , z1 , . . . , z′k−1 , zk ) ∈ Rd × ⋅ ⋅ ⋅ × Rd ; z1 ∈ Pz′ , z2 ∈ Pz′1 , . . . , zk ∈ Pz′ 0 k−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 and 󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z1 − z′1 󵄨󵄨󵄨 > η, 󵄨󵄨󵄨z2 − z′2 󵄨󵄨󵄨 ≤ η, . . . , |zk − y| ≤ η}, ⋅⋅⋅ and A2k+1 (η) = {(z′0 , z1 , . . . , z′k−1 , zk ) ∈ Rd × ⋅ ⋅ ⋅ × Rd ; z1 ∈ Pz′ , z2 ∈ Pz′1 , . . . , zk ∈ Pz′ 0 k−1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 and 󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 > η, 󵄨󵄨󵄨z1 − z′1 󵄨󵄨󵄨 > η, 󵄨󵄨󵄨z2 − z′2 󵄨󵄨󵄨 > η, . . . , |zk − y| > η}, and we shall classify those divisions into four cases: [A] = {Ai (η); |zk − y| > η}, 󵄨 󵄨 [B] = {Ai (η); 󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 > η and |zk − y| ≤ η}, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 [C] = {A1 (η)} ≡ {{󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z1 − z′1 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z2 − z′2 󵄨󵄨󵄨 ≤ η, . . . , |zk − y| ≤ η}}, and 󵄨 󵄨 [D] = {Ai (η); 󵄨󵄨󵄨x − z′0 󵄨󵄨󵄨 ≤ η and |zk − y| ≤ η} \ {A1 (η)}. We put I[N],t,η (ϵ, {s1 , . . . , sk }, x, y) =

∑

∫ {ps1 (ϵ, 0, x, z′0 )gϵ (z′0 , z1 )ps2 −s1 (ϵ, 0, z1 , z′1 )

Ai (η)∈[N] A (η) i

× gϵ (z′1 , z2 )ps3 −s2 (ϵ, 0, z2 , z′2 ) × ⋅ ⋅ ⋅ × gϵ (z′k−1 , zk )pt−sk (ϵ, 0, zk , y)}dzk dz′k−1 . . . dz1 dz′0 ,

(3.25)

[N] = [A], [B], [C], [D]. Then, in view of (3.13), we have pt (ϵ, {s1 , . . . , sk }, x, y) = (I[A],t,η + I[B],t,η + I[C],t,η + I[D],t,η ) (ϵ, {s1 , . . . , sk }, x, y),

(3.26)

since supp(gϵ (z′i−1 , ⋅)) ⊂ Pz′ for i = 1, . . . , k. i−1

Lemma 2.12. For any (x, y) ∈ (Rd × Rd ) \ △ (△ ≡ {(x, x); x ∈ Rd }), {s1 , . . . , sk }, η > 0, and any p > 1, there exists ϵ > 0 such that if 0 < ϵ′ < ϵ and t ≤ 1, then (I[A],t,η + I[B],t,η + I[D],t,η )(ϵ′ , {s1 , . . . , sk }, x, y) ≤ C(ϵ′ , η, k, p)t p . Proof. The proof is essentially the same as that in [142, Proposition III.4], but is a little more complicated.

2.3 Methods of finding the asymptotic bounds (II)

�

109

First, note that there exists ϵ > 0 such that if 0 < ϵ′ < ϵ and t ≤ 1, then 󵄨 󵄨 P{ sup 󵄨󵄨󵄨xs (ϵ′ , 0, x) − x 󵄨󵄨󵄨 > η} ≤ c (ϵ′ , η, p)t p 0≤s≤t

(3.27)

(see [142, Proposition I.4] and Lepeltier–Marchal [145, Lemme 17]). Then, we observe I[B],t,η (ϵ′ , {s1 , . . . , sk }, x, y) ≤ C1 (ϵ′ , η, k, p)t p . Next, let Gz : Rd → Rd , x 󳨃→ x + γ(x, z). If z ∈ int(supp h), then Gz defines a diffeõ z) ≡ Gz−1 (x) − x, z ∈ morphism by (A.2). Let Gz−1 denote its inverse mapping, and put γ(x, ∞ int(supp h). Since γ(x, z) is a bounded, C -function both in x and z, (A.1) and (A.2) imply ̃ z) is also bounded, and C ∞ in x ∈ Rd and z ∈ int(supp h). Note that γ(x, ̃ 0) = 0 that γ(x, since G0 (x) = x. For fixed ϵ > 0, put 󵄨 ̃ 󵄨 S̃ϵ = sup{󵄨󵄨󵄨γ(x, z)󵄨󵄨󵄨; x ∈ Rd , z ∈ int(supp(1 − ϕϵ ) ⋅ h) }. The following estimate, also obtained by Léandre [142, Proposition I.3], is used in the estimate of I[A],t,η (ϵ′ , {s1 , . . . , sk }, x, y): for every p > 1, and every η with S̃ϵ < η, lim sup sup (1/sp ) s→0

y∈Rd

∫

ps (ϵ, 0, x, y)dx < +∞.

(3.28)

{x;|x−y|>η}

Since S̃ϵ → 0 as ϵ → 0, it follows from (3.25), (3.28) that I[A],t,η (ϵ′ , {s1 , . . . , sk }, x, y) ≤ C2 (ϵ′ , η, k, p)t p . Using the inequality (3.27), we can prove a similar estimate for I[D],t,η (ϵ′ , {s1 , . . . , sk }, x, y). Noting Lemma 2.12, we only have to study I[C],t,η (ϵ′ , {s1 , . . . , sk }, x, y) for each small η > 0 and 0 < ϵ′ < ϵ. Put α(x, y) = κ (1 ≤ κ < +∞). Then, we have: Lemma 2.13. If η is small and 1 ≤ i < κ = α(x, y), then I[C],t,η (ϵ′ , St , x, y) = 0

for St ∈ 𝒮t,i ,

(3.29)

∫ I[C],t,η (ϵ′ , St , x, y)d P̃ t,ϵ′ (St ) = 0.

(3.30)

and hence

𝒮t,i

110 � 2 Perturbations and properties of the probability law Proof. Let Qx,η,i and Qx,i be as follows: Qx,η,i ≡ {z ∈ Rd ; ∃(z′0 , z1 , z′1 , . . . , z′i−1 , zi ) ∈ Rd × ⋅ ⋅ ⋅ × Rd , z1 ∈ Pz′ , z2 ∈ Pz′1 , . . . , zi ∈ Pz′ , 0 i−1 󵄨󵄨󵄨x − z′ 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z1 − z′ 󵄨󵄨󵄨 ≤ η, 󵄨󵄨󵄨z2 − z′ 󵄨󵄨󵄨 ≤ η, . . . , |zi − z| ≤ η}, 2 1 0 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨

Qx,i ≡ {z ∈ Rd ; ∃(z1 , . . . , zi−1 ) ∈ Rd × ⋅ ⋅ ⋅ × Rd , z1 ∈ Px , z2 ∈ Pz1 , . . . , zi−1 ∈ Pzi−2 , z ∈ Pzi−1 } = ⋃{Pzi−1 ; z1 ∈ Px , . . . , zi−1 ∈ Pzi−2 }.

Here, we put z0 = x. Then, Qx,i is a closed set in Rd since each Pzj is compact and zj 󳨃→ Pzj is continuous. Observe that Qx,η,i ⊃ Qx,i for all η > 0, and ⋂η>0 Qx,η,i = Qx,i . That is, y ∈ Qx,i if and only if y ∈ Qx,η,i for all η > 0. Since supp(gϵ′ (z′j , ⋅)) ⊂ Pz′ for j = 0, . . . , i − 1, j we observe I[C],t,η (ϵ′ , St , x, y) ≡ ∫ {ps1 (ϵ′ , 0, x, z′0 )gϵ′ (z′0 , z1 )ps2 −s1 (ϵ′ , 0, z1 , z′1 ) A1 (η)

× gϵ′ (z′1 , z2 )ps3 −s2 (ϵ′ , 0, z2 , z′2 ) × ⋅ ⋅ ⋅ × gϵ′ (z′i−1 , zi )pt−si (ϵ′ , 0, zi , y)}dzi dz′i−1 . . . dz1 dz′0 . (3.31) In view of the condition in Qx,η,i , we see that if y ∉ Qx,η,i then I[C],t,η (ϵ′ , St , x, y) = 0.

(3.32)

Recall that α(x, y) = κ and i < κ. By the definition of α(x, y) in Section 2.3.1, we have that y ∉ Qx,i , which implies y ∉ Qx,η,i for every sufficiently small η > 0. Hence, I[C],t,η (ϵ′ , St , x, y) = 0 for St ∈ 𝒮t,i if i < κ = α(x, y). Thus, ∫ I[C],t,η (ϵ′ , St , x, y)d P̃ t,ϵ′ (St ) = 0. 𝒮t,i

Lemma 2.14. Let 𝒦 be a class of nonnegative, equicontinuous, uniformly bounded functions whose supports are contained in a fixed compact set K. Let the constants η > 0 and ϵ′ > 0 be those appearing in I[C],t,η (ϵ′ , St , x, y) and (3.32), respectively. Then, for every δ > 0, there exists t0 > 0 such that sup s≤t

∫

f (z)pt−s (ϵ′ , 0, z, y)dz ≤ f (y) + δ

(3.33)

{z;|z−y|≤η}

for every f ∈ 𝒦, y ∈ K and every t ∈ (0, t0 ). Proof. For a given δ > 0, there exists η1 = η1 (K, 𝒦, δ) > 0 such that |f (z) − f (y)| < δ/4 for each |z − y| < η1 . We may assume η1 ≤ η by choosing small δ. Then, we have

2.3 Methods of finding the asymptotic bounds (II)

sup s≤t

f (z)pt−s (ϵ′ , 0, z, y)dz ≤ sup

∫

s≤t

{z;|z−y|≤η}

� 111

f (z)pt−s (ϵ′ , 0, z, y)dz

∫ {z;|z−y|≤η1 }

+ sup s≤t

∫

f (z)pt−s (ϵ′ , 0, z, y)dz.

(3.34)

{z;η1 0. As for the second term, since η > 0 is arbitrary in (3.14) and since all f ’s in 𝒦 are uniformly bounded, there exists t2 > 0 such that sup s≤t

∫

f (z)pt−s (ϵ′ , 0, z, y)dz ≤ δ/2

{z;η1 0, we have the assertion. ̄ Choose an arbitrary compact neighborhood U(x) of x and arbitrary compact sets ̄ K1 , . . . , Kκ−1 of Rd such that {z; |z − x| ≤ η} ⊂ U(x) and that Qx,η,i ⊂ Ki , i = 1, . . . , κ − 1. Set ̄ 𝒦 = {gϵ′ (z0 , ⋅), gϵ′ (z1 , ⋅), . . . , gϵ′ (zκ−1 , ⋅); z0 ∈ U(x), z1 ∈ K1 , . . . , zκ−1 ∈ Kκ−1 }. ′

′

′

′

′

′

To apply Lemma 2.14, we should be a little more careful since ϵ′ > 0 depends on the choice of η > 0 by Lemma 2.12. Since 𝒦 has the property in Lemma 2.14, for a given δ > 0, η > 0, ϵ′ > 0, there exists t0 > 0 such that for every 0 < t < t0 , sup s≤t

gϵ′ (z′i−1 , zi )pt−s (ϵ′ , 0, zi , z′i )dzi ≤ gϵ′ (z′i−1 , z′i ) + δ,

∫

(3.35)

{|zi −z′i |≤η}

̄ for z′0 ∈ U(x), z′i−1 ∈ Ki−1 , i = 2, . . . , κ, z′κ = y. From (3.35), we have, in view of (3.31), for 0 < t < t0 , I[C],t,η (ϵ′ , {s1 , . . . , sκ }, x, y) ≤ ∫ ps1 (ϵ′ , 0, x, z′0 )dz′0 ∫ dz′1 ⋅ ⋅ ⋅ ∫ dz′κ−1 {(gϵ′ (z′0 , z′1 ) + δ) ⋅ ⋅ ⋅ (gϵ′ (z′κ−1 , y) + δ)}. K1

̄ U(x)

Kκ−1

Hence, by (3.17), κ

t

t

κ

lim sup(1/t ) ∫ ⋅ ⋅ ⋅ ∫(∫ ϕϵ′ (z) h(z)dz) I[C],t,η (ϵ′ , {s1 , . . . , sκ }, x, y)dsκ ⋅ ⋅ ⋅ ds1 t→0

0

0

κ

≤ (∫ ϕϵ′ (z) h(z)dz)

(3.36)

112 � 2 Perturbations and properties of the probability law × ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzκ−1 {(gϵ′ (x, z1 ) + δ)(gϵ′ (z1 , z2 ) + δ) ⋅ ⋅ ⋅ (gϵ′ (zκ−1 , y) + δ)}. K1

K2

Kκ−1

(3.37) Since δ > 0 and K1 , . . . , Kκ−1 are arbitrary, we have, in view of Lemma 2.10 (with N = κ+1), Lemmas 2.11–2.13 and (3.10), (3.11), (3.26), that lim sup( t→0

≤

1 )p (x, y) tκ t

κ

1 (∫ ϕϵ′ (z) h(z)dz) ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzκ−1 {gϵ′ (x, z1 )gϵ′ (z1 , z2 ) ⋅ ⋅ ⋅ gϵ′ (zκ−1 , y)}. κ! Px

Pz1

Pzκ−2

(3.38)

Letting ϵ′ → 0, we have, in view of (3.12), lim sup( t→0

1 1 )p (x, y) ≤ ∫ dz1 ∫ dz2 ⋅ ⋅ ⋅ ∫ dzκ−1 {g(x, z1 )g(z1 , z2 ) ⋅ ⋅ ⋅ g(zκ−1 , y)}. (3.39) tκ t κ! Px

Pz1

Pzκ−2

The proof for the upper bound is now complete. The statement of Theorem 2.4 is immediate from (3.22) (with k = α(x, y)) and (3.39). The condition on the continuity of the Lévy measure well illustrates the nature of the polygonal method. 2.3.3 Example of Theorem 2.4 – easy cases In this subsection, we are concerned with the constant in the upper bound of pt (x, y) as t → 0, mentioned in Theorem 2.4. We shall restrict ourselves to a simple Lévy process to exhibit the virtue of the polygonal method. Example 2.2. Let d = m = 2, and let a smooth radial function η satisfy supp η = {x; |x| ≤ 1} and η(x) ≡ 1 in {x; |x| ≤ 1/2}. Put h(z) = η(z)|z|−2−α , α ∈ (0, 1),

for z ∈ R2 \ {0},

(3.40)

that is, h(z)dz is the Lévy measure for a truncated stable process (cf. [74, Section 3]) with index α ∈ (0, 1). Then, h satisfies (3.1) with c = 1 and (3.2). Let γ(x, z) = z, and let xt (x) be given by xt (x) = x + ∑ Δz(s). s≤t

(3.41)

Then, Px = x + supp h = x + B(1) (B(1) ≡ {x; |x| ≤ 1}), and g(x, z) is reduced to g(x, z) = h(z − x) (cf. (3.4)).

2.3 Methods of finding the asymptotic bounds (II)

�

113

Let x0 = (0, 0) and choose y0 = (e, 0) so that 1 < e < 2. We then have α(x0 , y0 ) = 2. The constant C(x0 , y0 , 2) is calculated as follows: C(x0 , y0 , 2) = ∫ g(x0 , z)g(z, y0 )dz = ∫ h(z − x0 )h(y0 − z)dz. Px0

(3.42)

B(1)

The integral (3.42) makes sense. Indeed, if z = 0, then h(y0 − 0) = h(y0 ) = 0. Since y0 ∈ {x; |x| > 1}, by the continuity of x 󳨃→ Px , there exists δ > 0 such that if |z − x0 | < δ, and then y0 ∉ supp(g(z, ⋅)). That is, g(z, y0 ) = 0 for z ∈ {z; |z − x0 | < δ}, and this implies that the integral exists. Example 2.3. Let m = d. Let z1 (t), . . . , zd (t) be an independent one-dimensional truncated stable processes (stationary martingales with jumps), whose common characteristic function is given by ψt (v) = exp[t ∫ (ei⟨z,v⟩ − 1 − i⟨z, v⟩)h(z)dz]. R\{0}

That is, t+

zj (t) = ∫ ∫ zÑ j (dsdz),

j = 1, . . . , d,

0 R\{0}

where Ñ j (dsdz)’s denote the compensated Poisson random measures on [0, +∞) × (R \ {0}), having the common mean measure h(z)dz×ds, i. e., Ñ j (dsdz) = Nj (dsdz)−h(z)dz×ds. Here, the Lévy measure h(z)dz is given by h(z) = η(z)|z|−1−α ,

α ∈ (0, 2), z ∈ R \ {0},

(3.43)

where η(z) ∈ C0∞ (R), 0 ≤ η(z) ≤ 1, supp η = {z; |z| ≤ c}, and η(z) ≡ 1 in {z; |z| ≤ c2 } for some 0 < c < +∞. Let e0 , e1 , . . . , ed be constant vectors in Rd such that ⟨e1 , . . . , ed ⟩ forms a basis of T0 Rd . Given x ∈ Rd , consider the jump process xt (x) given by d

c

xt (x) = x + ∑ ∑ ej Δzj (s) + e0 t = x + e1 z1 (t) + ⋅ ⋅ ⋅ + ed zd (t) + e0 t, j=1 s≤t

(3.44)

where ∑c denotes the compensated sum stated in Section 1.3. As a linear combination of truncated stable processes z1 (t), . . . , zd (t) and a uniform motion e0 t, xt (x) possesses a transition density pt (x, y) which is C ∞ with respect to y (y ≠ x) for t > 0 (cf. [143]). In [92], the lower bound of pt (x, y) as t → 0 for given distinct x, y is obtained for processes including the above type. However, the analysis to obtain the upper bound

114 � 2 Perturbations and properties of the probability law for pt (x, y) is somewhat complicated. This is because in order to get the upper bound, we have to track all the possible trajectories of xt (x), each having a Lévy measure which is singular with respect to the d-dimensional Lebesgue measure. To do this, we make use of a kind of homogeneity of the process, which is due to the vector fields e1 , . . . , ed being constant, and make a combinatorial discussion. For this purpose, we prepare the following notation. Let ℐ (d) = {(i1 , . . . , id ); i1 , . . . , id = 0 or 1}.

(3.45)

For I = (i1 , . . . , id ) ∈ ℐ (d), we put |I| = the length of I = i1 +⋅ ⋅ ⋅+id . We put I ′ = (i1′ , . . . , id′ ) ∈ ℐ (d) for I = (i1 , . . . , id ) in the following way: ij′ = 1 if ij = 0, and ij′ = 0 otherwise. We write I = (i1 , . . . , id ) ≤ J = (j1 , . . . , jd ) for I, J ∈ ℐ (d) if ir ≤ jr , r = 1, . . . , d. Consider a sequence b = (I1 , . . . , Iℓ ) of elements of I’s in ℐ (d). We denote by 𝒮 (k, ℓ) the following family of sequences: 𝒮 (k, ℓ) = {b = (I1 , . . . , Iℓ ); |I1 | + ⋅ ⋅ ⋅ + |Iℓ | = k, |Ij | ≥ 1, j = 1, . . . , ℓ}.

(3.46)

For b = (I1 , . . . , Iℓ ) ∈ 𝒮 (k, ℓ), we put the weight ω(b) and the α-weight ωα (b) of indices I1 , . . . , Iℓ as follows: ω(b) = |I1 | + ⋅ ⋅ ⋅ + |Iℓ |, ℓ

󵄨 󵄨 ωα (b) = ∑ (|Ir | − 󵄨󵄨󵄨Ir′ 󵄨󵄨󵄨/α). r=1

Consider the parallelepiped of Rd built on the basis ⟨e1 , . . . , ed ⟩ Q = {t1 e1 + ⋅ ⋅ ⋅ + td ed ; tj ∈ (supp h), j = 1, . . . , d}, and put Px = x + Q. Similarly, define sets QI = {ti1 e1 + ⋅ ⋅ ⋅ + tid ed ; tir ∈ (supp h) if ir = 1, tir = 0 if ir = 0} and PxI = x +QI for I = (i1 , . . . , id ) ∈ ℐ (d). Here PxI and QI are special forms on complexes, and continuous functions on it are denoted, e. g., by C(PxI ). For each y ∈ Rd (y ≠ x), we put β(x, y) ≡ ℓ0 (x, y) + 1.

(3.47)

Here, ℓ0 (x, y) denotes the minimum number of distinct points z1 , . . . , zℓ ∈ Rd such that z1 ∈ Px , z2 ∈ Pz1 , . . . , zℓ ∈ Pzℓ−1 , and y ∈ Pzℓ . This means that y ∈ Rd can be joined to x by a polygonal line with ℓ0 (x, y)-corners passing along the combination of the parallelepipeds Px ∪ Pz1 ∪ ⋅ ⋅ ⋅ ∪ Pzℓ (x,y) . 0

2.3 Methods of finding the asymptotic bounds (II)

�

115

Furthermore, we introduce jump weights to go to y from x ̃ y) = min{ω(b); b = (I , . . . , I ) ∈ (ℐ (d))ℓ , ℓ = β(x, y), β(x, 1 ℓ

such that z1 ∈ PxI1 , z2 ∈ PzI21 , . . . , zℓ−1 ∈ PzIℓ−1 ℓ−2

and y ∈ PzIℓℓ−1 for some z1 , . . . , zℓ−1 ∈ Rd },

(3.48)

̃ y)}. ρα (x, y) = min{ωα (b); b = (I1 , . . . , Iℓ ) ∈ (ℐ (d)) , ℓ = β(x, y), ω(b) = β(x, ℓ

(3.49)

̃ y) ≤ dβ(x, y) with β(x, y) < +∞ (since c > 0), and that ρ (x, y) ≤ We remark that β(x, α ̃β(x, y). In case c = +∞, we would always have β(x, y) = 1 (x ≠ y). We fix x ∈ Rd and I = (i1 , . . . , id ) ∈ ℐ (d) in what follows. We assume |I| = b, and denote by i1∗ , . . . , ib∗ the nonzero indices ir arrayed in numerical order. Consider the mapping b

HxI : (supp h)|I| → PxI , (ti1∗ , . . . , ti∗ ) 󳨃→ x + ∑ tir∗ eir∗ . b

(3.50)

r=1

This introduces the Lévy measure dμIx on PxI by ∫ f (z)dμIx (z) = ∫ ⋅ ⋅ ⋅ ∫ f (x + ti1∗ ei1∗ + ⋅ ⋅ ⋅ + ti∗ ei∗ )h(ti1∗ ) ⋅ ⋅ ⋅ h(ti∗ )dti1∗ ⋅ ⋅ ⋅ dti∗ , b

PxI

b

b

b

(3.51)

̄ for simplicity. Here, dz̄ denotes the volume for f ∈ C(PxI ). We abbreviate this by μIx (dz) I element induced on Px by ∗ 󵄨 ∗ 󵄨 dz̄ = 󵄨󵄨󵄨dei1 ∧ ⋅ ⋅ ⋅ ∧ deib 󵄨󵄨󵄨,

z ∈ PxI

(3.52)

where (dei1 , . . . , deib ) denotes the covector field on PxI corresponding to (ei1∗ , . . . , ei∗ ). ∗

∗

b

Since dμIx (z) is the measure transformed from h(t1 ) ⊗ ⋅ ⋅ ⋅ ⊗ h(tb )dt1 ⋅ ⋅ ⋅ dtb by the linear mapping HxI : t (t1 , . . . , tb ) 󳨃→ x + AI t (t1 , . . . , tb ),

(AI ≡ (ei1∗ , . . . , ei∗ )), it possesses the density with respect to dz,̄ which is C ∞ with respect b

to z ∈ PxI \ {x} and x ∈ Rd . We shall put

116 � 2 Perturbations and properties of the probability law

g I (x, z) =

dμIx (z) if z ∈ PxI \ {x}, ̄dz

g I (x, z) = 0

otherwise.

(3.53)

Explicitly, −1 󵄨 −1 󵄨 g I (x, z) = (h ⊗ ⋅ ⋅ ⋅ ⊗ h)((HxI ) (z))󵄨󵄨󵄨[JHxI ] 󵄨󵄨󵄨,

z ∈ PxI \ {x}.

(3.54)

Obviously, we have for I = (1, 1, . . . , 1), g I (x, z) = g(x, z), the density of the original Lévy measure of xt (x). Proposition 2.7. For each x, y ∈ Rd (y ≠ x), suppose that the numbers λ = β(x, y), κ = ̃ y), and ρ = ρ (x, y), defined as above, are finite. If κ ≥ d − 1, then we have β(x, α lim sup( t→0

1 )p (x, y) ≤ C. tρ t

The constant C has an expression C=

∑ (I1 ,...,Iℓ )∈𝒮(κ,ℓ), λ≤ℓ≤κ,I1 ≤⋅⋅⋅≤Iℓ

̄ ⋅ ⋅ ⋅ ∫ dz ̄ g I1 (x, z ) ⋅ ⋅ ⋅ g Iℓ (z , y)}. c(I1 , . . . , Iℓ ){ ∫ dz 2 ℓ 2 ℓ I

I

Px1

Pzℓ−1 ℓ−1

Here, we put c(I1 , . . . , Iℓ ) = (κ! ∏dj=1 (1/rj !))c∗ (κ, ℓ), where rj ≡ #{r ∈ {1, . . . , ℓ}; ir,j = 1} for Ir = (ir,1 , . . . , ir,d ) and c∗ (κ, ℓ) = lim sup( t→0

|I1 |−1−|I1′ |/α

{s1

t

t

t

0

s1

sℓ−1

1 ) ∫ ds1 ∫ ds2 ⋅ ⋅ ⋅ ∫ dsℓ tρ (s2 − s1 )|I2 |−1−|I2 |/α ⋅ ⋅ ⋅ (sℓ − sℓ−1 )|Iℓ |−1−|Iℓ |/α }. ′

′

The proof of this proposition depends on Léandre’s method in [142], and on some discussions in [92, 93]. A crucial idea in our proof is the (intuitive) notion of polygonal trajectories. A polygonal trajectory is a trajectory obtained from a piecewise linear curve whose segment from a corner point zi to the next corner point zi+1 is parallel to some combination of vectors contained in {e1 , . . . , ed }. It should be noted that in a small time interval, there is a tendency that the process xt (x) makes use of those trajectories which jump as few times as possible and stay at a point until just before the moment of the next jump. Intuitively, β(x, y) above can be interpreted as the minimum number of jumps by ̃ y) corresponds which the trajectory can reach y from x along a polygonal line, and β(x, to the weight of such a line. We need another notion of α-weight ρα (x, y) of a line for considering the factors which ‘stay’ in the same corner point at each jump moment, and this gives the real power in t. The decomposition of jumps to those factors which really jump and to those that stay in is carried out in terms of indices I and I ′ .

2.3 Methods of finding the asymptotic bounds (II)

� 117

Remark 2.7. Here xt (x) in (3.44) has a drift term c0 t. The conclusion of Theorem 1.4 is indifferent of it as it is an estimate as t → 0. If the support of Lévy measure is more general, e. g., it is composed of point masses, we face a more complex geometry on support of jumps; see Section 2.5.3. Examples of Proposition 2.7 We shall provide some simple examples of Proposition 2.7. Example 2.4. Let d = 2, c = 1, e1 = 𝜕x𝜕 , e2 = 𝜕x𝜕 , x0 = (0, 0), and y0 = (e, f ), 0 < e < 1 2 ̃ , y ) = 2, and we have 𝒮 (2, 1) = {((1, 1))}, and hence 1, 0 < f < 1. Clearly, β(x0 , y0 ) = 1, β(x 0 0 ρα (x0 , y0 ) = 2. We have C = C(x0 , y0 , 2, 1, 2) = c((1, 1))g (1,1) (x0 , y0 ) = h(e)h(f ) < +∞. In this case, we know exactly that pt (x0 , y0 ) = ℘t (e)℘t (f ) ∼ h(e)h(f )t 2 , as t → 0 (cf. [92]). Here ℘t (⋅) denotes the transition density of a 1-dimensional standard stable density of index α. Hence, the constant is the best possible. Example 2.5. Let d = 2, c = 1, e1 = 𝜕x𝜕 , e2 = 𝜕x𝜕 , x0 = (0, 0), and y0 = (e, 0), 0 < e < 1. 1 2 ̃ , y ) = 1, and we have 𝒮 (1, 1) = {((1, 0)), ((0, 1))} ≡ {(I ), (I )}, and Clearly, β(x0 , y0 ) = β(x 0 0 1 2 hence ρα (x0 , y0 ) = 1 − 1/α. We have C = C(x0 , y0 , 1, 1, 1 − 1/α) =

∑

c(I)g I (x0 , y0 )

(I)∈𝒮(1,1)

I1

I2

= c(I1 )g (x0 , y0 ) + c(I2 )g (x0 , y0 ) =(

1

1−

1 α

)h(e) < +∞

I

since y0 ∉ Px20 \ {x0 }. In this case, we know exactly pt (x0 , y0 ) = ℘t (e)℘t (0) ∼ as t → 0 (cf. [223, (2.2.11)]). Here, π1 Γ(1 + α1 ) = and hence our constant is overestimated.

Example 2.6. Let d = 2, c = 1, e1 = 𝜕x𝜕 , e2 = 1 ̃ , y ) = 2. We have Then, β(x , y ) = β(x 0

0

0

1 1 1 Γ(1 + )h(e)t 1− α , π α

1 1 Γ( 1 ) πα α 𝜕 , 𝜕x2


t 1/β }≍ log( ). t

(4.4)

Case that the coefficients are nondegenerate Proposition 2.10 ([95, 98]). (1) (off-diagonal estimate) (a) Assume y ∈ 𝒮 , that is, α(x, y) < +∞. Then, we have pt (x, y)≍t α(x,y)−d/β

as t → 0.

(4.5)

(b) Assume y ∈ 𝒮 ̄ \ 𝒮 (α(x, y) = +∞). Suppose b(x) ≡ 0 and let β′ > β. Then, log pt (x, y) is bounded from above by the expression of type Γ = Γ(t), N

Γ ≡ − min ∑ (wn log( n=0

1 1 1 ) + log(wn !)) + O(log( ) log log( )), tkn t t

and is bounded from below by the expression of type 2

1 1 −(c/β)(log( )) + O(log( )) t t

(4.6)

122 � 2 Perturbations and properties of the probability law with some c > 0 independent of β and y, as t → 0. Here, the minimum in (4.6) is taken with respect to all choices of a0 , . . . , aN by ξn for n = 1, 2, . . . , n1 and n1 ∈ N such that 󵄨 󵄨󵄨 1/β′ 󵄨󵄨y − An1 (x, ξ1 , . . . , ξn1 )󵄨󵄨󵄨 ≤ t

(4.7)

d+m×n where n1 = ∑Nn=0 wn for which wn = # of an in the choice. Here (An )∞ → Rd n=0 , An : R are functions defined by

A0 (x) = x,

{

An+1 (x, x1 , . . . , xn+1 ) = An (x, x1 , . . . , xn ) + γ(An (x, x1 , . . . , xn ), xn+1 ).

(4.8)

(2) (diagonal estimate) pt (x, x)≍t

− dβ

as t → 0 uniformly in x.

(4.2)′′

These results have been mentioned in Theorems 2.2(b), 2.3. Case that the coefficients are degenerate (The result has been obtained in the case that Xt is a canonical process as defined in Section 2.5.1. See [106] for a recent case of Itô process.) (1) (off-diagonal estimate) Consider, in particular, a singular Lévy measure m

m

j=1

j=1

μ = ∑ μ̃ j = ∑ Tj∗ μj .

(4.9)

Here, Tj∗ μj = μj ∘ Tj−1 , μj is a 1-dimensional Lévy measure of form ∑∞ n=0 kn δ{±ã n } (⋅) and Tj : zj 󳨃→ (0, . . . , 0, zj , 0, . . . , 0). We let ∞

μj (.) = ∑ kn (δ{ãn } (⋅) + δ{−ãn } (⋅)), n=0

j = 1, . . . , m,

(4.10)

where kn = pnβ ,

β ∈ (0, 2),

and

ã n = p−n .

Here, p denotes an arbitrary prime number. Proposition 2.11 ([97]). Let μ be given by (3.8) above. Let y ≠ x. Under certain assumptions, we have the following estimates for the density pt (x, y):

2.5 Auxiliary topics

� 123

(a) Assume y ∈ 𝒮 , that is, α(x, y) < +∞. Then, we have pt (x, y) ≤ Ct α(x,y)−d/β

as t → 0.

(b) Assume y ∈ 𝒮 ̄ \ 𝒮 (α(x, y) = +∞). Then, log pt (x, y) is bounded from above by the expression of type 1 1 Γ(t) + O(log( ) log log( )) t t

(4.11)

as t → 0. Here, α(x, y) is defined as above, and Γ(t) is given similarly to (4.6). Namely, the chain : Rd+m×n → Rd be a deterministic chain given by

∞ (An )∞ n=0 above is replaced by (Cn )n=0 given below: let Cn

C0 (x) = x,

Cn+1 (x; x1 , . . . , xn+1 ) = Cn (x; x1 , . . . , xn ) m

+ (∑ xn+1 Xj )(Cn (x; x1 , . . . , xn )) (j)

j=1

m

+ {Exp(∑ xn+1 Xj )(Cn (x; x1 , . . . , xn )) (j)

j=1

m

− Cn (x; x1 , . . . , xn ) − (∑ xn+1 Xj )(Cn (x; x1 , . . . , xn ))} (j)

j=1

m

= Exp(∑ xn+1 Xj )(Cn (x; x1 , . . . , xn )). (j)

j=1

(4.12)

(2) (diagonal estimate) pt (x, x)≍t

− βd

as t → 0 uniformly in x.

