Noncausal stochastic calculus 9784431565741, 9784431565765

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Shigeyoshi Ogawa

Noncausal Stochastic Calculus

Noncausal Stochastic Calculus

Shigeyoshi Ogawa

Noncausal Stochastic Calculus

123

Shigeyoshi Ogawa Department of Mathematical Sciences Ritsumeikan University Kusatsu, Shiga Japan

ISBN 978-4-431-56574-1 DOI 10.1007/978-4-431-56576-5

ISBN 978-4-431-56576-5

(eBook)

Library of Congress Control Number: 2017945261 © Springer Japan KK 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Japan KK The registered company address is: Chiyoda First Bldg. East, 3-8-1 Nishi-Kanda, Chiyoda-ku, Tokyo 101-0065, Japan

Preface

The aim of this book is to present an elementary introduction to the theory of noncausal stochastic calculus1 that arises as a natural alternative to the standard theory of stochastic calculus founded by Prof. Kiyoshi Itô. To be more precise, we are going to study in this book a noncausal theory of stochastic calculus based on the noncausal integral that was introduced by the author in 1979. We would like to show not only the necessity of such a theory of noncausal stochastic calculus but also its growing possibility as a tool for modelling and analysis in every domain of mathematical sciences. It was around 1944 that late Prof. Kiyoshi Itô first introduced a stochastic integral (with respect to Brownian motion), called nowadays the Itô integral in his name, and originated the theory of stochastic differential equations. Then not immediately but after the Second World War, the potential importance of his theory on stochastic calculus and stochastic differential equations was recognized by mathematicians worldwide and also by physicists and engineers. They all together welcomed Itô’s theory of stochastic differential equations and joined to develop the theory extensively in various directions following their disciplines. Since then the theory has been continuously developed and by virtue of their contributions the theory has grown to be one of the standard languages in mathematical sciences and engineering. However, during these 70 years of its history some problems were recognized to be out of the range of the standard theory of Itô calculus and some people have become aware of the necessity to develop an alternative theory to cover those irregular situations. Those are the problems of a noncausal nature. Let us remember that in the theory of Itô calculus, to fix our discussion let us take the calculus with respect to Brownian motion: the random functions f ðt; xÞ are supposed to be “causal” in the sense that, for each fixed “t”, the random variable f ðt; xÞ is not affected by the future behaviour of Brownian motion after “t”. But when we are not sure whether the functions satisfy this condition, we call such a situation “noncausal”

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The caligraphy presented in the previous page is the title in Japanese, given by Michi Ogawa.

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or “anticipative” and the problem involving such noncausal functions or events the author used to call “noncausal problems”. The causality being in the base of Itô’s theory, it can hardly apply to those genuine noncausal problems. We will show some typical noncausal problems in Chap. 1. As a solution to the necessity of such a stochastic calculus that is free from the causality constraint, there were attempts to develop the so-called noncausal (or anticipative) calculus. Grosso modo we have had two alternatives as follows: one is the calculus due to A. Skorokhod and other is the noncausal calculus introduced by S. Ogawa, the author of the present book. For the sake of distinction between them we like to call the former “anticipative calculus” and the latter “noncausal calculus”. The anticipative calculus was founded by A. Skorokhod in 1967 [54] on the basis of the so-called Skorokhod integral. This line of new calculus has been developed by a group of ex-Soviet mathematicians around him, including A. Seveljakov ([51], [52]), Yu. Daletskii and S. Paramanova ([2]), and later their studies were followed by many illustrative mathematicians such as; M. Zakai and D. Nualart ([60], [61]), S. Ustunel ([56]), P. Imkeller, E. Pardoux and P. Protter ([45]), M. Pontier ([46]), F. Russo ([48]) among others. The calculs de variation originated by P. Malliavin might be classified on this line. A little bit later from the introduction of the anticipative calculus, in 1979 S. Ogawa published a note in Comptes Rendus [26] on a probabilistic approach to Feynman’s path integral ([26], [25]), where he had introduced the idea of the noncausal stochastic integral and shown the possibility of an alternative way to stochastic calculus. That was indeed the beginning of the noncausal theory of stochastic calculus, the main subject of the present book. Unfortunately this attempt to the noncausal calculus had not attracted the attention of many people and during a certain period the research had been carried out by the author alone. However, after such a silent period the study has also begun to attract the interest of other mathematicians, among them the paper [60] by M. Zakai and D. Nualart is to be noted, where they introduced the author’s noncausal integral and referred to the relation with the Skorokhod integral. And now the research on our noncausal calculus is in a state of steady progress. These two attempts to develop a new calculus, “anticipative” or “noncausal”, were aiming at the same purpose of providing a stochastic calculus that can work without the assumption of causality, but as we will see in this book they are quite different from each other in many phases, for instance in their mathematical backgrounds, materials and domains of applications. The Skorokhod integral, and so also the anticipative calculus, is established in the framework of N. Wiener’s theory of homogeneous chaos [57], which was refined by K. Itô [13] to be the theory of multiple Wiener–Itô integrals. Hence it was quite natural as we saw in the epoch-making article by M. Zakai and D. Nualart ([60], 1987) that the research on the anticipative calculus was soon joined with the study on so-called Malliavin calculus and is still now on that path. On the other hand, another alternative calculus due to S. Ogawa, namely “the noncausal calculus”, is built on his stochastic integral of noncausal type which does not rely on the theory of homogeneous chaos and so is quite different from

Preface

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Skorokhod’s as tools of calculus. This means that these two theories of calculus constructed on different stochastic integrals should be different from each other. To explain the difference one would often like to say that the Skorokhod integral is a generalization of the Itô integral, while the author’s noncausal integral stands as a generalization of the symmetric integrals (e.g. the I1=2 -integral and Stratonovich– Fisk integral). But this is not a good explication. Tools should be compared by their usefulness; in which situations and to what purposes do they work better. By the difference between two theories, we mean to say that the application domains for each of these theories should not be the same. Anyhow these alternative theories of stochastic calculus have together grown up to cover many problems of noncausal nature and become recognized as the important counterparts to the standard theory. Hence for its growing importance the author thought of the necessity of providing an introductory textbook on this subject, more precisely a book with special emphasis on the “noncausal calculus”. The reasons for this idea are twofold as follows: (1) Our main interest in the application of the noncausal theory is the analysis of various functional equations arising as models for stochastic phenomena in physics or engineering and our noncausal calculus is better fit to such an objective than the anticipative calculus since, as we will see in this book, the mapping defined by the noncausal integral exhibits a natural linearity while the one by the Skorokhod integral does not. (2) All books published up to the present time concern Skorokhod and Malliavin calculus while few regard the “noncausal calculus”. In such situation we believe that the existence of a book with special emphasis on the noncausal calculus is itself desirable. We would add at this stage that what we intend to prepare here is not an exhaustive guidebook to general theory of noncausal stochastic calculus, but the first and introductory textbook on that subject. The author intends to achieve the purpose by presenting the theory with historical sketches and also various applications that are missed or hardly treated in the standard theory of causal calculus, as well as in the “anticipative calculus”. Such is the basic idea that exists behind the present book. The original project for the book was conceived in 2006, at the Workshop of Probability and Finance held at Firenze University where the author had met Dr. Catriona Byrne of Springer Verlag. There we talked about the plan of writing a monograph on the present subject. By her suggestion the author began to prepare the manuscript but the plan did not proceed in a straight way because of various difficult situations, official or personal, that the author would be facing in the coming years. It is only in the past two years that he has finally found time to accomplish the plan. Hence for the achievement of the book the author is grateful to his friends, Profs. Gerard Kerkyacharian, Dominique Picard, Huyen Pham at University Paris 7, with whom he could have many valuable discussions, especially during 2007 when he stayed at LPMA (Laboratoire de Probabilités et Modèlea Alétoires) of University Paris-7 and 6 for one year on his sabbatical leave from Ritsumeikan University. Special thanks go to the late Professor Paul Malliavin for his valuable advice and continual encouragement and to Prof. Chii-Ruey Hwang at Mathematical Institute

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of Academia Sinica in Taiwan who kindly invited the author many times to Academia Sinica for discussions and offered him many opportunities to give seminar talks about the main subject of the present book. The author wishes to thank Prof. Hideaki Uemura of Aichi-Kyoiku University for his collaboration in doing research on stochastic Fourier transformation, the contents of Chap. 8. He also wishes to thank the referees for the careful reading of the draft and many valuable comments and suggestions. For the realization of the publication plan the author is very grateful to the editors, Dr. Catriona Byrne of Springer Verlag Heidelberg and Mr. Masayuki Nakamura of Springer Tokyo, for their great patience and continual warm encouragements. The author sincerely recognizes that he has not been a good writer in forcing them to wait for such a long time. Kyoto, Japan July 2016

Shigeyoshi Ogawa

Contents

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Introduction – Why the Causality? . . . . . . . . . . . . . . . . . 1.1 Hypothesis of Causality . . . . . . . . . . . . . . . . . . . . . . 1.2 Noncausal Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Some Noncausal Problems . . . . . . . . . . . . . . . . . . . . 1.3.1 Problem in BPE Theory . . . . . . . . . . . . . . . 1.3.2 Convolution Product with the White Noise . 1.3.3 Vibration of a Random String . . . . . . . . . . . 1.4 Plan of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preliminary – Causal Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Some Properties of BM . . . . . . . . . . . . . . . . . . . . . 2.1.2 Construction of BM . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Itô Integral with Respect to BM . . . . . . . . . . . . . . . . . . . . . 2.2.1 Classes of Random Functions . . . . . . . . . . . . . . . . 2.2.2 Itô Integral for f 2 S . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Extension to f 2 M . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Linearity in Strong Sense . . . . . . . . . . . . . . . . . . . 2.2.5 Itô Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 About the Martingale Zt ðaÞ . . . . . . . . . . . . . . . . . . 2.3 Causal Variants of Itô Integral . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Symmetric Integrals . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Anti-Causal Function and Backward Itô Integral . . 2.3.3 The Symmetric Integral for Anti-Causal Functions 2.4 SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Strong Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Law of the Solution of SDE . . . . . . . . . . . . . . . . . 2.4.3 Martingale Zt and Girsanov’s Theorem . . . . . . . . .

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Noncausal Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Noncausal Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Question (Q1) – Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dependence on the Basis – About Q2 . . . . . . . . . . . . . . . . . . . . 3.4 Relation with Causal Calculus – About Q3 . . . . . . . . . . . . . . . . 3.4.1 Regular Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 More About the Integrability of Quasi-Martingales . . . . 3.4.3 Comments on Riemann-Type Integrals . . . . . . . . . . . . . 3.5 Miscellaneous Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Noncausal Integral for Random Fields. . . . . . . . . . . . . . 3.5.2 Case of Fractional Brownian Motion . . . . . . . . . . . . . . . 3.6 Appendix – Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . .

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Noncausal Integral and Wiener Chaos . . 4.1 Wiener–Itô Decomposition . . . . . . . . 4.2 Skorokhod Integral . . . . . . . . . . . . . . 4.3 Noncausal Integral in Wiener Space .

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Noncausal SDEs. . . . . . . . . . . . . . . . . . . . . 5.1 Symmetric SDE as Noncausal SDE . 5.2 SDE with Noncausal Initial Data . . . 5.3 Noncausal Itô Formula . . . . . . . . . . . 5.4 SDE in Noncausal Situation . . . . . . . 5.5 Examples . . . . . . . . . . . . . . . . . . . . .

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Brownian Particle Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Definition of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Method of Stochastic Characteristics . . . . . . . . . . . . . . . . . 6.2.1 Bridge to Parabolic Equations . . . . . . . . . . . . . . . . 6.2.2 A Small Extension. . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Nonlinear BPEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Stochastic Burgers’ Equation . . . . . . . . . . . . . . . . . 6.4.2 Numerical Solution of a Nonlinear Diffusion Equation – G. Rosen’s Idea . . . . . . . . . . . . . . . . . . 6.4.3 Girsanov’s Theorem? . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Noncausal Counterpart for Girsanov’s Theorem . .

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Noncausal SIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 SIE Driven by Brownian Motion . . . . . . . . . . . 7.1.1 Integration by Parts Method . . . . . . . . 7.1.2 A Stochastic Transformation Along Orthonormal Basis . . . . . . . . . . . . . . . . 7.2 SIE Driven by Brownian Sheet. . . . . . . . . . . . .

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Stochastic Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . 8.1 SFC and SFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Case of Causal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Case of Wiener Functionals . . . . . . . . . . . . . . . . . . 8.2.2 Causality Condition . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Scheme for Reconstruction of Kernels. . . . . . . . . . 8.2.4 Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Case of Noncausal Wiener Functionals . . . . . . . . . . . . . . . . 8.3.1 Extension to the Case of General Basis . . . . . . . . . 8.4 Case of Noncausal Wiener Functionals – 2 . . . . . . . . . . . . 8.4.1 Model and Results . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Method of Fourier Series for Volatility Estimation 8.5 Some Direct Inversion Formulae . . . . . . . . . . . . . . . . . . . . 8.5.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Natural SFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4 Another Scheme for Direct Inversion . . . . . . . . . . 8.5.5 A Variant of the Natural SFT . . . . . . . . . . . . . . . . 8.5.6 Remarks on SFTs of Different Types . . . . . . . . . . 8.5.7 About the Linearity of the SFT . . . . . . . . . . . . . . . 8.5.8 Extension to Noncausal Case. . . . . . . . . . . . . . . . .

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9

Appendices to Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Itô Integral with Respect to Martingales . . . . . . . . . . . . . . . 9.1.1 Procedure for Introduction . . . . . . . . . . . . . . . . . . . 9.1.2 Integral for f 2 M2;c . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Some Properties of Square Integrable Ft -Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Some Applications. . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Symmetric Integral with Respect to Martingale . . .

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10 Appendices 2 – Comments and Proofs . . . 10.1 Notes on Chap. 2 . . . . . . . . . . . . . . . 10.1.1 Martingale Inequalities . . . . 10.2 Statements in Chap. 2 . . . . . . . . . . . . 10.2.1 Proof of Proposition 2.7 . . . 10.2.2 Proof of Lemma 2.2 . . . . . . 10.2.3 Proof of Theorem 2.5 . . . . . 10.2.4 Gronwall’s Inequality . . . . . 10.2.5 Uniqueness of B-Derivative . 10.3 Note on Chap. 3 . . . . . . . . . . . . . . . . 10.4 Notes on Chap. 8 . . . . . . . . . . . . . . . 10.4.1 Proof of Theorem 8.2 . . . . .

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10.5 Proof of Theorem 8.4 . . . . . . . . . . . . . . . . . . . . 10.5.1 Sketch of Proof for Proposition 8.7 . . . 10.5.2 Dini’s Test. . . . . . . . . . . . . . . . . . . . . . 10.5.3 Proof of Proposition 8.9 . . . . . . . . . . . 10.5.4 Proof of Proposition 8.10 . . . . . . . . . .

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Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Chapter 1

Introduction – Why the Causality?

We are concerned with an alternative theory of stochastic calculus, referred to nowadays with the adjective noncausal, that contrasts to the standard theory of Itô calculus. We would like to give in this chapter a historical sketch of the noncausal theory so as to convince ourselves of the necessity and importance of such a theory. The net lecture on noncausal theory begins from Chap. 3, while in Chap. 2 we give as a preliminary for the main subject short reviews on such basic materials from the causal theory of Itô calculus as the Itô integral, as well as the symmetric integrals and B-derivatives.

1.1 Hypothesis of Causality As the notion of differential equations appeared with Newtonian mechanics, the SDE and SPDE (stochastic ordinary or partial differential equations respectively), including a random process, say X t , as the coefficient that represents random disturbances or input to the system, were expected to serve as mathematical models for those dynamical phenomena perturbed or driven by a random noise X t . In this order of consideration the evolution of such random phenomena should be represented by functionals of the basic noise process X t . In the history of mathematical sciences the random noise X t of most interest is the derivative X t = dtd Wt of the process Wt called Brownian motion or more generally X t = dtd Yt , the derivative of a Lévy process Yt . Since, in most cases, almost every sample function of such typical random processes, Wt or Yt , is not of bounded variation, those random equations could not be studied in the classical framework of differential calculus. A special calculus was needed to treat those random functional equations. The stochastic calculus originated by K. Itô is such a calculus. It works for random functions that appear as functionals of such basic stochastic processes.

© Springer Japan KK 2017 S. Ogawa, Noncausal Stochastic Calculus, DOI 10.1007/978-4-431-56576-5_1

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1 Introduction – Why the Causality?

