Shigeyoshi Ogawa Noncausal Stochastic Noncausal

Shigeyoshi Ogawa

Noncausal Stochastic Calculus

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

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

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 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 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”

1The caligraphy presented in the previous page is the title in Japanese, given by Michi Ogawa.

v vi Preface 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 . 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 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 vii

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 text- book 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 viii Preface 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 Shigeyoshi Ogawa July 2016 Contents

1 Introduction – Why the Causality? ...... 1 1.1 Hypothesis of Causality ...... 1 1.2 Noncausal Calculus ...... 2 1.3 Some Noncausal Problems ...... 3 1.3.1 Problem in BPE Theory ...... 3 1.3.2 Convolution Product with the ...... 6 1.3.3 Vibration of a Random String ...... 9 1.4 Plan of the Book ...... 10 2 Preliminary – Causal Calculus ...... 11 2.1 Brownian Motion ...... 12 2.1.1 Some Properties of BM ...... 13 2.1.2 Construction of BM...... 16 2.2 Itô Integral with Respect to BM ...... 19 2.2.1 Classes of Random Functions ...... 19 2.2.2 Itô Integral for f 2 S ...... 20 2.2.3 Extension to f 2 M ...... 24 2.2.4 Linearity in Strong Sense ...... 28 2.2.5 Itô Formula ...... 30 2.2.6 About the Martingale ZtðaÞ ...... 33 2.3 Causal Variants of Itô Integral ...... 36 2.3.1 Symmetric Integrals...... 36 2.3.2 Anti-Causal Function and Backward Itô Integral ...... 39 2.3.3 The Symmetric Integral for Anti-Causal Functions .... 42 2.4 SDE ...... 44 2.4.1 Strong Solution ...... 44 2.4.2 Law of the Solution of SDE ...... 48 2.4.3 Martingale Zt and Girsanov’s Theorem ...... 49

ix x Contents

3 Noncausal Calculus ...... 51 3.1 Noncausal Stochastic Integral ...... 52 3.2 Question (Q1) – Interpretation ...... 57 3.3 Dependence on the Basis – About Q2 ...... 60 3.4 Relation with Causal Calculus – About Q3 ...... 65 3.4.1 Regular Basis ...... 68 3.4.2 More About the Integrability of Quasi-Martingales .... 71 3.4.3 Comments on Riemann-Type Integrals ...... 75 3.5 Miscellaneous Cases ...... 76 3.5.1 Noncausal Integral for Random Fields...... 76 3.5.2 Case of Fractional Brownian Motion...... 79 3.6 Appendix – Proof of Lemma 3.4 ...... 79 4 Noncausal Integral and Wiener Chaos ...... 83 4.1 Wiener–Itô Decomposition ...... 84 4.2 Skorokhod Integral ...... 85 4.3 Noncausal Integral in Wiener Space ...... 88 5 Noncausal SDEs...... 91 5.1 Symmetric SDE as Noncausal SDE ...... 92 5.2 SDE with Noncausal Initial Data ...... 93 5.3 Noncausal Itô Formula ...... 98 5.4 SDE in Noncausal Situation ...... 100 5.5 Examples ...... 103 6 Brownian Particle Equation ...... 109 6.1 Definition of Solution ...... 109 6.2 Method of Stochastic Characteristics ...... 113 6.2.1 Bridge to Parabolic Equations ...... 115 6.2.2 A Small Extension...... 116 6.3 Nonlinear BPEs ...... 117 6.4 Applications ...... 119 6.4.1 Stochastic Burgers’ Equation...... 119 6.4.2 Numerical Solution of a Nonlinear Diffusion Equation – G. Rosen’s Idea...... 121 6.4.3 Girsanov’s Theorem?...... 123 6.4.4 Noncausal Counterpart for Girsanov’s Theorem ...... 124 7 Noncausal SIE ...... 127 7.1 SIE Driven by Brownian Motion ...... 128 7.1.1 Method ...... 129 7.1.2 A Stochastic Transformation Along Orthonormal Basis...... 132 7.2 SIE Driven by Brownian Sheet...... 135 Contents xi

8 Stochastic Fourier Transformation ...... 139 8.1 SFC and SFT ...... 141 8.2 Case of Causal Functions ...... 144 8.2.1 Case of Wiener Functionals...... 145 8.2.2 Causality Condition ...... 147 8.2.3 Scheme for Reconstruction of Kernels...... 148 8.2.4 Inversion Formula ...... 149 8.3 Case of Noncausal Wiener Functionals...... 150 8.3.1 Extension to the Case of General Basis...... 154 8.4 Case of Noncausal Wiener Functionals – 2 ...... 156 8.4.1 Model and Results...... 156 8.4.2 Method of Fourier Series for Volatility Estimation .... 157 8.5 Some Direct Inversion Formulae ...... 159 8.5.1 Preliminary ...... 159 8.5.2 Natural SFT...... 160 8.5.3 Proof of Theorem ...... 163 8.5.4 Another Scheme for Direct Inversion ...... 164 8.5.5 A Variant of the Natural SFT ...... 165 8.5.6 Remarks on SFTs of Different Types ...... 167 8.5.7 About the Linearity of the SFT ...... 167 8.5.8 Extension to Noncausal Case...... 168 9 Appendices to Chapter 2...... 171 9.1 Itô Integral with Respect to Martingales ...... 171 9.1.1 Procedure for Introduction...... 172 9.1.2 Integral for f 2 M2;c ...... 174 9.1.3 Some Properties of Square Integrable Ft-Martingales ...... 175 9.1.4 Some Applications...... 176 9.1.5 Symmetric Integral with Respect to Martingale ...... 177 10 Appendices 2 – Comments and Proofs...... 181 10.1 Notes on Chap. 2 ...... 181 10.1.1 Martingale Inequalities ...... 181 10.2 Statements in Chap. 2...... 183 10.2.1 Proof of Proposition 2.7 ...... 183 10.2.2 Proof of Lemma 2.2 ...... 184 10.2.3 Proof of Theorem 2.5 ...... 185 10.2.4 Gronwall’s Inequality ...... 187 10.2.5 Uniqueness of B-Derivative...... 187 10.3 Note on Chap. 3 ...... 188 10.4 Notes on Chap. 8 ...... 190 10.4.1 Proof of Theorem 8.2 ...... 190 xii Contents