(4.2)′′′

2.5 Auxiliary topics In this section, we state some topics which are related, but not directly connected, to the motif of Malliavin calculus of jump type. In Section 2.5.1, we state Marcus’ canonical processes. In Section 2.5.2, we state the absolute continuity of the infinitely divisible laws in an elementary way. In Section 2.5.3, we state some examples of chain movement approximations in Section 2.2.2. In Section 2.5.4, we state the support theorem for canonical processes. The proofs of the results in Sections 2.5.2, 2.5.4 are carried out by utilizing classical analytic methods, and we omit the details.

124 � 2 Perturbations and properties of the probability law 2.5.1 Marcus’ canonical processes In this section, we introduce Marcus’ canonical process [161], or geometric process, and the stochastic analysis on it. Let X1 , . . . , Xm be C ∞ , bounded vector-valued functions (viewed as the vector fields on Rd ), whose derivatives of all orders are bounded. We assume that the restricted Hörmander condition for X1 , . . . , Xm holds, that is, Lie (X1 , . . . , Xm )(x) = Tx (Rd )

(RH)

for all x ∈ Rd .

We also introduce the space of vectors Σk , k = 0, 1, 2, . . . by Σ0 = {X1 , . . . , Xm } and Σk = {[Xi , Y ]; i = 1, . . . , m, Y ∈ Σk−1 }. Here, [X, Y ] denotes the Lie bracket between X and Y . We say the vectors satisfy the uniformly restricted Hörmander condition if there exist N0 ∈ N and C > 0 such that N0

2

∑ ∑ (v, Y (x)) ≥ C|v|2

(URH)

k=0 Y ∈Σk

for all x ∈ Rd and all v ∈ Rd . Let Yt (x) be an Rd -valued canonical process given by m

dYt (x) = ∑ Xj (Yt− (x)) ∘ dzj (t), j=1

Y0 (x) = x

(5.1)

with z(t) = (z1 (t), . . . , zm (t)). The above notation can be paraphrased in Itô form as m

dYt (x) = ∑ Xj (Yt− (x))dzj (t) j=1

m

m

j=1

j=1

+ {Exp(∑ Δzj (t)Xj )(Yt− (x)) − Yt− (x) − ∑ Δzj (t)Xj (Yt− (x))}.

(5.2)

Here, the second term is a sum of terms of order O(|Δz(s)|2 ), which in effect takes into account all higher order variations of z(⋅) at time s = t−, and ϕ(t, x) ≡ Exp(tv)(x) is the exponential map by v, i. e., the solution flow of the differential equation

2.5 Auxiliary topics

dϕ (t, x) = v(ϕ(t, x)), dt

� 125

ϕ(0, x) = x.

In (5.1), (5.2), we have omitted the drift (and the diffusion) term(s) for simplicity. In a geometrically intrinsic form on a manifold M = Rd , the process Yt can be written by m

dYt = Exp(∑ dzj (t)Xj (Yt− ))(x), j=1

Y0 (x) = x.

According to this model, a particle jumps from x along the integral curve (geometric flow) described by ϕ(s, x) instantly, that is, the particle moves along ϕ(s, x) during 0 ≤ s ≤ 1 outside of the real time t and ends up at Exp(sv)(x)|s=1 . In this sense, a canonical process Yt is a real jump process with a fictitious auxiliary process, and Yt is connected to this continuous process through the fictitious time. The fictitious time can also be viewed as idealized time which one cannot observe. In this perspective, a canonical process is a continuous process on the idealized time interval which is partially observed. See [216] for this interpretation. This idealized situation appears in the stochastic theory of geometry [10, 62]. The equivalence between (5.1) and (5.2) implies that the canonical process is also an Itô process (refer to (5.3) below). In [137], the process Yt (x) is called ‘Stratonovich’ stochastic integral due to the last correction term on the right-hand side of (5.2). However, it is different from the original Stratonovich integral. In [137], they used this name since two integrals resemble each other in spirit. The difference lies in the fact that Yt (x) uses a geometric flow as a correction term instead of the algebraic average in the original one. We mimic their notation ∘dz(t) in this book. (Some papers use the notation ⬦dz(t) instead of it.) The SDE (5.1) has a unique solution which is a semimartingale [137, Theorem 3.2]. By pt (x, dy) we denote the transition function of Yt (x). The main point of introducing this process is that the solutions are the flows of diffeomorphisms in the sense explained in Section 1.3 (cf. [127, Theorem 3.1], [137, Theorem 3.9]). It is shown (cf. [137, Lemma 2.1]) that Yt (x) can be represented as m t

t

j=1 0

0

Yt (x) = x + ∑ ∫ Xj (Ys− (x))dzj (s) + ∫ h(s, Ys− (x))d[z]ds

(5.3)

where h(s, x) =

m Exp(∑m j=1 Δzj (s)Xj )(x) − x − ∑j=1 Δzj (s)Xj (x)

|Δz(s)|2

d is a Lipschitz process in s with a bounded Lipschitz constant, and [z]ds = ∑m j=1 [zj , zj ]s denotes the discontinuous part of the quadratic variation process of z(⋅). An intuitive meaning of this formula is as follows. At each jump time t of z, we open a unit length

126 � 2 Perturbations and properties of the probability law interval of fictitious time, over which the state Yt− (x) changes continuously along the integral curve Exp(r ∑m j=1 Δzj (t)Xj )(⋅), 0 ≤ r ≤ 1, to the state m

Yt (x) = Exp(∑ Δzj (t)Xj )(Yt− (x)). j=1

We see that the jump is not linear. Viewing this expression, the canonical process Yt seems to be near to the Fisk– Stratonovich integral of h (cf. [189, Chapter V.5]). However, it differs from the Fisk– Stratonovich integral in that the latter cares only for the continuous part of the quadratic variation process of z(⋅), whereas the former also cares for the discontinuous part of it. Hence, an intuitive view of the canonical process is that as the Itô integral takes care of, up to the second order of discontinuities, Δz(t) of the integrator, the canonical process also takes care of the higher orders (Δz(t))r , r = 1, 2, . . . of discontinuities in terms of the exponential function in h(t, x). This enables us to naturally treat the jump part geometrically. See the support theorems in Section 2.5.4. For the Jacobian process ∇Yt (x) and its inverse process (∇Yt (x))−1 , we refer to [100, Lemma 6.2] and [131, Lemma 3.1]. For this process, we have the following: Theorem 2.5 ([131, Theorem 1.1]). Under the uniformly restricted Hörmander condition (URH) on (Xj , j = 1, . . . , m) and a nondegeneracy condition (A.0) on μ, the density of the law of Yt exists for all x ∈ Rd and all t > 0: pt (x, dy) = pt (x, y)dy. Equation (5.2) is a coordinate-free formulation of SDEs on a manifold with jumps. See [137] for the precise definition for this type of integrals for semimartingales. We shall j call it a canonical SDE driven by a vector-field-valued semimartingale Y (t) = ∑m j=1 Z (t)Xj , according to Kunita [127, 128, 131]. See also [148]. 2.5.2 Absolute continuity of the infinitely divisible laws As it is called, the infinitely divisible law (ID) on R is the probability law which has power-roots of any order in the sense of convolutions. That is, π is an ID law if for any n, there exists a probability law πn such that π = πn ∗ ⋅ ⋅ ⋅ ∗ πn .

(5.4)

We recall the Lévy–Khintchine formula which characterizes the Fourier transform of an ID law. The law π is an ID law if and only if there exist a ∈ R, b ≥ 0, and a σ-finite measure μ satisfying ∫R\{0} (1 ∧ |x|2 )μ(dx) < ∞, for which π can be written as ̂ π(λ) = exp[iaλ −

b 2 λ2 + ∫ (eiλx − 1 − iλx1{|x|≤1} )μ(dx)]. 2 R\{0}

(5.5)

2.5 Auxiliary topics

� 127

By (5.5), this ID law is the law of X1 . Here, (Xt )t≥0 is a Lévy process. More precisely, this process is given by the Lévy–Itô decomposition t

t

̃ Xt = at + bWt + ∫ ∫ x N(dsdx) + ∫ ∫ xN(dsdx). 0 |x|≤1

(5.6)

0 |x|>1

Here, W is a real standard Brownian motion, N is a Poisson measure which has the density dsμ(dx) on R+ × R. Here, we see by the infinite divisibility that for a Lévy process Xt such that X1 ∼ π, every Xt has an infinitely divisible law πt , and it is given by π̂ t (λ) = exp t[iaλ −

b2 λ2 + ∫ (eiλx − 1 − iλx1{|x|≤1} )μ(dx)]. 2

(5.7)

R\{0}

As we shall see below, the parameter t is important to inquire into the absolute continuity of the ID law. The next theorem due to W. Döblin characterizes the continuity of the ID laws. Theorem 2.6. Let π be an ID law whose characteristic function is given by (5.5). Then π is continuous if and only if b ≠ 0 or μ is an infinite measure. Then, we have the following corollary. Corollary 1. Let π be an ID law whose characteristic function is given by (5.5). Then π is discrete if and only if b = 0 and μ is a finite or discrete measure. In what follows, we assume π contains no Gaussian part (b = 0). For a Lévy meā sure μ, let μ(dx) = (|x|2 ∧ 1)μ(dx). Theorem 2.7 (Sato [192]). Let π be an ID law whose characteristic function is given by (5.5). Assume b = 0 and that μ is an infinite measure. Suppose that there exists some p ∈ N such that μ̄ ∗p is absolutely continuous. Then, π is absolutely continuous. This theorem assumes a (very weak) smoothness for the Lévy measure. This assumption is used in stochastic calculus of variations developed by J.-M. Bismut (cf. [154]). A weak point of this theorem is that it excludes the discrete Lévy measure. The following theorem admits the case for a discrete Lévy measure and holds under a weaker assumption on small jumps. Theorem 2.8 (Kallenberg [113]). Let π be an ID law whose characteristic function is given by (5.5), and let b = 0. If ̄ lim ϵ−2 | log ϵ|−1 μ(−ϵ, ϵ) = +∞,

ϵ→0

then π has a C ∞ density whose derivatives of all orders decrease at infinity.

(5.8)

128 � 2 Perturbations and properties of the probability law We introduce a weaker assumption ̄ lim inf ϵ−2 | log ϵ|−1 μ(−ϵ, ϵ) > 2. ϵ→0

(5.9)

Since cos x ∼ 1 − x 2 /2(x → 0), it follows from the above calculation that 󵄨󵄨 ̂ 󵄨󵄨 −(1+ϵ)/2 , 󵄨󵄨π(z)󵄨󵄨 ≤ |z|

|z| → ∞,

and hence π is absolutely continuous by a Fourier lemma. Conversely, let π be the law of X1 . We do not know a priori if X1/2 is absolutely continuous under (5.9). Indeed, we only see by (5.7) that 󵄨󵄨 ̂ 󵄨 −(1+ϵ)/2 , 󵄨󵄨π1/2 (z)󵄨󵄨󵄨 ≤ |z|

|z| → ∞,

whereas π̂ 1/2 may not be integrable. As we shall see in Theorem 27.23 in [193], it happens that X1 is absolutely continuous and X1/2 is not. The condition (5.9) is very weak; apart from the condition on the logarithmic rate of ̄ convergence of μ(−ϵ, ϵ) to 0 as ϵ → 0, it is quite near to the infiniteness of μ. Indeed, if ̄ lim inf ϵ−2 μ(−ϵ, ϵ) > 0,

(5.10)

ϵ→0

then μ is an infinite measure. However, the converse of this assertion is false, and there exists an infinite measure which does not satisfy (5.10) or (5.8). Proposition 2.12. For each k, there exists an infinite measure μ such that ̄ lim inf ϵ−(2+k) μ(−ϵ, ϵ) = 0. ϵ→0

By considering a sequence which grows sufficiently rapidly, we can control the rate of convergence as we wish. The proposition seems to contradict the assertion of Kallenberg (cf. [113, Section 5]), however, it is repeatedly stated in the introduction in [165]. This is since the infinite measure satisfies ̄ lim ϵ−2 | log ϵ|r μ(−ϵ, ϵ) = +∞,

ϵ→0

r > 1.

Given Theorem 2.6, we can consider the following condition which is stronger with respect to the infiniteness of μ: ∫ |z|α μ(dz) = +∞,

α ∈ (0, 2).

|z|≤1

It seems that this condition may lead to the absolute continuity of π in the case that (5.10) fails. However, it is not true as we see in the following

2.5 Auxiliary topics

� 129

Proposition 2.13 (Orey [175]). For any α ∈ (0, 2), there exists a Lévy measure μ such that ∫ |z|α μ(dz) = +∞

and

∫ |z|β μ(dz) < ∞,

β>α

(5.11)

|z|≤1

|z|≤1

for which the ID law π given by (5.5) under the condition b = 0 is not absolutely continuous. It may seem that in Orey’s theorem a special relation between atoms in the Lévy measure and their weights might be preventing π from being absolutely continuous. In reality, there is no interplay between them. We can show the following counterexample. Proposition 2.14 (Watanabe [214]). Let c be an integer greater than 2, and let the Lévy measure be +∞

μ = ∑ an δ{c−n } . n=0

Here, we assume {an ; n ≥ 0} to be positive and bounded. Then, the ID law π, given by (5.5) and the condition b = 0, is not absolutely continuous. In Propositions 2.13, 2.14, the support of μ constitutes a geometric sequence, and the integer character c plays an important role. Proposition 2.14 holds valid even if c is a Pisot–Vijayaraghavan number [214]. This is the number u ∈ (1, ∞) such that for some integer-coefficient polynomial (with the leading coefficient being 1), it satisfies F(u) = 0, where the absolute values of all the roots of F = 0 except u are strictly smaller than 1. For example, u = (√5 + 1)/2 with F(X) = X 2 − X − 1. By a theorem of Gauss, Pisot– Vijayaraghavan numbers are necessarily irrational. In the case that the support of the Lévy measure consists of a geometric sequence, several families of algebraic numbers play an important role so that π becomes absolutely continuous. For a recent development of this topic, see [215]. Unfortunately, no necessary-and-sufficient condition is known for μ with respect to this family of numbers so that the corresponding ID law becomes absolutely continuous. The following criterion is known: Proposition 2.15 (Hartman–Wintner). Let μ be an infinite discrete Lévy measure, and let π be an ID law given by (5.5). Then, π is pure and is either singular continuous or absolutely continuous. For the proof of this theorem, see Theorem 27.16 in [193]. (It is a special case focussed on the discrete measure, and the proof is easy.) Some results in the multidimensional case To make representations simpler, we study ID laws having no Gaussian part. In (Rd , ‖⋅‖), let the Fourier transform π̂ of an ID law π be

130 � 2 Perturbations and properties of the probability law ̂ π(λ) = exp[i⟨a, λ⟩ + ∫ (ei⟨λ,x⟩ − 1 − i⟨λ, x⟩1{|x|≤1} )μ(dx)],

λ ∈ Rd .

(5.12)

Rd \{0}

Here, a ∈ Rd , and π is a positive measure such that ∫(|x|2 ∧ 1)μ(dx) < ∞, and ⟨⋅, ⋅⟩ is an inner product corresponding to |⋅|. Many results in the previous section also hold true in the multidimensional case since they do not employ the speciality that the dimension ̄ is one. For example, Theorems 2.6 and 2.7, where μ(dx) = (|x|2 ∧ 1)μ(dx), hold true in any dimensions. Conversely, under the assumption that ̄ ϵ ) = +∞, lim ϵ−2 | log ϵ|−1 μ(B

ϵ→0

(5.13)

where Bϵ denotes the ball with the center at the origin and radius ϵ, Theorem 2.8 is no longer true. This is because 1 󵄨 󵄨2 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨π(z)󵄨󵄨 ≤ exp −[ ∫ 󵄨󵄨󵄨⟨u, z⟩󵄨󵄨󵄨 μ(du)], 3 B1/z

and hence we cannot appropriately estimate ⟨u, z⟩ from below. However, if, instead of (5.13), we have the uniform angle condition lim ϵ−2

ϵ→0

󵄨󵄨 󵄨2 󵄨󵄨⟨v, x⟩󵄨󵄨󵄨 μ(dx) = +∞,

∫

uniformly in v ∈ 𝒮 d−1 ,

(5.14)

|⟨v,x⟩|≤ϵ

then again by Theorem 2.8, we can prove that π defined by (5.12) has a C ∞ -density whose derivatives of all orders vanish at infinity. A weak point of Theorem 2.8 is that it requires a certain isotropic property of the measure μ. Indeed, if there exists a hyperplane H such that μ is finite on Rd \ H, then π charges a hyperplane with positive probability and hence it is not absolutely continuous. We say that a measure ρ on S d−1 is radially absolutely continuous if the support of ρ is not contained in any hyperplane, and if for some positive function g on S d−1 × R+ , it holds that ∞

μ(B) = ∫ ρ(dξ) ∫ g(ξ, r)1B (rξ)dr S d−1

(5.15)

0

for all Borel sets B ∈ ℬ(Rd ). This decomposition applies for self-decomposable Lévy measures. For details on this important ID law, see [193, Chapter 15]. We say that μ satisfies the divergence condition if it holds that 1

∫ g(ξ, r)dr = +∞ 0

ρ-a. e.

(5.16)

2.5 Auxiliary topics

� 131

Note that (5.16) is weaker than (5.14) since it just asserts the infiniteness of μ with respect to each direction. Theorem 2.9 (Sato [192]). Let π be an ID law for which the characteristic function given by (5.12) satisfies (5.15), (5.16). Then, π is absolutely continuous. See [193, pp. 178–181] for the proof of this theorem. The proof uses induction with respect to the dimension and Theorem 2.7. The condition (5.15) is a very restrictive assumption since it requires the absolute continuity of the radial part of μ through g. We may consider whether this condition can be weakened. Another weak point of Theorem 2.9 is that it excludes the case that the support consists of curves which are not lines. In order to correspond to this case, we introduce the notion of generalized polar coordinates. Definition 2.3 (Yamazato [217]). We say that μ is absolutely continuous with respect to curves if there exist a finite measure ρ supported on S ⊂ 𝒮 d−1 and a positive function g such that for a Borel set B ∈ ℬ(Rd ), it holds that ∞

μ(B) = ∫ ρ(dξ) ∫ g(ξ, r)1B (φ(ξ, r))dr. S d−1

0

Here, φ : S × (0, +∞) → Rd \ {0} is a generalized polar coordinate in the sense that φ is a measurable bijection, and φ(ξ, 0+) ≡ 0, φ(ξ, 1) ≡ ξ, 𝜕r φ(⋅, ξ) exists and does not vanish for all ξ. For each vector space H, the set {r > 0; 𝜕r φ ∈ H} is either an empty set or (0, ∞). For Fk (ξ1 , . . . , ξk ) = {(r1 , . . . , rk ); (φ(ξi , ri )) being linearly independent and (𝜕r φ(ξi , ri )) being linearly dependent}, it holds that Leb(Fk (ξ1 , . . . , ξk )) = 0 for k ≥ 2 and ξ1 , . . . , ξk ∈ S. The next result is a partial generalization of Theorem 2.9. Theorem 2.10 ([217]). Let π be an ID law corresponding to a Lévy measure μ which is absolutely continuous with respect to curves. Then π is absolutely continuous in the following two cases: (1) μ is not supported on any hyperplane, and μ(H) = 0 or +∞ for each vector subspace H. (2) For each hyperplane P, μ(Rd \ P) is infinite, and μ(H) = 0 for each (d-2)-dimensional vector space H. ̃ Recently, a refinement has been obtained by Yamazato himself. Let μ(dx) = denotes the Lebesgue measure induced on the linear space H by restriction. |x|2 μ(dx). The symbol ΛH 1+|x|2

Proposition 2.16 ([218, Theorem 1.1]). Assume there exists an integer r ≥ 1 such that the following two conditions hold:

132 � 2 Perturbations and properties of the probability law (i) μ is represented as a sum μ = ∑∞ k=1 μk , where μk is concentrated on a linear subspace Hk and has no mass on any proper linear subspace of Hk and satisfies (μ̃ k )∗k 0 be chosen small and let N = N(t) be chosen as ≍ log( 1t ). A process (chain) starts a point x at t = 0 and approaches the point y at t, with an error satisfying (4.7). Here β′ > β is a parameter to control the approach of the chain to the target point y. The chain is composed as (4.7), where ξn represents some an among {a0 , . . . , aN }. Using an ’s for small n represents the jump steps (lengths) are large, and using an ’s for large n ≤ N represents the jump steps (lengths) are small. The chain can use each ‘step’ a0 , . . . , aN up to N 3 times for each ai , i = 0, . . . , N(t), but the ‘cost function’ ∑Nn=0 (wn log( tk1 ) + log(wn !)) + O(. . . ) in Γ(t) n should govern all on the condition (4.7), where kn is the coefficient of an (weight of the jumping point an ) derived from μ(⋅) and wn is the number of an ’s in the whole choice of an , n = 0, . . . , N. Although this chain is deterministic, the value of the cost function remains the same if we replace an in supp μ by the random choice ξn . The value of Γ(t) represents the upper bound pt (x, y) ≤ CeΓ(t) as t → 0 in case of Proposition 2.10(1)(b). We can imagine the asymptotic behavior of the density function associated with the processes below as t → 0, which is of rapid decrease. The corresponding values Cmin will represent the estimated values Γ(t) in the condition (4.7) for each t in the tables. (Here Cmin stands for ‘compensated minimum’.) In the tables, the last column w0 , . . . , wN denotes the number of an ’s for each n = 0, . . . , N used by the chain to approximate the trajectory to reach the point y up to the time t, where N = N(t) is given in (4.4). In this example we do not necessarily assume the order condition ((∗) in Section 2.1.2). Below we choose β′ > β to be β′ = 1.5β for the safe margin. Below we give tables for the following cases.

2.5 Auxiliary topics

� 133

(1) Let d = m = 1, γ(x, ζ ) = ζ (this means xt (x) is a Lévy process). Let x = 1, y = √2. −n nβ We assume μ = ∑∞ n=0 kn δan is given by an = (−3) , kn = 3 , β ∈ (0, 2), and assume β′ = 1.5β. We calculate Cmin ≡ min ∑Nn=0 (wn log( tk1 ) + log(wn !)) − log( 1t ) log log( 1t ) n

and the (wn )Nn=0 which attain it (Table 2.1).

Table 2.1: Finding the minimum in case (1). β

β′

t

N

Cmin

w� , . . . , wN

1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5

2.25 2.25 2.25 1.5 1.5 1.5 0.75 0.75 0.75

0.02 0.002 0.0002 0.02 0.002 0.0002 0.02 0.005 0.003

2 3 5 3 5 7 7 9 10

−1.696 3.499 7.067 4.701 10.813 22.582 12.854 23.758 30.031

003 0040 004002 0040 004100 00411100 12111001 1211110200 12111103000

1

(2) Let d = m = 2, γ(x, ζ ) = Aζ where A = ( 1.4142 1 1.4142

−1 1.4142 1 1.4142

). Let x = (1, 1), y = (√2, √3). We

−n −n 2 ′ assume μ = ∑∞ n=0 kn δan is given by an = ((−2) , (−3) ), kn = n , and β = 1.5β. N 1 1 1 We find Cmin ≡ min ∑n=0 (wn log( tk ) + log(wn !)) − log( t ) log log( t ), and the (wn )Nn=0 n which attains the minimum (Table 2.2).

Table 2.2: Finding the minimum in case (2). β

β′

t

N

Cmin

w� , . . . , wN

1.5 1.5 1.5 1.0 1.0 1.0

2.25 2.25 2.25 1.5 1.5 1.5

0.005 0.002 0.001 0.005 0.002 0.001

5 5 6 7 8 9

−13.9 −3.95 5.91 8.88 11.46 12.69

002064 002051 0010900 01501014 015000204 0150100014

(3) Let d = m = 2, γ(x, ζ ) = Aζ where 1 0

A=(

0 ). 1

Let x = (1, 1), y = (√2, √3). We assume that μ = ∑∞ n=0 kn δan is given by an = ((−2)−n , (−3)−n ), kn = n2 , and β′ = 1.5β. We find

134 � 2 Perturbations and properties of the probability law N

Cmin ≡ min ∑ (wn log( n=0

1 1 1 ) + log(wn !)) − log( ) log log( ), tkn t t

and the (wn )Nn=0 which attains the minimum (Table 2.3). Table 2.3: Finding the minimum in case (3). β

β′

t

N

Cmin

w� , . . . , wN

1.5 1.5 1.5 1.0 1.0 1.0

2.25 2.25 2.25 1.5 1.5 1.5

0.005 0.002 0.0001 0.005 0.002 0.001

5 5 8 7 8 9

−14.7 −3.08 27.77 7.09 23.44 29.76

110028 111009 112402020 11110602 015000204 1124020200

(4) Let d = m = 2, γ(x, ζ ) = A(x, ζ ) where cos θn sin θn

A(An , ζ ) = (

− sin θn )ζ cos θn

and θn = arctan(An,x /An,y ), where An,x , An,y are components of the value of point An in (4.8). Let x = (−2, −4), y = (√2, √3). We assume μ = ∑∞ n=0 kn δan is given by an = −n −n 2 ′ ((−2) , (−3) ), kn = n , and β = 1.5β. We find Cmin ≡ min ∑Nn=0 (wn log( tk1 ) + n

log(wn !)) − log( 1t ) log log( 1t ), and the (wn )Nn=0 which attains the minimum (Table 2.4).

Table 2.4: Finding the minimum in case (4). β

β′

t

N

Cmin

w� , . . . , wN

1.5 1.5 1.5 1.5 1.5

2.25 2.25 2.25 2.25 2.25

0.05 0.005 0.002 0.001 0.0001

2 5 5 6 8

21.258 25.908 34.384 33.852 45.999

506 602013 602013 6020004 602000202

Illustrations for case (4) in 2 dim We further give finer planar graphics in case (4). We try, given x = (x1 , x2 ) and y = (y1 , y2 ), to describe how the trajectory evolves from x and approaches y satisfying the cost function restrictions Cmin (Table 2.5). Although trajectories here are deterministic, they may help illustrate and comprehend the movements of jump process trajectories. The case (h) looks like the constellation Scorpius (viewed from the opposite direction), (l) the Draco, and the last case (w) looks like the Wain (Big Dipper).

2.5 Auxiliary topics

� 135

Table 2.5: Finding the minimum in case (4). Case

x

y

(a) (b) (c) (d) (e) (e) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) (u) (v) (v) (v) (w)

(−1, −1) (−1, −1) (−1, −1) (−1, 3) (−1, 3) (−1, 3) (−1, 3) (−1, 3) (−1, 3) (−1, −2) (−1, −2) (−1, −2) (−1, −2) (−1, −2) (−1, 1) (−1, 1) (−1, 3) (2, −1) (−2, 1) (−2, 1) (−2, 1) (−2, 1) (−1, 2) (2, −2) (2, −2) (2, −2) (−1, 2)

(0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (0.707, 3.464) (3.464, 0.707) (3.464, 0.707) (3.464, 0.707) ( 0.707, 3.464) (0.707, 0.577) (0.707, −0.69) (0.707, −0.69) (0.707, −0.69) (−0.707, 0.692) (−0.707, 0.692) (−3.464, −0.707) (−3.464, −0.707) (−3.464, −0.707) (3.464, −0.692)

β

β′

t

N

Cmin

1.3 1.3 1.3 1.0 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.0 1.3 1.3 1.3 1.0 1.3 1.3 1.3 1.0 1.3 1.3 1.3 1.3 1.3

1.95 1.95 1.95 1.5 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.5 1.95 1.95 1.95 1.5 1.95 1.95 1.95 1.5 1.95 1.95 1.95 1.95 1.95

0.1 0.05 0.005 0.005 1.0 0.5 0.002 0.01 0.005 0.5 0.01 0.002 0.001 0.001 0.05 0.01 0.1 0.001 0.1 0.005 0.05 0.001 0.002 0.5 0.1 0.05 0.001

2 3 5 7 0 1 6 5 5 7 5 6 7 9 3 5 2 9 2 5 3 9 6 1 2 3 7

12.519, 17.037, 24.089 18.844, 19.124, 23.366 25.341, 39.795, 44.41 28.006, 30.204, 30.897 1.791 1.666 32.735 23.238, 31.807, 53.376 23.946, 28.654, 30.128 10.003, 13.642, 24.392 23.653, 26.103, 57.24 37.244, 40.388, 50.791 54.324, 78.523, 87.126 57.495, 72.588, 73.666 12.579, 25.711, 35.318 18.4435, 28.775, 35.518 14.416, 22.283, 40.57 39.118, 40.217, 40.217 7.8706, 15.375, 17.444 22.442, 27.825, 33.674 9.5073, 11.662, 14.1 44.7533, 49.4114, 57.1017 32.009, 33.843, 35.271 9.8334 22.42 28.021 55.666, 56.401, 57.054

Below graphics show how trajectories approach the target point y. For the first example, the trajectory, starting from x = (−1, −1), approaches y = (√2/2, √3/2) ∼ ′ (0.707, 3.464) in 7 steps with the error t 1/β = (0.1)1/(1.95) using N(t) = 2 kinds of small jumps ai ’s for i = 0, 1, . . . , N(t) = 2. The distance between x and y decreases usually along further steps as the broken lines show. We put give examples of various x and y, t and β’s. Trajectories are ranked by the values of Cmins (the best-case value is 12.519, which happens in the first example, n = 1). We will show below the best trajectory, and sometimes also those for n = 2, 3, . . . in case the forms of graphs look interesting. The target zone around y, given by (2.14), is encircled by the dotted curve. The sequential line in graphs below inform the distances between the present position and the target (horizontal axis denotes the steps).

136 � 2 Perturbations and properties of the probability law

2.5 Auxiliary topics

� 137

138 � 2 Perturbations and properties of the probability law

2.5 Auxiliary topics

� 139

140 � 2 Perturbations and properties of the probability law

2.5 Auxiliary topics

� 141

142 � 2 Perturbations and properties of the probability law Numerical calculations and illustrations given here are to provide faint images for trajectories of solutions of SDEs and their approximations by Markov chains. For more concrete and realistic results, see the work by Yoshida et al. using a special tool called YUIMA [86]. 2.5.4 Support theorem for canonical processes In this section, we provide a support theorem for a canonical process Yt given in Secη tion 2.5.1. We construct ‘skeletons’ φt for Yt by using Marcus’ integral. The result is closely related to the trajectories of Markov chains treated in Section 2.2. We assume the condition (A.0) for μ and the restricted Hörmander condition (RH) for Xj (x)’s. In comparison with the proof of short-time asymptotic bounds of the transition density stated in the previous section, we can observe fine behaviors of the jump process. First, we construct skeletons as follows. Let u(t) = (u1 (t), . . . , um (t)), 0 ≤ t ≤ T, be an m R -valued, piecewise smooth, càdlàg functions having finite jumps. It is decomposed as u(t) = uc (t) + ud (t), where uc (t) is a continuous function and ud (t) is a purely discontinuous (i. e., piecewise constant except for isolated finite jumps) function. For η > 0, we put 𝒰η = {u ∈ D; u(t) = u

c,η

(t) + ud,η (t),

ud,η (t) = ∑ Δu(s) ⋅ 1{η 0 [P({ω; dT (Z, u) < δ}) > 0]}.

(5.21)

The support of the canonical process Y is similarly defined by supp Y. (x) = {φ ∈ D′ ; for all ϵ > 0 [P({ω; dT (Φ(x), φ) < ϵ}) > 0]}.

(5.22)

By 𝒮 x̄ , we denote the closure of 𝒮 x in (D′ , dT ). η We define the approximating processes Z η (t), Yt for each η > 0 as follows. Let t

̃ Z η (t) = ∫ ∫ zN(dsdz), 0 η 0 such that (5.28) does not hold, there exist γ = γη > 1 and a sequence {ηk } decreasing to 0 such that 󵄨 󵄨 αηηk = o(1/󵄨󵄨󵄨uηηk 󵄨󵄨󵄨), η

η

where αρ is the angle between the direction uρ and supp μ on {|z| = γη}, when it can be defined. η Notice first that it always holds in dimension 1 (with αηk = 0). Besides, it is verified to hold in higher dimensions whenever supp μ contains a sequence of spheres whose radii tend to 0 (in particular, a whole neighborhood of {0}), or when the intersection of supp μ with the unit ball coincides with that of a convex cone. For technical reasons, we also suppose that μ satisfies the following nondegeneracy and scaling condition:

2.5 Auxiliary topics

� 145

(H.2) There exist β ∈ [1, 2) and positive constants C1 , C2 such that for any ρ ≤ 1, C1 ρ2−β I ≤ ∫ zz∗ μ(dz) ≤ C2 ρ2−β I. |z|≤ρ

Besides, if β = 1, then we assume further 󵄨󵄨 󵄨󵄨 󵄨 󵄨 lim sup󵄨󵄨󵄨 ∫ z μ(dz)󵄨󵄨󵄨 < +∞. 󵄨󵄨 󵄨 η→0 󵄨 η≤|z|≤1

The inequalities above stand for the case when zz∗ are symmetric positive definite matrices. This means (H.2) demands both the nondegeneracy of the distribution of (point) masses around the origin, and the way of concentration of masses along the radius. We notice that if ⟨v, ⋅⟩ stands for the usual scalar product with v, they are equivalent to 󵄨 󵄨2 ∫ 󵄨󵄨󵄨⟨v, z⟩󵄨󵄨󵄨 μ(dz)≍ρ2−β |z|≤ρ

uniformly for unit vectors v ∈ S m−1 . In particular, β = inf{α; ∫|z|≤1 |z|α μ(dz) < +∞} (the Blumenthal–Getoor index of Z), and the infimum is not reached. Notice finally that the measure μ may be very singular and may have a countable support. The condition (H.2) implies the nondegeneracy condition for the matrix B in Section 3.5 with α = 2 − β. We now state our main theorem: Theorem 2.11. Under (H.1), (H.2), we have supp Y. (x) = 𝒮 x̄ . Here, ̄ means the closure in dT -topology. The condition (H.1) is crucial to the small deviations property: for any ϵ > 0 and t > 0, P(sup |Ys | < ϵ) > 0, s≤t

(5.29)

and is related to the distribution of points of mass in supp μ. This can be viewed as an adapted assertion to canonical processes of the support theorem due to Th. Simon [199] for Itô processes. For the proof of this theorem, see [96]. The support theorem helps us a lot in showing the strict positivity of the density function y 󳨃→ pt (x, y) (see [99, Section 4]).