As is well-known, the theory of Itô calculus is established on a hypothesis that every random function f (t, ω), that appears as the integrand of his stochastic integral, should be adapted to the past history of the basic process X s (s ≤ t); roughly speaking it means that the value of f (t, ω) at time t should not depend on the future behaviour {X u : u > t} “after time t” of the basic process X . In this context we like to call this hypothesis on the measurability of random functions the Causality Hypothesis. Throughout the book we call random functions satisfying this hypothesis causal. At the dawn of the theory of Itô calculus, this hypothesis did not seem to impose any serious restriction on the theory, partly because the first and the most important application of Itô theory was, and is still so now, the study of the SDE (i.e. stochastic differential equation) where the validity of causality hypothesis is viewed quite naturaly as the principle of causality in physics. For instance, with representation by SDE the phenomenon of diffusion is visibly understood as a physical phenomenon agitated by a mass of Brownian particles, thus in this mathematical description the behaviour in time “t” of each diffusive substance should be driven by the stochastic process, called Brownian motion, up to present time “t” and should not be affected by the future behaviour of Brownian motion. In other words the hypothesis of causality in the theory of Itô calculus has been admissible because of its natural concordance with the principle of causality in physics. It is easy to imagine why, from its beginning, Itô theory of SDEs has been accepted not only in physics but also in every domain of mathematical sciences as a standard language to describe and analyse random phenomena as well as systems with random mechanics. But in reality, Itô theory of stochastic calculus has profited much from the causality hypothesis because it allowed the theory to be developed along with the theory of the martingale, a powerful tool in the theory of stochastic processes. Remark 1.1 Hence, one may think that with an attempt to develop a noncausal calculus we would risk losing this advantageous situation. It is true in some sense, especially for the usage of the theory of the martingale, but as we will see in this book it is also true that our noncausal theory is established on the basis of the standard theory of Itô calculus. Nevertheless we have sufficient reasons to introduce such a noncausal theory as we will see by some typical examples presented below.

1.2 Noncausal Calculus We understand that, along with the theory of the martingale, Itô theory of stochastic calculus has grown up to be a standard language in all domains of mathematical sciences including physics and engineering. However, over time, the theory has been developed also as a calculus tool in stochastic theory and it began to face various situations where the hypothesis of causality is violated. Then some people began to be aware of the necessity of such an alternative theory that is free from the hypothesis of causality and already in the 1970s some attempts toward this aim were made by a few mathematicians. Among them the attempts by the following two groups are to

1.2 Noncausal Calculus

3

be noted: one is the calculus based on the so-called Skorokhod integral which was introduced by A. Skorokhod and studied by an ex-Soviet group around A. Skorokhod, Yu. Daletskii, S.J. Sevelyakov etc.; the other is the noncausal stochastic calculus that was introduced by S. Ogawa in the course of his study on some problems in mathematical physics, such as Brownian particle equations (cf. Chaps. 1 and 6), the square of the white noise (cf. Chaps. 1 and 10), stochastic Shrödinger equation (cf. Chap. 6), the stochastic integral equations of Fredholm type (cf. Chap. 7) that arose in the study of the boundary value problem to SDEs of the second order, etc. To make a clear distinction between these two models, in this book we call the former one anticipative calculus and the latter one noncausal calculus. These two forms of calculus were introduced independently of each other. The anticipative calculus was presented in the early 1970s, while the noncausal calculus of the author first appeared in 1979, in a note in Comptes Rendus [26] (see also [24, 25, 27]). As noted in the Preface, in this book we will mainly deal with the theory of noncausal calculus based on the author’s integral. One of the reasons for this plan is that, as far as the author knows, all publications until now about stochastic calculus of noncausal nature seem to be exclusively on Skorokhod and Malliavin calculus, while there are none for the noncausal theory of the author (the main subject of the book). Looking at such an unbalanced situation the author thought there should be a publication on the noncausal theory as well. We still find there many important problems left open but we believe that studying those problems will lead us to a new horizon of the stochastic theory.

1.3 Some Noncausal Problems For our motivation to study the noncausal calculus we would like to show in the rest of this chapter some typical problems of noncausal nature. The detailed discussions for each of those problems will be developed in several chapters after Chap. 3.

1.3.1 Problem in BPE Theory Let us imagine an ensemble of many particles moving on the line R 1 as if each particle behaves like a Brownian particle. There we may observe that the figure of the distribution of this mass at initial moment t = 0 is represented by a distribution function, say f (x), and we continue observing the movement of the mass to see the figure of distribution at time t > 0. We are interested in determining the distribution function of the particles at time t, say u(t, x). Hence we first ask ourselves what would be the mathematical model for this phenomenon. In mathematical physics, for such a transport phenomenon the evolution in time of the intensity of a physical quantity, such as the density, temperature or pressure

4

1 Introduction – Why the Causality?

etc., is represented by a PDE (partial differential equation) of either parabolic or hyperbolic type. Among them the PDE of the first order like ⎧ ⎨ ∂ u(t, x) + a(t, x) ∂ u(t, x) = A(t, x)u(t, x) + B(t, x), ∂t ∂x ⎩ (t, x) ∈ [0, ∞) × R 1 ,

(1.1)

might fit our present purpose. In fact, this equation represents a transport phenomenon that is exhibited by particles, each of which moves in space following the velocity field a(t, x), under the effect of amplification or attenuation factor A(t, x) and the external disturbance B(t, x). Applying this rigid idea to our phenomenon, which is stochastic in its nature, we may arrive at the following SPDE ⎧   d ∂ ⎨ ∂ u(t, x, ω) + a(t, x) + Wt (ω) u(t, x, ω) ∂t dt ∂x ⎩ = A(t, x)u(t, x, ω) + B(t, x), (t, x, ω) ∈ [0, ∞) × R 1 × Ω,

(1.2)

of course formally since it involves as coefficient the white noise dtd Wt , the derivative of Brownian motion Wt . We suppose that Wt (ω) is defined on a probability space (Ω, F , P) and ω stands for the generic point in Ω. Notice that the solution u(t, x, ω) is now considered to be a measurable function of three variables (t, x, ω). Later we will suppose some additional conditions of measurability on u(t, x, ω) so that the Eq. (1.2) becomes meaningful. Also notice that in this framework our problem of describing the evolution in time of the quantity u(t, x, ω), given its initial data f (x), is formulated as Cauchy problem with the initial condition, u(0, x, ω) = f (x) P − a.s.

(1.3)

This was the beginning of the theory of Brownian particle equations (BPEs), which is a class of SPDEs like (1.2) that describe the transport phenomena exhibited by Brownian particles (cf. [22–25] etc.). The Eq. (1.2) being formal, we need to give a firm definition to it, especially to the Cauchy problem (1.2)–(1.3). We should notice that the BPE (1.2) forces us to face an essential problem by the fact that the equation contains the direct product d Wt and of two random distributions (in the sense of L. Schwartz’s distribution) dt ∂ u. In the first article [22], this difficulty was avoided by applying the notion of ∂x weak solution in the theory of PDEs, which may read as follows: Definition 1.1 A real random function u(t, x, ω), defined on R+ × R 1 × Ω and measurable in (t, x, ω) with respect to the σ -field B R+ × B R 1 × F , is called the solution of the Cauchy problem (1.2)–(1.3) provided that:

1.3 Some Noncausal Problems

5

(s1)

for each fixed x the  random function (t, ω) −→ u(t, x, ω) is causal so that the stochastic integral u dWt is well-defined, and

(s2)

for any test function ϕ(t, x), having finite support in [0, T ] × R 1 , the function u(t, x, ω) satisfies the following equality (1.4) with probability one: 



T

dx R1

 {ϕt + (aϕ)x + A}u + Bϕ dt +

0  + ϕ(0, x)u 0 (x)d x = 0



T

dx R1

ϕx u(t, x, ω)dWt

0

(1.4)

P-a.s.,

R1

where ϕt = ∂t ϕ(t, x), ϕx = ∂x ϕ(t, x). Here the type of the stochastic integral is not yet specified. We remember that we have introduced the BPE as a microscopic model for such diffusion phenomena exhibited by Brownian particles, hence it is almost immediate to see that the Itô integral does not fit our present picture. In fact, if we employ the Itô integral then by taking the expectation on both sides of the weak solution formula (1.4) we easily find that the average u(t, x) := E[u(t, x, ·)] of the solution u(t, x, ω), if it exists, satisfies the following equality for any test functions ϕ(t, x): 



T

{ϕt + (aϕ)x + A}u + Bϕ dt +

dx R1

=0



0

 R1

ϕ(0, x)u 0 (x)d x

(1.5)

P-a.s.,

which states that the average u(t, x) = E[u(t, x, ·)] becomes a solution of the following PDE of the first order which has nothing common with the diffusion phenomenon: ∂ ∂ u(t, x) + a(t, x) u(t, x) = A(t, x)u(t, x) + B(t, x), ∂t ∂x (t, x) ∈ [0, ∞) × R 1 .

(1.6)

Remark 1.2 By the same reason, the Skorokhod integral (cf. Chap. 5) does not fit our picture at all. In reality the stochastic integral that should be employed for the definition of the solution is the so-called symmetric integral or the noncausal integral, another type of stochastic integral which is free from the causality restriction and is the main subject that we will study in this book. The noncausal problem arises when we consider a variant of BPE as follows:   d ∂ ∂ u(t, x, ω) + a(t, x) + Wt (ω) u(t, x, ω) ∂t dt ∂x d = A(t, x)u(t, x, ω) Wt + B(t, x), (t, x, ω) ∈ [0, ∞) × R 1 × Ω. dt

(1.7)

6

1 Introduction – Why the Causality?

The BPE being an SPDE of the first order in its form, for the construction of the solution of the Cauchy problem, we may intend to apply the method of characteristics which is familiar in the usual theory of PDEs. With a suitable modification to fit our case, since our equation is a stochastic PDE including the white noise as coefficient, the usual PDE theory might suggest that the solution will be constructed as being the solution of the following system of stochastic integral equations: 

u 0 (X 0(t,x) )

t

+ {Au(s, X s(t,x) , ω)dWs + B(s, X s(t,x) , ω)ds},  t 0 where X s(t,x) − x = − a(r, X r(t,x) )dr + Ws − Wt , 0 ≤ s ≤ t ≤ T.

u(t, x, ω) =

(1.8)

s

Here  t appears an essential problem of measurability in the stochastic integral term, A(s, X s(t,x) )u(s, X s(t,x) , ω)dWs . We know by definition (s2) that for each fixed 0

(s, x) the random variable u(s, x, ω) is measurable with respect to the field Fs , but on the other hand we see that for each fixed (t, x) the function X s(t,x) (ω) is adapted to the future field Fs,t = σ {Wt − Wr ; s ≤ r ≤ t}. The integrand A(s, X s(t,x) )u(s, X s(t,x) , ω) being a composite of those two random functions, we see that this  function fails to be causal or anti-causal and at this point t

the stochastic integral 0

Au(s, X s(t,x) , ω)dWs loses its meaning in the framework

of the causal theory of Itô calculus. Thus we see the necessity of an alternative theory of stochastic calculus which can be free from the hypothesis of causality. What we have seen above is merely an example of noncausal problems. There are in fact a lot of noncausal problems that we meet not only in the theory of stochastic processes but also in various domains of mathematical sciences, some of which we will pick up and study in this book after Chap. 4.

1.3.2 Convolution Product with the White Noise As we have pointed it, the importance of the causality hypothesis is mainly contained in the fact that about the under this hypothesis we can develop the discussion stochastic integral f (t, ω)dWt , or more generally the integral f (t, ω)d Mt with respect a martingale Mt , in a very clear and efficient way with the help of the theory of martingales. In other words, the hypothesis loses its merit when we deal with the integral f (t, ω)d X t with respect to a process X t which does not exhibit the martingale property. Such cases often occur in applications, for instance: the stochastic differential equation driven by a fractional Brownian motion (cf. A. Shiryaev [53],

1.3 Some Noncausal Problems

7

R. DelGade [3]), and the integral with respect to such a random field as Brownian sheet (cf. [33, 34]). Even in the case of the stochastic integral with respect to Brownian motion or to martingales, we meet instances where the hypothesis of causality is no more favourable. For instance, in the framework of the causal calculus, we see the difficulty of defining the convolution product f (t +s, ω)dWs of a random function f (t, ω), causal or not. We would like to add that the operation of making a convolution product of two functions or stochastic processes cannot be avoided when we use Fourier transformation. If we try to take the Fourier transformation of the product of two random distributions X t = f (t, ω) · W˙ t , we will meet the stochastic convolution  (F X )t =

f˜(t − s, ω)d W˜ s ,

where the symbols “F ” and “∼” (tilde) stand for the Fourier transform of the indicated functions. As we see below, we notice that F W˙ t becomes again a white noise. Here we will give a fictitious example concerning the convolution product of the white noise by itself. Example 1.1 (Square of the white noise) Let us imagine a particle moving along the real line R1 , the coordinate of which at time t ≥ 0 we denote by X (t), hence the Lagrangian of this one particle system becomes 

t

L(t; X ) := m

X˙ 2 (s)ds,

0

where m > 0 is the mass of the particle. By a standard textbook √ on quantum mechanics we know that the propagator of −1 L(t; X )}, where  is Heisenberg’s constant. This is the system is given by exp{  well-known. By the way, perhaps because the Schrödinger equation looks similar to an equation of parabolic type, I guess that many probabilists like to imagine how the story is going to be if an elementary particle behaves like Brownian motion! Then we √ −1 L(t; W (ω))} for the propagator of probability wave will have the form exp{  mentioned in quantum mechanics, and in this line of consideration we may be led to think about its meaning, or at least the meaning of the direct product of the white d noise W˙ t2 , W˙ t = Wt . It is of course absurd, for we know from a famous article dt by L. Schwartz [50] the impossibility of defining the direct product S · T of two distributions, say S, T ∈ S , as a distribution over some appropriate function space like Schwartz’s space S . Yes, it is absurd to think of the product (W˙ (ω))2 for each sample ω, so we may as well ask ourselves again, what is the white noise? Heuristically speaking, it is the

8

1 Introduction – Why the Causality?

derivative of a Brownian motion and the theory of random distributions, or generalized random functions, gives us another definition as follows: Definition 1.2 Let S (R) be the space of rapidly decreasing C ∞ -class functions f (x) endowed with the system of semi-norms, f m,n := supx∈R (1+|x|m )| f (n) (x)|. The white noise W˙ t (ω) is an S - valued random variable such that the characteristic functional is given in the following form:   √ 1 Φ(u) := E[exp{ −1(W˙ , u)}] = exp − u 2L 2 , 2

∀

u(x) ∈ S .

The existence of such a probability measure on the dual space S is a simple consequence of the general theorem by Bochner and Schwartz (cf. [7]) concerning the existence of a probability measure on the dual of any nuclear space of functions, which states that a continuous characteristic functional Φ(u) uniquely determines a probability measure on measurable space (S , C ) (C is a σ -field of subsets). Here we notice that the function space S in the above statements can be replaced by any other space of functions as long as it has the property of being nuclear. In other words, we see at this point an ambiguity in the definition of the white noise. Let L be a subspace of S , then as an immediate consequence of the definition of white noise on such L , we see from the equality below that the Fourier transform (in distribution sense) Z = F (W˙ ) of the white noise W˙ is again a white noise on L ; namely for u ∈ L we have √ √ E[exp{ −1(F W˙ , u)}] = E[exp{ −1(W˙ , F u)}] = exp{− 21 F u 2 } = exp{− 21 u 2 }, u ∈ L , where F W˙ and F u are the Fourier images of W˙ and u respectively. Let us continue this formal argument and suppose that the product (W˙ )2 is justified in some unknown sense. Then if we try to take formally the Fourier transformation of (W˙ )2 , we will have the following: F (W˙ )2 = Z ∗ Z , the convolution product where Z = F (W˙ ) is a white noise defined as an element of the dual of the nuclear apace L . Thus we find that if the convolution product Z ∗ Z of the white noise Z is well-defined, we may introduce the direct product (W˙ )2 as a random distribution on L . Following the customary argument, we may define the convolution product Z ∗ Z in the form  Z ∗ Z (u) := Z (u(t + ·))d Bt , u ∈ L, where B(t) is a Brownian motion induced  by the white noise Z on L , namely B(t) := Z (1[0,t] (·)). Since Z (u(t + ·)) = u(t + s)d Bs we have

1.3 Some Noncausal Problems

9

 Z ∗ Z (u) :=

 d Bt

u(t + s)d Bs .

We find that in this line of argument, the possibility of defining the direct product (W˙ )2 is reduced to the possibility of defining the convolution product with respect to the stochastic integral d B, which should be treated in the framework of our noncausal calculus. We will continue the discussion on this subject in Chap. 10 (in Example 10.1).

1.3.3 Vibration of a Random String To show another typical example of a noncausal problem, let us consider the boundary value problem for the second order SDE as follows: 

   d d dZ p(t) + q(t) X (t) = X (t) (t, ω) + A(t, ω) + B(t, ω) dt dt dt X (0) = x0 , X (1) = x1 ,

(1.9)

where x0 , x1 are real constant, and A(t, ω), B(t, ω) and Z t , t ∈ [0, 1] are arbitrary stochastic processes defined on the (Ω, F , P), with square integrable sample paths. Then as we do for the boundary value problem of ordinary differential equations, by using Green’s function K (t, s) corresponding to the above self-adjoint differential operator dtd p(t) dtd + q(t) with boundary condition of the same nature, we may get in a very formal way the following stochastic integral equation of Fredholm type:  X (t, ω) = f (t, ω) +

1

 L(t, s, ω)X (s)ds +

0

1

K (t, s)X (s)d Z (s),

0

1 where L(t, s, ω) = A(s, ω))K (t, s) and f (t, ω) = 0 B(s, ω)K (t, s)ds, and 1 0 K (t, s)X (s)d Z (s) represents a stochastic integral with respect to the process Z (t). This boundary value problem has its origin in a firm physical problem of the vibration of a string pinned at two sides x = 0, 1, whose section S(x) at point x ∈ [0, 1] varies in a stochastic way like S(x, ω) = C exp{Wx (ω)}. This is a very typical problem that cannot be treated in the framework of the causal theory of Itô calculus, since in such situations we can no more suppose that the solution X t is still causal (i.e. adapted) to the natural filtration generated by the underlying fundamental  process Z t . Hence the stochastic integral term

K (t, s)X (s) d Z s loses its meaning

in the framework of Itô’s theory. As a natural extension of such an SIE (stochastic integral equation) to the case where the Z t , X t are the random fields, namely the stochastic processes with multidimensional parameters t ∈ J = [0, 1]d , we can think of the SIE for the random

10

1 Introduction – Why the Causality?

fields. Imagine for example the case where the driving force process Z t is the Brownian sheet:   X (t, ω) = f (t, ω) + L(t, s, ω)X (s)ds + K (t, s, ω)X (s)d Z (s). J

J

These are the typical problems of noncausal nature that are to be studied in this book.

1.4 Plan of the Book The aim of the book is to give an introduction to the theory of noncausal calculus, mainly the calculus based on the author’s stochastic integral (except some parts in Chap. 8). Since the subject may not be familiar enough to most of the readers, even to a large number of specialists, we intend to develop the discussion by studying in each chapter various noncausal problems, so that we can clearly recognize the necessity and the efficiency of the new calculus. The book is organized in the following way: before entering into the lecture on the main subject, in Chap. 2 we give a short review on the causal calculus based on the Itô integral and its causal variant called the symmetric integral. This is because our noncausal stochastic calculus stands as a natural extension of the causal calculus based on the symmetric integral and somewhere in the book we need to review those materials as preliminaries. The extension to the integral with respect to martingales is also possible and will be referred to in later chapters, but to keep the main stream of discussions in Chap. 2 as clear as possible we have put that part in Chap. 9 as Appendices to Chap. 2. In Chap. 3, we introduce the noncausal integral and show its fundamental properties. The interpretation of the noncausal integral in the framework of homogeneous chaos is given in Chap. 4, where we will refer to another integral of noncausal type introduced by A. Skorokhod. Chapters 2–4 will serve as a theoretical introduction to the new calculus. In Chaps. 5–8, we will present discussions and results on the typical problems of noncausal nature, namely, in Chap. 5 we discuss the SDE of noncausal type, and in Chap. 6 we show some applications of our noncausal calculus to the theory of BPEs, more precisely we will study the Cauchy problem and show some basic properties of the solutions. Henceforth we will study in Chap. 7 the stochastic integral equation of Fredholm type and in Chap. 8 we discuss basic properties of the SFT (stochastic Fourier transformation) which was first introduced by the author in the study ([34]) of SIE of Fredholm type. Finally Chap. 10 is devoted to comments and verifications of those statements that are presented in the text without proofs.

Chapter 2

Preliminary – Causal Calculus

The theory of noncausal calculus is an alternative to the causal theory of Itô calculus but is not quite independent of it. As we will see in the main part of this book that starts from Chap. 3, our noncausal theory stands as a natural extension of the causal theory of Itô calculus, to be more precise, the causal theory based on the stochastic integral called symmetric integrals. We may emphasize that at this point our noncausal theory keeps a large part of its raisons d’être. Hence as preliminaries for the study of our noncausal theory of stochastic calculus we need to present in this chapter a necessary and minimum review on those materials and related facts from the causal calculus such as Brownian motion, the Itô integral, the symmetric integrals and the notion of the B-derivative of random functions. By doing this we also intend to prepare the list of symbols and terminologies concerning those materials that will be used throughout the book. We remark that what we intend to show in this chapter is not a standard review of Itô calculus but just a small note on it, thus for the details or further understanding of the causal calculus we would refer the reader to other standard textbooks on Itô calculus and some of the author’s articles (e.g. [20–25]). The presentation of these materials is in the following order: Brownian motion in Sect. 2.1, the Itô integral and related statements in Sect. 2.2, some elementary but important results concerning the SDE (stochastic differential equation) will be referred to in Sect. 2.3, while Sect. 2.4 is devoted to the note on variants of the Itô integral, where we repeat briefly the results concerning the B-derivative and the symmetric integrals, especially the integral I1/2 ( f ) that is introduced by the author ([20, 21]) and will be of frequent use in the discussions on our main theme. We also refer to the integral of symmetric type called the Stratonovich–Fisk integral ([14, 55]). Before entering into the discussion the author would like to have the reader’s attention on the symbols for stochastic integrals. As we are going to deal with plural stochastic integrals we need appropriate symbols to make clear distinctions between them. In particular, for the Itô integral we would assign f d0 Wt , by putting “0” at “dW ” to signify that the Itô integral is at the origin of the theory of stochastic calculus. © Springer Japan KK 2017 S. Ogawa, Noncausal Stochastic Calculus, DOI 10.1007/978-4-431-56576-5_2

11

12

2 Preliminary – Causal Calculus

2.1 Brownian Motion Definition 2.1 (1) A real-valued random variable X (ω) defined on a probability space (Ω, F , P) is called Gaussian if its characteristic function ϕ X (θ ) := E[exp{iθ X }] is given in the following form:   σ2 2 θ exp{iθ X (ω)}d P(ω) = exp imθ − E[exp{iθ X (ω)}] := 2 Ω 

∀

θ ∈ R1 ,

where m and σ ≥ 0 are real constants. (2) An n-tuple of real random variables X := (X 1 , . . . , X n ) ∈ Rn is called an Rn valued (or n-dimensional) Gaussian random variable provided that any linear com bination Y = nk=1 tk X k (ω) with ∀ (t1 , . . . , tn ) ∈ Rn is a real Gaussian random variable. By definition (2) above we see that: Proposition 2.1 The n components {X k , k = 1, n} of the Rn -valued Gaussian variable X = (X 1 , . . . , X n ) are independent provided that they are uncorrelated, namely, Cov(X i , X j ) = 0∀ i = j, where Cov(X i , X j ) = E[(X i − E X i )(X j − E X j )] is the covariance of (X i , X j ). Definition 2.2 (Gaussian process) A stochastic process X t (ω), t ∈ T ⊂ R is called Gaussian provided that for any n ∈ N and arbitrarily chosen n different points ti ∈ T, 1 = 1, . . . , n, the n-dimensional random variable (X ti (ω), . . . , X tn ) is Gaussian. We know that every finite dimensional Gaussian distribution is determined by a pair of parameters, namely, a mean vector m =t (m 1 , m 2 , . . . , m n ) ∈ Rn and an n × n-real symmetric positive definite matrix Γ called the covariance matrix, in the following way: f (x) = √

1 1 exp{− (Γ −1 (x − m), x − m)}, x =t (x1 , x2 , . . . , xn ) ∈ Rn . n 2 (2π ) |Γ |

Thus we notice that a Gaussian process X t , t ∈ T is completely determined by the pair of a mean function m(t) and a real kernel Γ (s, t), s, t ∈ T of positive definite type. Notice also that a real Gaussian process X t determined by these has the following properties: m(t) := E[X t ], Γ (s, t) := Cov(X s , X t ) = E[(X s − m(s))(X t − m(t))]. We introduce one of our principal materials, the Brownian motion (or the BM for short), in the following: Definition 2.3 (Brownian motion) (1) A real Gaussian process W.(ω) defined on (Ω, F , P) is called Brownian motion provided that:

2.1 Brownian Motion

(b1) (b2)

13

P{W0 = 0} = 1, E[Wt ] = 0, E[Ws Wt ] = s ∧ t for ∀ s,∀ t ≥ 0 where s ∧ t := min{s, t}.

(2) Let W1 (t), W2 (t), . . . , Wn (t) be n independent copies of the Brownian motion. The R n -valued Gaussian process W(t) =t (W1 (t), W2 (t), . . . , Wn (t)) is called the n-dimensional Brownian motion. Example 2.1 The following processes X t are all Brownian motions, where c is a positive constant: (1) X t = Wt+c − Wc , (2) X t = √1c Wct , (3) X t = t W1/t (t > 0) with convention X 0 = 0. A right continuous and increasing family {Ft , t ≥ 0} of sub σ -fields of F is called “filtration”:  Ft+h . Fs ⊂ Ft ⊂ F ∀ s < t, and Ft = h>0

For instance, GtW := σ {Ws | s ≤ t} or GtW ∨ σ {V } where V (ω) is a random variable independent of Brownian motion are filtrations. Definition 2.4 In this book, by natural filtration we understand a right continuous and increasing family of sub σ -fields {FtW , t > 0} such that FtW ⊃ GtW

∀

t

and that for any s ≤ t increment Wt − Ws is independent of FsW . Here we understand that every sub σ -field FtW is completed with all P-null sets.

2.1.1 Some Properties of BM The Brownian motion process, which is also called the Wiener process, was introduced by N. Wiener in 1930. Being one of the most important materials in the theory of stochastic processes, it has been studied extensively by many authors, and many books have been published. We do not intend to repeat in detail, even some parts of, its basic properties but we shall content ourselves in this subsection with listing only some of its remarkable properties which cannot be missed for our present purpose: (f1) From (b2) we see that E[(Wt − Ws )2 ] = |t − s| and that the random variable Wt − Ws follows the normal law, Wt − Ws ∼ N (0, |t − s|). (f2) The condition (b2) also implies that, for any 0 ≤ s ≤ t ≤ u ≤ v, we have E[(Wv − Wu )(Wt − Ws )] = E[Wv Wt − Wt Wu − Wv Ws + Wu Ws ] = t − t − s + s = 0. By virtue of Proposition 2.1 we see from (f2) that Brownian motion is a process of independent increments.

14

2 Preliminary – Causal Calculus

(f2)’ Or in other words, for any s ≤ t the increment Δs,t W := W (t) − W (s) is independent of the field GsW . This property implies that Brownian motion is a martingale along with the family of sub σ -fields {GtW }t>0 , that is, for any t ≥ s the following holds: E[Wt |GsW ] = Ws P − a.s. (f3) Brownian motion Wt is a martingale with respect to any natural filtration mentioned in Definition 2.4, E[Wt |FsW ] = Ws P − a.s. ∀ t ≥ s. (f4) For any fixed α ∈ R the process Z t = exp{αWt − martingale. In fact, for any t ≥ s we have

α2 t } 2

becomes an FtW -

   α 2 (t − s) |FsW E[Z t |FsW ] = E Z s exp αWt − 2    α 2 (t − s) = Z s E exp α(Wt − Ws ) − |FsW 2 = Z s P − a.s. since





 α 2 (t − s) α 2 (t − s) W E exp α(Wt − Ws ) − = E exp αWt−s − |Fs = 1. 2 2

(f5) The fact (f2) also implies that Brownian motion is a homogeneous Markov process having the following kernel as transition probability density:   (y − x)2 , x, y ∈ R, t > 0. exp − p(t, x, y) = √ 2t 2π t 1

(2.1)

Knowing the transition probability density we can construct the Markov process, so we confirm the existence of BM. Here are some important properties concerning the regularity of the sample path of BM. (f6) We notice the following property. Proposition 2.2 Almost every sample path of BM is continuous but is not of bounded variation on any finite interval. For the verification of this statement we appeal to the following result called Kolmogorov’s test, whose proof is omitted.

2.1 Brownian Motion

15

Theorem 2.1 If a real-valued stochastic process X t t ∈ [0, T ] satisfies the following condition for some positive constants α, β, C: E[|X t − X s |α ] ≤ C|t − s|1+β

∀

s, t ∈ [0, T ],

(2.2)

then almost every sample function of X t is continuous. Proof Now we verify the validity of Proposition 2.2. From the fact (f1) we have, E[(Wt − Ws )4 ] = 3(t − s)2 . Hence we see the continuity of the sample function by virtue of Kolmogorov’s test (Theorem 2.1) cited above. For the verification of the second assertion, we put Vn =

    n     W i + 1 − W i  .  n n  i=0

It suffices to show that lim Vn = ∞ P − a.s.

n→∞

Notice that the condition (b1) together with (f1) and (f2) implies the following inequality, n E[e−Vn ] = Πi=0 E[e−|W ( n )−W ( n ) |] = {E[e−|W ( n )| ]}n

  2 n     1  1  1   + W ≤ E 1 − W  n 2 n  n  1 1 ≤ 1− √ + −→ 0, (as n → ∞). 2n n i+1

i

1

Hence we see that limn→∞ Vn = ∞ almost surely and this implies the conclusion.  The properties (f7), (f8) below concern the regularity of sample paths of Brownian motion. The proofs can be found in every standard textbook (cf. [15]) and are omitted here for the sake of making the content of this chapter as compact as possible. (f7) Almost every sample path of the BM is not differentiable at almost every t ∈ [0, T ]. (f8) As for the modulus of continuity of W , we have the following result due to P. Lévy (cf. [15]):

16

2 Preliminary – Causal Calculus

⎡

⎤

⎢ ⎥ ⎢ ⎥ |W (t) − W (s)| ⎢  lim sup P⎢ = 1⎥ ⎥ = 1. ⎣0 ≤ s ≤ t ≤ 1 ⎦ 2h log h1 h =t −s ↓0

(2.3)

2.1.2 Construction of BM We would like to finish this section with a note on the existence of the Brownian motion, since we could find there a basic idea that leads us to the noncausal stochastic integral. We have already mentioned in (f5) how the BM is constructed as a Markov process. Here we shall show different ways for the construction. 1. Construction by a Fourier series. Let {ϕn (t)} be an orthonormal basis in L 2 (0, 1) and let {Ξn (ω)} be an i.i.d. family of random variables following the standard normal law N (0, 1). Given these, consider a sequence {X n (t, ω)}n of random functions defined in the following way:  t n  Ξn (ω) ϕk (s)ds. X n (t, ω) := 0

k=1

Notice that by Perseval’s equality we have 2 ∞  t     ϕk (s)ds  = ||1[0,t] (·)||2 2 = t, L   k=1

0

and notice that this convergence is uniform in t ∈ [0, 1]. Hence,  lim

1

m,n→∞ 0

  E |X n (t) − X m (t)|2 dt =



1

n 

0 k=m+1



t

|

ϕk (s)ds|2 dt = 0.

0

In other words the sequence {X n (t, ω)}n converges in L 2 ([0, 1] × Ω) to a limit, say X (t, ω). We see that E[X (t)] = limn E[X n (t)] = 0 and that Cov(X (s), X (t)) = lim Cov(X n (s), X n (t)) n→∞  t n  s  = lim ϕk(r )dr ϕk(r )dr = (1[0,s] (·), 1[0,t] (·)) L 2 n→∞

k=1

0

0

= s ∧ t, which shows that the limit X is a Brownian motion.

2.1 Brownian Motion

17

2. As for the convergence of the series X (·) we must refer to a much more general result due to K. Itô and M. Nisio: Theorem 2.2 (Itô and Nisio [12]) For an arbitrary orthonormal basis {ϕn (t)} in L 2 (0, 1) the series 

X (t.ω) =



t

Ξn ϕ˜n (t), where ϕ˜n (t) =

ϕn (s)ds

(2.4)

0

n

converges uniformly in t over [0, 1] with probability one. For the proof we would refer the reader to the article [12] cited above. 3. Instead we would like to show the result due to Ciesielski [1] which deals with the series (2.4) for a special basis and can be verified in an elementary way: Let {Hn,i , 0 ≤ i ≤ 2n−1 − 1, n ∈ N ∪ {0}} be the orthonormal system of Haar functions, namely H0,0 (t) = 1, t ∈ [0, 1], Hn,i (t) = 2

n−1 2

{1[2−n+1 i,2−n+1 (i+1/2)) (t) − 1[2−n+1 (i+1/2),2−n+1 (i+1)) (t)}, (2.5)

n ≥ 1, 0 ≤ i ≤ 2n−1 − 1, where 1 A (·) is the indicator function of set A. Given this we take a family of independent and identically distributed N (0, 1) random variables {Ξ0,0 , Ξn,i ; 0 ≤ i ≤ 2n − 1, n ∈ N} and consider the random series as follows: n ∞ 2 −1  Ξn,i H˜ n,i (t), (2.6) X (t) = Ξ0,0 t + n=1 i=0

where H˜ n,i (t) =



t

Hn,i (s)ds, t ∈ [0, 1].

0

Proposition 2.3 (Ciesielski [1]) The series X (t, ω) converges uniformly in t over [0, 1] with probability one and the sum X is a Brownian motion: ⎡

n 2 −1  k

P ⎣ lim

sup |

m,n→∞ t∈[0,1]

⎤ Ξk,i H˜ k,i (t)| = 0⎦ = 1.

k=m i=0

Proof Sketch of the proof: ∞  Yn (t), where Put X (t, ω) := Ξ0,0 t + n=1

Yn (t, ω) =

n 2 −1

i=0

Ξn,i H˜ n,i (t), n ≥ 1.