3 Analysis of Wiener–Poisson functionals You must take your chance, And either not attempt to choose at all Or swear before you choose, if you choose wrong Never to speak to lady afterward In way of marriage: therefore be advised. W. Shakespeare, The Merchant of Venice, Act 2, Scene 1

In this chapter, we consider the stochastic calculus on the Wiener–Poisson space. The Wiener–Poisson space is the product space of the Wiener and the Poisson spaces. By the Lévy–Itô decomposition theorem (cf. Proposition 1.1), a general Lévy process lives in this space. In Section 3.1, we study the Wiener space. Stochastic analysis on the Wiener space has been developed by Malliavin, Stroock, Watanabe, Shigekawa, and others, prior to that for the Poisson space. In Section 3.2, we study the Poisson space. Compared to the Wiener space, there exist several tools to analyse the Poisson space, and we shall introduce them accordingly. Section 3.3 is devoted to the analysis on the Wiener–Poisson space. We construct Sobolev spaces on the Wiener–Poisson space by using the norms induced by several derivatives on it. It turns out in Sections 3.5, 3.7 below that these Sobolev spaces play important roles in obtaining asymptotic expansions. In Section 3.4, we consider a comparison with the Malliavin operator (a differential operator) on the Poisson space. In Sections 3.5, 3.7, we state a theory on the composition of Wiener–Poisson variables with tempered distributions. This composition provides us with a theoretical basis regarding the content of Section 4.1. In Section 3.6, we state results on the existence of the density functions associated with the composition (concatenation) theory for Itô processes, and on an application of the (Hörmander’s) hypoellipticity result in the jumpdiffusion case.1 Throughout this chapter, we denote the Wiener space by (Ω1 , ℱ1 , P1 ) and the Poisson space by (Ω2 , ℱ2 , P2 ) in order to distinguish the spaces.

3.1 Calculus of functionals on the Wiener space In this section, we briefly present some basic facts of the Wiener space. We state the chaos expansion using the Hermite polynomials first. We state the results in this section quite briefly and confine ourselves to those notions which are necessary to state a general theory of stochastic calculus. Proofs for several propositions are omitted.

1 The phrase is not mathematically proper, but it is often used mainly in applications. We sometimes use the word, especially in the field of applications. https://doi.org/10.1515/9783110675290-004

3.1 Calculus of functionals on the Wiener space

� 147

Several types of Sobolev norms and Sobolev spaces can be constructed on the Wiener space. However, we will not go into detail on these Sobolev spaces here, and postpone it to Section 3.3. Below, we denote a function of 1-variable by f (t), and a function of n-variables by fn (t1 , . . . , tn ). If we write fi (t), it means that it is a function of the variable t with an index i. We denote by W (t) a 1-dimensional Wiener process (standard Brownian motion) on Ω1 with filtration (ℱ1,t )t∈T . For fn (t1 , . . . , tn ) ∈ L2 (Tn ), we denote by In (fn ), Jn (fn ) the following iterated integrals: In (fn ) = ∫ fn (t1 , . . . , tn )dW (t1 ) ⋅ ⋅ ⋅ dW (tn ), Tn

tn

t2

Jn (fn ) = ∫ ∫ ⋅ ⋅ ⋅ ∫ fn (t1 , . . . , tn )dW (t1 ) ⋅ ⋅ ⋅ dW (tn ). T 0

(1.1)

0

They are called Wiener chaos. The above Jn (fn ) is also written as ∫

fn (t1 , . . . , tn )dW (t1 ) ⊗ ⋅ ⋅ ⋅ ⊗ dW (tn ).

(1.2)

01/|v|} = 1 may occur if D̃ u F flounders violently for small |u|, u ∈ A(|v|−1/β ) for |v| > 1.

k ̃ Remark 3.10. If we restrict the range of u ∈ A(1)k = 𝕌k to u ∈ A(ρ(v)) in the norms ̃ to the decay order of |v| in the ‖ ⋅ ‖l,k,p;A and ‖ ⋅ ‖l,k,p,ρ̃ (used in [79]), we can relate ρ(v) ̃ is a function of v in the form of a negative power of right-hand side of (5.12), where ρ(v) |v|. Please, check [79, Lemma 4.4].

We have the following proposition (cf. [132, Lemma 2.14]): Proposition 3.7. Let F ∈ D∞ satisfy the Q-nondegeneracy condition. For any m, there exist l, k, p > 4, and Cm > 0 such that β󵄩 󵄩 ‖φ ∘ F‖′l,k,p;A ≤ Cm ( ∑ 󵄩󵄩󵄩(1 + |F|2 ) 󵄩󵄩󵄩l,k,p;A )‖φ‖−2m β≤m

(5.18)

for φ ∈ 𝒮 , A ⊂ A(1). Proof. We have ̂ E[φ(F)G] = E[∫ ei(v,F) φ(v)dv ⋅ G] i(v,F) ̂ ̂ = ∫ φ(v)E[e ⋅ G]dv = ∫ φ(v)ψ G (v)dv.

We divide the proof into 2 steps. (Step 1) We assert ψG ∈ 𝒮∞ . Indeed, we observe β

d

2

β

(1 − Δ) ψG (v) = E[G(1 + ∑(Fj ) ) ei(v,F) ]. j=1

Here, by Proposition 3.6, for each n, there exist l, k, p so that the right-hand side is dominated by β󵄩 󵄩󵄩 d 󵄩󵄩 󵄩󵄩 −nq /2 󵄩 2 C 󵄩󵄩󵄩(1 + ∑(Fj ) ) 󵄩󵄩󵄩 ‖G‖l,k,p/2;A (1 + |v|2 ) 0 󵄩󵄩 󵄩󵄩 j=1 󵄩 󵄩l,k,p/2;A −nq /2 󵄩󵄩 2 β󵄩 ≤ C 󵄩󵄩(1 + |F| ) 󵄩󵄩󵄩l,k,p;A ‖G‖l,k,p;A (1 + |v|2 ) 0 .

3.5 Composition on the Wiener–Poisson space (I) – general theory

� 215

Here, q0 ∈ (0, 1) is chosen independent of n in (3.52). Hence, β󵄩 󵄩 󵄨 󵄨󵄨 2 nq0 /2 (1 − Δ)β ψG (v)󵄨󵄨󵄨 ≤ C 󵄩󵄩󵄩(1 + |F|2 ) 󵄩󵄩󵄩l,k,p;A ‖G‖l,k,p;A . 󵄨󵄨(1 + |v| )

We take n ≤ 2m/q0 , then by (5.1), we observe β󵄩 󵄩 ‖ψG ‖2m ≤ Cm ( ∑ 󵄩󵄩󵄩(1 + |F|2 ) 󵄩󵄩󵄩l,k,p;A )‖G‖l,k,p;A , β≤m

m = 1, 2, . . . ,

with some l = lm , k = km , p = pm . This implies ψG ∈ 𝒮∞ . (Step 2) We check that the assertion (5.18) holds. We observe ‖φ ∘ F‖′l,k,p;A = ≤ ≤

󵄨 󵄨 sup 󵄨󵄨󵄨E[φ ∘ FG]󵄨󵄨󵄨

‖G‖l,k,p;A =1

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ̂ sup 󵄨󵄨󵄨∫ ψG (v)φ(v)dv 󵄨󵄨 󵄨󵄨 ‖G‖l,k,p;A =1󵄨󵄨 sup ‖G‖l,k,p;A =1

‖ψG ‖2m ‖φ‖̂ −2m

β󵄩 󵄩 ≤ Cm ( ∑ 󵄩󵄩󵄩(1 + |F|2 ) 󵄩󵄩󵄩l,k,p;A )‖φ‖−2m . β≤m

Hence, (5.18) follows. The above formula implies that for Φ ∈ 𝒮−2m and F ∈ D∞ , we can define the composition Φ ∘ F as an element of D′∞ . Since ⋃m≥1 𝒮−2m = 𝒮 ′ , we can define the composition Φ ∘ F for Φ ∈ 𝒮 ′ and F ∈ D∞ as follows. Definition 3.4. For a distribution T ∈ 𝒮 ′ , the composition T ∘ F is a linear functional on D∞ which is defined by ⟨T ∘ F, G⟩ = 𝒮 ′ ⟨ℱ T, E[Gei⋅F ]⟩𝒮 ,

G ∈ D∞ .

(5.19)

Due to the proof in Proposition 3.6, we have the following result for ψG (v). Proposition 3.8. Let for F ∈ D∞ assume sup|v|≥1 ‖QF (v)−1 ‖∗l,k,p < ∞ for all l, k, p, and let G ∈ D∞ . For any n ∈ N, there exist l, k ∈ N ∪ {0}, p > 2, and C > 0 such that 󵄨󵄨 󵄨 2 − 󵄨󵄨ψG (v)󵄨󵄨󵄨 ≤ C(1 + |v| )

q0 2

n

n −1 󵄩∗ k+1 󵄩 ̃ ∗ × ((‖DF‖∗l,k,p + (1 + ‖DF‖ ) sup󵄩󵄩󵄩QF (v) 󵄩󵄩󵄩l,k,p ) ‖G‖∗l,k,p . l,k,(k+1)p ) |v|≥1

(5.20)

216 � 3 Analysis of Wiener–Poisson functionals Choose G = 1. As stated in the introduction to Section 2.1, the above proposition implies that T ∘ F is well defined for T ∈ 𝒮 ′ and, in particular, F has a smooth density pF (y) as above. Remark 3.11. We have assumed that A0 (ρ), 0 < ρ < 1 is a family of open neighborhoods of the origin, whereas, in the limiting case of the above, we can also consider the case A0 (ρ) = {ρej , j = 1, . . . , d} which is not an open neighborhood, where ej , j = 1, . . . , d are distinct unit vectors in the direction j in Rd . We construct a measure μ(dz) = λ(dθ) ⋅ g(dr),

z = (r, θ)

so that d

λ(dθ) = ∑(pj δρej + qj δ−ρej )(dθ), j=1

g(dr) = g(±ej , dr) =

cj

|r|1+αj

dr,

where z = (r, θ) in polar coordinates, 0 ≤ pj , qj ≤ 1, pj + qj = 1, cj > 0, j = 1, . . . , d, and 0 < αj < 2, minj αj > 0. Even in this case, we have a smooth density for F; see [106, Section 2].

3.5.2 Sufficient condition for the composition Let F be a Wiener–Poisson functional. Condition (R). We say that F satisfies the condition (R) if for any positive integer k = 0, 1, 2, . . . and p > 1, the derivatives satisfy sup

t∈T,u∈A(1)

m

m

󵄨 󵄨p 󵄨 󵄨p E[∑ sup 󵄨󵄨󵄨𝜕zi D̃ t,z F ∘ εu+ 󵄨󵄨󵄨 + ∑ sup 󵄨󵄨󵄨𝜕zi 𝜕zj D̃ t,z F ∘ εu+ 󵄨󵄨󵄨 ] < +∞. k i=1 z∈A0 (1)

i,j=1 z∈A0 (1)

We recall φ(ρ) = ∫A (ρ) γ(z)2 μ(dz), and assume that the Lévy measure satisfies the 0 order condition in (3.5). Let B be the infinitesimal covariance, that is, a nonnegative symmetric matrix which satisfies (v, Bv) ≤ (v, Bρ v), v ∈ Rm \ {0} for 0 < ρ < ρ0 , for some ρ0 > 0. Here Bρ is the matrix

3.5 Composition on the Wiener–Poisson space (I) – general theory

Bρ =

1 ∫ zzT μ(dz). φ(ρ)

� 217

(5.21)

A0 (ρ)

We remark that such a matrix B exists (eventually, one may take B ≡ 0). However, we do not know if we can choose B > 0 (positive definite matrix) even if Bρ > 0 for each 0 < ρ < ρ0 . Nor B is unique in general; in this case, we choose one such matrix and fix it. One sufficient condition for the treatment of Bρ is due to Picard–Savona [182]: there exist 0 < α < 2 and c, C > 0 such that cρα |u|2 ≤ ∫ (u, z)2 μ(dz) ≤ Cρα |u|2

(5.22)

A0 (ρ)

holds for any u ∈ Rm and 0 < ρ < 1. Indeed, if we choose u = ej , j = 1, . . . , m (unit vectors), then mcρα ≤ ∫ |z|2 μ(dz) ≤ mCρα . A0 (ρ)

Hence (u, Bρ u) =

cρα |u|2 1 ≥ c′ |u|2 ∫ (u, z)2 μ(dz) ≥ φ(ρ) mCρα A0 (ρ)

for some c′ > 0 independent of ρ > 0. We assume the existence of B > 0 in what follows. Another (necessary and) sufficient condition for (3.5) is that there exist 0 < α < 2, c > 0, and r > 0 such that for |u| ≥ r, ∫ (1 − cos(u, z))μ(dz) ≥ c|u|2−α

(5.23)

Rm \{0}

with μ such that ∫|z|≥1 μ(dz) < ∞ (cf. [54, Condition (Hα ) and Corollary 1.2]). We now require sufficient conditions for the condition (ND). To this end, we introduce the following functionals derived from F: K̃ = ∫(𝜕D̃ t,0 F)B(𝜕D̃ t,0 F)T dt,

(5.24)

T

K̃ ρ = ∫ 𝜕D̃ t,0 FBρ (𝜕D̃ t,0 F)T dt,

ρ > 0.

T

Here, 𝜕D̃ t,0 F = 𝜕z D̃ t,z F|z=0 , the derivative with respect to z at z = 0.

(5.25)

218 � 3 Analysis of Wiener–Poisson functionals Furthermore, we put Q(v) = (v′ , where v′ =

v , |v|

1 F ̃ ′ ), (Σ + K)v T

Qρ (v) = (v′ ,

1 F (Σ + K̃ ρ )v′ ), T

(5.26)

v ∈ Rd , and ρ > 0.

Definition 3.5. (1) We say F satisfies the (ND2) condition if ΣF + K̃ is invertible and if, for any integer k and p > 1, we have sup

v∈Rd ,|v|≥1,u∈A(1)k

̃ ∘ ε+ v)−p ] < +∞. E[(v, (ΣF + K) u

(2) We say F satisfies the (ND3) condition if ΣF + K̃ ρ is invertible and if, for any integer k and any p > 1, there exists ρ0 > 0 such that sup

0 0, 󵄨 󵄨p 󵄨 󵄨p sup E[󵄨󵄨󵄨Tρ (v)−1 ∘ εu+ 󵄨󵄨󵄨 ] ≤ C sup E[󵄨󵄨󵄨Qρ (v)−1 ∘ εu+ 󵄨󵄨󵄨 ] ′ ′

0 1 such that for all v and u, P(τ ≤ ρ) ≤ cp ρ4p ,

0 < ρ < δ0 .

On the set E(ρ) = { sup |D̃ (t,z) F| < ρβ }, (t,z)∈A(ρ)

we have

220 � 3 Analysis of Wiener–Poisson functionals 󵄨󵄨 + +󵄨 󵄨󵄨Tρ∧τ ∘ εu − Qρ∧τ ∘ εu 󵄨󵄨󵄨 1 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨(v, D̃ t,z F)2 ∘ εu+ − (v, 𝜕D̃ t,0 F ⋅ z)2 ∘ εu+ 󵄨󵄨󵄨n(du), Tφ(ρ ∧ τ)

u = (t, z).

A(ρ∧τ)

Due to the condition (R), the integrand is dominated by the term |z|3 Φ(t, z, u), where Φ(t, z, u) is a nonnegative random variable satisfying sup sup E[ sup Φ(t, z, u)4p ] < +∞ t∈T γ(u)≤1

z∈A0 (δ0 )

for any p > 1. We put 󵄨 󵄨 Ψ = 󵄨󵄨󵄨Tρ∧τ ∘ εu+ − Qρ∧τ ∘ εu+ 󵄨󵄨󵄨. Then, the above implies that for any p > 1, there exists c1 > 0 E[Ψ4p ⋅ 1E(ρ) ] ≤ c1 ρ4p T 4p

(5.31)

if ρ < δ0 for any v and u. Then, combining the condition (ND3) and (5.31), we have E[(

4p

Ψ ) ] ≤ c2 ρ4p Qρ∧τ ∘ εu+

for ρ < δ0 for all u and v. Hence, by Chebyschev’s inequality, 1 P(E(ρ) ∩ (Ψ ≥ Qρ∧τ ∘ εu+ )) ≤ c3 ρ4p , 2

ρ < δ0 .

On the other hand, E(ρ)c = { sup |D̃ (t,z) F| ≥ ρβ }. (t,z)∈A(ρ)

Then, again by Chebyschev’s inequality, c

P(E(ρ) ) ≤ E[(

supu∈A(ρ) |D̃ u F| ρβ

2p′

)

] ≤ c4 ρ2(1−β)p ≤ c4 ρ4p ,

for some δ1 > 0 sufficiently small, where p′ = 4p

(R). This implies P(τ ≤ ρ) ≤ (c3 + c4 )ρ

for ρ
0 one has 󵄩󵄩 󵄩 −1 󵄩∗ −1 󵄩∗ 󵄩󵄩Q(v) 󵄩󵄩󵄩0,k,p ≤ c1 󵄩󵄩󵄩Sρ (v) 󵄩󵄩󵄩0,k,p for any ρ, v and k, p. Proof. We use the inequality |ei(v,Du F) − 1|2 ≤ |(v, D̃ u F)|2 . This implies Q(v) ∘ εu+ ≥ Sρ (v) ∘ εu+ a. e. dnj dP. Hence ̃

E[(Q(v) ∘ εu+ ) ] ≤ c−p E[(Sρ (v) ∘ εu+ ) ] −p

−p

a. e. dnj

for any ρ, v, j. We take the essential supremum with respect to u and sum up over j = 0, 1, . . . , k, and the assertion follows. ̃ Lemma 3.15. Suppose K(v) is invertible a. s. for any v. Then K̃ ρ (v) is also invertible a. s., and for 0 < ρ < δ0 , and for some c2 > 0 one has 󵄩󵄩 ̃ 󵄩 ̃ −1 󵄩󵄩∗ −1 󵄩∗ 󵄩󵄩Kρ (v) 󵄩󵄩󵄩0,k,p ≤ c2 󵄩󵄩󵄩K(v) 󵄩󵄩0,k,p for any ρ, v and k, p. Proof. We denote by λρ and Λρ the minimum and maximum eigenvalues of Bρ , respectively. Since lim infρ→0 Bρ ≥ B and B is nondegenerate, there exists 0 < ρ1 < ρ0 such that 0 < inf0 0. ′ We consider the case t = 0, Xt,T (x) = XT′ (x) only, since the discussion is similar in the other cases. We prove the first assertion only, as the proof of the second is similar. We prove this assertion as Claims III-i through III-iii as follows. We recall the original SDE for Xt′ , and denote it simply by t ′ Xt′ = x + ∫ 𝒳 (Xr− , dr). 0

Claim III-i. First, we assume σ(x)σ(x)T > 0. The Malliavin–Shigekawa derivative of XT′ (x) ∘ ϵu+ satisfies the following linear SDE: T

Dt XT′ (x)

∘

εu+

=

σ(Xt′ (x)

∘

εu+ )

′ ′ + ∫ ∇𝒴 (Xr− (x) ∘ εu+ , dr) ⋅ Dt Xr− (x) ∘ εu+ , t

230 � 3 Analysis of Wiener–Poisson functionals where 𝒴 (x) = 𝒳 ∘ εu+ ; see [24, (6-25)]. On the other hand, T ′ ′ ′ ∇Xt,T (Xt′ (x) ∘ εu+ ) = I + ∫ ∇𝒴 (Xr− (x) ∘ εu+ , dr)∇Xt,r− (x)(Xr′ (x) ∘ εu+ ). t

Comparing these two expressions, due to the uniqueness of the solution to the linear SDE, the solution to the above SDE is given by ′ Dt XT′ (x) ∘ εu+ = ∇Xt,T (x)(Xt′ (x) ∘ εu+ )σ(Xt′ (x) ∘ εu+ )

a. e. dtdP.

′ ′ (x), v ⊂ u, we estimate D̃ iu Dr × Here, as Dr Xs,t (x) ∘ εu+ is a linear sum of Dr D̃ v Xs,t ′ ′ + ̃ ̃ 𝜕z Du Xs,t (x) in place of Dr 𝜕z Du Xs,t (x) ∘ εu for each u, s < r < t. Hence we have, for any k ∈ N and p > 2, that there exists a positive constant C = Ck such that the solution Dt XT′ (x) ∘ εu+ satisfies

󵄨 󵄨p sup E[󵄨󵄨󵄨Dt XT′ (x) ∘ εu+ 󵄨󵄨󵄨 ] ≤ C. k

u∈A(δ0 )

Then we obtain the assertion for the case σ(x)σ(x)T > 0. This proves the intended assertion in view of the above. Claim III-ii. Next we consider the case det σ(x)σ(x)T = 0. Using the condition (ND), we proceed as follows. We will give the proof in the case |j| = 1 and k = 1 only. Let u1 = (r1 , z1 ). We consider the case where s ≤ r1 < t and z1 ∈ A0 (δ0 ) so that the vector 𝜕zi g(x, z1 ) and ⟨σ1 (x), . . . , σm (x)⟩ are linearly independent at x. As above, we ′ ′ ′ (x) is given by (x). This is since 𝜕z D̃ u X0,t estimate ∇Xs,t (x) ∘ εu+ in place of 𝜕z D̃ u X0,t ′ (x) = ∇Xr′1 ,T (Xr′1 − (x))𝜕zi g(Xr′1 − (x), z1 ) 𝜕zi D̃ u X0,t

(cf. [134, Lemma 6.4.1]). ′ ′ Then we can write Xs,t (x) ∘ εu+1 = Xr′1 ,t ∘ ϕr1 ,z1 ∘ Xs,r (x). Therefore, setting ∇xi = ∇, we 1− obtain ′ ′ ′ ∇Xs,t (x) ∘ εu+1 = ∇(Xr′1 ,t ∘ ϕr1 ,z1 ∘ Xs,r (x)) = JXr′ ,t ∇(ϕr1 ,z1 (Xs,r (x))), 1− 1− 1

where Jϕ is the Jacobian matrix of the map ϕ. ′ Since JXr′ ,t and Xs,r are independent, we have 1 1

󵄨 ′ 󵄨p 󵄨 󵄨p 󵄨 E[󵄨󵄨󵄨∇Xs,t (x) ∘ εu+1 󵄨󵄨󵄨 ] ≤ E[(E[󵄨󵄨󵄨JXr′ ,t (x ′ )󵄨󵄨󵄨 ]󵄨󵄨󵄨x ′ =ϕ r ,z 1

1 1

2 1/2 󵄨 E[󵄨󵄨󵄨∇ϕr1 ,z1 ′ (x) ) ] ∘Xs,r 1

󵄨2p 1/2 ′ ∘ Xs,r (x)󵄨󵄨󵄨 ] . 1

The right-hand side is bounded with respect to r1 , s, t, x, z1 because of [68] and the fact that ∇ϕr1 ,z1 (x) is bounded with respect to r1 , z1 , x. We apply the argument for ∇2 Xt′ (x) ∘ εu+ , . . . and obtain the assertion.

3.6 Smoothness of the density for Itô processes � 231

This proves the assertion for this case. Claim III-iii. In the mixed case, we decompose the space at each point into two, and apply the results in the two cases above for each direction. We make use of the jump ′ part of Xs,t (x), and apply Lemma 3.12. This proves the assertion in this case. ′ ′ Step IV. The cases for ∇j Xt,T (Xt− (x)) ∘ εu , |j| = 2, 3 are estimated in a similar way, as 2 ′ ′ we apply the argument for ∇ Xt (x)∘εu+ . For |j| = 2, 3, ∇j Xt,T (x)∘εu+ satisfies a linear SDE as j j in Step I above. We can apply a similar argument for D̃ i Dt 𝜕z X ′ (x) and Dt 𝜕z X ′ (x) ∘ ε+ .

This ends the proof of the lemma.

u

t,T

t,T

u

3.6.2 Big perturbations c Let u1 = (t1 , z1 ) ∈ 𝕌′′ 1 = T × ({z ∈ A0 (δ0 ) ; |z| ≤ 1}). Then,

Xt′ ∘ εu+1 = Xt′1 ,t ∘ ϕt1 ,z1 ∘ Xt′1 (x), where ϕt1 ,z1 (x) = x + g(x, z1 ). Here the operation by ϕ does not depend on t. We write ϕt,z to indicate the moment of ′′ the operation. For u = (u1 , . . . , un ) = ((t1 , z1 ), . . . , (tn , zn )) ∈ 𝕌′′ 1 × ⋅ ⋅ ⋅ × 𝕌1 , we put Xt′ ∘ εu+ = Xt′n ,t ∘ ϕtn ,zn ∘ Xt′n−1 ,tn ∘ ⋅ ⋅ ⋅ ∘ ϕt1 ,z1 ∘ Xt′1 . Based on this decomposition, we introduce the conditional randomization as follows. We write + Xt = Xt′ ∘ εq+ = Xt′ ∘ ε(q(u , 1 ),...,q(un ))

tn ≤ t < tn+1 ,

(6.15)

where q(u) = (q(u1 ), . . . , q(un )) is a Poisson point process with the counting measure N ′′ (dtdz) and tn = inf{t; N ′′ ((0, t] × (Rm \ {0})) ≥ n}. The reader can recall a similar decomposition of the density in Section 2.3.2 (3.16). As noted above, we first analyze the small-jump part X ′ of X, and then perturb it by big jumps due to q(u) by concatenations. Let F ′ = XT′ (x). Then, + ′ ′ F ′ ∘ εt,z = Xt,T ∘ ϕt,z ∘ Xt− (x).

Hence, ′ ′ ′ ′ D̃ t,z F ′ = Xt,T ∘ ϕt,z (Xt− ) − Xt,T (Xt− )

′ ′ ′ = ∇Xt,T (ϕt,θz (Xt− ))∇z ϕt,θz (Xt− )

232 � 3 Analysis of Wiener–Poisson functionals for some θ ∈ (0, 1). This implies F ′ ∈ D0,1,p by Lemma 3.18, and finally we observe F ′ ∈ D∞ . Furthermore, we have ′ ′ ′ 𝜕z D̃ t,z F ′ |z=0 = ∇Xt,T (Xt− )G(Xt− ).

Here, G(x) = ∇z g(x, z)|z=0 . In view of this, K̃ ρ , Σ in Section 3.5.2 turn into ′ ′ ′ ′ T ′ ′ T K̃ ρ′ = ∫ ∇Xt,T (Xt− )G(Xt− )Bρ G(Xt− ) ∇Xt,T (Xt− ) dt,

(6.16)

T

T

T

′ ′ ′ ′ ′ ′ Σ′ = ∫ ∇Xt,T (Xt− )σ(Xt− )σ(Xt− ) ∇Xt,T (Xt− ) dt,

(6.17)

T

respectively. Then, ′ ′ ′ ′ ′ T Σ′ + K̃ ρ′ = ∫ ∇Xt,T (Xt− )Cρ (Xt− )∇Xt,T (Xt− ) dt, T

where Cρ (x) = σ(x)σ(x)T + G(x)Bρ G(x)T . Furthermore, K̃ turns into ′ ′ ′ ′ T ′ ′ T K̃ ′ = ∫ ∇Xt,T (Xt− )G(Xt− )BG(Xt− ) ∇Xt,T (Xt− ) dt.

(6.18)

T

Then, T

′ ′ ′ ′ ′ Σ′ + K̃ ′ = ∫ ∇Xt,T (Xt− )C(Xt− )∇Xt,T (Xt− ) dt. T

Here, C(x) = σ(x)σ(x)T + G(x)BG(x)T . We remark that Cρ (x) ≥ G(x)Bρ G(x)T , and that Σ′ + K̃ ′ ≤ Σ′ + K̃ ρ′ ,

0 < ρ < ρ0 .

(6.19)

Let Q′ (v) = (v′ ,

1 ′ (Σ + K̃ ′ )v′ ), T

Qρ′ (v) = (v′ ,

1 ′ (Σ + K̃ ρ′ )v′ ), T

(6.20)

3.6 Smoothness of the density for Itô processes

corresponding to Q(v), Qρ (v) in Section 3.5 where v′ =

� 233

v . |v|

Definition 3.6. We say F satisfies the condition (NDB) if there exists C > 0 such that (v, C(x)v) ≥ C|v|2

(6.21)

holds for all x ∈ Rd . The condition (NDB) means the matrix C(x) is positive definite for all x. We make use of the functional F ′ = Xt′ derived from SDE (6.5). For u = (t, z), ′ ′ ′ ′ ′ D̃ (t,z) XT′ = Xt,T ∘ ϕt,z (Xt− ) − Xt′ = ∇Xt,T (ϕt,θz (Xt− ))∇z g(Xt− , θz)z

(6.22)

by the mean value theorem, where θ ∈ (0, 1). For u ∈ A(ρ), we choose the parametrization ρ = |v|−1/β in what follows. Then we have |u| = |z| ≤ |v|−1/β , and, due to conditions (R) and (D), as well as (6.22), 󵄨󵄨 ̃ ′󵄨 −1/β 󵄨󵄨D(t,z) XT 󵄨󵄨󵄨 ≤ C|z| ≤ C|v| for some constant C > 0. As 1/β ∈ [1, 2α1 ), we have |v|−1/β ≤ 1/|v| for |v| ≥ 1. This implies 1 that the condition |D̃ (t,z) XT′ | < C |v| holds for u = (t, z) ∈ A(ρ) a. e., and we have that the event {1|D̃ u F ′ |≤1/|v| = 1} holds for |v| large a. e. in the condition (ND) in (3.51). We now introduce a new condition (ND-n). Definition 3.7 (Condition (ND-n)). We say that a random variable F satisfies the condition (ND-n) if for all p ≥ 1, k ≥ 0, there exists β ∈ ( α2 , 1] such that sup

v∈Rd , |v|≥1,ρ=|v|−1/β

−1 󵄨󵄨 󵄨󵄨p 󵄨 󵄨 󵄨2 󵄨 sup E[󵄨󵄨󵄨(T −1 (v, Σv) + T −1 φ(ρ)−1 ∫ 󵄨󵄨󵄨(v, D̃ u F)󵄨󵄨󵄨 n(du)) ∘ ετ+ 󵄨󵄨󵄨 ] < +∞. (6.23) 󵄨 󵄨󵄨 k 󵄨 τ∈A (ρ) A(ρ)

This is equivalent to the condition (ND) for F ′ = Xt′ under the parametrization ρ = |v| . −1/β

Proposition 3.11. Assume that the condition (NDB) holds for F ′ . Then, the condition (ND-n) holds for F ′ . Due to this proposition and Proposition 3.7, the composition Φ ∘ F ′ of Φ ∈ 𝒮 ′ and F ′ is justified. The proof of this proposition consists of the following two propositions (Propositions 3.12, 3.13). Proposition 3.12. Assume the conditions (NDB) and (J.1), (J.2). Then, for each n ∈ N, the family {Xt′ (x)}x∈{|x|≤n} satisfies the condition (ND2) for p > 2. Proof. The matrix C(x) satisfies (6.21). By the expression of Σ′ + K̃ ρ′ , using Jensen’s int

t

equality (( 1t ∫0 f (s)ds)−1 ≤ 1t ∫0 f (s)−1 ds for f (⋅) > 0) and (6.19), we have

234 � 3 Analysis of Wiener–Poisson functionals

(v,

1 ′ (Σ + K̃ ′ )v) T

−p

∘ εu+ ≤

1 −1 󵄨2 󵄨 ′ (Xt′ (x)) ∘ εu+ v) 󵄨󵄨󵄨 dt, ∫󵄨󵄨(v, ∇x Xt,T CT 2 󵄨

0 < ρ < ρ0 .