18

2 Preliminary – Causal Calculus

We are going to show that the series

∞ 

Yn (t, ω) converges uniformly in t ∈ [0, 1]

n=1

with probability one. t The functions H˜ n,k (t) = 0 Hn.k (s)ds are just the functions called the Schauder n+1 basis, each of which has an equi-lateral triangular shape with height 2− 2 . Also we notice that (2.7) H˜ n,k (t) H˜ n, j (t) = 0 whenever k = j. Then by the property (2.7) we have the following estimate: 2n −1      Ξn,i (ω) H˜ n,i (t) ≤ max |Ξn,i |2−(n+1)/2 . sup |Yn (t)| =  i   t∈[0,1] i=0

Hence for an arbitrary positive α we get the inequality below:  P{sup |Yn (t)| ≥ α 2−n log 2n } t  ≤ P{max |Ξn,i |2−(n+1)/2 ≥ α 2−n log 2n } i   ≤ P{ maxn |Ξn.i | ≥ α 2 log 2n } ≤ 2n P{|Ξ0,0 | ≥ α 2 log 2n } 0≤i≤2 −1



≤ 2n 2

∞

e−x /2 d x. √ 2π 2

√ α 2n log 2

By the elementary inequality 

∞ A

e−x

2

/2

≤

1 −A2 /2 , e A

we get the following estimate, 

P{sup |Yn (t)| ≥ α 2−n log 2n } ≤ t

Since

 n

2 √1 2(1−α )n n

 P

 1 2 2 · √ 2(1−α )n . 2π log 2 n

< ∞ when α > 1, we see by Borel–Cantelli’s first lemma that

 sup |Yn (t)| < α 2−n n log 2 for all large enough n = 1.

t∈[0,1]

Since

 n

2−n n log 2 < ∞, we confirm that the series X (t, ω) converges uniformly

in t ∈ [0, 1] with probability one. It is immediate to see that the process X (t, ω) is Gaussian and that.,

2.1 Brownian Motion

19

E[X (t)] = 0, Cov(X (s), X (t)) = s∧t, 

hence we get the conclusion.

2.2 Itô Integral with Respect to BM Let Wt (ω) or W (t, ω), (t ≥ 1) be Brownian motion defined on a probability space (Ω, F , {FtW }, P), where {FtW } is the natural filtration mentioned in Definition 2.4. We review in this section how the Itô integral with respect to Brownian motion, the first stochastic integral, f (t, ω)d0 Wt is introduced for a certain class of random functions.

2.2.1 Classes of Random Functions We need to introduce some classes of random functions. First of all, by random function we understand in this book a real or complex-valued function f (t, ω) which is defined on the complete measure spaces (R1 × Ω, dt × d P) and is measurable in (t, ω) with respect to the product σ -field B R+ × F , where B R+ is the Borel field on R+ = [0, ∞). For the simplicity of argument and notations we restrict ourselves to the case of random functions f (t, ω) defined on the unit interval t ∈ [0, 1], but depending on the subject this restriction will be changed in a customary way to a case of random functions defined on a larger interval like functions on a finite interval [0, T ] or on R+ . Here is the list of symbols for classes of random functions which will be in frequent use throughout the book. • H: The totality of such random functions f (t, ω) that verify the condition  P

1

 | f (t, ω)|2 dt < ∞ = 1.

0

• M: Set of all such random functions f (t, ω) ∈ H that are adapted to the filtration {FtW }t>0 and, more precisely, are progressively measurable in (t, ω) with respect to the product field B[0,t] × FtW . We will call this constraint on the measurability of random functions the causality condition and call the random function of this class causal. • Note: When we say that a random function f (t, ω) is noncausal, it means that the function is not assured to be causal, in other words it simply means that f ∈ H. This may be an abuse of the word noncausal: nevertheless the word has been in use since the beginning of the theory, so also in this book we would like to follow this custom and hope that the reader will not be confused.

20

2 Preliminary – Causal Calculus

• M2 : The set of all causal random functions f (t, ω) that satisfy the condition 1 E[ 0 | f (t, ω)|2 dt] < ∞, i.e. M2 = M ∩ L 2 ([0, 1] × Ω, dt × d P). • M2c : The subset of M2 , consisting of all elements that are uniformly continuous in the mean sense, namely limh→0 supt E[| f (t + h) − f (t)|2 ] = 0. • S0 : The totality of such a random function f (t, ω) whose sample path is almost surely a step function in t ∈ [0, 1], that is, there exists a finite partition {0 = t0 < t1 < · · · < tn = 1} of [0, 1] and random variables { f i (ω), 0 ≤ i ≤ n − 1} such that (2.8) f (t, ω) = f i (ω), t ∈ [ti , ti+1 ), i = 0, . . . , n − 1. The random function of this class is called simple. • S: The set of all causal simple functions, that is, S = S0 ∩ M. Being “simple and causal” is equivalent to the fact that each random variable fi (ω) (i = 0, . . . , n − 1) . in the form (2.8) is measurable with respect to the σ -field FtW i • S2 : The set of all simple and causal random functions which are square integrable in (t, ω), namely S2 = S0 ∩ M2 . Itô’s stochastic integral of a causal random function f (t, ω) ∈ M with respect to Brownian motion is introduced step by step in the following way.

2.2.2 Itô Integral for f ∈ S Let f (t, ω) be an S-class random function. By definition of the class, there exists a partition {0 = t0 < t1 < · · · < tn = 1} of [0, 1] and a family of random variables { f i (ω), 0 ≤ i ≤ n − 1} such that f (t, ω) =

n−1 

f i (ω)1[ti ,ti+1 ) (t), t ∈ [0, 1],

(2.9)

i=0

Notice that each f i (ω) = f (ti , ω) is FtW measurable. i Definition 2.5

For a causal simple function f (t, ω) of the form (2.9) we put I ( f ) :=

n−1 

f i (ω)Δi W, Δi W = W (ti+1 ) − W (ti ).

i=0

We call I ( f ) the Itô integral of f (∈ S) with respect to Brownian motion and 1 denote it by 0 f (t, ω) d0 Wt . Here we notice that the representation form (2.9) of a simple function is not unique, indeed it can be represented along a different partition, but the above definition of the integral I ( f ) for f ∈ S does not depend on those representation forms.

2.2 Itô Integral with Respect to BM

21

For a sub-interval [a, b] ⊂ [0, 1], it is clear that the function 1[a,b] (·) f (·, ω) belongs to the class S. Hence the stochastic integral on the sub-interval [a, b] is well-defined in the following form: 

b

f (t.ω)d0 Wt := I (1[a,b] (·) f (·)).

a

We will denote the integral I (1[0,t] f ) also by It ( f ). The stochastic integral I ( f ) defines an application from S to L 0 (Ω, d P). Proposition 2.4 The integral I ( f ) (defined on S) has the following properties: (1)

Linearity: The application I ( f ) is linear, that is, for any functions f, g ∈ S and constants α, β the following equality holds: I (α f + βg) = α I ( f ) + β I (g).

(2)

Isometry: For a causal and square integrable simple function, f (t, ω) ∈ S2 , we have E[I ( f )] = 0 and 

1

E[|I ( f )| ] = E[ 2

0

| f (t, ω)|2 dt] = || f ||2L 2 ([0,1]×Ω) .

In other words, the Itô integral defines an isometry from S2 (⊂ L 2 ([0, 1] × Ω, dt × d P)) to L 2 (Ω, d P). Proof Property (1) is evident. As for the second equality in (2), we have ⎡ ⎤   E[|I ( f )|2 ] = E ⎣ { f i f j + f i f j }Δi W Δ j W + | f k |2 (Δk W )2 ⎦ =





i> j

E { f i f j + f i f j }Δ j W · E{Δi W |FtW } + i

i> j

=





 E[| f k |2 ](tk+1 − tk ) = E

k

k



  E | f k |2 E{(Δk W )2 |FtW } k

k 1

| f (t, ω)|2 dt,

0

and this implies the conclusion.



Proposition 2.5 Let f (t, ω) be a causal simple function. Then the stochastic process It ( f ) := I (1[0,t] (·) f (·)) has the following properties: (a) Almost all sample functions of It ( f ) are continuous in t. (b) The process It ( f ) is an FtW - martingale. t (c) When f ∈ S is real, the process Z t := exp{It ( f ) − 21 0 f 2 (s)ds} is a continuous FtW martingale and E[Z t ] = 1 ∀ t.

22

2 Preliminary – Causal Calculus

Proof (a) Let Δ be a partition associated to the simple function f (t, ω). Fix a t ∈ (0, 1] and denote by [tk , tk+1 ) the sub-interval that contains the t, then It ( f ) =

k−1 

f i (ω)Δi W + f k (ω){W (t) − W (tk )}

i=0

which shows the continuity in t of the path of process It ( f ). (b) Let s, t ∈ [0, 1] be such that t ≥ ti ≥ t j ≥ s ≥ t j−1 . Then we have i−1 

It ( f ) =

f k · Δk W + f i (Wt − Wti )

k=0 j−2 

=

f k · Δk W + f j−1 (Ws − Wt j−1 ) + f j−1 (Wt j − Ws )

k=0

+

i 

f k Δk W + f i (Wt − Wt j ),

k= j

Hence we get E[It ( f )|FsW ] =

j−2 

f k · Δk W + f j−1 (Ws − Wt j−1 )

k=0

⎡

+ E ⎣ f j−1 (Wt j − Ws ) +

i 

⎤  W f k · Δk W + f i (Wt − Wt j ) Fs ⎦

k= j



W = Is ( f ) + E f j−1 · E[Wt j − Ws |F j−1 ]

+

i 

⎤  f k E[Δk |FkW ] + f i E[Wt − Wt j |F jW ] FsW ⎦

k= j

= Is ( f ) P − a.s. (c) Let s, t be such that t ≥ ti ≥ s ≥ ti−1 . We have It ( f ) = Is ( f ) + f i−1 (Wti − Ws ) + f i (Wt − Wti ), and 



t



s

t

f (r )dr = f (r )dr + f 2 (r )dr 0 0 s  s 2 2 = f (r )dr + f i−1 (ti − s) + f i2 (t − ti ). 2

0

2

2.2 Itô Integral with Respect to BM

23

Combining these we get the following equality: Zt = Zs



× exp

 1 2 1 2 f i−1 (Wti − Ws ) − f i−1 (ti − s) + f i (Wt − Wti ) − f i (t − ti ) . 2 2

Hence  1 2 = Z s × E exp{ f i−1 (Wti − Ws ) − f i−1 (ti − s)} 2  1 ]FsW . ×E[exp{ f i (Wt − Wti ) − f i2 (t − ti )}|FtW i 2

E[Z t |FsW ]

Since E[exp{ f i (Wt − Wti ) −

1 2 f (t − ti )}|FtW ] = 1 P − a.s. i 2 i

we see that  1 2 (ti − s)|FsW E[Z t |FsW ] = Z s E exp{ f i−1 (Wti − Ws ) − f i−1 2 = Z s P − a.s. 

This completes the proof of (c). From property (c) in Proposition 2.5, we get the following result:

Proposition 2.6 Let f be a real and causal simple function. Then for any positive constants a, b, we have the following inequality:   a t 2 f (s, ω)ds} > b ≤ e−ab . P sup{It ( f ) − 2 0 t

(2.10)

Proof For a real f ∈ S, we put 1 Z t ( f ) := exp{It ( f ) − 2



t

f 2 (s, ω)ds}.

0

By property (c) we know that Z t (a f ) is an FtW -martingale. For the left-hand side of the inequality in (2.10) we have the following expression:

24

2 Preliminary – Causal Calculus

  a t 2 P sup{It ( f ) − f (s, ω)ds} > b 2 0 t   a2 t 2 f (s, ω)ds} > ab = P sup{It (a f ) − 2 0 t  = P sup Z t (a f ) > eab t

Hence by applying the submartingale inequality (Corollary 10.1 in Chap. 9) to the last term in the above inequality, we get   a t 2 P sup{It ( f ) − f (s, ω)ds} > b ≤ e−ab E[Z 1 (a f )] = e−ab . 2 0 t 

This completes the proof.

2.2.3 Extension to f ∈ M We have introduced the stochastic integral I ( f ) for the causal simple functions f ∈ S. We show that the domain of the integral is extended to the class M; the integral I ( f ) thus introduced for f ∈ M we call the Itô integral and denote by  f (t, ω)d0 Wt . There are many ways to achieve this aim. Here we basically follow the argument by H. McKean [15] which makes use of the exponential martingale Z t ( f ) ( f ∈ S). Let us begin with the following lemma, the verification of which may be found in every standard textbook on calculus and is left to the reader. Lemma 2.1 Let f (t) be a deterministic function which is square integrable over [0, 1]. We suppose that the function is extended over a larger interval in such way that; f (t) = 0 outside of [0, 1], then we have the following equality:  lim

h→0 0

1

| f (t + h) − f (t)|2 dt = 0.

The next statement plays a key rôle in the discussion. The proof will be given in the Appendices (see Chap. 10). Proposition 2.7 The class S is dense in M, that is: for any f ∈ M there exists a sequence { f n } in S such that  lim

n→∞ 0

1

| f (t, ω) − f n (t, ω)|2 dt = 0

P-a.s.

2.2 Itô Integral with Respect to BM

25

Now given a causal function f ∈ M and a positive α > 1, we consider a sequence of causal simple functions { f n } ∈ S such that 

1

 1 | f (t, ω) − f n (t, ω)| dt ≤ 2α for any large enough n = 1. 2n 2

P 0

(2.11)

Notice that the existence of such a sequence is assured by Proposition 2.7, and that the following equality holds: 

1

 2 | f n (t, ω) − f n−1 (t, ω)| dt ≤ 2α for all large enough n = 1. n 2

P 0

(2.12)

Let us consider the sequence of random functions {It ( f n )} defined by the integrals t It ( f n ) = 0 f n (s, ω)d0 Ws , then we have the following statement. Proposition 2.8 For the sequence { f n } ∈ S satisfying the condition (2.11) the sequence {It ( f n )} converges uniformly in t over [0, 1] with probability one, that is,  P

lim

sup |It ( f n ) − It ( f m )| = 0 = 1.

m,n→∞ t∈[0,1]

Proof We put gn = f n − f n−1 . For any fixed positive constants a, b, we have from Proposition 2.6 the following estimate:   a t 2 gn (s, ω)ds} > b ≤ e−ab . P sup{It (gn ) − 2 0 t Now fix another constant c > 1 and choose a, b in the above inequality in the following way:  c  a = n α log n, b = α log n, n then we get the following estimate:  P

a 2

sup {It (gn ) − t∈[0,1]

Since

 n

1 nc





t 0

1 gn (s)2 ds} > b ≤ e−c log n = c . n

< ∞, by Borel–Cantelli’s first lemma we get the following equality:

a P sup {It (gn ) − 2 t∈[0,1]



t

gn (s) ds} ≤ b for large enough n = 1, 2

0

which together with the estimate (2.12) implies that

26

2 Preliminary – Causal Calculus



 a 1 2 1 = P sup It (gn ) ≤ b + g (s)ds for large enough n 2 0 n t∈[0,1] √  2 log n for all large enough n . ≤ P sup It (gn ) ≤ nα t Repeating the same argument with −gn instead of the gn , we find that √  2 log n for all large enough n = 1. P sup |It (gn )| ≤ nα t Since

 √log n n

< ∞ and since It (gn ) = It ( f n ) − It ( f n−1 ), we confirm that the

nα

sequence {It ( f n )} almost surely converges in uniform topology. Hence we are done.  We remark at this stage that the limn I ( f n ) does not depend on the choice of the approximating sequence { f n }(∈ S), hence we arrive at the following definition of the Itô integral. Definition 2.6 (Itô integral) For the sequence {I ( f n )} constructed in Proposition 2.8 we call the limn I ( f n ) the Itô integral of the causal random function f ∈ M and 1 denote it by 0 f (t, ω)d0 Wt . Remark 2.1 (Integration over a general interval) It is only for the simplicity of the argument that we have limited our discussion to the integration over the interval [0, 1]. We clearly see that our argument works for the case of any finite interval [0, T ] T < ∞. Extension to the integration over the infinite interval [0, ∞) can be carried out in a similar way. ∞ 2 For a causal random function f (t, ω) such that 0 | f (t, ω)| dt < ∞ P-a.s. Since ∞ lim A→∞ A | f (t, ω)|2 dt = 0 (P-a.s.), we choose an increasing sequence of real numbers {cn } in the following way: 

∞

P cn

 1 | f (t, ω)| dt < 2α for large enough n = 1, 2n 2

and we put  c f n (t, ω) = 1[0,cn ] (t) f (t, ω). Then we see that for each n the Itô integral I ( f n ) = 0 n f (t, ω)d0 Wt is well-defined and that 

∞

P 0

| f (t, ω) − f n (t, ω)|2 dt
0, 0 | f (s, ω)|2 ds ≤ A}. Then I{t∧τ } ( f ) is an FtW -martingale. Proof The property (I–1) is evident from definition of the integral. For the verification of the first part of (I–2), we fix an f (t, ω) in M2 and take a sequence { f n } in S that converges to f in M2 , namely, 

1

lim E

n→∞

| f (t, ω) − f n (t, ω)|2 dt = 0.

0

We know from (2) in Proposition 2.4 that E[I ( f n )] = 0andE[|I ( f n )|2 ] = || f n ||2L 2 ([0, 1] × Ω). Letting n → ∞ on both sides of these equalities we see that E[I ( f )] = 0, E[|I ( f )|2 ] = || f ||2L 2 ([0,1]×Ω) . Now suppose that f ∈ M and { f n } ∈ S is such that  lim

n→∞ 0

1

| f (t) − f n (t)|2 dt = 0 P-a.s.

28

2 Preliminary – Causal Calculus

Then we have E[|I ( f )|2 ] = E[lim inf |I ( f n )|2 ] ≤ lim inf E[|I ( f n )|2 ] n→∞ n→∞  1  1  1 2 2 = lim E[ | f n | dt] ≤ E[ lim | f n | dt] = E[ | f |2 dt]. n→∞

0

n→∞ 0

0

Hence we see the validity of property (I–2). The property (I–3) being also immediate from the argument given in the proof of Proposition 2.8, we are going to verify the property (I–4). 1 (i) Let f ∈ M2 , then we can choose a sequence { f n } in S2 such that E[ 0 | f (t, ω) − f n (t, ω)|2 dt] = 0. For instance we may take   k k+1 k for t ∈ , . f n (t, ω) = f (ηn (t), ω) where ηn (t) = n n n For each f n the It ( f n ) being an FtW -martingale, we know that for any t ≥ s ≥ 0, the following holds: E[It ( f n )|FsW ] = Is ( f n )

P-a.s.

(2.13)

Fix an FsW -measurable random variable X in an arbitrary way, then we have from (2.13) the equality E[X · It ( f n )] = E[X · E{It ( f n )|FsW }] = E[X · Is ( f n )]. Letting n → ∞ on both sides of the above equality, we get the equality E[X · It ( f )] = E[X · Is ( f )], which states that the It ( f ) is an FtW -martingale. (ii) Notice that τ A (ω) is an FtW -stopping time and that the function 1{t≤τ A } (ω) f (t, ω) belongs to the class M2 . Since It∧τ A ( f ) = It (1{· ≤τ A } · f ) we confirm the validity of the statement. 

2.2.4 Linearity in Strong Sense We have seen how the Itô integral is introduced for causal functions in H and we have listed some of its basic properties. In the following sections we are going to show how the stochastic calculus based on the Itô integral works. Before that we like to close this section with the note on a remarkable character of the Itô integral, which is often missed in standard textbooks. • It is said that the Itô integral is a Riemann-type integral in the sense that the Itô integral can be defined as a limit of Riemann sums as follows:

2.2 Itô Integral with Respect to BM

29

Given a finite partition Δ = {0 = t0 < t1 < t2 < · · · < ti < · · · tn = 1} on the interval, that is [0, 1] in our present set-up, we consider the sum RΔ ( f ) :=



f (ti , ω)Δi W, where Δi W = W (ti+1 ) − W (ti ).

ti ∈Δ

1 and say that 0 f (t, ω)d0 Wt = lim|Δ|→0 RΔ ( f ), where |Δ| = max{ti+1 − ti ; ti ∈ Δ}. By looking at the proof of Proposition 2.7 we are sure that this is true at least for any f ∈ Mc . Remember that any f ∈ Mc is approximated by the sequence of causal simple functions f Δ as follows: f Δ (t, ω) = f (ηΔ (t), ω), where ηΔ (t) =



tk 1[tk ,tk+1 ) (t).

k

We are also sure that  1

f (t, ω)d0 Wt = lim RΔ ( f ) in the mean |Δ|→0

0

when f ∈ M2,c , i.e. when f is causal and continuous in the mean, lim sup E| f (t + h) − f (t)|2 = 0.

h→0 t∈[0,1]

In fact, we have I ( f ) − RΔ ( f ) =

n−1   i=0

ti+1

{ f (t) − f (ti )}d0 Wt ,

ti

hence lim E[|I ( f ) − RΔ ( f )|2 ] = lim

|Δ|→0

|Δ|→0

n   i=0

ti+1

E[| f (t) − f (ti )|2 ]dt = 0.

ti

Itô calculus was introduced for the theory of SDEs (stochastic differential equations) whose solutions in most cases belong to those classes cited above, hence the slogan saying “the Itô integral is a Riemann integral” has caused no problem. • Anyhow, the fact that the Itô integral I ( f ) for f ∈ M is defined along the sequence of approximate causal simple functions is important. Let f (t, ω) be a function in M and let α(ω) be an arbitrary random variable. Then we see that the function g(t, ω) := α(ω) f (t, ω) is no more causal and is excluded from the class M. But it does not prevent us from defining the Itô integral of a noncausal function of this particular form. In fact, let { f n } be a sequence of causal simple functions that

30

2 Preliminary – Causal Calculus

converges to f (t, ω), and let gn (t, ω) := α(ω) f n (t, ω). Then gn (t, ω) is simple but not causal, but still the sequence {gn } converges to g(t, ω) in H P-a.s. By the definition of the Itô integral for simple functions we have I (gn ) = α(ω)I ( f n ), hence we see that {I (gn )} converges to α(ω)I ( f ), namely we confirm that g(t, ω) is Itô integrable. • Generally speaking, once an integral is defined for functions in some subspace D of a function space, it defines at the same time an application from that domain D to a set in a metric space. In the case of the Itô integral the application I ( f ) is a mapping from M to L 0 (Ω, d P). On the other hand, in analysis it is widely believed that any application induced by an integral is linear. This is true for all stochastic integrals that we treat in this book, but the mappings, say T ( f ) for the moment, induced by the Itô integral or by the noncausal integral of the author (see Chap. 3) exhibit more strong linearity as follows: T (α(ω) f + β(ω)g) = α(ω)T ( f ) + β(ω)T (g), for any random variables α(ω), β(ω). This property we like to call the strong linearity.

2.2.5 Itô Formula Let X t be a stochastic process that accepts the following representation:  X t = ξ(ω) + 0

t



t

b(s, ω)ds +

a(s, ω)d0 Ws

(2.14)

0

where ξ(ω) is a random variable independent of the Brownian motion W., and a(·), b(·) are causal functions. Definition 2.7 Every causal stochastic process X t of the form (2.14) is called an Itô process and the totality of all such processes will be denoted by M I . In other words the Itô process is a special type of the Brownian semi-martingale. The expression (2.14) is also denoted by the following differential form: d X t = b(t, ω)dt + a(t, ω)d0 Wt , X 0 = ξ(ω). Remark 2.2 As a variant of this, the process of the following form (2.15) is called a quasi martingale: (2.15) d X t = d B(t, ω) + a(t, ω)d0 Wt ,

2.2 Itô Integral with Respect to BM

31

when a(t, ω) ∈ M and b(t, ω) is a process, causal or not, almost every sample of which is of bounded variation on [0, 1]. We will denote the totality of quasimartingales by M Q . The semi-martingale is the name for a causal quasi-martingale. Let X t be an Itô process and let F(t, x) be a real-valued function that is defined on R+ × R1 and is of C 1 -class in t and of C 3 -class in x with bounded derivatives. Theorem 2.3 (Itô formula) The stochastic process F(t, X t ) satisfies the following equality: d F(t, X t )   1 = Ft (t, X t ) + Fx (t, X t )b(t, ω) + Fxx (t, X t )b2 (t, ω) dt + Fx (t, X t )a(t, ω)d0 Wt , 2 where Ft = ∂t F, Fx = ∂x F, Fxx = ∂x2 F.

(2.16)

This equality is called the Itô formula. Proof Fix t and set tkn = nk t, k = 0, . . . , n. Then by the mean value theorem in calculus we have the following equality: F(t, X t ) − F(0, X 0 ) =

n−1  n n {F(tk+1 , X (tk+1 )) − F(tkn , X (tkn ))} k=0

=

n−1 

n n n n {F(tk+1 , X (tk+1 )) − F(tkn , X (tk+1 )) + F(tkn , X (tk+1 )) − F(tkn , X (tkn ))}

k=0

=

n−1  

n Ft (tkn + θk1 Δnk , X (tk+1 ))Δnk + Fx (tkn , X (tkn ))Δnk X

k=0

+

1  n F (t , X (tkn ) + θk2 Δnk X )(Δnk X )2 }, 2 xx k

where θk1 , θk2 (k = 0, . . . , n − 1) are constants in (0, 1) and n n − tkn , Δnk X = X (tk+1 ) − X (tkn ). Δnk = tk+1

By virtue of the smoothness of Ft , Fx and the continuity of the sample of X (t, ω), we see that lim

n→∞

n−1  k=0

Ft (tkn

 +

θk1 Δnk ,

n X (tk+1 ))Δnk

= 0

t

Ft (s, X (s))ds, P-a.s.

32

2 Preliminary – Causal Calculus

and n−1  1 {Fx (tkn , X (tkn ))Δnk X + Fxx (tkn , X (tkn ) + θk2 Δnk X )(Δnk X )2 } n→∞ 2 k=0    t 1   2 = Fx (s, X (s)) b(s, ω)ds + a(s, X (s))d0 Ws + Fx x (s, X (s))b (s)ds , 2 0 P-a.s.

lim



This completes the proof.

We know that Itô formula (2.16) is valid for a function F(t, x) with a less restrictive condition on regularity, but we do not enter into the details since we need not do so for our discussion in later chapters. Example 2.2 Applying the formula (2.16) to the case where  F(x) = e x , X t =

t

f (s, ω)d0 Wt −

0

1 2



t

f 2 (s, ω)ds,

0

we get the next equality for Z t = exp{X t },  d Zt = Zt

 1 2 1 f · d0 Wt − f dt + Z t f 2 dt = Z t f d0 Wt 2 2

from which we see that Z t is an FtW -martingale. The extension of the result (Theorem 2.3) to the case of multi-dimensional Itô process can be given as follows. p Let Xt =t (X t1 , X t2 , . . . , X t ) be a p-dimensional stochastic process, each comi ponent X t of which is generated by the following rule: d X ti = bi (t, ω)dt +

q 

aki (t, ω)d0 Wtk , 1 ≤ i ≤ p,

(2.17)

k=1

where a ij , bi (1 ≤ i ≤ p, 1 ≤ j ≤ q) are causal random functions and Wt =t (W 1 , W 2 , . . . , W q ) is the q-dimensional Brownian motion, namely the {W 1 , W 2 , . . . , W q } are independent Brownian motions. The Itô formula is extended in the following way. Theorem 2.4 (Itô formula 2) Let F(t, x) (t ∈ R+ , x =t (x1 , x2 , . . . , x p ) ∈ R p ) be a smooth function that is of C 1 -class in t and of C 3 -class in x with bounded derivatives. Then for the p-dimensional Itô process Xt given in (2.17) above we have the following equality: d F(t, Xt ) = L 1 F(t, Xt )dt + L 2 F(t, X t )d0 Wt ,

(2.18)

2.2 Itô Integral with Respect to BM

where L1 =

33

p p m  ∂ ∂ 1   i j ∂2 + bi (t, ω) + a a , ∂t ∂ xi 2 i, j=1 k=1 k k ∂ xi ∂ x j i=1

and L 2 F(t, X)d0 Wt =

p m  

bik (t, ω)

k=1 i=1

∂ F(t, X)d0 Wtk . ∂ xi

The demonstration of this statement is left to the reader. Example 2.3 Apply the Itôformula (2.18) to the case where F(x, y) = x y, X t = t t f (s, ω)d 0 Ws and Yt = 0 g(s, ω)d0 Ws where f, g ∈ M, then we find the fol0 lowing equality: 

t



t

f (s, ω)d0 Ws

g(s, ω)d0 Ws  t  t  s  s = f (s, ω)d0 Ws g(r.ω)d0 Wr + g(s, ω)d0 Ws f (r, ω)d0 Wr 0 0 0 0  t f (s, ω)g(s.ω)ds. + 0

0

(2.19)

0

2.2.6 About the Martingale Zt (α) Because of the remarkable property of the causal function 

1 Z t = exp It ( f ) − 2



t

 f (s, ω)ds

( f ∈ M),

2

0

we try to look at it from a different viewpoint. Let f ∈ M be a real causal function such that    E exp α 2



1

f 2 dt

0, x ∈ R} be the family of Hermite polynomials each element of which is defined by the following formula: h n (t, x) :=

 2  2 n ∂ (−1)n x x exp − exp n! 2t ∂ x n 2t

(n ≥ 0).

By this definition we see that the family of Hermite polynomials {h n (t, x)} has 2 exp{αx − α2 t } as its generating function, namely    ∞ α2 t exp αx − = α n h n (t, x). 2 n=0

(2.22)

t t Now by substituting 0 f (s, ω)d0 Ws and τ (t) = 0 f 2 (s, ω)ds for x and t respectively in the equality (2.22) we get the following equality:

2.2 Itô Integral with Respect to BM

35

  Z t (α) = exp α

  α2 t 2 f (s, ω)d0 Ws − f (s, ω)ds 2 0 0    t ∞  = α n h n τ (t), f (s, ω)d0 Ws . t

0

n=0

Comparing this with the equality (2.21), we find the following expressions: Lemma 2.3 For a real f ∈ M it holds that 

 sn−1  s1 f (s1 , ω)d0 Ws1 f (s2 , ω)d0 Ws2 . . . f (sn , ω)d0 Wsn 0 0 0    t = h n τ (t), f (s, ω)d0 Ws .

Z n (t) =

t

0

Hermite polynomial h n (t, x) is a polynomial of x of degree n which is explicitly computed by its definition. Therefore we find the formula mentioned Lemma 2.3 useful in n! t . to compute or estimate the n-th moment of the Itô integral E 0 f (s, ω)d0 Ws For instance: Lemma 2.4 For an f ∈ M ∩ L 4 the following estimate holds:

 E

t

4 f (s, ω)d0 Ws

 ≤ 36E

t

2 f (s, ω)ds

.

2

0

0

Proof We know that h 4 (t, x) = x 4 − 6t x 2 + 3t 2 . Since E[Z 4 (t)] = 0 we have t E[h 4 (τ (t), 0 f (s, ω)d0 Ws )] = 0, hence we find that 

t

E[(

 f d0 Ws )4 ] = E 6

0



t

f 2 ds 0

 ≤ 6E



2 

t

0

2 f 2 (s, ω)ds

0

t

f (s, ω)ds 2

f d0 Ws

"  ≤6 E

t

f d0 Ws )2 − 3(

0

0

From this we find the inequality.

t

0

t

4 #1/2 "  f d0 Ws E

2 #1/2

t 2

f ds

.

0



36

2 Preliminary – Causal Calculus

2.3 Causal Variants of Itô Integral 2.3.1 Symmetric Integrals As we have noticed in the preceding section, the Itô integral  t has such a remarkable f (s, ω)do Ws becomes a property that the process defined by the Itô integral X t = martingale and this fact is granted by the causality condition on the integrand f (t, ω) and by the particular form of Riemann sum. Hence any change in these situations would cause the loss of that nice property. Nevertheless we may think of the stochastic integral for the causal function by a Riemann sum of different form as follows: θ ( f ) := RΔ

n−1 

f (ti + θ Δi )Δi W,

(2.23)

i=0

where θ is a constant such that θ ∈ [0, 1]. We are also interested in the following Riemann sums: SΔθ ( f ) :=

n−1 

{ f (ti ) + θ Δi f }Δi W Δi f = f (ti+1 , ω) − f (ti , ω).

(2.24)

i=0 1/2

We may see that the Riemann sum SΔ ( f ) is just a stochastic variant of the trapezoidal formula in classic calculus. We are going to study the convergence of these Riemann sums. For this purpose we need to introduce a kind of regularity of the random function f (t, ω) with respect to the Brownian motion. Let {Δn } be a sequence of partitions Δn = {0 = t0n < · · · < tsn = 1} of [0, 1]. We call the sequence regular provided that Δn ⊂ Δn+1 and lim |Δn | = 0. n→∞

We say that a random function h(t, ω) is B-negligible if it exhibits the following property (W): (W) For each fixed t ∈ [0, 1] and for any regular sequence of partitions {Δn } it holds that lim

n→∞



(Δi h) · (Δi W ) = 0 (in P), where Δn (t) := Δn ∪ {t}.

ti ∈Δn (t)

In particular we call h(t, ω) strongly B-negligible if it satisfies the following condition (WS): (WS) For each fixed t ∈ [0, 1] and for any regular sequence of partitions {Δn } it holds that

2.3 Causal Variants of Itô Integral

lim

n→∞



37

|Δi h · Δi W | = 0 (in P), where Δn (t) := Δn ∪ {t}.

ti ∈Δn (t)

Example 2.4 A random function h(t, ω) that satisfies the following condition, for any regular sequence of partitions {Δn } of [0, 1], is strongly B-negligible: lim

n→0



|Δi h|2 = 0 in P.

ti ∈Δn

Thus any function, almost every sample of which is of bounded variation, is strongly B-negligible. Definition 2.8 (B-derivative) A random function f (t, ω) ∈ H is called B-differentiable (or strongly B-differentiable) if there  texists a causal function g(t, ω) ∈ M such that the function h(t, ω) = f (t, ω) − 0 g(s, ω)d0 Ws is B-negligible (or strongly B-neligible respectively). In this case the function g(t, ω) we call the ∂ f (t, ω). B-derivative of f (t, ω) and denote it by the symbol fˆ(t, ω) or by ∂ W t We see the uniqueness of the B-derivative in the following statement whose proof will be given in the last chapter “Appendices 2”: Proposition 2.10 The B-derivative of a B-differentiable function is uniquely determined. Remark 2.3 (B-differentiability [21, 43]) The notion of B-differentiability was first introduced in the study of the symmetric integral I1/2 and BPE (cf. [20–22]) where all random functions f (t, ω) are supposed to be causal. That was given in the following way. A causal random function f ∈ M is called B-differentiable (or differentiable with respect to Brownian motion) if there exists a causal random function, say fˆ(t, ω), that satisfies the following condition: lim sup

h↓0 0≤t≤1−h

 2  t+h   1  E  f (t + h) − f (t) − fˆ(s)d0 Ws  = 0. h t

It may be immediate to see that new definition of B-differentiablity is a refinement of this classic one. Example 2.5 Let X t be a random function of the form  X t = B(t, ω) +

t

a(s, ω)d0 Ws ,

0

where a(t, ω) is a causal function of the class M2 and B(t, ω) a function which is Hölder continuous in L 2 (Ω) sense, E[|B(t + h) − B(t)|2 ] = O(|h|α ) (1 < α). Then X is B-differentiable and ∂∂WXt = a(t, ω).