T

Hence, E[(v,

1 ′ (Σ + K̃ ′ )v) T

−p

∘ εu+ ] ≤

1 1 󵄨󵄨 ′ ′ + −1 󵄨󵄨2p 2 (v, ∇ X E[ (X (x)) ∘ ε ] dt. v) ∫ 󵄨 󵄨 x t,T t u 󵄨 󵄨 CT 2

T

Here, we have −1 󵄨2p 1 󵄨 ′ sup E[󵄨󵄨󵄨(∇x Xt,T (v, Xt′ (x)) ∘ εu+ v) 󵄨󵄨󵄨 ] 2 < +∞ k

u∈A(1)

(6.24)

for k = 1, 2, . . . Hence, we have the assertion. We prove (6.24) below. For that we introduce a matrix-valued process U x,t (u), t ≤ u ≤ T by u

u

u

t

t

t

′ ′ ′ ̃ U x,t (u) = ∫ ∇b(Xt,r− (x))dr + ∫ ∇σ(Xt,r− (x))dW (r) + ∫ ∫ ∇g(Xt,r− (x), z)N(drdz).

Then, the matrix-valued process ′ Φx,t (u) ≡ ∇Xt,u (x)

satisfies the linear SDE u

Φx,t (u) = I + ∫ dU x,t Φx,t (r−). t

That is, Φ is the Doléans-Dade exponential (stochastic exponential) ℰ (U)t,x . Since b, σ, and g are assumed to be Cb∞ -functions, the coefficients ∇b(x), ∇σ(x) of U x,t (u) are bounded, and |∇g(x, z)| ≤ K 1 (z), where K 1 (z) satisfies ∫ K 1 (z)p μ(dz) < +∞ for any p ≥ 2. Then, by the uniqueness of the Doléans-Dade exponential and by the direct expression of it (see [189, Section V.3]), sup

x∈Rd ,t∈T,t≤u≤T

󵄨 󵄨p E[󵄨󵄨󵄨Φx,t (u)󵄨󵄨󵄨 ] < +∞,

p > 2.

We observe ′ I + ΔU x,t (r) = I + ∇g(Xt,r (x), z),

z = q(r).

Therefore, the matrix I + ΔU x,t (r) is invertible by the conditions (J.1), (J.2).

3.6 Smoothness of the density for Itô processes

� 235

Let V x,t (u) be the process given by c

2

V x,t (u) = −U x,t (u) + [U x,t , U x,t ]u + ∑ (I + ΔU x,t (r)) (ΔU x,t (r)) . −1

t≤r≤u

(6.25)

Here, [U x,t , U x,t ]cu denotes the continuous part of the matrix-valued process (∑dk=1 [(U x,t )ik , (U x,t )kj ]u ). We introduce another matrix-valued process Ψx,t (u), t ≤ u ≤ T by the SDE x,t

u

Ψ (u) = I + ∫ Ψx,t (r−)dV x,t (r). t

That is, Ψx,t (u) = ℰ (V x,t )u . The process Ψx,t (u) is the inverse of Φx,t (u), that is, Ψx,t (u) = ′ (∇Xt,u (x))−1 (cf. [189, Corollary to Theorem II 38]). We can write Ψx,t (u) in the form u

u

′ ′ (x))dW (r) (x))dr + ∫ Ψx,t (r−)∇σ(Xt,r− Ψx,t (u) = ∫ Ψx,t (r−)∇b′ (Xt,r− t

t

u

′ ̃ + ∫ ∫ Ψx,t (r−)∇h(Xt,r− (x), z)N(drdz), t

where ∇b′ is the drift part determined by (6.25), and h is given by the relation I + h = (I + g)−1 . As the coefficients ∇b′ (x), ∇σ(x), and ∇h(x, z) are bounded functions (due to the conditions (J.1), (J.2)), we have sup

x∈Rd ,t∈T,t 0 such that 1 󵄨 󵄨󵄨 2 − nq0 󵄨󵄨E[Gev (F)]󵄨󵄨󵄨 ≤ C(1 + |v| ) 2

(6.26)

for all |v| ≥ 1, where q0 > 0 is as in Proposition 3.8. Choosing G ≡ 1, this assertion implies that F has a smooth density function pF (⋅) under the condition (NDB); see the introduction of Section 2.1. To show Proposition 3.14, we first establish the following proposition, where F ′ = Xt′ is derived from an SDE for which the jump coefficient satisfies the conditions (J.1), (J.2). Here F ′ satisfies the condition (D). Definition 3.8. We define RF ∼ (ρ) = ′

1 T ∫ D̃ u F ′ (D̃ u F ′ ) n(du), φ(ρ) A(ρ)

RFρ (v) = ′

′ 1 ′ F′ ′ 1 (v , Σ v ) + 2 (v, RF ∼ (ρ)v) T |v| T

where ρ = |v|−1/β . Using Proposition 3.6, we have the following result for F ′ = Xt′ (x) which is a version of Proposition 3.6 in Section 3.5.1 using a light norm | ⋅ |∗l,k,p . Proposition 3.15. Assume the condition (NDB) for F ′ . We have then that for each n there exist l, k, m = m(l, k) ∈ N, p > 2, and C > 0 such that for |v| > 1, 1 󵄨󵄨 ′ 󵄨 2 − nq0 󵄨󵄨E[Gev (F )]󵄨󵄨󵄨 ≤ C(1 + |v| ) 2 n ′ k+1 m(l,k) 󵄨 󵄨∗ 󵄩 󵄩∗ 󵄩 ̃ ′ 󵄩󵄩∗ sup󵄨󵄨󵄨(1 + QF (v)−1 )󵄨󵄨󵄨0,k,p ) × ((󵄩󵄩󵄩DF ′ 󵄩󵄩󵄩l,k,p + (1 + 󵄩󵄩󵄩DF 󵄩󵄩l,k,(k+1)p ) )

|v|≥1

×

(6.27)

‖G‖∗l,k,p .

Here m(l, k) is an integer depending on l, k. Proof. Our objective below is to show the finiteness of the right-hand side of (6.27) under the condition (NDB). If (6.27) holds, choosing G ≡ 1, we have the exponential decay of the characteristic function. We recall QF (v) = ′

where v′ =

v ,v |v|

1 ′ ′ ′ 1 (v , Σ v ) + −1 T 2 |v| φ(|v| β )T

∈ Rd , ρ > 0.

̃ ′ 󵄨 󵄨2 ∫ 󵄨󵄨󵄨ei(v,Du F ) − 1󵄨󵄨󵄨 n(du), A(|v|

−1 β

)

3.6 Smoothness of the density for Itô processes � 237

We start from Proposition 3.6: 1 󵄨󵄨 2 − nq0 ′ 󵄨 󵄨󵄨E[Gev (F )]󵄨󵄨󵄨 ≤ C(1 + |v| ) 2 k+1 󵄩 󵄩∗ 󵄩 ̃ ′ 󵄩󵄩∗ 󵄩 F ′ −1 󵄩∗ n × ((󵄩󵄩󵄩DF ′ 󵄩󵄩󵄩l,k,(k+1)p + (1 + 󵄩󵄩󵄩DF 󵄩󵄩l,k,(k+1)p ) ) sup󵄩󵄩󵄩Q (v) 󵄩󵄩󵄩l,k,p )

|v|≥1

× ‖G‖∗l,k,p

(6.28)

for |v| ≥ 1, due to Lemmas 3.7, 3.12. ′ We need bounds for E[QF (v)−1 ]. As the (NDB) condition implies the (ND-n) condi′ ′ tion, we have, since QF (v) ≥ c RFρ (v) for some 0 < c < 1, ′ 󵄨 ′ 󵄨 sup E[QF (v)−p ] ≤ c−p sup E[󵄨󵄨󵄨RFρ (v)−p 󵄨󵄨󵄨],

|v|≥1

− β1

where ρ = |v| As

(6.29)

|v|≥1

, and the right-hand side is finite by the (ND-n) condition.

(l+1)(2k ) 󵄩󵄩 F ′ −1 󵄩󵄩∗ 󵄩 ′ 󵄩∗ 󵄩 ̃ ′ 󵄩󵄩∗ 󵄩󵄩Q (v) 󵄩󵄩l,k,p ≤ cl,k,p (1 + 󵄩󵄩󵄩DF 󵄩󵄩󵄩l,k,2p + 󵄩󵄩󵄩DF 󵄩󵄩l,k,2(l+1)(2k+1 )p )

by Lemma 3.12, we have the assertion in case q(u) = 0. Next we consider the case |q(u)| ≥ 1 (case l ≥ 0, k ≥ 0). ′ Suppose that RρF (v) are invertible a. s. for any ρ, v. Then 󵄨󵄨 F ′ −1 󵄨󵄨∗ 󵄨 F ′ −1 󵄨∗ 󵄨󵄨Q (v) 󵄨󵄨0,k,p ≤ c󵄨󵄨󵄨Rρ (v) 󵄨󵄨󵄨0,k,p for 0 < ρ < |v|−1/β , k = 0, 1, 2, . . . , p ≥ 2. ̃ ′ This follows from the inequality |ei(v,Du F ) −1|2 ≥ c|(v, D̃ u F ′ )2 |, which holds with some constant 0 < c < 1 on the set |D̃ u F ′ | < |v|−1 , due to (ND-n). Then the inequalities QF (v) ∘ εu+ ≥ cRFρ (v) ∘ εu+ ′

′

hold a. s. if 0 < ρ ≤ |v|−1/β and u ∈ A(1)k . Therefore, if QF (v) ∘ εu+ are invertible a. s., then ′

RFρ (v)|ρ=|v|−1/β ∘ εu+ are also invertible a. s., and satisfy ′

E[(QF (v) ∘ εu+ ) ] ≤ c−p E[(RρF (v) ∘ εu+ ) ], ′

′

−p

−p

for all ρ ≤ |v|−1/β < 1, u ∈ A(1)k . We use the finiteness of right-hand side due to (NDB). And as 󵄨 ′ 󵄨 󵄨 ′ 󵄨∗ sup󵄨󵄨󵄨QF (v)−1 󵄨󵄨󵄨0,k,p;A(ρ) ≤ sup󵄨󵄨󵄨QF (v)−1 󵄨󵄨󵄨0,k,p , |v|≥1

|v|≥1

we have the assertion for l ≥ 0, k ≥ 0. This completes the proof due to the Proposition 3.6 and Lemma 3.12.

238 � 3 Analysis of Wiener–Poisson functionals 3.6.4 Concatenation (II) – the case that (D) may fail Next, we consider the case that F may not satisfy the condition (D); however, we still assume that the conditions (J.1), (J.2) hold, that is, we have the case q(u) ≠ 0 in (6.15). We still assume that supp μ is compact in A(1) and that T < +∞. To verify (6.29) in this general case, we construct a process X̃ ′ (x, St , t) as follows. Here again, we encounter the methodology in Section 2.3.2. By St = {s1 , . . . , sk ′ }, we denote the jump times of z′′ s in [0, t]. Let 𝒮t,k ′ = [0, t]k / ∼, where ∼ means the identification of (s1 , . . . , sk ′ ) by permutation. Let 𝒮t = ∐k ′ ≥0 𝒮t,k ′ . The law P̃ t of the moments of the big jumps related to z′′ s is given by ′

∫ f (St )d P̃ t (St ) k′

= (t ∫ μ′′ (dz))

k′

= (∫ μ (dz)) ′′

1 1 exp(−t ∫ μ′′ (dz)) × ′ ∫ ⋅ ⋅ ⋅ ∫ f (s1 , . . . , sk ′ )ds1 ⋅ ⋅ ⋅ dsk ′ k′! tk

1 exp(−t ∫ μ′′ (dz)) × ∫ ⋅ ⋅ ⋅ ∫ f (s1 , . . . , sk ′ )ds1 ⋅ ⋅ ⋅ dsk ′ k′!

(6.30)

if #St = k ′ , where f is a symmetric function on [0, t]k . Let St ∈ 𝒮t . We introduce a new process X̃ ′ (x, St , t) by the SDE ′

t

X̃ ′ (x, St , t) = x + ∫{b(X̃ ′ (x, St , r)) − ∫ g(X̃ ′ (x, St , r), z)μ′′ (dz)}dr t

0

t

+ ∫ σ(X̃ ′ (x, St , r))dW (r) + ∫ ∫ g(X̃ ′ (x, St , r−), z)Ñ ′ (drdz) 0

+

0

∑

si ∈St ,si ≤t

g(X̃ ′ (x, St , si −), ξi ).

Here, (ξi ) are independent and identically distributed, obeying the probability law μ′′ 1 (dz) =

δ0 ∧ |z|2 μ′′ (dz) . ∫ δ0 ∧ |z|2 μ′′ (dz)

We put F̃ ′ = X̃ ′ (x, St , t), identifying zi in q(u) = ((t1 , z1 ), . . . , (tk ′ , zk ′ )) with (St , ξi ). Then, F̃ ′ can be regarded as + F ′ ∘ εq(u) . By using this notation, we can identify Qρ′ (v)∘εq+ with Qρ′ (v)(St ). Hence, our objective is to show

3.6 Smoothness of the density for Itô processes

� 239

Proposition 3.16. Assume the condition (NDB). Then, 󵄨 󵄨2(k+1)p′ mA (du)]dP′′ (q)) < +∞, sup(∫ E ′ [∫󵄨󵄨󵄨Qρ′ (v)−1 (St ) ∘ εu+ 󵄨󵄨󵄨 |v|≥1

A ⊂ A(1), u ∈ A(1)k , 0 < ρ < 1,

k = 1, 2, . . . ,

(6.31)

for each even number p′ ≥ 2. Here, the expectation with respect to the measure P′ is denoted by E ′ . Proof of Proposition 3.16. Step 1. In case k = 1, with u1 = (t1 , z1 ), 0 ≤ t1 ≤ t, F̃ ′ ∘ ε(t+ 1 ,z1 ) = X̃ t′1 ,t ∘ ϕt1 ,z1 ∘ X̃ t′1 (x), where ϕt,z (x) = x + g(x, z). We remark ϕt,0 (x) = x,

∇z ϕt,z (x) = ∇z g(x, z).

Hence, D̃ t1 ,z1 X̃ ′ = X̃ t′1 ,t ∘ ϕt1 ,z1 ∘ X̃ t′1 − (x) − X̃ t′1 ,t ∘ X̃ t′1 − (x) = ∇X̃ t′1 ,t (ϕt1 ,θz1 (X̃ t′1 − ))∇z ϕt1 ,θz1 (X̃ t′1 )z1

for θ ∈ (0, 1), and 𝜕D̃ t1 ,0 X̃ ′ = ∇X̃ t′1 ,t1 (X̃ t′1 − )∇z g(X̃ t′1 , 0) = ∇X̃ t′1 ,t (X̃ t′1 − )G(X̃ t′1 ). We see X̃ ′ ∈ D0,1,p;A . We also have X̃ ′ ∈ D1,0,p;A , and finally X̃ ′ ∈ D∞ . Hence, Qρ′ (v)(St ) ∘ εu+1 − Qρ′ (v)(St )

1 1 T T ≥ (v′ , ∫{∇X̃ t′1 ,t G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 2 T T

T

T

× zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t ) }dtv′ )

a. e.,

|z1 | < 1.

240 � 3 Analysis of Wiener–Poisson functionals Then, since Qρ′ (v)(St ) ≥ 0, Qρ′ (v)(St )−1 ∘ εu+1 ≤ c(v′ , (

1 T T ∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) T T

−1

T T × zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt)v′ ) ,

a. e.

Here, we have −1 󵄨p 󵄨 󵄨󵄨 󵄨󵄨 ′ 󵄨󵄨E[󵄨󵄨(∇X̃ t1 ,t (x)) 󵄨󵄨󵄨 ]󵄨󵄨󵄨 ≤ C exp{cp (t − t1 )}.

(6.32)

Indeed, let ϕs,t denote the stochastic flow generated by X̃ ′ (x, St , t). Consider the reversed process Zs = ϕ−1 (t−s)−,t . Then, Zs (St ) satisfies the (backward) SDE between big jumps si+1

Zs (St ) = y − ∫ {b(Zu (St ))}du s si+1

si+1

− ∫ σ(Zu (St ))dWu + ∫ ∫ g0 (Zu− (St ), z)Ñ ′ (dudz), s

Zsi+1 − = y,

s

si < s < si+1 .

Here, g0 is such that x 󳨃→ x + g0 (x, z) is the inverse map of x 󳨃→ x + g(x, z) for μ′ a. e. z. Then, [68, Theorem 1.3] implies the assertion (6.32); see also [45]. The finiteness of the term 󵄨󵄨 1 T T 󵄨 E ′ [∫󵄨󵄨󵄨(v′ , ( ∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 󵄨󵄨 T T

−1 󵄨p′

×

T T zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt)v′ )

󵄨󵄨 󵄨󵄨 mA (du1 )] 󵄨󵄨 󵄨

for all |v′ | = 1 holds as follows. It is sufficient to show the finiteness of 󵄨󵄨 1 T T 󵄨 E ′ [∫󵄨󵄨󵄨( ∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 󵄨󵄨 T T

−1 󵄨p 󵄨󵄨 T T × zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt) 󵄨󵄨󵄨 mA (du1 )]. 󵄨󵄨 ′

3.6 Smoothness of the density for Itô processes � 241

We have 1 T T T T ∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) )zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt) T

−1

(

T

≤

1 T T T T −1 ∫(∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) )zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) ) dt. T T

Here, we used the Jensen’s inequality ( T1 ∫T f (s)ds)−1 ≤ T1 ∫T f (s)−1 ds for a positive function f . We integrate the right-hand side with respect to mA . Then, we have 󵄨󵄨 1 T T 󵄨 ∫󵄨󵄨󵄨( ∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 󵄨󵄨 T T

−1 󵄨 T T 󵄨󵄨 × zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt) 󵄨󵄨󵄨mA (du1 ) 󵄨󵄨 1 T T ≤ ∫ ∫(∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )BG(X̃ t′ ) ) T

×

T T −1 T z1 (∇X̃ t′1 ,t (x)G(X̃ t′1 )) ) dtmA (du1 ).

Let λ1 > 0 denote the minimum eigenvalue of C(x). Due to (NDB), we may assume it does not depend on x. Then, E′ [

1 T T ∫ ∫(∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )BG(X̃ t′ ) ) T T

T −1 × zT1 (∇X̃ t′1 ,t (x)G(X̃ t′1 )) ) dtmA (du1 )]

≤ E ′ [∫

1 2 ∫ λ−1 1 |z1 | dtmA (du1 )] CT T

1 ≤ λ−1 C ′ TE ′ [1] = C ′ λ−1 1 < +∞ T 1

a. e. P′′

uniformly in v by (NDB). Here, C ′ > 0 does not depend on ρ. From this, it follows that 󵄨󵄨 T T 󵄨 E ′ [∫󵄨󵄨󵄨(∫ ∇X̃ t′1 ,t (x)G(X̃ t′1 )z1 (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 󵄨󵄨 T

−1 󵄨p′

×

T T zT1 G(X̃ t′1 ) (∇X̃ t′1 ,t (x)) dt)

uniformly in v for p′ ≥ 2.

󵄨󵄨 󵄨󵄨 mA (du1 )] < +∞ 󵄨󵄨 󵄨

a. e. P′′

242 � 3 Analysis of Wiener–Poisson functionals This implies 󵄨 󵄨p′ E ′ [sup ∫󵄨󵄨󵄨Qρ′ (v)(St )−1 ∘ εu+1 󵄨󵄨󵄨 mA (du1 )] ≤ C(q(u))mA (A(1)) < +∞ v

a. e. P′′ ,

where p′ ≥ 2, 0 < t ≤ T. We shall integrate the right-hand side with respect to the measure d P̃ t (St ) ⊗ μ′′ 1 . We ′′ put C1 (t) = E μ1 [C(q(u))]. Then, C1 (t) ≤ C1 (T) < +∞ since μ′′ is a bounded measure and 1 supp μ is compact. For the general case #St = k ′ , we proceed as above by replacing μ′′ 1 (dz1 ) with ′′ ′ μ1 (dz1 ) ⊗ ⋅ ⋅ ⋅ ⊗ μ′′ (dz ). Since (ξ ) are independent and identically distributed, having i k 1 a bounded law, we have ′′ ⊗k ′

E (μ1 ) [C(q(u))] ≤ C1 (t)k , ′

k ′ = 1, 2, . . . ,

where q(u) = ((t1 , z1 ), . . . , (tk ′ , zk ′ )), 0 < t1 < ⋅ ⋅ ⋅ < tk ′ < t. Summing up with respect to the Poisson law, we have, in view of (6.30), ∞ 1 1 k ′ −tc k′ k′ k′ c e C (t) t = (tcC1 (t)) e−tc ∑ 1 ′ ′ k! k! k ′ =0 k ′ =0 ∞

∑

1 k′ (tcC1 (t)) e−tcC1 (t) etcC1 (t) ′ k! k ′ =0 ∞

≤ ∑

≤ etcC1 (t) ≤ eTcC1 (T) ,

where we put c = μ′′ 1 ({|z| ≥ 1}). Hence, we have the assertion (6.31) for k = 1. Step 2. Similarly, in the general case, for u = (u1 , . . . , uk ) = ((t1 , z1 ), . . . , (tk , zk )), ′ F̃ ′ ∘ εu+ = X̃ t′k ,t ∘ ϕtk ,zk ∘ X̃ t′k−1 ,tk ∘ ⋅ ⋅ ⋅ ∘ ϕt1 ,z1 ∘ X̃ 0,t . 1

Hence, D̃ u F̃ ′ = F̃ ′ ∘ εu+ − F̃ ′ = ∇X̃ t′k ,t (ϕtk ,θk zk (Ik−1 ))∇z ϕtk ,θk zk (Ik−1 )

× ⋅ ⋅ ⋅ × ∇X̃ t′2 ,t3 (ϕt2 ,θ2 z2 (I1 ))∇z ϕt2 ,θ2 z2 (I1 )

× ∇X̃ t′1 ,t2 (ϕt1 ,θ1 z1 (X̃ t′1 − ))∇z ϕt1 ,θ1 z1 (X̃ t′1 − )z1 ⊗ ⋅ ⋅ ⋅ ⊗ zk ,

for θi ∈ (0, 1), i = 1, . . . , k. Here, I1 = ∇X̃ t′1 ,t2 (ϕt1 ,θ1 z1 (X̃ t′1 − ))∇z ϕt1 ,θ1 z1 (X̃ t′1 − ) and Ii are defined inductively by Ii = ∇X̃ t′i ,ti+1 (ϕti ,θi zi (Ii−1 ))∇z ϕti ,θi zi (Ii−1 ),

i = 2, . . . , k.

3.6 Smoothness of the density for Itô processes

� 243

Hence, 𝜕D̃ (t,0) F̃ ′ = 𝜕z D̃ u F̃ ′ |z=0 = ∇X̃ t′k ,t (Jk−1 )G(Jk−1 ) ⋅ ⋅ ⋅ ∇X̃ t′2 ,t3 (J1 )G(J1 )∇X̃ t′1 ,t2 (X̃ t′1 − )G(X̃ t′1 − ). Here, J1 = ∇X̃ t′1 ,t2 (X̃ t′1 − )G(X̃ t′1 − ), and Ji is defined inductively as Ji = ∇X̃ t′i ,ti+1 (Ji−1 )G(Ji−1 ),

i = 2, . . . , k.

We have Qρ′ (v)(St ) ∘ εu+ − Qρ′ (v)(St )

1 T T ≥ (v′ , ∫{𝜕z D̃ u F̃ ′ |z=0 z1 ⊗ ⋅ ⋅ ⋅ ⊗ zk (σ(X̃ t′ )σ(X̃ t′ ) + G(X̃ t′ )Bρ G(X̃ t′ ) ) 2 T

T × (𝜕z D̃ u F̃ ′ |z=0 z1 ⊗ ⋅ ⋅ ⋅ ⊗ zk ) }dtv′ )

a. e., |z1 | < 1, . . . , |zk | < 1.

Then, since Qρ′ (v)(St ) ≥ 0, Qρ′ (v)(St )−1 ∘ εu+ ≤ ck (v′ , (

1 Tk

∫ 0 1 there exists a Cp > 0 such sup P(E γ (ϵ) ∩ F γ (ϵ)c ) < Cp ϵp γ

(6.53)

for any ϵ > 0. The proof of this lemma is due to Komatsu–Takeuchi’s estimate [120] and a subtle calculation. We give the proof at the end of this subsection. The following lemma is used to prove the assertion (6.51) above. Lemma 3.20 ([133]). Let N > 1, k ∈ N. Then for any h ∈ N there exists a Ch > 0 such that sup

|x|≤N,u∈A(1)k

for 0 < ϵ < 1.

P(Th (x) ∘ εu+ < ϵ) ≤ Ch ϵh

3.6 Smoothness of the density for Itô processes

� 255

The proof of this lemma depends on the decomposition of the event according to the measurement of displacement Xt′ (x) − x or ∇Xt′ (x) − I. See [133, Lemma 3.1] for the proof. We continue the proof of Proposition 3.17 (i). (Step 2) Case k ≥ 1. We consider first the case k = 1. Let u = (s1 , z1 ) and put ȲV (t) = vT Xt∗−1 V (x, t) ∘ εu+ , n1 Ȳ (t) = ∑k=0 ∑V ∈ϒ′ YV (t). We introduce the events k

−2i 󵄨 󵄨2 Ē i = { ∑ ∫󵄨󵄨󵄨ȲV (t)󵄨󵄨󵄨 dt < ϵq },

i = 0, 1, . . . , n1 ,

V ∈ϒ′i T

and observe ȲV (t) = vT Xt∗−1 V (x, t),

if s1 > t,

ȲV (t) = vT Xt∗−1 ϕ∗−1 u V (x, t),

if s1 < t.

Here if s1 < t then Xt∗−1 ϕ∗−1 u V (x, t) satisfies t

Xt∗−1 ϕ∗−1 u V (x, t)

=

Xs∗−1 V (x, s1 )

+ ∫ Xr∗−1 ℒ′ {ϕ∗−1 u V (x, r)}dr s1

m t

j + ∑ ∫ Xr∗−1 ϕ∗−1 u [Vj (r), V (r)](x)dW (r) j=1 s

1

t ∗−1 ∗−1 ̃ + ∫ ∫ Xr∗−1 {ϕ∗−1 r,z ϕu V (x, r) − ϕu V (x, r)}N(drdz) s1 |z|≤δ t

∗−1 ∗−1 + ∫ ∫ Xr∗−1 {ϕ∗−1 r,z ϕu V (x, r) − ϕu V (x, r)}N(drdz). s1 |z|>δ

Then as in Step 1 we can show c sup sup P(Ē i ∩ Ē i+1 ) ≤ Cϵh ,

|x|≤N v∈S d−1

i = 1, . . . , n1 − 1,

(6.54)

for 0 < ϵ < ϵ1 . We put F̄ = Ē 1 ∩ ⋅ ⋅ ⋅ ∩ Ē n1 . We shall prove sup sup P(F)̄ ≤ Cϵq |x|≤N

v∈S d−1

−2n1 h

.

(6.55)

256 � 3 Analysis of Wiener–Poisson functionals ̄ v) = ∫ |Ȳ (t)|2 dt. Then one can write To this end, set K(x, T s1

T

0

s1

̄ v) = ∫ vT X ′ ∗−1 H(x, t)(X ′ ∗−1 )T vdt + ∫ vT X ′ ∗−1 ϕ∗−1 H(x, t)(X ′ ∗−1 ϕ∗−1 )T vdt. K(x, t t t u t u By the assumption (MUH)δ , there exists c1 > 0 such that T

vT H(x, t)v ≥ c1 |v|2 ,

∗−1 vT ϕ∗−1 v ≥ c1 |v|2 , u H(x, t)ϕu

for t ∈ T, |x| ≤ Mh , u ∈ A(1). Here H(x, t) and Mh are as above. Then we have (6.55) as in Step 1. Using (6.54), (6.55), we can show sup sup P(Ē 0 ) < C ′′ ϵq

−2n1

h

|x|≤N v∈S d−1

for 0 < ϵ < ϵ1 . We have the assertion (6.47) for k = 1 as in Step 1. We can verify the case k ≥ 2 similarly. Part II. Next we consider the case Xt (x) = Xt′ (x) ∘ εq+ . We write Π(x) = Π′ (x) ∘ εq+ . We show the case k = 0 only, since the cases for k ≥ 1 are similar to the above. We have c

sup sup P(Ei ∘ εq+ ∩ (Ei+1 ∘ εq+ ) ) ≤ Cϵh ,

|x|≤N v∈S d−1

i = 1, . . . , n1 − 1

(6.56)

as in (6.50) (first part). We have also sup sup P(F ∘ εq+ ) ≤ Cϵq

−2n1

|x|≤N v∈S d−1

h

,

0 < ϵ < ϵ1 .

(6.57)

n

1 Indeed, put K ′ (x, v) = ∑i=0 ∫T ∑V ∈ϒ′ |YV (t)|2 dt. Then by (MUH)δ , i

󵄨 󵄨2 K ′ (x, v) ≥ C1 ∫󵄨󵄨󵄨vT ∇Xt′ 󵄨󵄨󵄨 dt. T

Hence K ′ (x, v)−1 ≤

1 󵄨 󵄨2 ∫󵄨󵄨∇X ′ 󵄨󵄨 dt C1 T 2 |v|2 󵄨 t 󵄨 T

by Jensen’s inequality, and E[K ′ (x, v)−1 ∘ εq+ ] ≤

h

1 󵄨 󵄨2 E[(∫󵄨󵄨󵄨∇Xt′ ∘ εq+ 󵄨󵄨󵄨 dt) ]ϵh 2 2 h (C1 T |v| ) T

3.6 Smoothness of the density for Itô processes � 257

for h > 1 by Chebyschev’s inequality. −2n1 Since F ⊂ {K ′ (x, v) < (n1 + 1)ϵq }, P(F ∘ εq+ ) ≤ P(K ′ (x, v) ∘ εq+ < (n1 + 1)ϵq

−2n1

) ≤ C ′ ϵq

for |x| < N, v ∈ S d−1 . Here we used P(K ′ (x, v)−1 ∘ ϵq+ > ((n1 + 1)ϵq we have (6.57). By (6.56), (6.57), we have in view of (6.49) sup sup E[(v, Θ′ (x)v)

−p

|x|≤N v∈S d−1

∘ εq+ ] < C ′ ϵq

−2n1

h

−2n1

−2n1

h

)−1 ) ≤ C ′ ϵq

−2n1

h

. Hence

,

0 < ϵ < ϵ1 ,

(6.58)

,

0 < ϵ < ϵ1 ,

(6.59)

for p < hq−2n1 /2. Hence sup sup E[(v, Π′ (x)v)

−p

|x|≤N v∈S d−1

∘ εq+ ] < C ′ ϵq

−2n1

h

for p < hq−2n1 /2. Indeed, (6.58) implies sup |x|≤N,q∈A(1)

󵄨 󵄨−2p/d + E[󵄨󵄨󵄨det Θ′ (x)󵄨󵄨󵄨 ∘ εq ] < +∞.

As | det Π′ (x)| = | det ∇X ′ (x)|2 | det Θ′ (x)|, and since sup |x|≤N,q∈A(1)

󵄨 󵄨−4p/d + E[󵄨󵄨󵄨det ∇X ′ (x)󵄨󵄨󵄨 ∘ εq ] < +∞

due to the flow property, we have the assertion (6.59). As Π(x) = Π′ (x) ∘ εq+ a. s., we have the assertion (6.47) for p < hq−2n1 /2. Since h ∈ N is arbitrary, we have the assertion for all p > 1. Proof of Proposition 3.17(ii). The assertion follows as in Proposition 3.13 with Θ(x, t) replacing C(x). This ends the proof of Theorem 3.6. Proof of Lemma 3.19. In order to prove this lemma, we need the following estimate due to Komatsu–Takeuchi. To apply it, we introduce below a parameter v, and assume it satisfies 161 < v < 81 temporary, due to a technical reason. (We believe there occurs no confusion by using the same character v as above.) After the sublemma below is established with this restriction, we then choose instead 241 < v < 121 , 361 < v < 181 , and so on, and we apply the (modified) sublemma consecutively. Finally, we see that the assertion is valid for all 0 < v < 81 .