38

2 Preliminary – Causal Calculus

In particular, every Itô process X t ; d X t = b(t, ω)dt + a(t.ω)d0 Wt with a(t, ω) ∈ M, is B-differentiable. After the introduction of the notion of B-differentiablity we find it convenient to denote by H1 the totality of all B-differentiable functions causal or not, and by M1 the set of all B-differentiable causal functions, i.e. M1 := H1 ∩ M. We have the following result whose proof is given in Chap. 10 (Appendices–2); Theorem 2.5 ([21]) Let {Δn } be an arbitrary sequence of partitions in [0, 1] such that, Δn ⊂ Δn+1 and limn→∞ ||Δn || = 0. Then for every B-differentiable function θ ( f ), SΔθ n ( f ) f ∈ M1 and a fixed θ ∈ [0, 1], the two sequences of Riemann sums RΔ n 1 converge in probability to the same limit Iθ ( f ) which we also denote by 0 f dθ Wt and call the θ -integral. The integral Iθ ( f ) is related to the Itô integral I0 ( f ) in the following form; 

1

Iθ ( f ) = I0 ( f ) + θ

fˆ(t, ω)dt,

0

∂f . fˆ = ∂ Wt

Among these integrals Iθ , the two I0 and I1/2 are of particular importance, the former is of course the Itô integral and the latter we call the symmetric integral. Example 2.6 (a formula concerning the white noise) Let X t be a causal function defined by the symmetric integral as follows: 

t

Xt = X0 +

f (s, ω)dWs ,

f (t, ω) ∈ M1

0

for some B-differentiable function f (t, ω). Notice that we can verify the validity of the following expression in the sense of L. Schwartz’s distribution: d X t = f (t, ω)W˙ . X˙ t := dt On the other hand, by Theorem 2.5 we see that   t 1 ˆ E[X t ] = E f (s, ω)ds , 2 0

∀

t.

We will often find it convenient to write this fact in the following form: E[ f (t, ·)W˙ ] =

1 E[ fˆ(t, ·)]. 2

(2.25)

The importance of the symmetric integral is simply explained by the following fact. For a semi-martingale X t = at + bWt + c, (a,b,c: consts) and a smooth function F(t, x), we have by the Itô formula

2.3 Causal Variants of Itô Integral

 dt F =

Ft

39

+

a Fx

 b2  + Fx x dt + bFx d0 Wt . 2

With the symmetric integral this equality is expressed in the more simple form dt F = Ft dt + Fx {adt + bdWt } = Ft dt + Fx d X t , ∂ Fx (t, X t ) = Fxx b. In other words, with the symmetric integral since we have ∂ W t the differential formula in classic calculus is conserved. But the application of this property to a more general case must be done with a special care to the notion of the B-derivative as we see below.

2.3.2 Anti-Causal Function and Backward Itô Integral This and the following Sects. 2.3.2 and 2.3.3, treat some special subjects which will be related only to a problem discussed in Chap. 7. Hence an impatient reader can skip these two subsections. Looking back to the definition of the Itô integral I ( f ), f ∈ S, we recognize that the causality condition together with the employment of the special form of Riemann sum  f (ti )Δi W, Δ = {0 ≤ t1 < t2 < · · · < tn ≤ T } RΔ ( f ) := ti ∈Δ

ist essential in endowing the martingale property to the stochastic process It ( f ) = 0 f (s, ω)d0 Ws defined by the Itô integral. A similar result might occur in a retrograde situation as we see below. Let G t := σ {Wv − Wu : t ≤ u ≤ v} and let F t be a decreasing family of σ -fields such that • Ft ⊃ G t, • F t is independent of Gt := σ {Wv − Wu : u ≤ v ≤ t}. The σ -field F t presents the future behaviour of the Brownian motion after time t, and we call the random function f (t, ω) ∈ H anti-causal when it is adapted to the filtration {F t }t . We will denote by M the totality of anti-causal random functions, namely M := { f ∈ H : f (t, ω) is anti-causal}, and by M2 its subset M ∩ L 2 ([0, 1] × Ω, dt × d P). We will also denote by M2,c the subset of M2 consisting of all elements which are continuous in the mean, lim h→0 E[| f (t + h) − f (t)|2 ] = 0. Given an anti-causal function f (t, ω) and a partition Δ = {0 = t0 < t1 < · · · < tn = 1} of [0, 1], we consider a retrograde Riemann sum R Δ ( f ) :=

n−1  i=0

f (ti+1 , ω)Δi W, where Δi W = W (ti+1 ) − W (ti ).

(2.26)

40

2 Preliminary – Causal Calculus

We notice that for a causal function f ∈ M this is just the Riemann sum which leads to the integral I1 ( f ) as |Δ| → 0. But in the present case we consider the sum for an anti-causal function. If we write f Δ (t, ω) := f (ηΔ (t), ω), where ηΔ (t) = ti+1 when t ∈ (ti , ti+1 ], then f Δ (t, ω) is a simple anti-causal function and the retrograde Riemann sum in (2.26) is expressed in the I1 -integral form; R Δ ( f ) = I1 ( f Δ ). The sum has the following property the verification of which is almost immediate and is omitted. Proposition 2.11 When f ∈ M2 , we have E[R Δ ( f )] = 0 and Δ

E[R ( f ) ] = 2

n−1 

 E[ f (ti+1 , ω)](ti+1 − ti ) = E[

1

2

| f Δ (t, ω)|2 dt].

(2.27)

0

i=0

Now let {Δn } be an increasing family of partitions such that, Δn ⊂ Δn+1 (as sets) and limn→∞ |Δn | = 0. Given this and an anti-causal random function f (t, ω) ∈ M2 we put f n (t, ω) := f Δn (t, ω), n ∈ N. Notice that for each n, f n (t, ω) ∈ M2 . We have the following statement. Proposition 2.12 For an f (t, ω) ∈ M2,c , the sequence {I1 ( f n )} converges in the mean sense. Proof By the continuity of f (t, ω) we see that limn→∞  f − f n 2 = 0. On the other hand, by the isometry property (2.27), we find that lim E[|I1 ( f m ) − I1 ( f m )|2 ] = lim  f n − f m 2 = 0.

m,n→∞

m,n→∞



This completes the proof.

Definition 2.9 For an anti-causal function f ∈ M2,c , the limit in the mean limn→∞ I1 ( f n ) of the sequence {I1 ( f n )} in Proposition 2.12, we denote by I1 ( f ) 1 or by 0 f d1 Wt , and call it the backward Itô integral. 1 Remark 2.4 The same symbol 0 f (t, ω)d1 Wt is used for the different cases, namely for the causal or noncausal functions. They are quite different from each other; for the causal function it means the sum of the Itô integral with the additional term,  I1 ( f ) = I0 ( f ) + 0

1

fˆ(t, ω)dt,

2.3 Causal Variants of Itô Integral

41

∂ where fˆ(t, ω) = ∂ W f , while for the anti-causal function f (t, ω) the integral t 1 0 f d1 Wt is just the backward Itô integral. Hence when we see this notation we must be careful on the causality of the integrand.

Let [a, b] be a sub-interval in [0, 1] and Δ = {0 = t0 < · · · < tn = 1} be a partition of [0, 1]. For an anti-causal function f ∈ M2c we observe that g Δ (t, ω) = 1[a,b] (t) f Δ (t, ω) is a simple anti-causal function and its backward Itô integral is well-defined as follows: Δ

I1 (g ) = f (t ){W (t ) − W (a)} +

r −1 

f (ti+1 )Δi W + f (tr +1 ){W (b) − W (tr )},

i=

where t = min{ti ≥ a; ti ∈ Δ}, tr = max{ti ≤ b; ti ∈ Δ}. We see by this formula that   E |I1 (1[a,b] · f Δ )|2 = E



b

| f Δ (t)|2 dt ,

a

consequently the convergence in the mean of the sequence {I1 (1[a,b] · f Δ )} as n → b ∞, the limit we denote by a f (t, ω)d1 Wt . In particular for the case [a, b] = [t, 1] (0 ≤ t ≤ 1) we have 

1

f Δ (s, ω)d1 Ws

t

= f (tn ){W (tn ) − W (tn−1 )} +

n−1 

f (ti )Δi W + f (t ){W (t ) − W (t)},

i=+1

1 by which we see that the function (t, ω) → t f Δ (s, ω)d1 Ws is adapted to the decreasing family of σ -fields {F t }t and that the equality 

1



1

f (r, ω)d1 Wr | F ] = s

E[ t

f (r, ω)d1 Wr , P − a.s.

(2.28)

s

holds for any 0 ≤ t ≤ s. The integral I1 ( f ) having been defined as the limit in the mean of the sequence 1 ( f )}, we have reached the following statement: of retrograde Riemann sums {RΔ n Proposition 2.13 For an anti-causal function f ∈ M2,c , the function defined by the 1 retrograde Itô integral, t f d1 Wr exhibits the martingale property of retrograde type (2.28). Remark 2.5 Let f ∈ M2,c and let Z t ( f ) be an anti-causal process defined by

42

2 Preliminary – Causal Calculus

 Z t ( f ) = exp{ t

1

1 f (r, ω)d1 Wr − 2



1

f 2 (r, ω)dr }.

t

Then by a similar argument we may confirm that the equality E[Z t ( f )|F s ] = Z s ( f ) holds P-a.s. for any t ≤ s. For the later discussion we prepare an Itô formula of backward type. Let X t1 , X t2 be anti-causal Itô processes defined by  X ti

1

=



1

f i (s, ω)d1 Ws +

t

gi (s, ω)ds, i = 1, 2,

t

where f i , gi (i = 1, 2) are anti-causal functions belonging to the class M2,c . Then following the same argument as in the case of causal calculus it is almost immediate to establish the next result. Proposition 2.14 (backward Itô formula) For a smooth function F(x, y) and an interval [a, b] ⊂ [0, 1], the next equality holds: F(X b1 , X b2 ) − F(X a1 , X a2 )  b  1 b {Fx d1 X t1 + Fy d1 X t2 } − {Fx x f 12 + Fyy f 22 + 2Fx y f 1 f 2 }dt. =− 2 a a 1 1 Example 2.7 Let X t1 = t f d1 Ws and X t2 = t gd1 Ws , then noting X 11 = X 12 = 0 we have  1  1  1  1 { f (s) g(r )d1 Wr + g(s) f (r )d1 Wr }d1 Ws + f (s)g(s)ds. X t1 X t2 = t

s

s

t

2.3.3 The Symmetric Integral for Anti-Causal Functions For an anti-causal function X (t, ω) its symmetric integral (of backward type) can be defined similarly to the case of causal functions. Given a partition Δ = {0 = t0 < t1 < · · · < tn−1 < tn = 1} we consider for a fixed θ ∈ [0, 1] the Riemann sum as follows: n  θ ( f ) := X (tk + θ Δk )Δk W, (2.29) RΔ k=1

where, Δk = tk+1 − tk , Δk W = W (tk+1 ) − W (tk ). 1 ( f ) converges as |Δ| → 0, and to assure the We know that the sequence RΔ convergence for the case θ < 1 we need some assumption on the regularity of the integrand X t , namely a kind of B-differentiability. But for the simplicity of discussion

2.3 Causal Variants of Itô Integral

43

we suppose that the anti-causal function X t is given in the formula of the backward Itô integral, 

1

Xt =



1

f (s, ω)d1 Ws +

t

g(s, ω)ds

f, g ∈ M2,c .

t

We may call such a process the backward Itô process. Proposition 2.15 Let {Δn } be an increasing family of partitions of [0, 1]. Then for θ ( f )} converges each fixed θ ∈ [0, 1] the sequence of retrograde Riemann sums {RΔ n 1 in the mean as |Δn | → 0, and the limit which we denote by 0 X s dθ Ws is expressed in the following form: 

1

 X s dθ Ws =

0

1



1

X s d1 Ws + θ

0

f (s, ω)ds.

(2.30)

0

θ Proof Let us write Rn (θ ) = RΔ ( f ) and tk (θ ) = tk + θ Δk , then we have n

Rn (θ ) − Rn (1) =

n  {X (tk + θ Δk ) − X (tk+1 )}Δk W k=1

=

n−1  tk+1  k=0

tk (θ)

f (s, ω)d1 Ws · Δk W.

By the formula in Example 2.7 we find 

tk+1

f (s, ω)d1 Ws · Δk W

tk (θ)



=

tk+1 tk (θ)

 f (s, ω){W (tk+1 ) − W (s)}d1 Ws +

tk+1

 X s d1 Ws +

tk

tk+1

tk (θ)

= T1 (k) + T2 (k), where T1 (k) = T2 (k) =



tk+1

tk (θ)  tk+1 tk (θ)

 f (s, ω){W (tk+1 ) − W (s)}d1 Ws +

tk+1

X s d1 Ws ,

tk

f (s.ω)ds.

It is routine to verify that lim

n→∞

n−1  k=0

T1 (k) = 0 in L 2 (Ω, d P)

f (s.ω)ds}

44

2 Preliminary – Causal Calculus

and that lim

n→∞

Since limn→∞ Rn (1) =

1 0

n 



1

T2 (k) = θ

f (s, ω)ds.

0

k−1

X s d1 Ws , this completes the proof.



2.4 SDE A stochastic functional equation for an unknown process X t as follows is called a stochastic integral equation,  X t = x0 +

t

 b(s, X s )ds +

0

t

a(s, X s )d0 Ws ,

0 ≤ t ≤ T,

(2.31)

0

where x0 ∈ R and a(t, x), b(t, x) are real functions measurable in (t, x). In the condition that the unknown process X (t, ω) is limited to be causal (with respect to the filtration {FtW }) the equation becomes meaningful in the framework of the Itô integral. It is customary to represent this Eq. (2.31) by the following symbolic form which is called the stochastic differential equation (or SDE for short) of Itô type: d X t = b(t, X t )dt + a(t, X t )d0 Wt ,

X 0 = x0 .

(2.32)

The discussion on SDEs based on Itô calculus is not our principal subject in this book. So we do not give here a detailed review about it, but we intend to give only some elementary results for the reference in later chapters.

2.4.1 Strong Solution Definition 2.10 A continuous stochastic process X t (ω), (t ≥ 0), defined on the same probability space (Ω, F , P) as Brownian motion Wt (ω) and adapted to the filtration {FtW , t ≥ 0}, is called the strong solution of the SDE (2.32) provided that the couple (W, X ) satisfies the Eq. (2.31) with probability one for all t ∈ [0, T ]. As for the fundamental properties of the strong solution we have the following statement. Theorem 2.6 Let a(t, x), b(t, x) be real and smooth functions with bounded derivatives in x, i.e. |∂x b(t, x)|, |∂x a(t, x)| 0, we have the following inequality: P{ max |X n+1 (t) − X n (t)| > 2M} t∈[0,T ]  t       ≤ P max  αn (s)ds  > M + P max t∈[0,T ]

0

 t      βn (s)d0 Ws  > M .  t∈[0,T ] 

(2.39)

0

As for the first term on the right hand side, we have  t     T    max  αn (s)ds  > M ≤ P T αn2 (s)ds > M 2 t∈[0,T ] 0 0  T L2T (C1 T )n ≤ E dn−1 (s)ds ≤ C3 , M2 n!M 2 0 

P

2

where C3 = C2CL1 T . As for the second term, by applying Doob’s submartingale inequality (see Theorem 10.2 in Appendices) we get the following estimate:

2.4 SDE

47

  t   2 T   1   P max  βn (s)d0 Ws  ≥ M ≤ 2 E βn (s)d0 Ws t∈[0,T ] 0 M 0  (C1 T )n L2 T dn−1 (s)ds ≤ C4 , ≤ 2 M 0 n!M 2 

where C4 = max{C3 , (C2 L 2 /C1 )}.

1 Combining these two estimates with the inequality (2.39) and putting M = √ , 4 n! we find  (C1 T )n 2 P max |X n+1 (t) − X n (t)| > √ ≤ 2C . √ 4 4 t∈[0,T ] n! n!

 n The series n (C√1 Tn!) being convergent, by virtue of Borel- Cantelli’s first lemma we get from the above inequality the following result:  P

2 for large enough n = 1. max |X n+1 (t) − X n (t)| ≤ √ 4 t∈[0,T ] n!

This implies that the sequence {X n (t, ω)} converges to a limit X ∞ (t) almost surely and uniformly in t ∈ [0, T ]. Now letting n → ∞ on both sides of the equation (2.34) we confirm that the limit X ∞ (t) solves the SDE. What is left is the verification of the uniqueness of strong solution. So let Y (t, ω) be another strong solution of the SDE, for which we have the following equality:  X (t) − Y (t) =

t



t

{b(s, X (s)) − b(s, Y (s))}ds+

0

{a(s, X (s)) − a(s, Y (s))}d0 Ws .

0

Put d(t) = E[|X (t) − Y (t)|2 ] then, following the same argument as we have done, we find that  t d(t) ≤ 2L 2 (1 + T )

d(s)ds. 0

Since d(0) = 0 the application of Gronwall’s inequality (see the subject in Chap. 10) shows us that d(t) = 0 for any t, hence P{X (t) = Y (t)} = 1 for any t. This completes the proof.



Remark 2.6 The solutions X (t), Y (t) being continuous, we see by separability of those processes that P{X (t) = Y (t) ∀ t ∈ [0, T ]} = 1.

48

2 Preliminary – Causal Calculus

2.4.2 Law of the Solution of SDE We have shown a statement on the existence and uniqueness of the strong solution of the Cauchy problem for the SDE (2.32). It would be intuitively clear that this solution X t is a Markov process, for the following two reasons: (1) Because of the uniqueness as the solution of the SDE, the value of the process after “t” {X u , u ≥ t} depends only on the final data “X t ” and the increments {Wu − Wt , u ≥ t} of the driving force W.; besides (2) the increments {Wu − Wt , u ≥ t} are independent of W . the past history Ft− Hence we are interested in the transition probability of the solution X of the SDE, P(s, x, t, dy) = P{X t ∈ dy|X s = x}, s ≤ t, x ∈ R1 . Suppose for the simplicity of discussion that the transition kernel has the density P(s, x, t, dy) = p(s, x, t, y)dt. Now fix a smooth function f (x) ∈ Cb2 with finite support and consider the expectation  E[ f (X t )|X s = x] =

R1

f (y) p(s, x, t, y)dy.

By the Itô formula we have the equality f (X t ) = f (x) +

 t s

f  (X r ){b(r, X r )ds + a(r, X r )d0 Wr } +

 1 t  f (X r )b2 (r, X r )ds, 2 0

from which we see that,  E[ f (X t )|X s = x] = f (x) + E s

t

1  2 { f (X r )b(r, X r ) + f (X r )b (r, X r )}ds . 2 

By changing the order of integrations we find the following equality:  f (y) p(s, x, t, y)dy  t  1 dr p(s, x, t, y)dy{ f  (y)b(r, y) + f  (y)b2 (r, y)}. = f (x) + 2 R1 s R1

Thus by taking into account the fact that f (x) is of compact support and by applying the integration by parts formula to this, we get the following:  R1



 dr ∂r p(s, x, r, y) + ∂ y {b(r, y) p(s, x, r, y)} s 1 2 2 − ∂ y {b (r, y) p(s, x, r, y)} = 0. 2

f (y)dy

t

2.4 SDE

49

The test function f (x) being arbitrary, by virtue of Weyl’s lemma (see for example [15]) we get from this equality the equation for u(t, x) := p(0, x0 , t, x) as follows: ∂ ∂ 1 ∂2 2 {b (t, x)u(t, x)}, u(t, x) = − {b(t, x)u(t, x)} + ∂t ∂x 2 ∂x2 u(0, x) = δx0 (x).

(2.40)

This is the so-called Kolmogorov forward equation. The backward equation can be obtained by taking u(t, x) = E[ f (X T )|X t = x] and applying a similar argument based on the Itô formula, which would read as follows: ∂ ∂ b2 (t, x) ∂ 2 u(t, x) + b(t, x) u(t, x) + u(t, x) = 0, t ≤ T, ∂t ∂x 2 ∂x2 u(T, x) = f (x). (2.41) In terms of the transition probability density p(t, x, T, y) the function u(t, x) is writ ten as u(t, x) = R1 f (y) p(t, x, T, y)dy, hence again by applying Weyl’s lemma we see that from equations in (2.41) the following equations hold: ∂ b2 (t, x) ∂ 2 ∂ + b(t, x) + } p(t, x, T, y) = 0, t ≤ T, ∂t ∂x 2 ∂x2 p(T, x, T, y) = δ y (x).

{

2.4.3 Martingale Zt and Girsanov’s Theorem For a nice real function f (t, ω) belonging to the class M, we have introduced the causal function Z t by the following form: 

t

Z t = exp 0

1 f (s, ω)d0 Ws − 2



t

 f (s, ω)ds . 2

0

We have seen that this positive function is the unique strong solution of the Itô SDE d Z t = f (t, ω)Z t d0 Wt ,

Z 0 = 1.

By this fact we notice that the Z t is an FtW -martingale with E[Z t ] = E[Z 0 ] = 1. Moreover since Z t > 0 we see that d Q = Z t d P becomes another probability on the same measurable space (Ω, F ). We denote the expectation with respect to this new probability by E Q [·]. It is interesting to ask how the Brownian motion Wt looks like under new probability d Q. For this aim we consider a stochastic process Yt as follows: dYt = dWt − f (t, ω)dt.

50

2 Preliminary – Causal Calculus

Then by Itô formula we obtain the equality, d(Yt Z t ) = Z t {d0 Wt − f (t, ω)dt} + Yt f (t, ω)Z t d0 Wt + Z t f (t, ω)dt = Z t {1 + Yt f (t, ω)}d0 Wt , which shows that Yt Z t is an FtW -martingale under the original measure d P. Hence for an arbitrary event A ∈ FsW , we have, E Q [1 A Yt ] = E[1 A Yt Z t ] = E[1 A E[Yt Z t |FsW ]] = E[1 A Ys Z s ] = E Q [1 A Ys ], in other words, E Q [Yt |FsW ] = Ys Q-a.s.. This means that Yt is an FtW -martingale under the measure d Q. On the other hand, we easily see that the quadratic variation of Yt is d[Y ]t = dt. Consequently by Lemma 9.3 we get the following result: Proposition 2.16 (Girsanov’s Theorem) Under the measure d Q = Z t d P the process Yt ; dYt = dWt − f (t, ω)dt becomes a Brownian motion.

Chapter 3

Noncausal Calculus

We have seen in the previous chapter that the theory of Itô calculus was established after the introduction of the stochastic integral called the Itô integral and that this causal integral has two important features as follows: 1. Hypothesis of causality: The Itô integral (with respect to the Brownian motion Wt , or a square integrable martingale) is defined for such a causal random function f (t, ω) that is almost surely square integrable in t ∈ [0, 1] and is adapted to the natural filtration {FtW , t ≥ 0} associated to Brownian motion or, in general case, a square integrable martingale. 2. Riemann integral: The Itô integral is defined as a Riemann integral 0 ( f ), lim RΔ

|Δ|→0

with the Riemann sums of particular form as follows: 0 RΔ ( f ) :=

n−1 

f (ti , ω){W (ti+1 ) − W (ti )}.

i=0

It is by this causality dependent formalism that we like to call Itô calculus causal calculus or causal theory in this book, where we are going to develop an alternative theory of such stochastic calculus that is free from the hypothesis of causality. At this stage it may be opportune to remark that the two properties, being Causal and being Riemann integral mentioned above as characteristic features of the Itô integral, are independent of each other. The second property could help the Itô integral to cover a certain case of noncausal functions. The reader will readily be convinced of the importance of this remark when we see the Itô integrability of such a noncausal random function of the form f (t, ω) = α(ω) · g(t, ω), the product of a causal and Itô integrable function g(t, ω) with an arbitrary random variable α(ω). © Springer Japan KK 2017 S. Ogawa, Noncausal Stochastic Calculus, DOI 10.1007/978-4-431-56576-5_3

51

52

3 Noncausal Calculus

In this chapter we are going to introduce the noncausal stochastic integral and study fundamental properties of the stochastic calculus based on our noncausal integral. For its immense importance in the theory of stochastic calculus, causal or not, as in the previous chapter we will restrict our discussion mainly to the stochastic integral with respect to the real Brownian motionWt (ω), which we suppose defined on such a complete probability space (Ω, F , {FtW }, P) endowed with the natural filtration {FtW , t ≥ 0} that makes Brownian motion be a martingale. Then as the second step we will study the case of integration with respect to square integrable martingales and finally the noncausal integral with respect to other stochastic processes which are not martingales. We understand by ( f, g), the inner product of two elements f (t), g(t) ∈ L 2 ([0, 1], dt), the quantity 

1

( f, g) =

f (t)g(t)dt, g(t) = complex conjugate of g(t).

0

As an abuse but convenient usu of this notation we will write by (W˙ , f ) :=



1

f (t)dWt (Wiener integral)

0

the inner product for the white noise W˙ = and an f ∈ L 2 ([0, 1], dt).

d W, dt t

the derivative of Brownian motion,

3.1 Noncausal Stochastic Integral Let Wt (ω), t ≥ 0 be real Brownian motion defined on a complete probability space (Ω, F , {FtW }, P) with the natural filtration {FtW }. Given a random function f (t, ω) ∈ H and an orthonormal basis {ϕn } in the Hilbert space L 2 (0, 1) we consider the formal random series Sϕ ( f ) =

∞ 

( f, ϕn ) · (ϕn , W˙ )

(3.1)

n=1

where (ϕn , W˙ ) = (W˙ , ϕn ) =



1

ϕn (t)dWt .

(3.2)

0

Definition 3.1 (Noncausal integral) (1) The random function f ∈ H is called integrable with respect to the basis {ϕn }, or ϕ-integrable for short, provided that the series 1 Sϕ ( f ) converges in probability and in that case the sum is denoted by 0 f dϕ Wt .

3.1 Noncausal Stochastic Integral

53

(2) When the function f (t, ω) is noncausal integrable with respect to any basis 1 and the value of the noncausal integral 0 f dϕ Wt does not depend on the basis, the function f ∈ H is called universally integrable (or simply u-integrable). Remark 3.1 Under the convention (3.2), we might see in (3.1) that the form of defining random series Sϕ ( f ) would reflect Perseval’s equality in the theory of Fourier series; here the random series stands for the formal inner product of the two random elements f (t, ω), W˙ = dtd Wt . But for simplicity of notation we suppose throughout the discussion that every orthonormal basis {ϕn }, appearing in Definition 3.1, and random functions are real unless the contrary is mentioned. Hence the Fourier coefficient ( f, ϕn ) of a function f (t) ∈ L 2 ([0, 1], dt) (or f (t, ω) ∈ H) with respect to the basis {ϕn } is given by  ( f, ϕn ) =

1

f (t, ω)ϕn (t)dt.

0

We have given the definition of our noncausal integral with respect to Brownian motion W.(ω), but the first thing to notice about this is that the objective of Definition 3.1 is not limited to the integral with respect to Brownian motion or more generally to other square integrable martingales. With our formalism we may go much further so as to introduce the stochastic integral with respect to any other stochastic processes say Z (t) (t ∈ R), or even with respect to any random field Z(t), t ∈ Rd , as ˙ (the Fourier coefficient of the random long as the relevant quantity (ϕn , Z˙ ) or (ϕn , Z) ˙ distribution Z ) is well-defined. Because of its fundamental importance we are going to focus our discussion on the noncausal integral with respect to Brownian motion; we will discuss the latter two cases later in this chapter. Now let us get back to the noncausal integral with respect to Brownian motion. 1 t We will understand by 0 f dϕ Ws for t ∈ [0, 1] the integral 0 1[0,t] (s) f (s, ω)dϕ Ws . But notice that we do not know yet whether the ϕ-integrability on [0, 1] assures the integrability on the sub-interval [0, t] ⊂ [0, 1]. Below we will take some simple examples to see how the noncausal integral works. Example 3.1 Brownian motion W. is u-integrable on any sub-interval [s, t] ⊂ [0, 1] and we have the following equality: 

t

Wr dϕ Wr =

s

1 2 {W − Ws2 }, 2 t

where {ϕn } is an arbitrary basis in L 2 ([0, 1], d x). In fact, we have the equality Wt =

 n

Z n (ω)ϕ˜n (t), 0 ≤ t ≤ 1,

54

3 Noncausal Calculus

where



1

Zn =



t

ϕn (t)dWt , ϕ˜n (t) =

0

ϕn (s)ds.

0

Moreover we know by the Itô–Nisio theorem (Theorem 2.2) that the mode of convergence of the series is uniform in t ∈ [0, 1] with probability one. Hence, we get the equality below: t  m,n Z m Z n s ϕm (r )ϕ˜n (r )dr    = 21 { n Z n ϕ˜n (t)}2 − { n Z n ϕ˜n (s)}2

t

W (r )dϕ Wr =

s

= 21 (Wt2 − Ws2 ). Example 3.2 Let f (t, ω) ∈ H be a random function almost all samples of which are of bounded variation. Then f (t, ω) is u-integrable and we have for any basis {ϕn } the equality 

1



1

f (t, ω)dϕ Wt = f (1, ω)W (1) −

0

W (t)d f (t),

0

because we have  1  f (t, ω)ϕn (t)dt = f (1, ω) ( f, ϕn ) = 0

1

 ϕn (s)ds −

0

1



t

d f (t)

0

ϕn (s)ds.

0

On the other hand, we know by the Itô–Nisio theorem that the series  n

1

0



t

ϕn (r )dWr

ϕn (s)ds

0

converges uniformly in t ∈ [0, 1] with probability one, hence we get the conclusion. Example 3.3 (Stochastic convolution product) Let {T } = {T0 (t), Ti,n (t); i = 1, 2, n ∈ N} be the system of trigonometric functions, T0 (t) = 1, T1,n (t) =

√ √ 2 cos 2π nt, T2,n (t) = 2 sin 2π nt.

(3.3)

For convenience, we will denote this system by {Ti,n (t); i = 1, 2, n ∈ N ∪ 0} with the following easy convention: T1,0 (t) = T0 (t) = 1, T2,0 (t) = 0. Let K (t) = k0 + 2  n i=1

∞  2 

ki,n Ti,n (t) be a periodic function with period 1 such that

n=1 i=1

|ki,n | < ∞. Then the stochastic process X t , defined by the Wiener integral as

3.1 Noncausal Stochastic Integral

55

follows:



1

Xt =

K (t − s)dWs ,

0

is integrable with respect to the basis {T }, in other words the convolution product 11 0 0 K (t − s)dWs dT Wt is well-defined. In fact, we have  k1,n {T1,n (t)T1,n (s) + T2,n (t)T2,n (s)} n=1  + k2,n {T2,n (t)T1,n (s) − T1,n (t)T2,n (s)}

K (t − s) = k0 +

∞ 

√1 2

(3.4)

so we have X t = k0 Z 0 ∞    √1 k1,n {T1,n (t)Z 1,n + T2,n (t)Z 2,n } + k2,n {T2,n (t)Z 1,n − T1,n (t)Z 2,n } , + 2 n=1

where

 Z 0 = W (1),

Z i,n =

1

Ti,n (s)dWs (1 = 1, 2, n ≥ 1).

0

Hence by definition of the noncausal integral we find that ST (X ) = k0 Z 02 +

∞  k1,n 2 2 }, √ {Z 1,n + Z 2,n 2 n=1

and that the series ST (X ) almost surely converges under the prescribed condition, 2   |ki,n | < ∞. n i=1

The next example also shows that our noncausal integral can apply to a noncausal function. Example 3.4 For a fixed s ∈ R we put W s (t) = W (t + s, ω) − W (s, ω). Then W s (t) is integrable with respect to the system of trigonometric functions {Ti,n (t), i = 1, 2, n ∈ N ∪ {0}}. Here we give a proof of the statement in Example 3.4 for the case that |s| > 1; the verification for the other case is left to the reader. Proof By the representation form of BM and the definition of the shifted process W s (t) we have  t ∞  2  s Z i,n Ti,n (r )dr, W s (t) = Z 0s (ω)t + n=1 i=1

0

56

3 Noncausal Calculus

where

1 Z 0s = 0 dW s (r ) = W s (1) − W s (0), 1  1+s s = 0 Ti,n (r )dW s (r ) = s Ti,n (r − s)dWr . Z i,n

s Here we notice that when |s| > 1 the variables {Z i,n , Z i,n : i = 1, 2, n ≥ 0} are independent. Now taking the following equalities into account:

 0

t

T2,n (t) T1,n (s)ds = , 2π n



t

T2,n (s)ds =

0

1 √ { 2 − T1,n (t)} ∀ n ≥ 1, 2π n

we find that W s (t) = Z 0s t +

∞  √ 1  s s Z 1,n T2,n (t) + Z 2,n { 2 − T1,n (t)} . 2π n n=1

Thus we have (W s (·), T1,0 (·)) =

∞ s 1 s  Z 2,n Z0 + , √ 2 2π n n=1

(3.5)

and for n ≥ 1 we have Zs

2,n (W s (·), T1,n ) = − 2πn ,

(W s (·), T2,n ) =

√

1 {− 2πn

s 2Z 0s + Z 1,n }.