258 � 3 Analysis of Wiener–Poisson functionals Sublemma 3.1 ([120, Theorem 3]). Let v be an arbitrary number such that 0 < v < 81 . There exist a positive random variable ℰ (λ, γ) satisfying E[ℰ (λ, γ)] ≤ 1 and positive constants C, C0 , C2 , C2 independent of λ, γ such that the following inequality holds on the set {θγ ≤ λ2v } for all λ > 1 and γ: λ4 ∫{|Y γ (t) |2 ∧ T

1 1 }dt + v log ℰ (λ, γ) + C λ λ2

󵄨 󵄨2 󵄨 󵄨2 ≥ C0 λ1−4v ∫󵄨󵄨󵄨aγ (t)󵄨󵄨󵄨 dt + C1 λ2−2v ∫󵄨󵄨󵄨f γ (t)󵄨󵄨󵄨 dt T

T

1 + C2 λ2−2v ∫ ∫{|g γ (t, z) |2 ∧ 2 }dtμ(dz). λ

(6.60)

T

To prove Lemma 3.19, we choose 0 < η < 1 so small that 2 − 2v − α(1 + η) >

4 q

holds, where q is as in the statement of Lemma 3.19. Then we can choose an r > 1 such that q 1 1 >r> ∨ . 4 2 − 2v − α(1 + η) 1 − 4v The restriction above of the value of v reflects here the restriction on the value of α, so that 0 < α < 2(1 − v) < 158 < 2. However, as we noted above, v can be chosen smaller consecutively, and the argument is valid for all given α such that 0 < α < 2. First, we consider the last term of right-hand side of (6.60). As g γ (t, 0) = 0, the function z 󳨃→ g γ (t, z) can be written as g γ (t, z) = 𝜕g γ (t)z + r γ (t, z) for some r γ (t, z). Then 1󵄨 γ 2 󵄨󵄨 γ 󵄨2 1 󵄨2 1 γ 󵄨󵄨g (t, z)󵄨󵄨󵄨 ∧ 2 ≥ 󵄨󵄨󵄨g (t, z)󵄨󵄨󵄨 ∧ 2 − (r (t, z)) , 2 λ λ and hence for 0 < κ < λ, 󵄨 󵄨2 1 󵄨 󵄨2 1 ∫(󵄨󵄨󵄨g γ (t, z)󵄨󵄨󵄨 ∧ 2 )μ(dz) ≥ ∫ (󵄨󵄨󵄨g γ (t, z)󵄨󵄨󵄨 ∧ 2 )μ(dz) λ λ κ |z|≤ λ

≥

2 󵄨󵄨 1 z 󵄨󵄨󵄨 1 󵄨 ∫ (󵄨󵄨󵄨𝜕g γ (t) 󵄨󵄨󵄨 ∧ 2 2 )|z|2 μ(dz) 󵄨 󵄨 2 |z| 󵄨 |z| λ 󵄨 κ |z|≤ λ

2

κ −( ) λ

2

∫ ( |z|≤ κλ

|r γ (t, z)| ) |z|2 μ(dz) |z|2

3.6 Smoothness of the density for Itô processes

� 259

2 󵄨󵄨 1 z 󵄨󵄨󵄨 1 κ 󵄨 ≥ φ( ) ∫ (󵄨󵄨󵄨𝜕g γ (t) 󵄨󵄨󵄨 ∧ 2 )μ̄ κ/λ (dz) 󵄨 󵄨 2 λ |z| 󵄨 κ 󵄨 κ |z|≤ λ

󵄩󵄩 r γ 󵄩󵄩󵄩2 󵄩 󵄩󵄩 󵄩󵄩 |z|2 󵄩󵄩󵄩 , 󵄩 󵄩

2󵄩

κ κ − φ( )( ) λ λ

where φ(ρ) = ∫|z|≤ρ |z|2 μ(dz) and μ̄ ρ (dz) is a probability measure given by μ̄ ρ (dz) = |z|2 1 μ(dz). φ(ρ) {|z|≤ρ}

We put λ = e−r and κ = eηr . Then κ = ϵ(1+η)r , λ

(6.61) γ

r 4 and φ( κλ ) ≥ C4 ϵα(1+η)r by the order condition. Since ‖ |z| ≤ λ2v on {θγ ≤ λ2v }, the last 2‖ term of (6.60) dominates

󵄨 󵄨 C4 ϵα(1+η)r 󵄨󵄨󵄨𝜕g γ (t)Bϵ(1+η)r 𝜕g γ (t)T 󵄨󵄨󵄨 − C5 ϵ(2+α)(1+r)r , on {θγ ≤ λ2v } due to (6.61). Here Bρ is the covariant matrix of the Lévy measure μ on {|z| ≤ ρ}. Hence (6.60) implies ϵ−4r ∫{|Y γ (t) |2 ∧ϵ2r }dt + ϵrv log ℰ (ϵ−r , γ) + C T

󵄨2 󵄨 󵄨2 󵄨 ≥ C0 ϵ−r(1−4v) ∫󵄨󵄨󵄨aγ (t)󵄨󵄨󵄨 dt + C1 ϵ−r(2−2v) ∫󵄨󵄨󵄨f γ (t)󵄨󵄨󵄨 dt T

T

+ C2 C4 ϵ

−r(2−2v) α(1+η)r

ϵ

󵄨 󵄨 ∫󵄨󵄨󵄨𝜕g γ (t)B𝜕g γ (t)T 󵄨󵄨󵄨dt − C2 C5 ϵ−(2−2v) ϵ(2+α)(1+r)r . T

Here B denotes the infinitesimal covariance with respect to {Bρ }. We set ρ = r min{1 − 4v, 2 − 2v − α(1 + η)} − 1 according to the choice of r above. Then ρ > 0 and the above inequality yields ϵ−4r ∫{|Y γ (t) |2 ∧ϵ2r }dt + ϵrv log ℰ (ϵ−r , γ) + C T ′ 󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨2 ≥ C6 ϵ−(1+ρ) ∫{󵄨󵄨󵄨aγ (t)󵄨󵄨󵄨 + 󵄨󵄨󵄨f γ (t)󵄨󵄨󵄨 + 󵄨󵄨󵄨g̃ γ (t)󵄨󵄨󵄨 }dt − C7 ϵ−(1+ρ) ϵr .

T

on {θγ ≤ ϵ−rv }, where r ′ = 2(1 + η)r > 1.

260 � 3 Analysis of Wiener–Poisson functionals We put γ

G1 = {θγ > ϵ−rv } = {θγ/rv > ϵ−1 }, γ

G2 = {θγ ≤ ϵ−rv } ∩ {∫{|Y γ (t−) |2 ∧ϵ2r }dt < ϵq } T

󵄨2 󵄨2 󵄨 󵄨 󵄨 ∩ {∫{󵄨󵄨󵄨aγ (t)󵄨󵄨󵄨 + 󵄨󵄨󵄨f γ (t)󵄨󵄨󵄨 + 󵄨󵄨󵄨g̃ γ (t)󵄨󵄨󵄨 }dt > ϵ}. 󵄨2

T

Then γ

γ

E γ (ϵ) ∩ F γ (ϵ)c ⊂ G1 ∪ G2 . γ

γ

Therefore the left-hand side of (6.53) is dominated by supγ {P(G1 ) + P(G2 )}. We will calculate them separately. We have by Chebyschev’s inequality p/rv

γ

sup P(G1 ) ≤ ϵp E[(sup θγ ) γ

γ

] ≤ cp ϵp .

On the other hand, due to (6.60), ϵrv

γ

G2 ⊂ {ℰ (ϵ−r , γ)

≥ exp(−ϵq−4r + C6 ϵ−ρ − C7 ϵ−ρ ϵr −1 − C)}. ′

By Chebyschev’s inequality again, γ

ϵrv

sup P(G2 ) ≤ eC exp(ϵq−4r − C6 ϵ−ρ + C7 ϵ−ρ ϵr −1 ) × E[ℰ (ϵ−r , γ) ]. ′

γ

We have that ϵq−4r + C7 ϵ−ρ ϵr −1 < ′

C6 −ρ ϵ 2 rv

holds for ϵ < ϵ0 , for some ϵ0 > 0. Since E[ℰ (ϵ−r , γ)ϵ ] ≤ 1 with 0 < ϵrv < 1, we have for any p > 1, γ

sup P(G2 ) ≤ eC exp(− γ

C6 −ρ ϵ ) ≤ cp′ ϵp 2

for any 0 < ϵ < ϵ0 . This proves the assertion (6.53). Remark 3.15. In the above expression, the symbol ℰ (λ, γ) may look like a phantom. In reality it is written as an exponential of Wiener and Poisson integrals; see [120, Lemma 5.1]. The idea of dilation is explained in [203] in a lucid way. Lemma 3.19 above is based on Sublemma 6.1 by Komatsu–Takeuchi [120]. This lemma can be regarded as a substitution of Norris-type lemma [164] prepared for jump

3.7 Composition on the Wiener–Poisson space (II) – Itô processes

� 261

and diffusion processes. A summary on Norris-type lemma is given also in Cass [45]. A result on the existence of the density under such a delicate geometrical condition may take an active role in a neuron model in [105]. Density in 𝒮 By Proposition 3.14, F has a smooth density pF (y) under the condition (NDB). In case that F ∈ D∞ and under the condition (ND-n), we can show that the density is also rapidly decreasing at infinity. Indeed, let φ(v) = E[ev (F)] be the characteristic function of F. Then, φ(v) is represented by φ(v) = ∫ ei(v,y) pF (y)dy. Hence, pF (y) = (

d

1 ) ∫ e−i(v,y) φ(v)dv, 2π

and j

(1 + |y|2 ) ∇α pF (y) = (

d

1 ) (−i)|α| ∫ e−i(v,y) (1 − Δ)j vα φ(v)dv, 2π α

(6.62)

α

if the right-hand side is finite. Here, α is a multiindex, vα = v1 1 ⋅ ⋅ ⋅ vdd and j = 0, 1, 2, . . . Since 𝜕vα ev (F) = i|α| F α ev (F) and j

(1 − Δ)j vα φ(v) = E[(1 − Δ)j vα ev (F)] = ∑ j Ci E[(−Δ)i {vα ev (F)}], i=0

we have 󵄨󵄨 󵄨 j α 2 −nq0 /2+|α| 󵄨󵄨(1 − Δ) v φ(v)󵄨󵄨󵄨 ≤ C(α, j)(1 + |v| ) by Proposition 3.10 (G ≡ 1) and since F ∈ Lp for all p > 1. Choosing a sufficiently large n (depending on α and j), we see that the right-hand side of (6.62) is finite. This implies that (1 + |y|2 )j ∇α pF (y) is rapidly decreasing as |y| → +∞ for any j, α. Hence pF (y) is in 𝒮 . We continue the analysis on the density in Section 4.1.2.

3.7 Composition on the Wiener–Poisson space (II) – Itô processes Let F = Xt , where Xt is a general Itô process given in Section 3.6.1. We again assume the order condition for the Lévy measure μ. Formula (6.26) suggests that for any n ∈ N, there exist some l = ln , k = kn , p = pn , and C > 0 such that

262 � 3 Analysis of Wiener–Poisson functionals − 󵄨 󵄨 sup 󵄨󵄨󵄨E[Gev (F)]󵄨󵄨󵄨 ≤ C(1 + |v|2 ) ∗

q0 2

n

(7.1)

.

‖G‖l,k,p =1

Indeed, calculating as in Theorem 3.2 in [79] for F = F ′ ∘ εq+ , and putting G ≡ 1, we can show Proposition 3.18. For any n ∈ N, there exist some l, k ∈ N, p > 2 such that 󵄩󵄩 󵄩∗′ ′ 2 − 󵄩󵄩ev (F)󵄩󵄩󵄩l,k,p ≤ C (1 + |v| )

q0 2

n

(7.2)

′ for some C ′ = Cl,k,p,n under the condition (NDB).

We give a sketch of the proof. Let F ′ be a jump and diffusion process composed of diffusion and small jump parts, as in (6.42). We put Ut′ (v) = Vu′ (v)

=

−(i |v|v 2 , Dt F ′ ) QF (v)

,

′

− β1 −1

|v|−2 T −1 φ(|v|

) (ev (−D̃ u F ′ ) − 1)1A(ρ) QF (v) ′

,

where QF (v) = ′

1 1 ′ ′ 󵄨 󵄨2 (v , Σv ) + ∫ 󵄨󵄨󵄨ev (D̃ u F ′ ) − 1󵄨󵄨󵄨 n(du), 1 − T |v|2 Tφ(|v| β ) A(ρ(v))

(7.3)

v , ρ(v) = |v|−1/β . where v′ = |v| As γ(τ) ≤ 1 and |q(u)| > 1,

QF (v)(St ) ∘ ετ+ = QF (v) ∘ εq+ ∘ ετ+ . ′

+ Hence, to show (7.2), we first write F = F ′ ∘ εq(u) and show the estimate for F ′ viewing q(u) as a parameter, and then integrate it with respect to the big jumps q(u) = (St , (ξi )). Here, we have

Lemma 3.21. Suppose that F ′ satisfies (NDB). Then, for any l, k ∈ N and p ≥ 2, 󵄩 󵄩∗ sup(|v|󵄩󵄩󵄩U ′ (v)󵄩󵄩󵄩l,k,p ) < +∞, |v|≥1

− β1

sup(|v|Tφ(|v| |v|≥1

󵄩 󵄩∗ )󵄩󵄩󵄩V ′ (v)󵄩󵄩󵄩l,k,p ) < +∞.

This lemma follows as Lemma 4.4 in [79].

(7.4)

3.7 Composition on the Wiener–Poisson space (II) – Itô processes

� 263

Let Z′ = U ′ ⊕ V ′. Then, we have ̃ ) − 1))ev (F ). 𝒟(t,u) ev (F ) = ((iv, DF ) ⊕ (ev (DF ′

′

′

′

Hence, one can check the identity ev (F ′ ) = ⟨U ′ ⊕ V ′ , 𝒟(t,u) ev (F ′ )⟩K , ̂ This leads to where K = L2 (T, Rm ) ⊕ L2 (𝕌, N). ′ ̄ E P [Gev (F ′ )] = E P [⟨GZ ′ , 𝒟(t,u) ev (F ′ )⟩K ] = E P [δ(GZ )ev (F ′ )]. ′

′

′

Iterating this argument, we have ′ ′ ̄ ̄ ′ δ(Z ̄ ′ ⋅ ⋅ ⋅ δ(Z ̄ ′ δ(GZ ̄ E P [Gev (F ′ )] = E P [δ(GZ )ev (F ′ )] = E P [δ(Z ))))ev (F ′ )]. ′

′

′

Hence, in order to show (7.2), it is sufficient to show that for each n, there exist l, k ∈ N, an even number p ≥ 2, l′ , k ′ ∈ N, an even number p′ ≥ 2, and C > 0 such that n

sup

‖G‖∗′

l ,k ′ ,p′

q − 0n 󵄩󵄩⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 󵄩∗ ′ ̄ ′ ∘ ε+ δ(Z ̄ ′ ∘ ε+ ⋅ ⋅ ⋅ δ(Z ̄ ′ ∘ ε+ δ(GZ ̄ ∘ εq+ ))))󵄩󵄩󵄩l,k,p ≤ C(1 + |v|2 ) 2 . 󵄩󵄩δ(Z q q q

(7.5)

=1

Indeed, for n = 1, we have by the inequality (Theorem 3.5) 󵄩󵄩 ̄ 󵄩∗ ′ 󵄩󵄩δ(GZ (v))󵄩󵄩󵄩l,k,p 1 1 −1 󵄩 󵄩∗ 󵄩 󵄩∗ ≤ C(󵄩󵄩󵄩GU ′ (v)󵄩󵄩󵄩l+p−1,k,p + T 2 φ(|v| β ) 2 󵄩󵄩󵄩GV ′ (v)χA(ρ(v)) 󵄩󵄩󵄩l,k+p−1,p ) 1 1 −1 󵄩 󵄩 󵄩∗ 󵄩∗ ≤ C‖G‖∗l+p−1,k,2p (󵄩󵄩󵄩U ′ (v)󵄩󵄩󵄩l+p−1,k+p,2p + T 2 φ(|v| β ) 2 󵄩󵄩󵄩V ′ (v)χA(ρ(v)) 󵄩󵄩󵄩l+p−1,k+p−1,2p ) ≤ C ′ ‖G‖∗l+p−1,k+p−1,2p |v|−q0 , where in the second inequality, we have used the Cauchy–Schwarz inequality, and in the last we have used the previous lemma (Lemma 3.21) and the order condition. Hence, we have the assertion for l′ = l + p − 1, k ′ = k + p − 1 and p′ = 2p.

264 � 3 Analysis of Wiener–Poisson functionals Suppose that the assertion holds for n − 1. Then, we have n

󵄩∗ 󵄩󵄩⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ′ ̄ ′ δ(Z ̄ ′ ⋅ ⋅ ⋅ δ(Z ̄ ′ δ(GZ ̄ ))))󵄩󵄩󵄩l,k,p 󵄩󵄩δ(Z 1 1 −1 󵄩∗ 󵄩 󵄩 ̃ ′ 󵄩󵄩 ≤ C(󵄩󵄩󵄩δ(U ′ )󵄩󵄩󵄩l+p−1,k,2p + T 2 φ(|v| β ) 2 󵄩󵄩󵄩δ(V )󵄩󵄩l,k+p−1,2p )‖G‖∗l′ ,k ′ ,p′ n−1

󵄩⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ 󵄩∗ ′ ̄ ′ ⋅ ⋅ ⋅ δ(Z ̄ ′ δ(GZ ̄ × 󵄩󵄩󵄩δ(Z )))󵄩󵄩󵄩l+p−1,k+p−1,2p

−q0 |v|−q0 (n−1) ≤ C|v|−q0 n , ≤ C‖G‖∗n l′ ,k ′ ,p′ |v|

where the second inequality follows from the assumption of the induction. Hence, we have the assertion for F ′ by induction. + In case F = F ′ ∘ εq(u) , since P̃ t,1 (St ) ⊗ (μ′′ )⊗#St is a bounded measure, we integrate the above inequality with respect to this measure and obtain (7.5). This proves the assertion (7.2) for F. Hence we can define the composition Φ∘F of F with Φ ∈ 𝒮 ′ under (NDB) due to (7.2). This leads to the asymptotic expansion of the density (Section 4.1.2) in the next chapter under a condition equivalent to (NDB).

4 Applications Incessant is the change of water where the stream glides on calmly: the spray appears over a cataract, yet vanishes without a moment’s delay. But none of them has resisted the destructive work of time. Kamo no Chomei (1155–1216), Hojoki/The Ten Foot Square Hut (Japanese classic)

In this chapter, we provide some applications of jump processes and their stochastic analysis given in the previous sections. In Section 4.1, we give an asymptotic expansion of functionals which can be viewed as models for financial assets. Namely, we apply the composition techniques developed in Section 3.5 to a functional F(ϵ) of the solution Xt = St (ϵ) to the canonical SDE with parameter ϵ > 0. A notion of asymptotic expansion in the space D′∞ is introduced, and it is applied to F(ϵ) as ϵ → 0. We take Φ in the space 𝒮 ′ of tempered distributions. We shall justify the asymptotic expansion Φ ∘ F(ϵ) ∼ f0 + ϵf1 + ϵ2 f2 + ⋅ ⋅ ⋅

in D′∞ .

They are based on formal Taylor expansions in powers of ϵ. However, they are justified quantitatively using the three-parameter norms introduced in Section 3.3. Taking the expectation, the above gives an asymptotic expansion for the value E[Φ ∘ F(ϵ)] ∼ E[f0 ] + ϵE[f1 ] + ϵ2 E[f2 ] + ⋅ ⋅ ⋅ . Especially in case Φ(x) = ev (x) = ei(v,x) , this result gives the asymptotic expansion of the density function p(ϵ, y) ∼ p0 (y) + ϵp1 (y) + ϵ2 p2 (y) + ⋅ ⋅ ⋅ , which is closely related to the short-time asymptotic estimates stated in Theorems 2.2, 2.4 and Proposition 2.7 in Sections 2.2, 2.3. We treat this topic in Section 4.1.2. In Section 4.1.3, we provide some examples of asymptotic expansions which often appear in applications in the theory of mathematical finance. In Section 4.1, we denote the solution of the SDE by St or St (ϵ) instead of Xt , according to the custom in financial theory. Section 4.2 is devoted to the application of a finance theory for jump processes to the optimal consumption problem. Although the problem we treat here is one-dimensional, it provides us with a good example of the interplay between probability theory (stochastic analysis) and the PDE theory (boundary value problems).

4.1 Asymptotic expansion of the SDE In this subsection, we apply the composition techniques developed in Sections 3.5, 3.7 to a functional F(ϵ) of the solution St (ϵ) to the canonical SDE with parameter ϵ > 0. https://doi.org/10.1515/9783110675290-005

266 � 4 Applications In economic models, the processes which represent the asset prices are often complex. We would like to analyze the behavior of them by inferring from simple ones. In our model below, St (1) = St (ϵ)|ϵ=1 is the target process, and ϵ > 0 is an approximation parameter (coefficient to random factors). We expand in Section 4.1.1 the composition Φ ∘ St (ϵ) with Φ ∈ 𝒮 ′ by Φ ∘ St (ϵ) ∼ f0 + ϵf1 + ϵ2 f2 + ⋅ ⋅ ⋅ . Each term fj = fj,t in the above has a proper meaning according to the order j. Based on this expansion, the expansion of the transition density function (Section 4.1.2) will be helpful in practical applications. We consider an m-dimensional Lévy process Zt given by t

t

̃ + ∫ ∫ zN(dsdz), Zt = σW (t) + ∫ ∫ zN(dsdz) 0 |z|>1

0 |z|≤1

where σ is an m × m matrix such that σσ T = A. Let ϵ ∈ (0, 1] be a small parameter. Let S(ϵ) be a d-dimensional jump process depending on ϵ, given as the solution to the canonical SDE m

dSt (ϵ) = a0 (St (ϵ)) + ϵ ∑ ai (St (ϵ)) ∘ dZti , i=1

S0 (ϵ) = x.

(1.1)

That is, St (ϵ) is given by t

m

t

St (ϵ) = S0 (ϵ) + ∫ a0 (Ss− (ϵ))ds + ϵ ∑ ∫ ai (Ss− (ϵ))σij ∘ dW j (s) t

+ ϵ{∫ ∫ t

i,j=1 0

0

0 |z|≤1

(ϕz1 (Ss− (ϵ))

t

̃ − Ss− (ϵ))N(dsdz) + ∫ ∫ (ϕz1 (Ss− (ϵ)) − Ss− (ϵ))N(dsdz) 0 |z|>1

m

̂ + ∫ ∫ (ϕz1 (Ss− (ϵ)) − Ss− (ϵ) − ∑ zi ai (Ss− (ϵ)))N(dsdz)}. 0 |z|≤1

i=1

(1.2)

Here, ai (x), i = 0, 1, . . . , m are bounded C ∞ -functions from Rd to Rd with bounded derivatives of all orders, and ∘dW j (s) denotes the Stratonovich integral with respect to the Wiener process W j (t). Here, ϕzs denotes a one-parameter group of diffeomorphisms (inm z z tegral curve) generated by ∑m i=1 zi ai (⋅), and ϕ1 = ϕs |s=1 . We may also write Exp(t ∑i=1 zi ai ) z z for ϕt . Obviously, the coefficient function ϕ1 (x) depends on ai (x)’s. See Section 2.5.1 for details on canonical SDEs.

4.1 Asymptotic expansion of the SDE

� 267

We remark that our asset model above adapts well to the geometric Lévy model (cf. [69, Section 2], [173, Section 2.2]). The jump term represents the discontinuous fluctuation of the basic asset with parameter ϵ. In our model, each jump size of the driving Poisson random measure N is not necessarily small, but the total fluctuation given as the stochastic integral of N is controlled by the small parameter ϵ > 0. Remark 4.1. By transforming the Stratonovich integral into an Itô integral, (1.2) can be written simply as t

m

t

St (ϵ) = S0 (ϵ) + ∫ ã 0 (Ss− (ϵ))ds + ϵ ∑ ∫ ai (Ss− (ϵ))σij dW j (s) t

i,j=1 0

0

̃ + ϵ ∫ ∫(ϕz1 (Ss− (ϵ)) − Ss− (ϵ))N(dsdz),

(1.3)

0

where 1 m ã 0 (x) = a0 (x) + ϵ ∑ ∇ai (x)ai (x)σij2 2 i,j=1 m

+ ϵ{ ∫ (ϕz1 (x) − x)μ(dz) + ∫ (ϕz1 (x) − x − ∑ zi ai (x))μ(dz)}. |z|>1

|z|≤1

i=1

We also remark that our model is slightly different from those studied in [117, 135] and from [219], where the drift coefficient ã 0 also depends on ϵ when transformed into the Itô form. It is known that (1.2) has a unique global solution for each ϵ. We sometimes omit ϵ and simply write St for St (ϵ). One of our motivations in mathematical finance is to make a composition Φ(ST (ϵ)) = Φ ∘ ST (ϵ) with some functional Φ appearing in the European call or put options, and obtain an asymptotic expansion Φ(ST (ϵ)) ∼ f0 + ϵf1 + ϵ2 f2 + ⋅ ⋅ ⋅ in some function space. Note that the above is not a formal expansion, but it is an expansion with an intrinsic sense in terms of the Sobolev norms introduced in Section 3.3. As expanded above, errors from the deterministic factor are arranged in terms of the order of the power of ϵ. As ϵ → 0, each coefficient of ϵj will provide a dominant factor at the power j in the model. We recall m × m matrices Bρ and B, namely

268 � 4 Applications Bρ =

1 ∫ zzT μ(dz), φ(ρ) |z|≤ρ

B = lim inf Bρ , ρ→0

provided that the Lévy measure satisfies the order condition as in Chapters 2 and 3. We set B = 0 if μ does not satisfy the order condition. The Lévy process is called nondegenerate if the matrix A + B is invertible, where A = σσ T . We define a d × m matrix C(x) α by (a1 (x), . . . , am (x)). We choose the same quantity q0 = 1 − 2β (1 + k1 ) > 0 as in (3.52) 0 (Section 3.3) and fix it.

4.1.1 Analysis on the stochastic model We introduce a basic assumption on A + B. Assumption (AS) There exists c > 0 such that for all x ∈ Rd , it holds that (v, C(x)(A + B)C(x)T v) ≥ c|v|2 .

(1.4)

Remark 4.2. (1) This assumption corresponds to (NDB) in Definition 3.6 (Section 3.6). (2) Assume d = 1. If S0 (ϵ) = x > 0 and A > 0, then the (Merton) model stated in (1.2) can be interpreted to satisfy the assumption (AS) with C(x) = x and g(x, z) = ϕz1 (x) − x. Indeed, in this case, the process St (ϵ) hits {0} only by the action of the drift and the diffusion factors, and once it hits {0}, it stays there afterwards. We denote by St (ϵ) the solution to (1.2) starting from S0 (ϵ) = x at t = 0. It is known that it has a modification such that St (ϵ) : Rd → Rd is a C ∞ -diffeomorphism a. s. for all t > 0 (cf. Fujiwara–Kunita [68]). In order to simplify the notation, we introduce the following assumption: Assumption (AS2) 󵄨 󵄨 ∫ 󵄨󵄨󵄨ϕz1 (x) − x 󵄨󵄨󵄨μ(dz) < +∞

for all x ∈ Rd .

(1.5)

|z|>1 d Then, we have ∫Rm \{0} {ϕz1 (x) − x − ∑m i=1 zi ai (x)1{|z|≤1} }μ(dz) < +∞ for all x ∈ R . t

put

We note that St (0) is given by the ODE St (0) = S0 (0) + ∫0 a0 (Ss (0))ds, S0 (0) = x. We 1 Ft (ϵ) = (St (ϵ) − St (0)), ϵ

and let

4.1 Asymptotic expansion of the SDE

�

269

F(ϵ) = FT (ϵ), where T < +∞. In the sequel, we sometimes suppress ϵ in St (ϵ) for notational simplicity. We can calculate the stochastic derivatives of F easily. Indeed, m

Dt F = (∇St,T (St− )(∑ ai (St− )σij ), j = 1, . . . , m), i=1

a. e. dtdP,

where ∇St,T (x) is the Jacobian matrix of the map St,T (x) : Rd → Rd , and Ss,t = St ∘ Ss−1 . Further, since d d S (x + ϵ(ϕzs (x) − x)) = ∇St,T (x + ϵ(ϕzs (x) − x)) (x + ϵ(ϕzs (x) − x)) ds t,T ds m

= ∇St,T (x + ϵ(ϕzs (x) − x))ϵ ∑ zi ai (ϕzs (x)), i=1

we have 1 D̃ t,z F = (St,T (St− + ϵ(ϕz1 (St− ) − St− )) − St,T (St− )) ϵ m

= ∇St,T (St− + ϵ(ϕzθ (St− ) − St− )) ∑ zi ai (ϕzθ (St− )), i=1

by the mean value theorem, where θ ∈ (0, 1). Due to the above, we have 𝜕D̃ t,0 F(ϵ) = 𝜕z D̃ t,z F(ϵ)|z=0 = ∇St,T (St− )(a1 (St− ), . . . , am (St− )),

a. e. dtdP.

Repeating this argument, we find F ∈ D∞ . (See Section 3.3 for the definition of the Sobolev space D∞ .) Due to [189, Theorem V.60], we have Lemma 4.1 ([100, Lemma 6.2]). Assume (AS) and (AS2). The Jacobian matrix ∇Ss,t is invertible a. s. Both |∇Ss,t | and |(∇Ss,t )−1 | (norms in matrices) belong to Lp for any p > 1. Further, there exist positive constants cp and Cp such that 󵄨 󵄨p E[|∇Ss,t |p ] + E[󵄨󵄨󵄨(∇Ss,t )−1 󵄨󵄨󵄨 ] ≤ Cp exp cp (t − s)

(1.6)

holds for any x and s < t, and ϵ > 0 sufficiently small. Furthermore, we have Lemma 4.2. For any k ∈ N, there exist positive constants cp′ and Cp′ such that −1 󵄨p 󵄨 sup E[󵄨󵄨󵄨(∇Ss,t ∘ εu+ ) 󵄨󵄨󵄨 ] ≤ Cp′ exp cp′ (t − s), k

u∈A(1)

(1.7)

270 � 4 Applications with any p ≥ 2, holds for any x and s < t, and ϵ > 0 sufficiently small. Remark 4.3. In [100], we have assumed that the support of μ is compact. However, the assertion can be extended to the whole space under the assumption (AS2). Proof of Lemma 4.2. We will show supu∈A(1)k |(∇Ss,t ∘ εu+ )−1 | ∈ Lp , p ≥ 2. For k = 1, we write (∇Ss,t ∘ εu+ )

−1

= (∇Ss,t )−1 ∘ εu+ = D̃ u (∇Ss,t )−1 + (∇Ss,t )−1 .

For general u = (u1 , . . . , uk ), (∇Ss,t )−1 ∘ εu+ is a sum of terms D̃ ui ,...,ui (∇Ss,t )−1 , where 1 l i1 , . . . , il ∈ {1, . . . , k}, l ≤ k. As |(∇Ss,t )−1 | ∈ Lp by Lemma 4.1, we will show that supu∈A(1)k |D̃ u (∇Ss,t )−1 | ∈ Lp , u = (u1 , . . . , uk ) for k = 1, 2, . . . We recall that the inverse matrix (∇Ss,t )−1 satisfies a. s. the SDE t

(∇Ss,t )−1 = I − ∫(∇Ss,r− )−1 ∇ã 0 (Sr− )dr s

t

m

+ ϵ ∑ ∫(∇Ss,r− )−1 ∇ai (Sr− )σij dW j (r) i,j=1 s t

+ ϵ ∫ ∫(∇Ss,r− )−1 {(∇ϕz1 (Sr− ))

−1

s

̃ − I}N(drdz).

(1.8)

Let k = 1 and let u = (s1 , z1 ). Since D̃ u (∇St− )−1 = 0 if t < s1 , we assume t ≥ s1 . Then, D̃ u (∇St− )−1 should satisfy t

−1 D̃ u (∇St )−1 = (∇ϕz1 (Ss1 )) − I − ∫ D̃ u ((∇Sr− )−1 ⋅ ∇ã 0 (Sr− ))dr s1

m

t

+ ϵ ∑ ∫ D̃ u ((∇Sr− )−1 ⋅ ∇ai (Sr− ))σij dW j (r) i,j=1 s

1

t

̃ + ϵ ∫ ∫ D̃ u {(∇Sr− )−1 (∇ϕz1 (Sr− )−1 − I)}N(drdz).

(1.9)

s1

We will show |D̃ u (∇St )−1 | ∈ Lp . Let τn , n = 1, 2, . . . be a sequence of stopping times such that τn = inf{t > s; |D̃ u (∇Ss,t )−1 | > n} (= T if the set {. . . } is empty). In the following discussion, we denote the stopped process D̃ u (∇Ss,t∧τn )−1 as D̃ u (∇St )−1 . However, constants appearing in the inequalities do not depend on n or on u.