(3.6)

Summing up equalities (3.5), (3.6) and taking the independence of the variables, s {Z 1,n , Z i,n ; i = 1, 2, n ≥ 0} for |s| > 1, into account we confirm the almost sure convergence of the defining series ST (W s ), hence the integrality of the function W s (t) (at least for |s| > 1) with respect to the system {Ti,n } of trigonometric functions.  Here are some basic questions that may arise immediately after the definition; (Q1) What is the meaning of noncausal integrability with respect to a basis {ϕn }? (Q2) How does the notion of the noncausal integral depend on the choice of basis? (Q3) What is the relation with causal theory of stochastic calculus? Does our noncausal integral have any relation with the Itô integral or other variants of causal type? (Q4) How is the integral characterized in the framework of homogeneous chaos? (Q5) How does our noncausal integral work in various problems of stochastic calculus? In what follows, we are going to give answers to, or will study, these questions; especially we will study the questions (Q1)–(Q3) in this chapter. Question (Q4) we will study in Chap. 4, and as for the question (Q5), we will show in Chaps. 5–8 how our noncausal calculus works, by taking various topics of noncausal nature.

3.2 Question (Q1) – Interpretation

57

3.2 Question (Q1) – Interpretation It may be seen that the essence of Definition 3.1 lives in Perseval’s equality in 1 harmonic analysis, that is, we intend to understand the integral 0 f (s, ω)dWs as the inner product ( f, W˙ ) which, by formal application of Perseval’s equality, would give us the random series Sϕ ( f ) in (3.1). We may as well remember the well-known Itô–Nisio theorem [12] (Theorem 2.2 in Chap. 2) about the uniform and almost sure convergence of the following random series: W (t, ω) =

 n

1



t

ϕn (s)dWs ·

0

ϕn (s)ds.

0

We will show in this section some interpretations or modifications of its definition as preparation for further discussions. For an arbitrarily fixed orthonormal basis {ϕn } and the Brownian motion W. we introduce the sequence of approximate processes, Wnϕ (t, ω) =

n  (ϕk , W˙ )ϕ˜k (t),

(3.7)

k=1

where (ϕk , W˙ ) =



1



0

Then, since E[(ϕk , W˙ )W (t)] =

t

ϕk (s)dWs , ϕ˜k (t) =

ϕk (s)ds.

0

t 0

ϕk (s)ds, we see that for each t

lim E[|W (t) − Wnϕ (t)|2 ] = lim {t −

n→∞

n→∞

n 

ϕ˜k (t)2 } = 0.

k=1

Hence by the Itô–Nisio theorem we confirm that: ϕ

Proposition 3.1 For any basis {ϕn } in L 2 (0, 1) the sequence {Wn (·, ω)} almost surely converges to the Brownian motion W. uniformly in t ∈ [0, 1], namely

P

 lim sup |W (t) − Wnϕ (t)| = 0 = 1.

n→∞ 0≤t≤1

ϕ

As it is immediate to see that each approximate process Wn (t, ω) almost surely has a smooth sample path, we notice that for any random function f (t, ω) ∈ H the 1 ϕ integral 0 f (t, ω)dWn (t) is well-defined as a Stieltjes integral. Hence we get the next interpretation.

58

3 Noncausal Calculus

Proposition 3.2 The random function f ∈H is ϕ-integrable if and only if the 1 ϕ sequence of Stieltjes integrals, Sn ( f, {ϕ}) = 0 f (t, ω)dWn (t), converges in probability as n tends to infinity. In this case we have 

1

lim

n→∞ 0

f (t, ω)dWnϕ (t) =



1

f (t, ω)dϕ Wt .

0

For a given orthonormal basis {ϕn } we form the sequence of kernels Dnϕ (t, s) =



ϕk (t)ϕk (s).

(3.8)

k≤n

Then the statement of Proposition 3.2 can be expressed in a different way as follows: Proposition 3.3 For the random function f ∈ H the following equality holds provided that one of terms on both sides exists in probability: 

1

 f dϕ Wt = lim

n→∞ 0

0



1

1

f (t, ω)Dnϕ (t, s)dWs (in pr obabilit y).

dt 0

(3.9)

Remark 3.2 Remembering that for any orthonormal basis {ϕn } the sequence {Dnϕ } is a δ-sequence, we may give a slightly generalized definition for the noncausal integral. Given a δ-sequence {K n (t, s)}, we introduce the noncausal integral with respect to this sequence in the following way: 

1

 f (t, ω)d K Wt := lim

n→∞ 0

0



1

1

dt

f (t, ω)K n (t, s)dWs (in probability).

0

For later use, we remember the complete orthonormal system of Haar functions {H0,0 (t), Hn,i (t) (0 ≤ i ≤ 2n−1 − 1, n ≥ 1)} and repeat its definition below: ⎧ ⎪ ⎨ H0,0 (t) = 1,n−1 t ∈ [0, 1], Hn,i (t) = 2 2 {1[2−n+1 i,2−n+1 (i+1/2)) (t) − 1[2−n+1 (i+1/2),2−n+1 (i+1)) (t)}, ⎪ ⎩ n ≥ 1, 0 ≤ i ≤ 2n−1 − 1,

(3.10)

where 1 A (·) is the indicator function of the set A. Taking this as the defining basis, we introduce the approximate process of the Brownian motion in the following form: n 2 −1   k−1

WnH (t)

:= W (1)t +

k=1

i=0

Then we have the following statement.

0

t

 Hk,i (s)ds · 0

1

Hk,i (s)dWs .

3.2 Question (Q1) – Interpretation

59

Lemma 3.1 We have WnH (t) = W (2−n i) + {W (2−n (i + 1)) − W (2−n i)}2n (t − 2−n i), for t ∈ [2−n i, 2−n (i + 1)).

(3.11)

In other words, the approximate process WnH (t) is just a Cauchy polygonal approximation of the process W (t, ω) passing through the following dyadic lattice points {(2−n i, W (2−n i, ω)), 0 ≤ i ≤ 2n − 1}. Proof Let {χn,i (t)} be a family of rectangular functions such that n

χn,i (t) = 2 2 1 In,i (t), n ≥ 0, 0 ≤ i ≤ 2n − 1,

(3.12)

where {In,i } are dyadic sub-intervals; In,i := [2−n i, 2−n (i + 1)). Notice that for any m < n the following relation holds: ∀

(Hn,i , χm, j ) = 0 Hence χn, j (t) =

i,

∀

j.

(n,0) 

(χn, j , Hk,i )Hk,i (t),

(3.13)

(k,i)

where the term on the right hand side means the sum as follows: n 2 −1  (χn, j , H0,0 ) + (χn, j , Hk,i )Hk,i (t). k

k=1 i=0

Also notice that n 2 −1

(χn, j , Hk,i )(χn, j , Hg,h ) = δk,g δi,h , for any k, g ≤ n.

(3.14)

j=0

On the other hand, by the relation (3.13) we obtain the equality below: “The right hand side of the equality (3.10)” n  1 2 −1  t = χn, j (s)ds χn, j (s)dWs j=0

0

2 −1(n,0) 

0



n

=

j=0

=

t

(χn, j , Hk,i ) 0

(k,i)

(n,0)  (n,0)  t (k,i) (g,h) 0

Hk,i (s)ds

(n,0) 



(g,h)



1

Hk,i (s)ds 0

Hg,h (s)dWs

1

(χn, j , Hg,h )

n −1 2

j=0

Hg,h (s)dWs

0

(χn, j , Hk,i )(χn, j , Hg,h ).



60

3 Noncausal Calculus

Hence by the equality (3.14), we see that “The right hand side of the equality (3.10)”  1 (n,0)  t Hk,i (s)ds Hk,i (s)dWs = WnH (t), at any t ∈ [0, 1] = (k,i)

0

0

and the proof is completed. The next example is significant since it shows that our noncausal integral includes a Riemann-type definition of the stochastic integral. Example 3.5 (Case of Haar functions) The noncausal integral with respect to the system of Haar functions is given by the following form: 

1

f (t, ω)d H W (t)

0

:= lim

n→∞

n −1 2

k=0

 2n

2−n (k+1) 2−n k

f (s, ω)ds · {W (2−n (k + 1)) − W (2−n k)} (in P). (3.15)

This is just a Riemann-type definition of the stochastic integral where the mean  2−n (k+1) value 2n 2−n k f (s, ω)ds is employed as the representative value of the integrand f (t, ω) in each sub-interval [2−n k, 2−n (k + 1)).

3.3 Dependence on the Basis – About Q2 We would like to emphasize the point that our noncausal integral depends on the choice of the basis. As we have seen in the previous paragraph (in Example 3.5 and in the equality (3.15)) by a suitable choice of the basis our noncausal integral can include the Riemann-type definition as an example. However, up to the present this question of dependence on the basis has not yet been fully studied and we know little except the followings. Let {ϕn } be a basis and {ψn } be another one obtained by rearrangement of the elements ϕn of the first basis. Suppose that a random function f (t, ω) is integrable with respect to the basis {ϕn } and ask whether f (t, ω) is also integrable with respect to the second basis {ψn }. What we can see immediately from the definition is that, even in such a special case, the ϕ-integrability does not necessarily assure the integrability f (t, ω) is ψ-integrable, with respect to the basis {ψn } and that, even if the function  dψ Wt coincide with each we do not know whether the two integrals dϕ Wt , other. Let us consider the problem for this special case. Let us call {ψn } explained above a rearrangement of the basis {ϕn }, and let us say that a random function f (t, ω)

3.3 Dependence on the Basis – About Q2

61

is “universally ϕ-integrable” when the function is integrable with respect to any 1 rearrangement of the basis {ϕn } with the same sum to 0 f dϕ Wt . We see easily that when the defining series Sϕ ( f ) in (3.1) absolutely converges, i.e., ∞  |( f, ϕn ) · (ϕn , W˙ )| < ∞, (3.16) n=1

then the function f (t, ω) is “universally ϕ-integrable”. In fact, in this case we notice that the series (3.16) converges almost surely, hence for almost all ω the series absolutely converges and this implies the “universal ϕ- integrability”. Remark 3.3 But we are not sure at this moment whether the “universal ϕ-integrability” implies the absolute convergence mentioned in (3.16). This still remains as one of open problems in noncausal theory. As a general question  between noncausal integrals with respect  about the relation to two different bases, f dϕ Wt and f dψ Wt , we have the following result. 1 Proposition 3.4 ([28]) Let f (t, ω) be integrable, √ in L (d P)√sense, with respect to the system {T } of trigonometric functions {1, 2 sin 2π nt, 2 cos 2π nt}, namely we suppose that the following series converges in L 1 (d P):

S{T } ( f ) :=

∞ 2  

( f, Ti,n )Z i,n ,

i=1 n=0

where T1,n (t) =

√

2 cos 2π nt, T2,n =

√

2 sin 2π nt, n ∈ N,

with the convention: T1,0 (t) = 1[0,1] (t), T2,0 (t) = 0, and 

1

Z i,n :=

Ti,n (t)dWt , for n ≥ 1 and Z 2.0 = 0.

0

Then the function f (t, ω) is integrable with respect to the system of Haar functions {Hn,i } (given in (2.5)) and the following equality holds: 

1 0

 f d H Wt =

1

f d T Wt .

0

For the demonstration of this proposition we need the following simple lemma. Lemma 3.2 For the system of trigonometric functions {Tα,n , α = 1, 2, n ≥ 0} the following equalities hold:

62

3 Noncausal Calculus n 2 −1

(χn,k , Tα,l )(χn,k , Tβ,m )  −n 2 2 mπ ⎪ ⎨ δα,β δl,m sin2−n , for m ≥ 1, mπ = 1, for n = l = m = 0, α = β = 1, ⎪ ⎩ 0, otherwise, k=0⎧

(3.17)

where α, β = 1, 2 and {χn,k } are those given in (3.12). The verification of this statement is not difficult and is left to the reader. Below we are going to give a proof for Proposition 3.4. 1 Proof By Proposition 3.2 it suffices to show that limn→∞ 0 f (t, ω)dWnH (t) exists in probability. We see by Example 3.5 the following expression for the Stieltjes integral: 

1 0

=

f (t, ω)dWnH (t)

n 2 −1



( f, χn,i )(χn,i , W˙ )

i=0

⎫ ⎧ ∞ 2 ⎨  ⎬ ( f, Tα,m )(Tα,m , χn,i ) (χn,i , Tβ,h )(Tβ,h , W˙ ) ⎩ ⎭

∞  2  m=0 α=1

i=0

=

n 2 −1

h=0 β=1

= I1 + I2 + I3 , where ⎫⎧ ⎫ n22n  2 ⎬ ⎨ ⎬ I1 = ( f, Tα,m )(Tα,m , χn,i ) (χn,i , Tβ,h )(Tβ,h , W˙ ) ⎭⎩ ⎭ ⎩ m=0 α=1 i=0 h=0 β=1 ⎫⎧ ⎫ ⎧ n 2 −1 ⎨ n22n  ∞ 2 2 ⎬⎨  ⎬  I2 = ( f, Tα,m )(Tα,m , χn,i ) (χn,i , Tβ,h )(Tβ,h , W˙ ) ⎭⎩ ⎭ ⎩ m=0 α=1 i=0 h=n22n +1 β=1 ⎫ ⎧ n 2 −1 ⎨  ∞ 2 ⎬  I3 = ( f, Tα,m )(Tα,m , χn,i ) (χn,i , W˙ ). ⎭ ⎩ 2n ⎧

n 2 −1 ⎨ n22n  2

i=0

m=n2 +1 α=1

First we are going to verify that limn I2 = 0, limn I3 = 0 in probability. In fact for the term I2 by Lemma 3.2 we get the inequality below: ⎡ |I2 | ≤ ⎣

⎧

n 2 −1 ⎨ n22n  2

i=0

⎡

×⎣

⎩

m=0 α=1

⎧

2 −1 ⎨ n

i=0

⎩

⎫⎤1/2 ⎬ ( f, Tα,m )(Tα,m , χn,i ) ⎦ ⎭

∞ 

⎫⎤1/2 2 ⎬  ( f, Tβ,m )(Tβ,m , χn,i ) ⎦ ⎭

m=n22n +1 β=1

3.3 Dependence on the Basis – About Q2

63

 2  2 1/2  sin 2−n mπ 2 ≤ ( f, Tα,m ) 2−n mπ α=1 ⎫2 ⎤1/2 ⎡ n ⎧ 2 −1 ⎨  ∞ 2 ⎬  ×⎣ (χn,i , Tα,m )(Tα,m , W˙ ) ⎦ ⎭ ⎩ 2n m=n2 +1 α=1

i=0

 ≤

1

1/2 f 2 (t, ω)dt

⎡ ⎣

0

⎧

n 2 −1 ⎨

i=0

⎩

∞ 

⎫2 ⎤1/2 2 ⎬  (χn,i , Tα,m )(Tα,m , W˙ ) ⎦ . ⎭

m=n22n +1 α=1

On the other hand we have ⎫2 ⎧ n 2 −1 ⎨  ∞ 2 ⎬  E (χn,i , Tα,m )(Tα,m , W˙ ) ⎭ ⎩ 2n m=n2 +1 α=1

i=0

=

n 2 −1

∞ 

∞ 

2  (χn,i , Tα,m )2 = 2

i=0 m=n22n +1 α=1

m=n22n +1



sin 2−n mπ 2−n mπ

2

C ≤ (C : const.), n hence we see that limn→∞ I2 = 0 in probability. By a similar argument we can verify that limn→∞ I3 = 0 in probability. As for the evaluation of the last term I1 , we decompose it into three parts as follows:  2 n2

 sin 2−n mπ 2  2n

I1 =

m=0

2−n mπ

where we understand

( f, Tα,m )(Tα,m , W˙ ) = I4 + I5 + I6 ,

α=1

sin 2−n mπ = 1 for m = 0, 2−n mπ

and n2  2  I4 = ( f, Tα,m )(Tα,m , W˙ ), 2n

m=0 α=1 [n/2] 2

I5 = −

m=0

and



sin 2−n mπ 2−n mπ

2

 −1

2  ( f, Tα,m )(Tα,m , W˙ ), α=1

64

3 Noncausal Calculus n2  2n

I6 = −

m=2[n/2] +1



sin 2−n mπ 2−n mπ



2

2  −1 ( f, Tα,m )(Tα,m , W˙ ). α=1

By assumption of the theorem we know that limn→∞ I4 exists in probability and 1 equals the integral 0 f (t, ω)dT Wt . We are going to verify that as n tends to infinity , I6 converge to 0 in probability. the remaining terms  I5 2 ˙ Put S(k) = ∞ m=k α=1 ( f, Tα,m )(Tα,m , W ), then the term I6 can be written in the following form: n2 



2n

I6 =

m=2[n/2] +1

[S(m) − S(m + 1)]

sin 2−n mπ 2−n mπ



2

−1



2

2  sin 2−n mπ sin 2−n (m − 1)π = S(m) − + S(n22n + 1) −n mπ −n (m − 1)π 2 2 m=2[n/2] +1  

2 sin 2[n/2]−n π − 1 S(2[n/2] + 1). + 2[n/2]−n π n2  2n

Now let ak (0 = a0 < a1 < a2 < · · · ) be the point where the function { sinπ πx x }2 takes its k-th maximum value. Then the function { sinπ πx x }2 being monotone in the sub-intervals [k, ak ), [ak , (k + 1)), (k = 1, 2, . . .), we obtain the following estimate for the term I6 : E|I6 | ≤E|S(2[n/2] + 1)| + E|S(n22n + 1)| !   n22n !

 !! sin 2−n mπ 2 sin 2−n (m − 1)π 2 !! + max E|S(m)| − ! ! ! 2−n mπ ! 2−n (m − 1)π 2[n/2]