4.1 Asymptotic expansion of the SDE

�

271

Drift term We have D̃ u ((∇Sr− )−1 ⋅ ∇ã 0 (Sr− )) = D̃ u (∇Sr− )−1 ⋅ ∇ã 0 (Sr− ) + (∇Sr− )−1 ⋅ D̃ u ∇ã 0 (Sr− ) + D̃ u (∇Sr− )−1 ⋅ D̃ u ∇ã 0 (Sr− ) and D̃ u (∇ã 0 (Sr− )) = (D̃ u Sr− )T ∇2 ã 0 (Sr− + θD̃ u Sr− ) for some θ ∈ (0, 1) by the mean value theorem. Since ∇ã 0 is a bounded function, we have the inequality (by Jensen’s inequality) 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 −1 ̃ 󵄨 ̃ E[󵄨󵄨∫ Du (∇Sr− ) ∇a0 (Sr− )dr 󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 p t t 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ 󵄨 󵄨p −1 󵄨󵄨 󵄨󵄨󵄨 ′ ≤ cE[󵄨󵄨∫󵄨󵄨Du (∇Sr− ) 󵄨󵄨dr 󵄨󵄨 ] ≤ c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr. 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 s Similarly, since ∇2 ã 0 is a bounded function, 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 −1 ̃ −1 󵄨󵄨󵄨󵄨 ̃ 󵄨 󵄨 󵄨 󵄨 󵄨 E[󵄨󵄨∫(∇Sr− ) Du ∇ã 0 (Sr− )dr 󵄨󵄨 ] ≤ cE[󵄨󵄨∫󵄨󵄨(∇Sr− ) 󵄨󵄨󵄨󵄨Du Sr− 󵄨󵄨dr 󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 󵄨s 󵄨 t

󵄨 󵄨p ≤ c′ ∫ E[󵄨󵄨󵄨(∇Sr− )−1 󵄨󵄨󵄨 |D̃ u Sr− |p ]dr s

t

p/p 󵄨 󵄨p p/p ≤ c ∫(E[󵄨󵄨󵄨(∇Sr− )−1 󵄨󵄨󵄨 1 ]) 1 (E[|D̃ u Sr− |p2 ]) 2 dr ′′

s

where

1 p1

+

1 p2

= p1 , p1 > 1, p2 > 1. Since E[|D̃ u Sr− |p2 ] < +∞ for all p2 ≥ 1, we have t 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨 󵄨p E[󵄨󵄨󵄨∫(∇Sr− )−1 D̃ u ∇ã 0 (Sr− )dr 󵄨󵄨󵄨 ] ≤ c′′′ ∫ E[󵄨󵄨󵄨(∇Sr− )−1 󵄨󵄨󵄨 ]dr. 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 s

As for the third term, 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨󵄨 −1 󵄨󵄨󵄨󵄨 ̃ −1 ̃ ̃ ̃ 󵄨 󵄨 󵄨 󵄨 ̃ E[󵄨󵄨∫ Du (∇Sr− ) Du ∇a0 (Sr− )dr 󵄨󵄨 ] ≤ cE[󵄨󵄨∫󵄨󵄨Du (∇Sr− ) 󵄨󵄨󵄨󵄨Du Sr− 󵄨󵄨dr 󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 󵄨s 󵄨 t

󵄨 󵄨p ≤ c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 |D̃ u Sr− |p ]dr. ′

s

272 � 4 Applications Hence, we have t 󵄨󵄨p 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨 󵄨p E[󵄨󵄨󵄨∫ D̃ u (∇Sr− )−1 D̃ u ∇ã 0 (Sr− )dr 󵄨󵄨󵄨 ] ≤ c′′ ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr 󵄨󵄨 󵄨󵄨 󵄨 󵄨s s

as above. Diffusion term We have D̃ u ((∇Sr− )−1 ⋅ ∇ai (Sr− )) = D̃ u (∇Sr− )−1 ⋅ ∇ai (Sr− ) + (∇Sr− )−1 ⋅ D̃ u ∇ai (Sr− ) + D̃ u (∇Sr− )−1 ⋅ D̃ u ∇ai (Sr− ). Then, t 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨 󵄨p −1 j E[󵄨󵄨󵄨∫ D̃ u (∇Sr− ) ⋅ ∇ai (Sr− )σij dW (r)󵄨󵄨󵄨 ] ≤ c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr. 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 s

We have D̃ u (∇ai (Sr− )) = (D̃ u Sr− )T ∇2 ai (Sr− + θ′ D̃ u Sr− ) for some θ′ ∈ (0, 1) as above. Hence, t 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 󵄨 󵄨p −1 j ′ ̃ 󵄨 󵄨 E[󵄨󵄨∫(∇Sr− ) ⋅ Du ∇ai (Sr− )σij dW (r)󵄨󵄨 ] ≤ c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr 󵄨󵄨 󵄨󵄨 󵄨s 󵄨 s

as above. The third term is bounded similarly. Jump term We have D̃ u {(∇Sr− )−1 (∇ϕz1 (Sr− )−1 − I)} = D̃ u (∇Sr− )−1 ⋅ (∇ϕz1 (Sr− )−1 − I)

+ (∇Sr− )−1 D̃ u (∇ϕz1 (Sr− )−1 − I)

+ D̃ u (∇Sr− )−1 ⋅ D̃ u (∇ϕz1 (Sr− )−1 − I). Then, 󵄨󵄨 t 󵄨󵄨p 󵄨󵄨 󵄨󵄨 −1 z −1 ̃ ̃ 󵄨 E[󵄨󵄨∫ ∫ Du (∇Sr− ) (∇ϕ1 (Sr− ) − I)N(drdz)󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨s 󵄨

4.1 Asymptotic expansion of the SDE

� 273

󵄨󵄨p/2 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨2 ̂ 󵄨󵄨 ̃ −1 z −1 󵄨 ≤ cE[󵄨󵄨∫ ∫󵄨󵄨Du (∇Sr− ) (∇ϕ1 (Sr− ) − I)󵄨󵄨 N(drdz)󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨 󵄨s t

󵄨p ̂ 󵄨 + c E[∫ ∫󵄨󵄨󵄨D̃ u (∇Sr− )−1 (∇ϕz1 (Sr− )−1 − I)󵄨󵄨󵄨 N(drdz)]. ′

(1.10)

s

We remark that −1 󵄨 󵄨2 sup ∫󵄨󵄨󵄨(∇ϕz1 (x)) − I 󵄨󵄨󵄨 μ(dz) < +∞, x

−1 󵄨 󵄨p sup ∫󵄨󵄨󵄨(∇ϕz1 (x)) − I 󵄨󵄨󵄨 μ(dz) < +∞. x

t Hence, the right-hand side of (1.10) is dominated by c′′ ∫s E[|D̃ u (∇Sr− )−1 |p ]dr. We note that also (∇ϕz1 )−1 = ∇((ϕz1 )−1 ) = ∇(ϕ−z 1 ) by the inverse mapping theorem.

Hence,

−z −z −z D̃ u (∇ϕz1 (Sr− )−1 − I) = D̃ u (∇ϕ−z 1 (Sr )) = ∇ϕ1 (Ss1 ,r ∘ ϕ1 (Sr )) − ∇ϕ1 (Ss1 ,r ) m

−z −z −z = ∇2 ϕ−z 1 (Ss1 ,r ∘ ϕθ′′ (Sr )) ⋅ ∇Ss1 ,r (ϕθ′′ (Sr )) ⊗ ∑(−zi )ai (ϕθ′′ (Sr )) i=1

T

m

−z 2 −z −z = − (∇Ss1 ,r (ϕ−z θ′′ (Sr ))) ∇ ϕ1 (Ss1 ,r ∘ ϕθ′′ (Sr )) ⋅ ∑ zi ai (ϕθ′′ (Sr )) i=1

for some θ′′ ∈ (0, 1). From this expression, we obtain t

󵄨 󵄨p ̂ E[∫ ∫󵄨󵄨󵄨(∇Sr− )−1 D̃ u (∇ϕz1 (Sr− )−1 − I)󵄨󵄨󵄨 N(drdz)] s

t

󵄨 󵄨p 󵄨 󵄨󵄨p ≤ cE[∫ ∫󵄨󵄨󵄨(∇Sr− )−1 󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨∇Ss1 ,r (ϕ−z θ′′ (Sr ))󵄨󵄨 s

󵄨󵄨 m 󵄨󵄨p 󵄨󵄨 󵄨 󵄨󵄨p 󵄨󵄨󵄨 −z −z 󵄨 ̂ × 󵄨󵄨󵄨∇2 ϕ−z (S ∘ ϕ (S )) ⋅ z a (ϕ (S )) ∑ 󵄨 󵄨 ′′ ′′ s1 ,r r 󵄨 󵄨󵄨 i i θ r 󵄨󵄨󵄨 N(drdz)]. 1 θ 󵄨󵄨i=1 󵄨󵄨 p We have E[supy |∇Ss1 ,r (y)|p ] < +∞, supx ∫ |∇2 ϕ−z 1 (x)| μ(dz) < +∞, and

󵄨󵄨 m 󵄨󵄨p 󵄨󵄨 󵄨󵄨 −z sup ∫󵄨󵄨󵄨∑ zi ai (ϕθ (x))󵄨󵄨󵄨 μ(dz) < +∞, 󵄨󵄨 󵄨󵄨 x,θ 󵄨i=1 󵄨

p ≥ 2. t

So the right-hand side of the above formula is dominated by c′ ∫s E[|(∇Sr− )−1 |p ]dr.

274 � 4 Applications Finally, 󵄨󵄨p 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 ̂ 󵄨󵄨 ] E[󵄨󵄨󵄨∫ ∫ D̃ u (∇Sr− )−1 D̃ u (∇ϕz1 (Sr− )−1 − I)N(drdz) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨s t 󵄨󵄨p/2 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ̃ 󵄨󵄨2 ̂ 󵄨󵄨 −1 ̃ z −1 ≤ cE[󵄨󵄨∫ ∫󵄨󵄨Du (∇Sr− ) Du (∇ϕ1 (Sr− ) − I)󵄨󵄨 N(drdz)󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 󵄨 󵄨s t

󵄨 󵄨p ̂ + cE[∫ ∫󵄨󵄨󵄨D̃ u (∇Sr− )−1 D̃ u (∇ϕz1 (Sr− )−1 − I)󵄨󵄨󵄨 N(drdz)]. s

Due to arguments similar to those above, both terms are dominated by t

󵄨 󵄨p c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr. s

Since we know E[|(∇Sr− )−1 |p ] < +∞ for all p > 1, all the terms are dominated by t

󵄨 󵄨p c ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr. s

Hence, by Lemma 4.1, t

󵄨 󵄨p 󵄨 󵄨p E[󵄨󵄨󵄨D̃ u (∇St− )−1 󵄨󵄨󵄨 ] ≤ c + ϵc ∫ E[󵄨󵄨󵄨D̃ u (∇Sr− )−1 󵄨󵄨󵄨 ]dr. s

Therefore, 󵄨 󵄨p E[󵄨󵄨󵄨D̃ u (∇St− )−1 󵄨󵄨󵄨 ] ≤ c exp(ϵc(t − s))

(1.11)

by the Gronwall’s lemma. As the constants can be chosen not depending on n or on u, we let n → +∞ to obtain the assertion for k = 1. Repeating this argument, we have the assertion of the lemma. By the differentiability of z 󳨃→ ϕzs , we can see that D̃ t,z F is twice continuously differentiable with respect to z = (z1 , . . . , zm ) a. s. Writing St (ϵ) = Xt in the form considered in Section 3.6, σ(x), b(x), and g(x, z) are given by σ(x) = ϵ(a1 (x), . . . , am (x))σ, ã 0 (x) and g(x, z) = ϵ(ϕz1 (x) − x) = ϵ(Exp(∑m i=1 zi ai )(x) − x), respectively. Since all the derivatives of ai (x)’s are bounded and satisfy the linear growth condition, we are in the same situation as in Section 3.6.1. The condition (NDB) is satisfied if (AS) holds, and St (ϵ) enjoys the flow property since it is a canonical process. Regarding F(ϵ) = ϵ1 (ST (ϵ) − ST (0)) where ST (0) = Exp(Ta0 )(x)

4.1 Asymptotic expansion of the SDE

� 275

is deterministic, we see, by Lemma 3.18 from Chapter 3, that the condition (R) holds for F(ϵ) for each ϵ > 0. That is, for any positive integer k and p > 1, the derivatives satisfy, for 0 < ϵ < 1, sup

m

m

󵄨 󵄨p 󵄨 󵄨p E[∑ sup 󵄨󵄨󵄨𝜕zi D̃ t,z F(ϵ) ∘ εu+ 󵄨󵄨󵄨 + ∑ sup󵄨󵄨󵄨𝜕zi 𝜕zj D̃ t,z F(ϵ) ∘ εu+ 󵄨󵄨󵄨 ] < +∞. k

t∈T,u∈A(1)

i=1 |zi |≤1

i,j=1 |z|≤1

(1.12)

We write the Malliavin covariance R(ϵ) related to the Wiener space depending on ϵ > 0 as T

R(ϵ) = ∫ Dt F(ϵ)(Dt F(ϵ)) dt.

(1.13)

T

The Malliavin covariance R(ϵ) is then represented by R(ϵ) = ∫ ∇St,T (St− )C(St− )AC(St− )T ∇St,T (St− )T dt

a. s. dP.

(1.14)

T

We put T ̃ K(ϵ) = ∫(𝜕D̃ t,0 F(ϵ))B(𝜕D̃ t,0 F(ϵ)) dt.

(1.15)

T

̃ Then, the covariance K(ϵ) is written as T ̃ K(ϵ) = ∫ ∇St,T (St− )C(St− )B(C(St− )) ∇St,T (St− )T dt

a. s. dP.

(1.16)

T

Furthermore, we put T K̃ ρ (ϵ) = ∫ 𝜕D̃ t,0 F(ϵ)Bρ (𝜕D̃ t,0 F(ϵ)) dt

a. s. dP

(1.17)

T

for ρ > 0. Note that K̃ ρ (ϵ) can be written as K̃ ρ (ϵ) = ∫ A(ρ)

1 T ̂ ∇S (S )C(St− )zzT (C(St− )) ∇St,T (St− )T N(du). φ(ρ) t,T t−

(1.18)

We introduce the following condition to guarantee the existence of the composition Φ ∘ F(ϵ) with Φ ∈ 𝒮 ′ uniformly in ϵ. Definition 4.1. We say that F(ϵ) = (F 1 , . . . , F d ) satisfies the condition (UND) (uniform nondegeneracy condition) if for all p ≥ 1 and any integer k, there exists β ∈ ( α2 , 1] such that

276 � 4 Applications 󵄨󵄨 1 󵄨 lim sup sup sup ess supu∈Ak (ρ) E[󵄨󵄨󵄨( (v, R(ϵ)v) 󵄨󵄨 T ϵ→0 ρ∈(0,1) v∈Rd , |v|=1

󵄨2 󵄨 ̂ + T −1 φ(ρ)−1 ∫ 󵄨󵄨󵄨(v, D̃ u F(ϵ))󵄨󵄨󵄨 1{|D̃ u F(ϵ)|≤ρβ } N(du)) A(ρ)

−1

󵄨󵄨p 󵄨 ∘ εu+ 󵄨󵄨󵄨 ] < +∞, 󵄨󵄨

where R(ϵ) = (Rij (ϵ)) and Rij (ϵ) = ∫T Dt F i (ϵ)Dt F j (ϵ)dt. In [79], we put 1{|(v,D̃ u F(ϵ))|≤ρβ } in place of 1{|D̃ u F(ϵ)|≤ρβ } . Here, we have strengthened the condition. We are treating a sequence of random variables with parameter ϵ defined on the Wiener–Poisson space. We set the topology among them by using the order of ϵ as it tends to 0 in terms of the ‖ ⋅ ‖l,k,p norms. Here and in what follows, norms ‖ ⋅ ‖l,k,p stand for ‖ ⋅ ‖l,k,p;A with A = A(1). We define the asymptotic expansion in D∞ and D′∞ in terms of the order of ϵ. Definition 4.2. Let G(ϵ) be a Wiener–Poisson variable. We write G(ϵ) = O(ϵm ) in D∞ if G(ϵ) ∈ D∞ for ϵ ∈ (0, 1], and if for all l, k ≥ 0 and for any p ≥ 2, it holds lim sup

‖G(ϵ)‖l,k,p

ϵ→0

ϵm

< ∞.

Definition 4.3. (1) We say that F(ϵ) has the asymptotic expansion ∞

F(ϵ) ∼ ∑ ϵj fj j=0

in D∞

(1.19)

if it holds that (i) F(ϵ), f0 , f1 , . . . ∈ D∞ for any ϵ. (ii) For any nonnegative integer m, m

F(ϵ) − ∑ ϵν fν = O(ϵm+1 ) ν=0

in D∞ .

(2) We say that Φ(ϵ) ∈ D′∞ has an asymptotic expansion ∞

Φ(ϵ) ∼ ∑ ϵj Φν j=0

in D′∞

(1.20)

if for any m ≥ 0, there exist l = l(m) ≥ 0, k = k(m) ≥ 0, and p ≥ 2 such that for some Φ(ϵ), Φ0 , Φ1 , Φ2 , . . . , Φm ∈ D′l,k,p , it holds that

4.1 Asymptotic expansion of the SDE

lim sup ϵ→0

�

277

󵄩󵄩′ 󵄩 m 󵄩󵄩 1 󵄩󵄩󵄩󵄩 j 󵄩 < +∞. Φ(ϵ) − ϵ Φ ∑ 󵄩 ν󵄩 󵄩󵄩 ϵm+1 󵄩󵄩󵄩󵄩 󵄩󵄩l,k,p ν=0

Remark 4.4. In (1), the topology for (F(ϵ)) is a weak topology given as an inductive limit in m of those given by the strong convergence with respect to the ‖ ⋅ ‖l,k,p -norm up to the order m = 1, 2, 3, . . ., where the index (l, k, p) is arbitrary. On the other hand, in (2), the topology for (Φ(ϵ)) is a weak topology given as an inductive limit in m of those given by the ∗-weak topology induced on D′l,k,p (the topology on D′l,k,p of pointwise convergence with respect to the ‖ ⋅ ‖′l,k,p -norm) up to the order m = 1, 2, 3, . . . , where the index (l, k, p) is as above. Lemma 4.3. (1) If F(ϵ) ∈ D∞ and G(ϵ) ∈ D∞ , ϵ ∈ (0, 1], are such that ∞

F(ϵ) ∼ ∑ ϵj fj j=0

in D∞

and ∞

G(ϵ) ∼ ∑ ϵj gj j=0

in D∞ ,

then H(ϵ) = F(ϵ)G(ϵ) satisfies ∞

H(ϵ) ∼ ∑ ϵj hj j=0

in D∞

and hj ’s are obtained by the formal multiplication: h0 = g0 f0 ,

h1 = g0 f1 + g1 f0 ,

h2 = g0 f2 + g1 f1 + g2 f0 ,

(2) If F(ϵ) ∈ D∞ and Φ(ϵ) ∈ D′∞ , ϵ ∈ (0, 1], such that ∞

F(ϵ) ∼ ∑ ϵj fj j=0

in D∞

and ∞

Φ(ϵ) ∼ ∑ ϵj Φj j=0

in D′∞ ,

then Ψ(ϵ) = F(ϵ)Φ(ϵ) satisfies ∞

Ψ(ϵ) ∼ ∑ ϵj Ψj j=0

in D′∞

...

278 � 4 Applications and Ψj ’s are obtained by the formal multiplication: Ψ0 = f0 Φ0 ,

Ψ1 = Φ0 f1 + Φ1 f0 ,

Ψ2 = Φ0 f2 + Φ1 f1 + Φ2 f0 ,

...

(1.21)

The proof of this lemma is similar to that of [87, Proposition 9.3(i), (iii)]. The following is the main assertion. Theorem 4.1 ([79, Theorem 5.5]). Suppose F(ϵ) satisfies the condition (UND), and that j ′ ′ F(ϵ) ∼ ∑∞ j=0 ϵ fj in D∞ . Then, for all Φ ∈ 𝒮 , we have that Φ ∘ F(ϵ) ∈ D∞ has the asymp′ totic expansion in D∞ , namely 1 n n (𝜕 Φ) ∘ f0 ⋅ (F(ϵ) − f0 ) n! m=0 |n|=m ∞

Φ ∘ F(ϵ) ∼ ∑ ∑

∼ Φ0 + ϵΦ1 + ϵ2 Φ2 + ⋅ ⋅ ⋅

in D′∞ .

Here, Φ0 , Φ1 , Φ2 are given by the formal expansions: Φ0 = Φ ∘ f0 ,

d

Φ1 = ∑ f1i (𝜕xi Φ) ∘ f0 , i=1

d

Φ2 = ∑ f2i (𝜕xi Φ) ∘ f0 + i=1 d

Φ3 = ∑ f3i (𝜕xi Φ) ∘ f0 + i=1

1 d i j 2 ∑ f f (𝜕 Φ) ∘ f0 , 2 i,j=1 1 1 xi xj 1 d i j k 3 2 d i j 2 Φ) ∘ f0 , ∑ f1 f2 (𝜕xi xj Φ) ∘ f0 + ∑ f f f (𝜕 2! i,j=1 3! i,j,k=1 1 1 1 xi xj xk

... In case m = d = 1, the condition (UND) holds under the assumption that infx∈R a(x) > 0; see [79, Theorem 7.4]. Before applying this theorem directly, we introduce another nondegeneracy condition. ̃ is invertible and Definition 4.4. We say F(ϵ) satisfies the condition (UND2) if R(ϵ) + K(ϵ) if for any integer k and p > 1, lim sup ϵ→0

sup

|v|=1,u∈A(1)k

−p ̃ E[(v, (R(ϵ) + K(ϵ)) ∘ εu+ v) ] < +∞.

We claim: Proposition 4.1. The assumptions (AS), (AS2) imply the condition (UND2). Proof. By (1.14), (1.16), and (AS), we have 󵄨 󵄨2 ̃ (v, (R(ϵ) + K(ϵ)) ∘ εu+ v) ≥ C ∫󵄨󵄨󵄨vT (∇St,T ∘ εu+ )󵄨󵄨󵄨 dt. T

(1.22)

4.1 Asymptotic expansion of the SDE

� 279

Since we have, by the Jensen’s inequality, t

t

−1

1 ( ∫ f (s)ds) t

≤

0

1 ∫ f (s)−1 ds t 0

for a positive function f (⋅), 1 ̃ (R(ϵ) + K(ϵ)) ∘ εu+ v) T

−1

(v,

≤

1 󵄨 󵄨−2 ∫󵄨󵄨󵄨(v, (∇St,T ∘ εu+ ))󵄨󵄨󵄨 dt 2 CT T

1 −1 󵄨2 󵄨 ≤ ∫󵄨󵄨(∇St,T ∘ εu+ ) 󵄨󵄨󵄨 dt. CT 2 |v|2 󵄨 T

In the last inequality, we used the inequality |vT ∇St,T ∘ εu+ |−1 ≤ |(∇St,T ∘ εu+ )−1 |/|v|. Hence, by taking v with |v| = 1, sup

|v|=1,u∈A(1)k

≤

E[(v,

−p 1/p

1 ̃ (R(ϵ) + K(ϵ)) ∘ εu+ v) ] T

1 −1 󵄨2p 1/p 󵄨 ∫ sup E[󵄨󵄨󵄨(∇St,T ∘ εu+ ) 󵄨󵄨󵄨 ] dt. 2 CT u∈A(1)k T

The last term is dominated by a positive constant Cp,k > 0 by Lemma 4.2. Furthermore, we have Proposition 4.2. The condition (UND2) for F(ϵ) implies the condition (UND) for F(ϵ). Combining Propositions 4.1 and 4.2, we can use the theory of composition given in Section 3.5, to obtain the asymptotic expansion using Theorem 4.1 under the nondegeneracy of the coefficients. To prove the above proposition, we introduce the following notion. ̃ is invertible Definition 4.5. We say that F(ϵ) satisfies the condition (UND3) if R(ϵ) + K(ϵ) and if for any integer k and any p > 1, there exists ρ0 > 0 such that lim sup sup ϵ→0

0 0.

0

This implies ̃ ′ (z)) −βψ(z) + L0 ψ(z) + U(ψ r R 1 1 − r r−1 ≤ −β zr + σ 2 R(r − 1)zr + R(f (z) − μz)zr−1 + R zr . r 2 r The right-hand side of the above is dominated by 1 R 1 {−β + σ 2 r(r − 1) + (κ − μ)r + (1 − r)R r−1 }zr ≤ 0 r 2

on (0, +∞). Hence, the condition (2.20) is met. Theorem 4.2. We assume (B.3), (2.2), (B.4), and (2.20). Then, u = uM ∈ ℬ of (2.14) is a viscosity solution of (2.11) and is concave on [0, ∞).

4.2 Optimal consumption problem �

315

Proof. In this proof, we assume the dynamic programming principle (Bellman principle) that u(z) is continuous on [0, ∞) and satisfies τz ∧τ

1 1 1 u(z) = sup E[ ∫ e−(β+ ϵ )t {U(c(t)) + u(z(t))}dt + e−(β+ ϵ )τz ∧τ u(z(τz ∧ τ))] ϵ c∈𝒞M

(∗)

0

for any bounded stopping time τ. By the above, we assume u is in C([0, ∞)). Then, we prove that u is a viscosity solution of (2.11) in the following way. (a) u is a viscosity supersolution. Let z ∈ O ⊂ (0, ∞), where O is an open interval. Choose any ϕ ∈ C 2 (O) such that ϕ(z) = u(z) = c0 and u ≥ ϕ in O. We can assume c0 = 0 without loss of generality. Let m ∈ [0, ∞). Choose ϵ > 0 so that z − ϵm ∈ O. Since the jumps of z(t) are only rightward, z 󳨃→ u(z) is nondecreasing. Hence, ϕ(z) = u(z) ≥ u(z − ϵm) ≥ ϕ(z − ϵm). This implies −mϕ′ (z) ≤ 0. Hence, F(z, u, p, q, B1 (z, v, p), B1 (z, ϕ, p)) ≤ 0 for p = ϕ′ (z), q = ϕ′′ (z). Take c(t) ≡ c ∈ 𝒞M . Let τr be the exit time of z(t) from Ō r (z) = {z′ ; |z − z′ | ≤ r} ⊂ (0, ∞). By the dynamic programming principle (∗), t∧τr

1 1 1 u(z) ≥ E[ ∫ e−(β+ ϵ )s {U(c(s)) + u(z(s))}ds + e−(β+ ϵ )t∧τr u(z(t ∧ τr ))]. ϵ

(2.27)

0

We replace u by ϕ as u(z) = ϕ(z) and use the ordering u ≥ ϕ in O. Then, we have, by the Itô’s formula and (2.11), t∧τr

1 1 1 0 ≥ E[ ∫ e−(β+ ϵ )s {U(c(s)) + u(z(s))}ds + e−(β+ ϵ )t∧τr u(z(t ∧ τr ))] − ϕ(z) ϵ

0

t∧τr

1 1 ≥ E[ ∫ e−(β+ ϵ )s {U(c) + Lϕ(z(s)) + ϕ(z(s)) − cϕ′ (z(s))}ds] ϵ

0

1 1 1 ≥ min [Lϕ(y) + ϕ(y) − cϕ′ (y) + U(c)] × E[ (1 − e−(β+ ϵ )(t∧τr ) )]. 1 ϵ y∈Ō r (z) β+

ϵ

Dividing the last inequality by t and taking successively the limits as t → 0 and r → 0, we have Lϕ(z) − cϕ′ (z) + U(c) ≤ 0. Maximizing the right-hand side over c ∈ 𝒞M , we have

316 � 4 Applications F(z, u, p, q, B1 (z, v, p), B1 (z, ϕ, p)) ≤ 0,

z>0

̃ u, p, q, B1 (z, v, p), B1 (z, ϕ, p)) ≤ 0, F(z,

z>0

and

with p = ϕ′ (z), q = ϕ′′ (z). Hence, u is a subsolution of (2.11). (b) u is a viscosity subsolution. The proof is by reduction to absurdity. Let z ∈ O ⊂ (0, ∞). Let ϕ ∈ C 2 (O) be any function such that ϕ(z) = u(z) and u ≤ ϕ on O. Assume that the subsolution inequality for ϕ fails at z. Then, there exists an ϵ > 0 such that F(ϕ) = F(z, u, p, q, B1 , B1 ) ≤ −ϵ and −ϕ(y) ≤ −ϵ on some ball Ō r (z) ⊂ O with p = ϕ′ (z), q = ϕ′′ (z). We use that there exists a constant η > 0 and t0 > 0 such that E[e

−(β+ ϵ1 )t

t∧τ r,c

1 1 ϕ(z(t ∧ τ )) + ∫ e−(β+ ϵ )s {U(c(s)) + u(z(s))}ds] ≤ ϕ(z) − ηt ϵ

r,c

0

holds for all t ∈ (0, t0 ] and any c ∈ 𝒞 (cf. [112, Lemma 4.4.3], [104, Proposition 2.5]). Here, τ r,c denotes the exit time of z(t) from Ō r (z) by c. Fix t ∈ (0, t0 ). By the dynamic programming principle (∗), we have that there exists c ∈ 𝒞M such that u(z) ≤ E[e

−(β+ ϵ1 )t

t∧τ r,c

1 1 1 ϕ(z(t ∧ τ )) + ∫ e−(β+ ϵ )s {U(c(s)) + u(z(s))}ds] + ηt ϵ 2

r,c

0

for each τ = τ r,c . As u ≤ ϕ in O, we have 1 1 u(z) ≤ ϕ(z) − ηt + ηt = ϕ(z) − ηt. 2 2 This is a contradiction. To see the concavity of u, we also recall TM h(z) of (2.24). We prove below that TM h(z) n is concave if h is concave. Moreover, by induction, TM h is concave for any n ≥ 1. By Lemma 4.10, we have n TM h→u

as n → ∞.

We can choose h(z) = 0(z) ∈ ℬφ , where 0(z) denotes the function identically equal to zero. Then, we can conclude the assertion. To this end, we fix h ∈ ℬφ and assume it is concave.

4.2 Optimal consumption problem

� 317

Choose distinct points z1 , z2 in (0, ∞), and choose processes z1 (t), z2 (t) starting from these points, respectively. For each δ > 0, we choose c1 (t), c2 (t) in 𝒞M , corresponding to z1 (t), z2 (t), respectively, satisfying τi

1 1 TM h(zi ) − δ < E[∫ e−(β+ ϵ )t {U(ci (t)) + h(zi (t))}dt], ϵ

i = 1, 2.

(2.28)

0

Let 0 < λ < 1, and we put cλ (t) = λc1 (t) + (1 − λ)c2 (t). We shall show below that cλ (t) ∈ 𝒞M , and that τz∘

1 1 E[ ∫ e−(β+ ϵ )t {U(cλ (t)) + h(z∘ (t))}dt] ϵ

0

τz1

1 1 ≥ λE[ ∫ e−(β+ ϵ )t {U(c1 (t)) + h(z1 (t))}dt] ϵ

0

τz2

1 1 + (1 − λ)E[ ∫ e−(β+ ϵ )t {U(c2 (t)) + h(z2 (t))}dt]. ϵ

0

Here, z∘ (t) is the solution to the following SDE: ̃ dz∘ (t) = (f (z∘ (t)) − μz∘ (t) − cλ (t))dt − σz∘ (t)dW (t) + z∘ (t) ∫(eζ − 1)N(dt dζ ), z∘ (0) = λz1 + (1 − λ)z2 . We put zλ (t) = λz1 (t) + (1 − λ)z2 (t), and τzλ = τz1 ∨ τz2 . We then have 0 ≤ zλ (t) ≤ z∘ (t),

a. s.

Indeed, using the concavity and Lipschitz continuity of f (⋅), t

E[(zλ (t) − z (t)) ] ≤ E[∫(f (zλ (s)) − f (z∘ (s))) ⋅ 1{zλ (s)≥z∘ (s)} ds] ∘

+

0

t

≤ Cf ∫ E[(zλ (s) − z∘ (s)) ]ds. +

0

Here, Cf denotes the Lipschitz constant. By Gronwall’s inequality, we have the assertion. This implies cλ (t) ∈ 𝒞M .

318 � 4 Applications By this comparison result, using the monotonicity and the concavity of U(⋅) and h(⋅), τz∘

1 1 TM h(λz1 + (1 − λ)z2 ) ≥ E[ ∫ e−(β+ ϵ )t {U(cλ (t)) + h(z∘ (t))}dt] ϵ

0 τz

λ

1 1 ≥ E[ ∫ e−(β+ ϵ )t {U(cλ (t)) + h(zλ (t))}dt] ϵ

0

τz

λ

1 1 ≥ λE[ ∫ e−(β+ ϵ )t {U(c1 (t)) + h(z1 (t))}dt] ϵ

0

τz

λ

1 1 + (1 − λ)E[ ∫ e−(β+ ϵ )t {U(c2 (t)) + h(z2 (t))}dt]. ϵ

0

This implies that TM h(z) is concave, yielding therefore that u is concave. Remark 4.9. The proof of the Bellman principle (∗) in the general setting (i. e., without continuity or measurability assumptions) is not easy. Occasionally, one is advised to decompose the equality into two parts: (i) τz′ ∧τ

1 1 1 u(z) ≤ sup E[lim sup ∫ e−(β+ ϵ )t {U(c(t)) + u(z(t))}dt + e−(β+ ϵ )τz′ ∧τ u(z(τz′ ∧ τ))], ϵ c∈𝒞M z′ →z

0

(ii)

and the converse τz′ ∧τ

1 1 1 u(z) ≥ sup E[lim inf ∫ e−(β+ ϵ )t {U(c(t)) + u(z(t))}dt + e−(β+ ϵ )τz′ ∧τ u(z(τz′ ∧ τ))]. ′ ϵ z →z c∈𝒞M

0

The proof for (i) is relatively easy. The proof of (ii) is difficult if u(z) is not assumed to be continuous. This principle is proved in the diffusion case in [112, Theorem 4.5.1], and in a twodimensional case for some jump-diffusion process in [101, Lemma 1.5]. The next result provides a verification theorem for finding the value function in terms of the viscosity solution for (2.6). Theorem 4.3. We assume (B.3), (2.2), (B.4), and (2.20). Then, there exists a concave viscosity solution v of (2.6) such that 0 ≤ v ≤ φ. Proof. Let 0 < M < M ′ . By (2.24), we have TM h ≤ TM ′ h for h ∈ ℬ. Hence, uM must have a weak limit, which is denoted by v,

319

4.2 Optimal consumption problem �

uM ↑ v

as M → ∞.

It is clear that v is concave and then continuous on (0, ∞). We shall show v(0+) = 0. It follows from (2.24) and (2.21) that τz

1 1 v(z) ≤ sup E[∫ e−(β+ ϵ )t {U(c(t)) + v(z(t))}dt] ≤ φ(z), ϵ c∈𝒞

z ≥ 0.

(2.29)

0

Hence, for any ρ > 0, there exists c = (c(t)) ∈ 𝒞 which depends on z such that τz

v(z) − ρ < E[∫ e

−(β+ ϵ1 )t

0

τz

1 1 U(c(t))dt] + E[∫ e−(β+ ϵ )t v(z(t))dt]. ϵ

(2.30)

0

First, let t

t

0

0

1 ̃ N(t) ≡ z exp((κ + μ)t − σW (t) − σ 2 t + ∫ ∫ ζ N(dsdζ ) − ∫ ∫(eζ − 1 − ζ )μ(dζ )ds). 2 Since z(t) ≤ N(t) for each t > 0, by the Gronwall’s inequality, and since E[ sup N 2 (s)] < +∞ 0≤s≤t

for each t > 0, we see by (2.10) that 1

E[e−(β+ ϵ )(τz ∧s) z(τz ∧ s)] τz ∧s

1 = z + E[ ∫ e−(τz ∧s)t {−(β + )z(t) + f (z(t)) − μz(t) − c(t)}dt]. ϵ 0

By (B.4), we then choose ϵ > 0 so that f ′ (0+) + μ < β + 1/ϵ. Then, by letting s → +∞ and z → 0, we get τz

E[∫ e 0

−(β+ ϵ1 )t

τz

1

c(t)dt] ≤ z + E[∫ e−(β+ ϵ )t f (z(t))dt] 0 τz

1

≤ z + E[∫ e−(β+ ϵ )t κN(t)dt] → 0. 0

On the other hand, by the concavity (B.3) of U(⋅), for each ρ > 0, there exists Cρ > 0 such that U(x) ≤ Cρ x + ρ

(2.31)

320 � 4 Applications for x ≥ 0. Then, letting z → 0 and then ρ → 0, we obtain τz

τz

1

1

E[∫ e−(β+ ϵ )t U(c(t))dt] ≤ Cρ E[∫ e−(β+ ϵ )t c(t)dt] + 0

0

ρ

β+

1 ϵ

→ 0.

Thus, by the dominated convergence theorem, ∞

1 1 1 v(0+) ≤ E[ ∫ e−(β+ ϵ )t v(0+)dt] ≤ v(0+), ϵ βϵ + 1

(2.32)

0

which implies v(0+) = 0. Thus, v ∈ C([0, ∞)). By Dini’s theorem, uM converges to v locally and uniformly on [0, ∞). Therefore, by the standard stability results (cf. [17, Proposition II.2.2]), we deduce that v is a viscosity solution of (2.10) and then of (2.6). Comparison (monotonicity) We give a comparison theorem for the viscosity solution v of (2.6). Theorem 4.4. Let fi , i = 1, 2 satisfy (2.2) and let vi ∈ C([0, ∞)) be the concave viscosity solution of (2.6) for fi in place of f such that 0 ≤ vi ≤ φ, i = 1, 2. Suppose f1 ≤ f2 .

(2.33)

Then, under (B.3), (2.2), (B.4), and (2.20), we have v1 ≤ v2 . Proof. The proof follows again by reduction to absurdity. Let, for ν > 1, φν (z) ≡ zν + B. By (2.15), (B.4), we first note that there exists 1 < ν < 2 such that 1 − β + σ 2 ν(ν − 1) + κν − μν < 0. 2

(2.34)

Hence, by (2.15), (2.16), φν (z) satisfies 1 ′ ′ − βφν (z) + σ 2 z2 φ′′ ν (z) + (f2 (z) − μz)φν (z) − ∫{φν (z) ⋅ γ(z, ζ )}μ(dζ ) 2 1 ≤ (−β + σ 2 ν(ν − 1) + κν − μν)zν + Aνzν−1 − βB < 0, z ≥ 0 2 for a suitable choice of B > 0.

(2.35)

4.2 Optimal consumption problem

that

� 321

Suppose that v1 (z0 ) − v2 (z0 ) > 0 for some z0 ∈ (0, ∞). Then, there exists η > 0 such sup[v1 (z) − v2 (z) − 2ηφν (z)] > 0.

(2.36)

z≥0

Since, by the concavity of v, it holds that v1 (z) − v2 (z) − 2ηφν (z) ≤ φ(z) − 2ηφν (z) → −∞

as z → +∞.

This implies that we can find some z̄ ∈ (0, ∞) such that sup[v1 (z) − v2 (z) − 2ηφν (z)] = v1 (z)̄ − v2 (z)̄ − 2ηφν (z)̄ > 0. z≥0

(2.37)

Define Ψn (z, y) = v1 (z) − v2 (y) −

n |z − y|2 − η(φν (z) + φν (y)) 2

(2.38)

for each n ≥ 1. It is clear that Ψn (z, y) ≤ φ(z) + φ(y) − η(φν (z) + φν (y)) → −∞

as z + y → +∞.

Hence, we can find (zn , yn ) ∈ [0, ∞) × [0, ∞) such that Ψn (zn , yn ) = sup{Ψn (z, y) : (z, y) ∈ [0, ∞) × [0, ∞)}. Since n |z − yn |2 − η(φν (zn ) + φν (yn )) 2 n ≥ v1 (z)̄ − v2 (z)̄ − 2ηφν (z)̄ > 0,

Ψn (zn , yn ) = v1 (zn ) − v2 (yn ) −

(2.39)

we can deduce n |z − yn |2 ≤ v1 (zn ) − v2 (yn ) − η(φν (zn ) + φν (yn )) 2 n ≤ φ(zn ) + φ(yn ) − η(φν (zn ) + φν (yn )) < M for some M > 0. Thus, we deduce that the sequences {zn + yn } and {n|zn − yn |2 } are bounded by some constant, and |zn − yn | → 0

as n → ∞.

(2.40)

Moreover, we can find zn → ẑ ∈ [0, ∞)

and

yn → ẑ ∈ [0, ∞)

as n → ∞

322 � 4 Applications for some ẑ ∈ [0, +∞), taking a subsequence if necessary. By the definition of (zn , yn ), we recall that n |z − yn |2 − η(φν (zn ) + φν (yn )) 2 n ≥ v1 (zn ) − v2 (zn ) − 2ηφν (zn ),

Ψn (zn , yn ) = v1 (zn ) − v2 (yn ) −

and thus n |z − yn |2 ≤ v2 (zn ) − v2 (yn ) + η(φν (zn ) − φν (yn )) → 0 2 n

as n → ∞.

(2.41)

Hence, n|zn − yn |2 → 0 as n → ∞. Passing to the limit in (2.39), we get v1 (z)̂ − v2 (z)̂ − 2ηφν (z)̂ > 0

and ẑ > 0.

(2.42)

Next, let V1 (z) = v1 (z) − ηφν (z) and V2 (y) = v2 (y) + ηφν (y). Applying Ishii’s lemma (cf. [52], [63, Section V.6]) to Ψn (z, y) = V1 (z) − V2 (y) −

n |z − y|2 , 2

we obtain q1 , q2 ∈ R such that ̄ V1 (zn ), (n(zn − yn ), q1 ) ∈ J 2,+

̄ V2 (yn ), (n(zn − yn ), q2 ) ∈ J 2,− 1 0

−3n (

0 q )≤( 1 1 0

0 1 ) ≤ 3n ( −q2 −1

−1 ) 1

(2.43)

where J

2,± ̄

{ { Vi (z) = {(p, q) : { {

∃zr → z, } } , ∃(pr , qr ) ∈ J 2,± Vi (zr ), } } (Vi (zr ), pr , qr ) → (Vi (z), p, q)}

i = 1, 2,

and the inequality in matrices is defined so that the difference of the two makes a nonnegative matrix. Recall that 2,+ J 2,+ v1 (z) = {(p + ηφ′ν (z), q + ηφ′′ ν (z)) : (p, q) ∈ J V1 (z)},

2,− J 2,− v2 (y) = {(p − ηφ′ν (y), q − ηφ′′ ν (y)) : (p, q) ∈ J V2 (y)}.

Hence, by (2.43), 2,+ ̄ (p1 , q̄ 1 ) := (n(zn − yn ) + ηφ′ν (zn ), q1 + ηφ′′ ν (zn )) ∈ J v1 (zn ),

2,− ̄ (p2 , q̄ 2 ) := (n(zn − yn ) − ηφ′ν (yn ), q2 − ηφ′′ ν (yn )) ∈ J v2 (yn ).

4.2 Optimal consumption problem �

323

By the definition of viscosity solutions, we have 1 ̃ 1 ) ≥ 0, − βv1 (zn ) + σ 2 z2n q̄ 1 + (f1 (zn ) − μzn )p1 + B1 (zn , v1 , p1 ) + B1 (zn , ϕ1 , ϕ′1 ) + U(p 2 1 ̃ 2 ) ≤ 0. − βv2 (yn ) + σ 2 y2n q̄ 2 + (f2 (yn ) − μyn )p2 + B1 (yn , v2 , p2 ) + B1 (yn , ϕ2 , ϕ′2 ) + U(p 2 Here, the C 2 -functions ϕ1 , ϕ2 are such that ϕ′1 (zn ) = p1 = n(zn − yn ) + ηφ′ν (zn ), v1 − ϕ1 has a global maximum at zn , and ϕ′2 (yn ) = p2 = n(zn − yn ) − ηφ′ν (yn ), v2 − ϕ2 has a global minimum at yn , respectively. Furthermore, due to [11, Lemma 2.1], there exists a monotone sequence of functions (ϕ1,k (z)), ϕ1,k (z) ∈ C 2 such that v1 (zn ) = ϕ1,k (zn ), and v1 (z) ≤ ϕ1,k (z) ≤ ϕ1 (z), ϕ1,k ↓ v1 (z) (k → ∞) for z = zn + γ(zn , ζ ), ζ ≥ 0. We can also take a sequence (ϕ2,k )(z) of C 2 -functions such that v2 (z) ≥ ϕ2,k (z) ≥ ϕ2 (z), ϕ2,k ↑ v2 (z) (k → ∞) for z = yn + γ(yn , ζ ), ζ ≥ 0. Putting these inequalities together, we get 1 β[v1 (zn ) − v2 (yn )] ≤ σ 2 (z2n q̄ 1 − y2n q̄ 2 ) 2 ̃ 1 ) − U(p ̃ 2 )} + {(f1 (zn ) − μzn )p1 − (f2 (yn ) − μyn )p2 } + {U(p + ∫ {(v1 (zn + γ(zn , ζ )) − v2 (yn + γ(yn , ζ ))) − (v1 (zn ) − v2 (yn )) |ζ |>1

+ (p1 ⋅ γ(zn , ζ ) − p2 ⋅ γ(yn , ζ ))}μ(dζ ) + ∫ {(ϕ1 (zn + γ(zn , ζ )) − ϕ2 (yn + γ(yn , ζ ))) |ζ |≤1

+ (ϕ1 (zn ) − ϕ2 (yn )) − (ϕ′1 (zn ) ⋅ γ(zn , ζ ) − ϕ′2 (yn ) ⋅ γ(yn , ζ ))}μ(dζ )

≡ I1 + I2 + I3 + I4 + I5 ,

say.

We consider the case where there are infinitely many zn ≥ yn ≥ 0. By (2.41) and (2.43), it is easy to see that 1 2 ′′ 2 2 ′′ I1 ≤ σ 2 {3n|zn − yn |2 + η(z2n φ′′ ν (zn ) + yn φν (yn ))} → σ ηẑ φν (z)̂ 2 By monotonicity and as p1 ≥ p2 , I3 ≤ 0. By (2.33), we have f1 (zn )(zn − yn ) ≤ f2 (zn )(zn − yn ). Hence, I2 ≤ (f2 (zn ) − f2 (yn ))n(zn − yn ) − μn(zn − yn )2

+ η{(f1 (zn ) − μzn )φ′ν (zn ) + (f2 (yn ) − μyn )φ′ν (yn )}

̂ ′ν (z)̂ as n → ∞. → η(f1 (z)̂ + f2 (z)̂ − 2μz)φ

as n → ∞.

324 � 4 Applications As for the term I4 , we have I4 ≤ ∫ {(ϕ1 (zn + γ(zn , ζ )) − ϕ1 (zn )) − (ϕ2 (yn + γ(yn , ζ )) − ϕ2 (yn )) |ζ |>1

+ (p1 ⋅ γ(zn , ζ ) − p2 ⋅ γ(yn , ζ ))}μ(dζ ) → − 2ηφ′ν (z)̂ ∫ γ(z,̂ ζ )μ(dζ ). |ζ |>1

Similarly, I5 ≤ ∫ {(ϕ1 (zn + γ(zn , ζ )) − ϕ1 (zn )) − (ϕ2 (yn + γ(yn , ζ )) − ϕ2 (yn )) |ζ |≤1

+ (γ(zn , ζ )ϕ′1 (zn ) − γ(yn , ζ )ϕ′2 (yn ))}μ(dζ ) ̂ z,̂ ζ )μ(dζ ). → − 2η ∫ φ′ν (z)γ( |ζ |≤1

Thus, by (2.35), 1 ̂ ≤ 2η{ σ 2 ẑ2 φ′′ ̂ ′ν (z)̂ − ∫ φ′ν (z)̂ ⋅ γ(z,̂ ζ )μ(dz)} β[v1 (z)̂ − v2 (z)] ν (z)̂ + (f2 (z)̂ − μz)φ 2 ≤ 2ηβφν (z)̂ due to (2.34). This contradicts (2.42). Suppose that there are infinitely many yn ≥ zn ≥ 0. By concavity, we have vi (yn ) ≥ vi (zn ), i = 1, 2, and thus v1 (yn ) − v2 (zn ) ≥ v1 (zn ) − v2 (yn ). Hence, the maximum of Ψn (z, y) is attained at (yn , zn ). Interchanging yn and zn in the above argument, we again get a contradiction as above. Thus, the proof is complete. Remark 4.10. In view of (2.5), we observe that the value of v(z) will increase in the presence of the jump term (i. e., r̃ > 0). 4.2.3 Regularity of solutions In this subsection, we show the smoothness of the viscosity solution v of (2.6) in case σ > 0, the presence of the diffusion factor. Theorem 4.5. Under (B.3), (2.2), (B.4), (2.20), and for σ > 0, we have v ∈ C 2 ((0, ∞)) and v′ (0+) = ∞. Proof. We divide the proof into six steps.

4.2 Optimal consumption problem �

325

Step 1. By the concavity of v, we recall a classical result of Alexandrov (cf. [4] and [34, 63]) to see that the Lebesgue measure of (0, ∞) \ 𝒟 is zero, where 𝒟 = {z ∈ (0, ∞) : v is twice differentiable at z}.

By the definition of twice differentiability, we have (v′ (z), v′′ (z)) ∈ J 2,+ v(z) ∩ J 2,− v(z),

∀z ∈ 𝒟,

and hence ̃ ′ (z)). βv(z) = L0 v + U(v Let d ± v(z) denote the right and left derivatives, respectively: d + v(z) = lim y↓z

v(y) − v(z) , y−z

d − v(z) = lim y↑z

v(y) − v(z) . y−z

(2.44)

Define r ± (z) by 1 βv(z) = σ 2 z2 r ± (z) + (f (z) − μz)d ± v(z) 2 ̃ ± v(z)) + ∫{v(z + γ(z, ζ )) − v(z) − d ± v(z) ⋅ γ(z, ζ )}μ(dζ ) + U(d for all z ∈ (0, ∞).

(∗∗)

Since d + v = d − v = v′ on 𝒟, we have r + = r − = v′′ a. e. by the formula above. Furthermore, d + v(z) is right continuous by definition, and so is r + (z). Hence, it is easy to see by the mean value theorem that y

v(y) − v(z) = ∫ d + v(s)ds, z s

d + v(s) − d + v(z) = ∫ r + (t)dt, z

s > z.

Thus, we get 1 R(v; y) := {v(y) − v(z) − d + v(z)(y − z) − r + (z)|y − z|2 }/|y − z|2 2 y

= ∫(d + v(s) − d + v(z) − r + (z)(s − z))ds/|y − z|2 z y

s

z

z

= ∫{∫(r + (t) − r + (z))dt}ds/|y − z|2 → 0

as y ↓ z.

326 � 4 Applications ̃ Step 2. We show that U(z) is strictly convex. Suppose it is not true. Then, ̃ 0 ) = ξ U(z ̃ 1 ) + (1 − ξ)U(z ̃ 2) U(z

(2.45)

for some z1 ≠ z2 , 0 < ξ < 1 and z0 = ξz1 + (1 − ξ)z2 . Let mi , i = 0, 1, 2 be the maximizer of ̃ i ). By a simple manipulation, mi = (U ′ )−1 (zi ). It is clear (since m0 ≠ mi ) that U(z ̃ i ) = U(mi ) − mi zi ≥ U(m0 ) − m0 zi , U(z

i = 1, 2.

(2.46)

By (2.45), we observe that the equality in (2.46) must hold. Hence, we have m0 = m1 = m2 and then z1 = z2 . This is a contradiction. Step 3. We claim that v(z) is differentiable at z ∈ (0, ∞) \ 𝒟. It is a well-known fact in convex analysis (cf. [49, Example 2.2.5]) that 𝜕v(z) = [d + v(z), d − v(z)],

∀z ∈ (0, ∞),

(2.47)

where 𝜕v(z) denotes the generalized gradient of v at z and d + v(x), d − v(x) are given above. Suppose d + v(z) < d − v(z), and set p̂ = ξd + v(z) + (1 − ξ)d − v(z), r̂ = ξr + (z) + (1 − ξ)r − (z),

0 < ξ < 1.

We put R(v; y) = {v(y) − v(z) − d + v(z)(y − z) − 21 r + (z)|y − z|2 }/|y − z|2 . If it were true that lim sup R(v; y) > 0, y→z

then we could find a sequence yn → z such that limn→∞ R(v; yn ) > 0. By virtue of the result in Step 1, we may consider that yn ≤ yn+1 < z for every n, taking a subsequence if necessary. Hence, by the definition of R(v; y), v(yn ) − v(z) − d + v(z)(yn − z) ≥ 0. n→∞ |yn − z| lim

This leads to d + v(z) ≥ d − v(z), which is a contradiction. Thus, we have lim supy→z R(v; y) ≤ 0, and (d + v(z), r + (z)) ∈ J 2,+ v(z). Similarly, we observe that (d − v(z), r − (z)) ∈ J 2,+ v(z). By the convexity of J 2,+ v(z), we get (p,̂ r)̂ ∈ J 2,+ v(z). Now, by Step 2, we see that ̃ p)̂ < ξ U(d ̃ + v(z)) + (1 − ξ)U(d ̃ − v(z)), U( and hence, by (∗∗),

4.2 Optimal consumption problem

� 327

1 ̃ p). ̂ βv(z) > σ 2 z2 r̂ + (f (z) − μz)p̂ + ∫{v(z + γ(z, ζ ) − v(z) − p̂ ⋅ γ(z, ζ )}μ(dζ ) + U( 2 On the other hand, by the definition of the viscosity solution, 1 ̃ βv(z) ≤ σ 2 z2 q + (f (z) − μz)p + ∫{v(z + γ(z, ζ ) − v(z) − p ⋅ γ(z, ζ )}μ(dζ ) + U(p), 2 ∀(p, q) ∈ J 2,+ v(z). This is a contradiction. Therefore, we deduce that 𝜕v(z) is a singleton, and this implies v is differentiable at z. Step 4. We claim that v′ (z) is continuous on (0, ∞). Let zn → z and pn = v′ (zn ) → p. Then, by the concavity v(y) ≤ v(z) + p(y − z),

∀y.

Hence, we see that p belongs to D+ v(z), where D+ v(z) = {p ∈ R : lim sup y→z

v(y) − v(z) − p(y − z) ≤ 0}. |y − z|

Since 𝜕v(z) = D+ v(z) and 𝜕v(z) is a singleton by Step 3, we deduce p = v′ (z). This implies the assertion. Step 5. We set u = v′ . Since it holds that 1 βv(zn ) = σ 2 z2n u′ (zn ) + (f (zn ) − μzn )u(zn ) 2 ̃ + ∫{v(zn + γ(zn , ζ ) − v(zn ) − u(zn ) ⋅ γ(zn , ζ )}μ(dζ ) + U(u(z n )),

zn ∈ 𝒟,

the sequence {u′ (zn )} converges uniquely as zn → z ∈ (0, ∞) \ 𝒟, and u is Lipschitz near z (cf. [49]). We recall that 𝜕u(z) coincides with the convex hull of the set, that is, 𝜕u(z) = {q ∈ R : q = lim u′ (zn ), zn ∈ 𝒟, zn → z}. n→∞

Then, 1 βv(z) = σ 2 z2 q + (f (z) − μz)u(z) 2 ̃ + ∫{v(z + γ(z, ζ )) − v(z) − u(z) ⋅ γ(z, ζ )}μ(dζ ) + U(u(z)),

∀q ∈ 𝜕u(z).

Hence, we observe that 𝜕u(z) is a singleton, and then u(z) is differentiable at z. The continuity of u′ (z) follows immediately as above. Thus, we conclude that u ∈ C 1 ((0, ∞)), and hence (0, ∞) \ 𝒟 is empty.

328 � 4 Applications Step 6. Suppose v′ (z) ≤ 0. Then, by (B.3), we have ̃ ′ (z)) = ∞. U(v This contradicts to the equality (∗∗) in view of the conclusion in Step 3. Thus, we deduce v′ (0+) > 0, and by (B.3) again, v′ (0+) = ∞. 4.2.4 Optimal consumption In this subsection, we combine the optimal consumption policy c∗ with the optimization problem (2.3). We assume σ > 0 throughout this subsection in order to guarantee the existence of a C 2 -solution. Lemma 4.11. Under (B.3), (2.2), (B.4), and (2.20), we have lim inf E[e−βt v(z(t))] = 0

(2.48)

t→∞

for every c ∈ 𝒞 . Proof. By (2.18), we can choose 0 < β′ < β such that 1 − β′ φ(z) + σ 2 z2 φ′′ (z) + (f (z) − μz)φ′ (z) 2 ̃ ′ (z)) ≤ 0, + ∫{φ(z + γ(z, ζ )) − φ(z) − φ′ (z) ⋅ γ(z, ζ )}μ(dζ ) + U(φ Here, φ(z) = z + B. Hence, Itô’s formula gives t

E[e

−βt

φ(z(t))] ≤ φ(z) + E[∫ e−βs (−β + β′ )φ(z(s))ds], 0

from which ∞

(β − β )E[ ∫ e−βs φ(z(s))ds] < ∞. ′

0

Thus, lim inf E[e−βt φ(z(t))] = 0. t→∞

By Theorem 4.3, v(z) ≤ φ(z), which implies (2.48).

z ≥ 0.

(2.49)

329

4.2 Optimal consumption problem �

Now, we consider the equation ̃ dz∗ (t) = (f (z∗ (t)) − μz∗ (t) − c∗ (t))dt + σz∗ (t)dW (t) + z∗ (t−) ∫(eζ − 1)N(dtdζ ), z∗ (0) = z > 0,

(2.50)

where c∗ (t) = (U ′ ) (v′ (z∗ (t−))) ⋅ 1{t≤τz∗ } .

(2.51)

−1

Here, c∗ (t) is the consumption rate which maximizes the Hamilton function (associated with L) applied to v. For a typical case U(c) = r1 cr (r ∈ (0, 1)), it is given by 1

c∗ (t) = (v′ (z∗ (t−))) r−1 ⋅ 1{t≤τz∗ } . Lemma 4.12. Under (B.3), (2.2), (B.4), and (2.20), there exists a unique solution z∗ (t) ≥ 0 of (2.50). Proof. Let G(z) = f (z+ ) − μz − (U ′ )−1 (v′ (z+ )). Here, z+ = max(z, 0). Since G(z) is Lipschitz continuous and G(0) = 0, there exists a solution χ(t) of ̃ dχ(t) = G(χ(t))dt + σχ(t)dW (t) + χ(t−) ∫(eζ − 1)N(dtdζ ),

χ(0) = z > 0.

(2.52)

Define z∗ (t) ≡ χ(t ∧ τχ ) ≥ 0. We note that by the concavity of f , G(z) ≤ C1 z+ + C2 for some C1 , C2 > 0. Then, we apply ̂ given by the comparison theorem (Lemma 4.9) to (2.52) and z(t) ̃ ̂ = (C1 z(t) ̂ + + C2 )dt + σ z(t)dW ̂ ̂ d z(t) (t) + z(t−) ), ∫(eζ − 1)N(dtdζ

̂ = z > 0, (2.53) z(0)

̂ for all 0 < t < τz∗ ∧ τẑ . Furthermore, it follows from the to obtain 0 ≤ z∗ (t) ≤ z(t) definition that τz∗ = τχ , and hence ̃ dz∗ (t) = 1{t≤τz∗ } [f (z∗ (t)) − μz∗ (t) − c∗ (t)dt + σz∗ (t)dW (t) + z∗ (t−) ∫(eζ − 1)N(dtdζ )]. Therefore, z∗ (t) solves (2.50). ̃ be another solution of (2.50). We notice that the funcTo prove uniqueness, let z(t) tion z 󳨃→ G(z) is locally Lipschitz continuous on (0, ∞). By using Theorem 4.4 in Section 2.2, we can get ̃ ∧ τz∗ ∧ τz̃ ). z∗ (t ∧ τz∗ ∧ τz̃ ) = z(t

330 � 4 Applications Hence, τz∗ = τz̃ ,

̃ z∗ (t) = z(t)

for t ≤ τz∗ .

̃ for all 0 ≤ t < τz∗ . This implies z∗ (t) = z(t) We assume the SDE (2.50) has a strong solution. As it is expected from (2.6), the optimal consumption policy is given as follows. Theorem 4.6. Let the assumptions of Theorem 4.5 hold. Then, an optimal consumption c∗ = {c∗ (t)} is given by (2.51). Proof. We set ζ = s ∧ τz∗ for any s > 0, and let 󵄨 󵄨 1 󵄨 󵄨 ζn = ζ ∧ inf{t > 0; 󵄨󵄨󵄨z∗ (t)󵄨󵄨󵄨 < or 󵄨󵄨󵄨z∗ (t)󵄨󵄨󵄨 > n}. n By (2.6), (2.50), and Itô’s formula, we have E[e−βζn v(z∗ (ζn −))] ζn

1 = v(z) + E[∫ e−βt {−βv(z) + v′ (z)1{t 0, and Kt corresponds to the activity to invest. The expected utility is measured up to time T in terms of Zt by

332 � 4 Applications

E[∫ e−βt dZt ], T

instead of the consumption rate c(t) composed in the utility function. The paper [55] studies, in a similar framework, the control in switching between paying dividends and investments. The paper [75] studies the same model Xt , but the expected utility is measured by E[U(∫ e−βt dZt )], T

where U(⋅) is a utility function. Here, one measures the gross amount of dividends up to time T by U(⋅) instead of the consumption rate c(t), which makes some difference in the interpretation. In these papers, the main perturbation term is the diffusion (Brownian motion). On the other hand, Framstad [66] has studied a model (Xt ) given by ̃ dXt = Xt− (μ(Xt )dt + σ(Xt )dW (t) + ∫ η(Xt− , ζ )N(dtdζ )) − dHt in the Wiener–Poisson framework. Here, Ht denotes a càdlàg process corresponding to the total amount of harvest up to time t, and σ(⋅) ≥ 0. Under the setting that the expected utility is measured by ∞

E[ ∫ e−βt dHt ], t0

he describes that the optimal harvesting policy is given by a single value x ∗ , which plays a role of a barrier at which one reflects the process downward. The paper [75] also leads to a similar conclusion. More recent works are [5, 12–14], which mention the way to construct optimal controls, and others. A paper has appeared with the viewpoint that the value function is a viscosity solution (see [141]). The issue is extensively researched, especially by people in insurance; see the papers in, e. g., the journal ‘Insurance: Mathematics and Economics’, volume 43–44, 2009. For recent development concerning the maximum principle in the jump-diffusion setting, see [174]. We do not know results like Theorems 4.5, 4.6 in case σ may degenerate. In case one considers the operator L = (−Δ)s , 0 < s < 1, it is written as Lu(z) = cs ∫{u(z + γ(z, ζ )) − u(z)}μ(dζ ), where γ(z, ζ ) = z,

μ(dζ ) = c|z|−(d+2s) .

(2.55)

4.2 Optimal consumption problem

� 333

It is known (called, antilocality) to have the property that, for any smooth function u, if u = Lu = 0 in some open set in X then u ≡ 0 in X. Equivalently, supp u ∪ supp(−Δ)s u = X

(2.56)

where X denotes a whole space (see [73, Theorem 1.2] for u ∈ H −r , r ∈ R, or theorems in [88–90] for u ∈ C0∞ ). The problem for such an operator is called the fractional Calderón problem after [43]. The proof requires the microlocal analysis techniques for the hyper-functions by Sato–Kashiwara–Kawai, cf. [116]. On the other hand, putting d = 1, we let, as in text (cf. Section 4.2), u(x) = sup E[φ(zτ ) : τ < ∞] τ

(2.57)

where zt is a stochastic process starting from x with infinitesimal generator L = (−Δ)s , and τ is a stopping time. Let φ denote a smooth function with a compact support in X. Then u(x) is a solution to the obstacle problem s

u≥φ

(−Δ) u ≥ 0 s

in X,

(2.58)

in X,

(2.59)

(−Δ) u = 0

(2.60)

for x such that u(x) > φ(x) and lim u(x) = 0,

|x|→∞

(2.61)

(see [201, Proposition 3.3] for details). Here φ(x) is called an obstacle, and the problem is related with the inverse problem (cf. [43]). This problem is also deeply related with the above problem, for example, with the financial mathematics when dealing with a pricing model for American options (see [50, Chapter 12], [38]). A typical case is case φ(x) = (K − ex )+ . In the general case where L depends on x, by using the Bellman principle, the solution u(x) is proved to be a viscosity solution of an integro-differential equation as in Section 4.2.

A Appendix A friend of Diagoras tried to convince him of the existence of the gods, by pointing out how many votive pictures tell about people being saved from storms at sea by dint of vows to the gods, to which Diagoras replied that there are nowhere any pictures of those who have been shipwrecked and drowned at sea. Marcus Tullius Cicero (106–43 BC), De natura decorum III 89

A.1 Notes and trivial matters A.1.1 Notes For basic material which is supposed to be preliminary in this book, the reader can see [39] and [191]. Chapter 1 The composition of Chapter 1 is compact. For details and proofs of basic aspects concerning stochastic processes, stochastic integrals, and SDE of diffusion type, see the classical [87]. The results in Section 1.1 are well known; see [193]. The results in Section 1.2 are mainly due to [130, 189] and [53]. The results in Section 1.3 are also due to [130]. For details on the stochastic flow appearing in Section 1.3, see [9, Section 6.4]. Additionally, the equivalence between (3.14) and (3.15) is justified by Theorem 6.5.2 in [9]. For SDEs on manifolds, see [9, 61]. For the SDE of diffusion type, the Lipschitz condition can be relaxed to a local Lipschitz condition [59, (5.3.1)]. In a recent paper, an estimate of the density under a non-Lipschitz condition [57] is obtained. The theory of martingales and quadratic variations has been developed by P.-A. Meyer, D. Williams, Kunita–Watanabe, etc. An explanation carrying over the original atmosphere in these days can be found in Rogers–Williams, vol. 2 [191]. Based on fundamental works in real analysis–probability around 1960, P.-A. Meyer, Kunita– Watanabe (1967) [126] gave a characterization of martingales. After that Stroock and Varadhan (1971) have given lots of applications to the theory of SDEs. These days have been decisive moments in modern probability theory. A professor of function theory, P. Malliavin in Paris has developed an analytic theory of path-wise trajectories in 1976 [153]. In the 1990s, the results in stochastic analysis have been applied to mathematical finance in an essential way, and have been implemented in software. With respect to Itô’s formula (Proposition 1.3, Theorem 1.3), a (north-) Korean mathematician J. Cho has published a note [48] (in Korean) ‘Stochastic integral equation’ (1963), which contains ‘Itô’s formula with diffusion and jump part’. The note was published in North Korea and has not been well known around the world, but a copy has been kept in the library of K. Itô whom Cho’s previous professor Kim ( ) at Seoul University (supposedly) was a classmate in Tokyo Imperial University, and was found by N. Ikeda who https://doi.org/10.1515/9783110675290-006

336 � A Appendix has taken care of the heritage of K. Itô. This note was translated recently into Japanese by Daehong Kim (not published), and some part of it has been translated into English by Y. Ishikawa and A. Takeuchi. The content of the note seems to be influenced by the author’s habilitation thesis under A. V. Skorokhod in Moscow [202]. Chapter 2 There are two approaches to stochastic processes. One is based on a macroscopic point of view using functional-analytic approaches by, e. g., Kolmogorov, Malliavin–Stroock; the other is based on a microscopic view tracking each trajectory by, e. g., Itô, Bismut, and Picard. We adopted the latter. Section 2.1.1 is due to [94]. Section 2.1.2 is due to [178, 179] and [100]. Picard’s method was a completely different kind of perturbation, which originally came across the Pyrenees. A part of Section 2.1.3 is due to [60] and to [15, 16]. Many people had interest in the integration-by-parts setting on the Poisson space in the 1990s, especially in control theory and application fields. For example, Carlen–Pardoux [47] (1990) and Elliot–Tsoi [60] (1993). Historically, the Bismut–Graczyk method stated in Section 2.1.3 was developed before that stated in Section 2.1.1. It has evolved to cover the variable-coefficient case in a fairly general setting; see [120]. In the time shifts method in Section 2.1.3, the choice of (πi ), the localization factor (‘weight’), is important. We may regard it as a patchwork art, as the weights π look like patches. There are some attempts to analyze jump processes as boundary processes (e. g., [30]), however, these results have not been fruitful in a significant way. Sections 2.2, 2.3 are based on [93, 95, 143, 180] and [181]. Basic methods and techniques to find upper and lower bounds of the density functions which appear in Sections 2.2.2–2.2.3 can be regarded, from today’s viewpoint, as approximation methods under the outer clothing of probability, which are familiar in the control theory. In effect we need perturbations by allowance functions (φn ) given at Lemma 2.7 for the precise upper bound of the density. The same point of view and an attitude to the problem can be sensed also in [134, Chapter 3]. Examples in Section 2.5.3 can be regarded in this context. Historically, the polygonal method stated in Section 2.3 was developed before the Markov chain method in Section 2.2. In those days the author did not know the idea to separate the big jump part of the driving Lévy process from the small by the magnitude O(t 1/β ), and used simply the constant threshold. The proof of Sublemma 2.3 in Section 2.2 is closely related to the strict positivity of the densities for infinitely divisible random variables, mentioned at the end of Section 2.5.4. The upper bound of the density for a not necessarily short, but still bounded, time interval is given in [182]. Section 2.5.2 is adapted from [102]. As for numerical results stated in Section 2.5.3, calculating the constant C = C(x, y, κ) in Theorem 2.4 numerically is not an easy task, in general. Especially if the function g(x, y) has singular points at the border of the integral region, numerical calculation of iterated integrals in (3.6) is known to be heavy. Section 2.5.4 is due to [96, 97].

A.1 Notes and trivial matters

� 337

Chapter 3 It can be said that P. Malliavin’s theory is an application of function theory to an infinitedimensional space, with background on Mecanique aleatoire by J.-M. Bismut. Malliavin was a specialist in function theory and Fourier analysis. As it is understood, Malliavin calculus uses Fourier analysis in its essential part. So it can also be viewed as a probabilistic version of the Fourier theory. Stochastic calculus of variations (Malliavin calculus) seems to be born based on his ingenious idea and his mind grabbing the essence. For quantum probabilists, on the other hand, the Wiener space seems to be a reality. For details on the iterated integral expressions of Wiener (–Poisson) variables treated in Section 3.1, see [186]. For the Charlier polynomials stated in Section 3.2, also see [186]. As for the well-definedness of multiple integrals with respect to Lévy processes, see [115]. For the Clark–Ocone formula on the Poisson space (Section 3.2.3), the origin may be traced back to A. Ju. Sevljakov [195]. The assertion for the Clark–Ocone formula can be extended to the Wiener–Poisson space in L2 (Ω). The contents in Section 3.3 is mainly based on [100] and [106]. The definition of the norms ‖F‖l,k,p;A , ‖U‖l,k,l;A , ‖Vu ‖l,k,p;A in Section 3.3.3 is based on the idea that we take the projection of the Wiener factor of F first, and then measure the remaining Poisson factor, both in the sense of (weighted) Lp norm. We can consider other norms as well on the Wiener–Poisson space. Indeed, by taking another Sobolev norm for the Wiener ̃ ) factor, it is possible that the framework and procedure of showing the estimate for δ(V can be carried over to show the estimate of δ(U), where it has been carried out using the Meyer-type inequality in Lemma 3.10. Up to present, there seems to exist no decisive version of the definition for the norms on the Wiener-Poisson space. Section 3.4 is due to [24]. The difference operator D̃ does not satisfy the local property, as is seen in (3.15) in Chapter 3, and this makes the estimate of the adjoint operator δ̃ complicated due to extra terms. This may be a motif for several theories using time shift method stated in Section 2.1.3, as things could be used as local operators then, thus avoiding nonlocal operations. Our Lemma 3.5 may serve so that naturally nonlocal operators D̃ and δ̃ can be treated slightly easier. A basic idea for our original ‖⋅‖0,1,p;U norm does come from the naive idea to measure the difference |F(ω ∘ εu ) − F(ω)| divided by the length |u|. The background is norms such 1 as ‖ϕ‖ = ∫r | ϕ(t) |dt, r > 0 and ‖f ‖ = ∫ |f −E[f ]|dP used in the real analysis (e. g., BMO-type t norms). The form used in the text (‖ ⋅ ‖0,1,p;A ) is that where dP(⋅) is deformed to dP(⋅|A), which may not provide a direct intuition on jumps but is for easily measuring micro activities on jumps. The new norm | ⋅ |∗0,k,p is associated associated to maximal functions. Our theory on stochastic analysis stated in Sections 3.5–3.7 and basic properties of Lévy processes stated in Chapter 1 are deeply related with the theory of Fourier analysis. The contents of Sections 3.5, 3.7 is due to [79, 100, 106, 132]. In Section 3.5.1, the iteration procedure using the functional ev (F) = ei(v,F) works effectively. This iteration procedure

338 � A Appendix in the Poisson space will be another virtue of Malliavin calculus, apart from a perspective for the integration-by-parts formula procedure in the Wiener space. Indeed, employing the iteration method, it has turned out that the integration-by-parts formula is not a unique tool for showing the smoothness of the transition density. The origin of the theory stated in Sections 3.5–3.7 is [178]. Between [100] and [79] there has been a quest for finding proper objects in the Poisson space, which may correspond to Malliavin matrix in the Wiener space. The theory is also closely related to [166, Section 2.1.5]. A related topic is treated in [172, Chapter 14]. The proof of the continuity property of the Skorohod integral (Theorem 3.4) is based on [106], in place of [100]. The contents of Section 3.6 is inspired by [132]. The result is for the case of processes of variable coefficients, t

t

t

̃ Xt = ∫ b(Xr , r)dr + ∫ σ(Xr , r)dWr + ∫ ∫ g(Xr− , r, z)N(drdz). 0

0

0

The paper [132] tries to analyze SDEs of this type. The expression in the main estimate of Proposition 3.15 has been tamed in a new setting. The method used in Section 3.6.4 is new. The version of reforming the Lévy measure of fixed type given at the beginning of Section 3.6.5 to another type is an attempt to use the theory of nonlocal Dirichlet forms in [35, 36]. The former part of Section 3.6.5 is mainly due to [120] and [133]. This part was added in the second edition, so the SDE here is based on the setting that the vector fields depend on t. There are cases where the density function exists despite the fact that the stochastic coefficients are fairly degenerate and the (ND) condition or other similar nondegenerate conditions fail for the solution of SDEs. We are often required delicate estimates in such cases, see, e. g., [56]. The extension of the theory for the SDE stated in the former half of Section 3.6.5 to the case of other types of Lévy measures is not well known. The inclusion of this subsection is also due to author’s recent interest in Itô processes on manifolds, as Itô processes with big jumps do not satisfy the conditions (J.1) and (J.2), in general. Chapter 4 Section 4.1 is based on [79]. A related work is done by Yoshida [220]. Indeed, in this aspect, there are several works by Tokyo group around N. Yoshida; Yoshida [219, 220], Masuda [86], Kunitomo [135], Takahashi [136], Shiraya [198], and others, some of them concern mathematical finance theory. The authors have even made a package for financial calculations related to stochastic processes of jump type [86]. The result in Section 4.1.2 is new. Section 4.1.3 is also new. In those days of making script in calculation of this section, financial mathematics using Malliavin calculus (diffusion processes or jump-diffusions) was a kind of world-wide boom. After nowaday pandemic and the Zoom days I am thinking what kind of mathematics is really for happiness of human beings.

A.1 Notes and trivial matters

� 339

Section 4.2 is based on [101, 104]. The existence of ‘trace’ at the boundary for the value function can be justified analytically using the theory of pseudodifferential operators [205]. For the dynamic programming principle, see [101] Section 4. The control theory with respect to jump processes is developing. See, e. g., [112] and Kabanov’s recent papers. During 2015–2018, the author and Dr. Yamanobe considered an application of Malliavin calculus to a stochastic action potential model in axons [105]. Some part of the result is cited in Appendix A.1.3. For a practicable application of the result, one needs a powerful resource for numerical analysis, which requires money.

A.1.2 Ramsey theory We are concerned with a one-sector model of optimal economic growth under uncertainty in the neoclassical context. It was originally presented by Ramsey [190] and was refined by Cass [44]; see also [64]. Define the following quantities: y(t) = labor supply at time t ≥ 0, x(t) = capital stock at time t ≥ 0, λ = constant rate of depreciation (written as β in the text), λ ≥ 0, F(x, y) = production function producing the commodity for the capital stock x ≥ 0 and the labor force y ≥ 0. We now state the setting in the model. Suppose that the labor supply y(t) at time t and the capital stock x(t) at t are governed by the following stochastic differential equations: dy(t) = ry(t)dt + σy(t)dW (t),

y(0) = y > 0,

r ≠ 0, σ ≥ 0,

(A.1)

̃ dx(t) = (F(x(t), y(t)) − λx(t) − c(t)y(t))dt + x(t−) ∫ (eζ − 1)N(dtdζ ) |ζ | 0.

(A.2)

|ζ |≥1

An intuitive explanation is as follows. Suppose there exists a farm (a fazenda, or a company, a small country, …) with economical activities based on labor and capital. This model could be used for a feudal demesne in medieval Western Europe (after Carolingian dynasty). At time t, the farm initiates production which is expressed by F(x(t), y(t)). At the same time, he or she has to consume the capital at the rate c(t), and the capital may depreciate as time goes by. The second and third terms on the right-hand side of (A.2) correspond to continuous or discontinuous random fluctuations in capital.

340 � A Appendix Under this interpretation, we may even assume that the labor supply y(t) is constant temporary, and consider the optimal consumption under the randomly fluctuating capital. This assumption has a projected economical meaning since the labor supply is not easy to control (i. e., decrease) in short time. This type of problem is called the stochastic Ramsey problem. For reference, see [71]. We denote by V (x, y) the value function associated with x(t) and y(t). The HJB equation associated with this problem reads as follows: 1 βV (x, y) = σ 2 y2 Vyy (x, y) + ryVy (x, y) + (F(x, y) − λx)Vx (x, y) 2

+ ∫{V (eζ x, y) − V (x, y) − Vx (x, y) ⋅ x(eζ − 1)}μ(dζ ) ̃ x (x, y)y), + U(V

V (0, y) = 0, x > 0, y > 0.

(A.3)

We seek for a solution V (x, y) in a special form, namely, V (x, y) = v(z),

z=

x . y

(A.4)

Then, (A.3) turns into the form (2.6)–(2.7) in Section 4.2 with f (z) = F(z+ , 1) and μ = r − r̃ + λ − σ 2 . Here, we assume the homogeneity of F with respect to y, F(x, y) = F( xy , 1)y. We can show that V (x, y) defined as above is a viscosity solution of (A.3), and that if σ > 0 then it is in C 2 ((0, ∞) × (0, ∞)). This yields the optimal consumption function as treated in Section 4.2.4. The condition z(t) ≥ 0, t ≥ 0 a. s. implies that the economy can sustain itself, temporarily. Although it is not implied in general that we can find the solution of (A.3) as in (A.4) in the setting (A.1)–(A.2), under the same assumption as above (i. e., the labor supply is constant), the problem will correspond to what we have studied in the main text. That is, we can compare the value function with the form v(z) = V (z, 1).

A.1.3 A small application to neural cell theory A.1.3.1 Physiological background As an application of the theory of asymptotic expansion, we briefly treat in this section a nerve cell model mathematically. Namely, we study the excitement of nerve cells. Neural field equations describe the activity of neural cells at a microscopic level. Furthermore, nerve systems have some noise sources, which affect the information processing in them. Ion channel is an entity to make electrical signals in membranes of neurons. The stochastic nature of the ion channel is an intrinsic source of noise in neurons. On the other hand, chemical synapse, that is, a place where a neuron communicates with other neurons chemically, is a source of extrinsic noise in neurons. Chemical synapses release vesicles containing neurotransmitter into the synaptic cleft, and the release of

A.1 Notes and trivial matters

� 341

the synaptic vesicles is random (quantal release). The spontaneous release of the vesicles is modeled by a Poisson process in case of low probability of release of vesicles [209, 210]. A nerve cell makes a depolarization (increase of action potential) if it receives an electrical stimulation above the threshold level. This phenomenon was mathematically modeled by Hodgkin–Huxley as follows [70, 76, 80]. A nerve cell is surrounded by a special envelope, so that the interior has resting potential which is negative (−50 ∼ −60 mV) against the outside. Across the envelope, the ion compositions of cell fluid are different: the inside contains more K+ (potassium) ions and the outside contains more Na+ (sodium) and Cl+ (chlorine) ions. A nerve cell envelope has several kinds of ion channel. On the resting envelope, the K+ ion channel has high permeability. This is due to the K+ leakage channel, which opens and closes randomly. When it opens, the K+ ions flow out of the cell according to the concentration gradient across the envelope, up to the level where electro-chemical potential is balanced. As a result, a slightly negative membrane potential is realized inside the cell. This resting potential, denoted by Es , is called Nernst potential. A cell envelope can be modeled as an electric circuit consisting of a capacitor, resistor, and battery. The current Iion which passes the ion channel is given by Iion = −C

dV , dt

(A.5)

where C denotes the capacity of the capacitor and V denotes the membrane potential. On the other hand, V = rIion + Es , where r denotes the resistance. Let g = Iion =

1 r

(A.6)

(the conductance). Then

1 ⋅ (V − Es ) = g ⋅ (V − Es ). r

(A.7)

By (A.5), (A.7), C

dV = −g(V − Es ). dt

(A.8)

Hodgkin and Huxley (1952) have proposed a neurophysiological model based on the giant axon of squid. The model has proved to be valid also for other kinds of nerve cells. On the surface of the cell, there exist several ion channels. The authors paid attention to Na+ and K+ channels. Other ion channels are treated as one channel and regarded as a leak channel. Then by (A.8), C

dV = −gNa (V − ENa ) − gK (V − EK ) − gL (V − EL ) + Iapp . dt

(A.9)

342 � A Appendix Here gNa , gK , gL are conductances of Na+ , K+ , and the ‘leak’ ions, respectively, ENa , EK , EL are the corresponding Nernst potentials, and Iapp denotes the current appended to the cell from the exterior. The formula (A.9) is the base equation in the H-H model.1 Next we handle the potassium channel conductance gK . Suppose it satisfies the equation of the form dgK = f (V , t). dt

(A.10)

We assume the K+ channel has two states, O (open) and C (closed). We denote by α(V ) and β(V ) the velocities of successions from C to O and from O to C, respectively. Let the probability that the channel is open be n(V ), then the probability that it is closed is 1 − n(V ). This leads to the equation dn(V ) = αK (V )(1 − n(V )) − βK (V )n(V ). dt

(A.11)

We carry out a similar calculation for the probability m(V ) that the sodium channel is open, to have dm(V ) = αNa (V )(1 − m(V )) − βNa (V )m(V ). dt

(A.12)

By these calculations and by experiments, it is known that gK , gNa are well modeled as gK = ḡK n4 ,

gNa = ḡNa m3 h,

(A.13)

respectively. Here ḡK , ḡNa are constants, and h(V ) denotes a function describing an auxiliary effect. Due to these calculations, the Hodgkin–Huxley equation is written as dV dt dn(V ) dt dm(V ) dt dh(V ) dt C

= −ḡNa m3 h(V − ENa ) − ḡK n4 (V − EK ) − ḡL (V − EL ) + Iapp , = αK (V )(1 − n(V )) − βK (V )n(V ), = αNa (V )(1 − m(V )) − βNa (V )m(V ), = αh (V )(1 − h(V )) − βh (V )h(V ).

(A.14)

A.1.3.2 Stuart–Landau oscillator It is known that the solution of the Hodgkin–Huxley equation has a Hopf bifurcation.

1 Technically, we assume the nerve cell is in the ‘space-clamped’ state.

A.1 Notes and trivial matters

� 343

Stuart–Landau equation (oscillator) has been developed by L. Landau and T. Stuart as a model of turbulence in 1944–1958. It can also be viewed as a mathematical model of the Hodgkin–Huxley equation which well describes the Hopf bifurcation [7, 176]. The Stuart–Landau oscillator is described by dA = αA − β|A|2 A, dt

(A.15)

where A is the complex amplitude of the equation, while α, β ∈ C. The evolution of the disturbance is given by the real part of A. We set Re α > 0, Re β > 0. In this setting, the Stuart–Landau oscillator shows stable oscillation. To see this, we rewrite it. If we set A = r0 exp(i2πθ0 ), in polar coordinates, the oscillator is given as follows: dr0 (t) = κr0 (t)(1 − c1 r0 (t)2 ), dt

dθ0 (t) = (c0 − c2 r0 (t)2 )/2π, dt

(A.16)

where r0 (t) ∈ R, θ0 (t) ∈ [0, 1), κ = Re α, c0 , c1 , c2 ∈ R and c0 = Im α, c1 = Re β/ Re α, c2 = Im β. If we set x0 (t) = r0 (t) cos(2πθ0 (t)), y0 (t) = r0 (t) sin(2πθ0 (t)), then the oscillator is given by dx0 (t) = κx0 (t){1 − c1 (x0 (t)2 + y0 (t)2 )} − y0 (t){c0 − c2 (x0 (t)2 + y0 (t)2 )}, dt dy0 (t) = κy0 (t){1 − c1 (x0 (t)2 + y0 (t)2 )} + x0 (t){c0 − c2 (x0 (t)2 + y0 (t)2 )}, dt and x0 (0) = x0 , y0 (0) = y0 .

(A.17)

If c1 = 1, it has a stable limit cycle which is the unit circle with period 2π/(c0 − c2 ). As t → ∞, trajectories starting from any initial point would converge to the limit cycle. Let us consider a stochastic version of (A.17), which we call the stochastic Stuart– Landau oscillator: dXtϵ = b1 (Xtϵ , Ytϵ )dt + ϵσ(t, Xtϵ , Ytϵ )dW (t) + ϵdJt , dYtϵ = b2 (Xtϵ , Ytϵ )dt, and

X0ϵ = x0 , Y0ϵ = y0 .

(A.18)

Here b1 (x, y) = κx{1 − c1 (x 2 + y2 )} − y{c0 − c2 (x 2 + y2 )},

b2 (x, y) = κy{1 − c1 (x 2 + y2 )} + x{c0 − c2 (x 2 + y2 )},

(A.19)

where ϵ ∈ (0, 1] is a sufficiently small noise parameter, W (t) is a standard Wiener process, Jt is a compound Poisson process Jt = ∑N(t) Yj , where Yj are i. i. d. random variables j=1 obeying the normal law N(m, 1), m ∈ R, and N(t) is a Poisson process with intensity λ > 0. We assume W (t) and Jt in (A.18) are mutually independent and X0ϵ = x0 , Y0ϵ = y0

344 � A Appendix are initial values not depending on ϵ, respectively. When ϵ = 0, the solution of (A.18) is denoted by (x0 (t), y0 (t)). We remark that Xtϵ , Ytϵ are stochastic processes with jumps, where stochastic parts are supposed to represent the emission of small vesicles. If, in the Stuart–Landau oscillator, the drift coefficient is locally Lipschitz continuous in the space variable, we can show that the oscillator has a strong solution for any initial condition (X0ϵ , Y0ϵ ) satisfying E[‖(X0ϵ , Y0ϵ )‖2 ] < +∞. A similar recent development in another case will be the Jansen and Rit neural mass model, which also provides a model for stochastic bifurcation [1]. A.1.3.3 Stochastic analysis We set 𝒮t,k = ([0, t]k / ∼) and 𝒮t = ∐k≥0 𝒮t,k , where ∐k≥0 denotes the disjoint sum and ∼ means the identification of the coordinates on the product space [0, t]k by permutation. The distribution P̃ t of the moments (instants) of jumps related to Jt in [0, t] is given on 𝒮t,k by ∫ {#St =k}

t

t

0

0

1 1 f (St ) d P̃ t (St ) = {λkt ( ) exp(−λt )} × k ∫ ⋅ ⋅ ⋅ ∫ f (u1 , . . . , uk ) du1 ⋅ ⋅ ⋅ duk , (A.20) k! t

where λt = λt, St ∈ 𝒮t,k , and f is a function on 𝒮t,k (a symmetric function on [0, t]k ). Here {. . .} denotes the Poisson law probability to have k jumps in [0, t], and the remaining part is the uniform law probability of jump arrival times in [0, t] given that there are k jumps. In case k is fixed, the law of S1 , . . . , Sk is uniform (with Sk denoting the kth jump time). On the other hand, the number of u1 , . . . , uk is given by the Poisson law. Let J be a random variable whose law is N(0, 1) and choose a family of independent random variables Ji , i = 1, 2, . . . , having the same law as J. Suppose there are k jumps in [0, t]. For a fixed St = {S1 , S2 , . . . , Sk } (S1 ≤ S2 ≤ ⋅ ⋅ ⋅ ≤ Sk ), we consider the solution of the following SDE with jumps: Xsϵ (St )

s

= x0 +

∫ b1 (Xrϵ (St ), Yrϵ (St ))dr 0 s

s

ϵ ϵ + ∫ ϵσ(r, Xr− (St ), Yr− (St ))dW (r) + ϵ

Ysϵ (St ) = y0 + ∫ b2 (Xrϵ (St ), Yrϵ (St ))dr.

0

∑

Si ∈St ,Si ≤s

Ji , (A.21)

0

Here we remark the law of Ysϵ (St ), 0 < s < t is determined through Xsϵ (St ) by W (s) and (Ji , si ), 0 < si < t in (A.21). We denote by P(Xsϵ (St ) ∈ dx, Ysϵ (St ) ∈ dy) the joint law of (Xsϵ (St ), Ysϵ (St )). Then we have the marginal distribution

A.1 Notes and trivial matters

� 345

∫ P(Xsϵ (St ) ∈ dx, Ysϵ (St ) ∈ dy) = P(Xsϵ (St ) ∈ dx), which we denote by Ps (ϵ, St , dx). By Malliavin–Hörmander’s theorem, there exists a density pt (ϵ, St , x, y; x0 , y0 ) locally on small time intervals directed by St . We integrate out with respect to St , and denote the density by pt (ϵ, x, y, x0 , y0 ). Namely pt (ϵ, x, y, x0 , y0 ) = ∫ pt (ϵ, St , x, y, x0 , y0 ) d P̃ t (St ) 𝒮t ∞

= ∑ ∫ pt (ϵ, St , x, y, x0 , y0 |{#St = k}) d P̃ t (St ) k=0 𝒮

t,k

∞

:= ∑ pt (ϵ, x, y, x0 , y0 |k). k=0

(A.22)

Here pt (ϵ, St , x, y, x0 , y0 |{#St = k}) denotes the conditional density of Xtϵ given #St = k. Between consecutive jumps, the process is driven by the Wiener process with coefficient ϵσ(⋅) and with the drift term, whose density is expanded in the powers of ϵ. On the other hand, the jump part ϕ of the density is determined by the properties of the jumps. Indeed, the trajectories of Ut = (Xtϵ , Ytϵ ) are woven by diffusion and jump parts, Ut = ξτn ,t ∘ ϕτn ∘ ξτn−1 ,τn ∘ ⋅ ⋅ ⋅ ∘ ϕτ1 ∘ ξ0,τ1 , where τk s are a sequence of jump times of the Poisson process Nt , and ∘ denotes the composition of maps. Here the diffusion part ξt is a stochastic process generated by the continuous SDE (A.21), ξs,t is the point of ξ at t starting at s for s < t, and ϕτ (x, y) = (x + 1, y), τ ∈ {τ1 , . . . , τn }. Summing up, we have Theorem ([105]). The transition density of Xtϵ is asymptotically given for t ∈ (0, T] by the iterated integral ∞

∞ t sk

s2

pt (ϵ, x, x0 , y0 ) = ∑ λk exp(−λt) ∫ ∫ ∫ ⋅ ⋅ ⋅ ∫ pt (ϵ, {s1 , . . . , sk }, x, y, x0 , y0 )dyds1 ⋅ ⋅ ⋅ dsk . k=0

−∞ 0 0

0

(A.23)

Here sk (k = 1, 2, . . . ) stands for the kth jump time and satisfies 0 < s1 < ⋅ ⋅ ⋅ < sk < t. Let #St = k and suppose the jump times are given by {s1 , . . . , sk } (s1 ≤ ⋅ ⋅ ⋅ ≤ sk ). Then the density pt (ϵ, {s1 , . . . , sk }, x, y, x0 , y0 ) is given by the iterated integral

346 � A Appendix pt (ϵ, {s1 , . . . , sk }, x, y, x0 , y0 ) = ∫ dxk ∫ dxk′ ∫ dxk−1 ⋅ ⋅ ⋅ ∫ dx2 ′ ∫ dx1 ∫ dx1 ′ ∫ ⋅ ⋅ ⋅ ∫ R

R

R

R

R

R

R

R

fs1 (ϵ, x1 ′ , y1 ; 0, x0 , y0 ) ⋅ ϕ(x1 ; x1 ′ + ϵm, ϵ2 ) ⋅ fs2 (ϵ, x2 ′ , y2 ; s1 , x1 , y1 ) × ⋅ ⋅ ⋅ × fsk (ϵ, xk ′ , yk ; sk−1 , xk−1 , yk−1 ) ⋅ ϕ(xk ; xk ′ + ϵm, ϵ2 ) × ft (ϵ, x, y; sk , xk , yk )dy1 dy2 ⋅ ⋅ ⋅ dyk .

(A.24)

Here ϕ(x; ϵm, ϵ2 ) denotes the 1-dimensional normal distribution with mean ϵm and variance ϵ2 , and fs (ϵ, x, y; xi , yi ) denotes the density for the diffusion part. We write fsi +v (ϵ, x, y; xi , yi ) = φv (ϵ, x, y; xi , yi ) for 0 < v < si+1 − si (i = 1, 2, . . . , k − 1). Here φv (ϵ, x, y; xi , yi ) is asymptotically given by φv (ϵ, x, y; xi , yi )dy = ϕ((x, y); (x0 (v), y0 (v)), ϵ2 Σv (xi , yi ))dy + ϵ∇Hdy + O(ϵ2 ), where the term H is some complex function (cf. [105]).

(A.25)

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List of symbols (symbol, first appeared section)

Norms |F|0,k,p;A , 3.3.2 ‖F‖′l,k,p;A , 3.3.3 |F|∗0,k,p;A , 3.3.2 ‖F‖l,k,p;A , 3.3.1 ‖F‖∗l,k,p , 3.3.3 γ(u), 3.3.2 |φ|2m , 3.5.1 |φ|∼2m , 3.5.1 ‖φ‖2m , 3.5.1 ‖ψ‖−2m , 3.5.1 ‖U‖l,0,p , 3.3.1 ‖U‖l,k,p;A , 3.3.3 ‖U‖∗l,k,p , 3.3.3 ‖V ‖l,k,p;A , 3.3.3 ‖V ‖∗l,k,p , 3.3.3 ‖X‖l,0,p , 3.3.1 ‖Z‖l,k,p;A , 3.3.3 ‖Z‖∗l,k,p , 3.3.3

Spaces T, 1.1.1 A(ρ), 2.1.2, 3.3.2 ℬ, ℬφ , 4.2.2 C̃ −2m , 3.5.1 𝒞, 4.2.1 𝒞M , 4.2.2 D, 1.1.3 D, 1.2.2 , 3.1.1 D(1) 1,2 , D(2) 1,2

3.2.3

𝒟 , 3.3.1 𝒟(2) , 3.3.2 𝒟,̄ 3.3.3 Dl,k,p;A , 3.3.3 D∗l,k,p , 3.3.3 (1)

D′l,k,p , 3.3.3 D∞ , 3.3.3 D′∞ , 3.3.3

https://doi.org/10.1515/9783110675290-008

𝒟, 4.2.3 𝒟,̄ 3.3.3 𝒟∞ , 3.3.3 H2m , 3.5.1 H−2m , 3.5.1 K1 , 3.3.1 K2 , 3.3.2 K, 3.3.3 Ω1 , 2.1.1 Ω2 , 2.1.1 𝒫, 1.2.2, 2.1.3 P, 3.3.3 𝒫1 , 3.3.1 𝒫2 , 3.3.2 𝒮, 2.1.2, 2.2.1 𝒮∞ , 3.3.3 𝒮,̄ 3.3.3 𝒮, 3.3.1, 3.5.1 𝒮2m , 3.5.1 𝒮 x , 2.5.4 𝒮 ′ , 3.5.1 𝕌, 2.1.1 𝒵, 3.3.2 𝒵∞ , 3.3.3 𝒵,̄ 3.3.3

Functions ev (x) = ei(v,x) , 3.5.1 φ(ρ) = ∫|z|≤ρ |z|2 μ(dz), ψG (v), 3.5.1 Ut , Vu , 3.3.3 Z F (v), 3.5.1 Z ̃ F (v), 3.5.1

Operators C(x), Cρ (x), 3.6.2 Dt , 2.1.2, 3.3.1 D̃ u , 2.1.2

3.5.2, 3.3.2

358 � List of symbols

𝒟(t,u) , 3.3.3 𝒟̄ (t,z) , 3.4 DNt , 3.2.1 D̃ t , 3.2.1 δ, 3.1.2 δ,̃ 2.1.2 δ,̄ 3.3.3 εu+ , εu− , 2.1.2 K ̃ , 3.5.2 Kρ̃ , 3.5.2 K ̃ ′ , 3.5.2 Kρ̃ ′ , 3.5.2 K ̃ (ϵ), 4.1.1 Kρ̃ (ϵ), 4.1.1 [M], 1.2.2 QF (v), 3.5.2 Q(v), 3.5.2 Qρ (v), 3.5.2 Σ, 3.5.2 R(ϵ), 4.1.1 Tρ (v), 3.5.2 [X, Y ], 1.2.2

Processes ℰ(M)t , 1.2.4 St (ϵ), 4.1 xt (x), 1.3 xs (r, St , x), 2.2.2 Xt , 1.1.1 Xt′ , 3.6.1 X ̃ ′ (x, St , t), 3.6.4 z(t), 1.1.1 Zs (St ), 3.6.4

Measures μ(dz), 1.1.1 ̄ M(du), 3.3.2 ̂ M(du), 3.3.2 N(dsdz), 1.1.1 ̂ N(dsdz), 1.1.1 ̃ N(dsdz), 1.1.1 π, 2.5.1 n(du), 3.2.2

Index Q-nondegenerate 209 α-weight 114 (A.0) 40 (A.1) 40 (A.2) 41 (A.3) 79 (AS) 268 (AS2) 268 (B.1) 303 (B.2) 303 (B.3) 304 (B.4) 308 (D) 224 (FV) 15 (H.1) 144 (H.2) 145 (MUH)δ 248 (ND) 203 (ND-n) 233 (ND2) 218 (ND3) 218 (NDB) 233 (R) 216 (RH) 124 (SH) 246 (UH) 246, 247 (UND) 275 (UND2) 278 (UND3) 279 (URH) 124

Clark–Ocone formula 152, 165 compensated Poisson random measure 7 conditional quadratic variation 24

accessible point 81 adapted 17 angle bracket 25 antilocality 333 asymptotic expansion 265 asymptotically accessible point 81

index (of Lévy measure) 36 infinitely divisible 126 integration-by-parts 48, 55, 63, 67, 73, 75, 153, 193, 205 Ishii’s lemma 322 Itô process 35 Itô’s formula 11, 30

Barlow’s theorem 39 Blumenthal–Getoor index 14 càdlàg 6 càglàd 20 canonical process 124 carré-du-champ operator 204 characteristic function 36 Charlier polynomial 158 https://doi.org/10.1515/9783110675290-009

difference operator 62 dilation 250 divergence operator 153, 174 Doléans–Dade equation 166 Doléans–Dade exponential 31 Donsker’s delta function 298 dual previsible projection 25 dynamic programming principle (Bellman principle) 315 elementary process 19 F-differentiable 52 filtration 17 finite variation 14 flow condition 224 flow of diffeomorphisms 41 geometric Lévy 267 Girsanov transformation 32 gradient operator 149 growth function 302 Hermite polynomials 148 HJB equation 304 Hodgkin–Huxley equation 341

jump time 73 Kunita–Watanabe representation theorem 29 Lépingle–Mémin condition 32 Lévy measure 7 Lévy process 5 Lévy–Itô decomposition 127

360 � Index

Lévy–Khintchine representation 7 local martingale 18 Lyapunov function 308 Malliavin matrix 203 Malliavin operator 49 Malliavin–Shigekawa derivative 148 Markov chain approximation 78 martingale 17 martingale representation theorem 29 measurable process 17 Meyer type inequality 168, 198 modification 17 multi-dimensional stable process 10 Novikov condition 34 order condition 61, 170 Ornstein–Uhlenbeck semigroup 154 pathwise uniqueness 37 pathwise uniqueness theorem 38 Pisot–Vijayaraghavan number 129 Poisson counting measure 8 Poisson point process 8 Poisson process 9 Poisson random measure 6 Poisson space 50 polygonal geometry 101 predictable 19 previsible 19 previsible projection 25 progressively measurable 17 quadratic variation 23 renewal process 9 restricted Hörmander condition 124

semimartingale 18 skeletons 142 Skorohod integral 153 Skorohod metric 16 Skorohod space 16 small deviations 144 stable process 9 stochastic flow of diffeomorphism 79 stochastic integral 19 stopping time 18 strong solution 37 submartingale 17 supermartingale 18 trace 305 transmission property 305 truncated stable process 112 ucp-topology 20 uniformly restricted Hörmander condition 124 utility function 303 value function 303 viscosity solution 305 viscosity subsolution 306 viscosity supersolution 306 weak drift 224 weak solution 37 weak uniqueness theorem 38 weight 114 Wiener kernel method 148 Wiener process 10 Wiener space 50 Wiener–Poisson space 191 Yamada–Watanabe-type theorem 38

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