10.5 Proof of Theorem 8.4 ...... 191 10.5.1 Sketch of Proof for Proposition 8.7 ...... 197 10.5.2 Dini’s Test...... 199 10.5.3 Proof of Proposition 8.9 ...... 200 10.5.4 Proof of Proposition 8.10 ...... 200 Epilogue...... 203 References ...... 205 Index ...... 209 Chapter 1 Introduction – Why the Causality?

We are concerned with an alternative theory of stochastic calculus, referred to nowa- days 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), includ- ing a random process, say Xt , 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 Xt . In this order of consideration the evolution of such random phenomena should be represented by functionals of the basic noise process Xt . In the history of mathematical sciences the = d random noise Xt of most interest is the derivative Xt dt Wt of the process Wt called = d Brownian motion or more generally Xt dt 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 1 S. Ogawa, Noncausal Stochastic Calculus, DOI 10.1007/978-4-431-56576-5_1 2 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 Xs (s ≤ t); roughly speaking it means that the value of f (t,ω)at time t should not depend on the future behaviour {Xu : 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. stochas- tic 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 , 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](seealso[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 R1 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,sayu(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 (1.1) (t, x) ∈[0, ∞) × R1, 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) + W (ω) u(t, x,ω) ∂ t ∂ ⎩ t dt x (1.2) = A(t, x)u(t, x,ω)+ B(t, x), (t, x,ω)∈[0, ∞) × R1 × Ω,

d of course formally since it involves as coefficient the white noise dt Wt , the derivative of Brownian motion Wt . We suppose that Wt (ω) is defined on a (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 of two random distributions (in the sense of L. Schwartz’s distribution) W and dt t ∂ 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: 1 Definition 1.1 A real random function u(t, x,ω), defined on R+ × R × Ω and ( , ,ω) σ B × B × F measurable in t x with respect to the -field R+ R1 , 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 udWt is well-defined, and (s2) for any test function ϕ(t, x), having finite support in [0, T ]×R1, the function u(t, x,ω)satisfies the following equality (1.4) with probability one:     T  T dx {ϕt + (aϕ)x + A}u + Bϕ dt + dx ϕx u(t, x,ω)dWt R1 0 R1 0 (1.4)

+ ϕ(0, x)u0(x)dx = 0 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 dx {ϕt + (aϕ)x + A}u + Bϕ dt + ϕ(0, x)u0(x)dx R1 0 R1 (1.5) = 0 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 (1.6) (t, x) ∈[0, ∞) × R1.

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) + W (ω) u(t, x,ω) ∂ t ∂ t dt x (1.7) d = A(t, x)u(t, x,ω) W + B(t, x), (t, x,ω)∈[0, ∞) × R1 × Ω. dt t 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:  t ( , ,ω)= ( (t,x)) + { ( , (t,x),ω) + ( , (t,x),ω) }, u t x u0 X0 Au s Xs dWs B s Xs ds  0 t (1.8) (t,x) − =− ( , (t,x)) + − , ≤ ≤ ≤ . where Xs x a r Xr dr Ws Wt 0 s t T s

Here appears an essential problem of measurability in the stochastic integral term, t ( , (t,x)) ( , (t,x),ω) A s Xs u s Xs 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 ( , ) (t,x)(ω) on the other hand we see that for each fixed t x the function Xs is adapted to the future field Fs,t = σ{Wt − Wr ; s ≤ r ≤ t}.

( , (t,x)) ( , (t,x),ω) The integrand A s Xs u s Xs 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 ( , (t,x),ω) the stochastic integral Au s Xs dWs loses its meaning in the framework 0 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 con- tained in the fact that under this hypothesis we can develop the discussion about the stochastic integral f (t,ω)dWt , or more generally the integral f (t,ω)dMt 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)dXt with respect to a process Xt which does not exhibit the mar- tingale 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 Xt = f (t,ω)· Wt , we will meet the stochastic convolution  ˜ ˜ (F X)t = f (t − s,ω)dWs , where the symbols “F ” and “∼” (tilde) stand for the Fourier transform of the indi- ˙ cated functions. As we see below, we notice that F Wt 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)}  the system is given by exp  , where is Heisenberg’s constant. This is well-known. By the way, perhaps because the Schrödinger equation looks similar to an equa- tion 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))} will have the form exp  for the propagator of probability wave 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˙ 2, W˙ = W . It is of course absurd, for we know from a famous article t t dt t by L. Schwartz [50] the impossibility of defining the direct product S · T of two dis- tributions, 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,sowemay 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 general- ized random functions, gives us another definition as follows: Definition 1.2 Let S (R) be the space of rapidly decreasing C∞-class functions ( ) := ( +| |m )| (n)( )| f x endowed with the system of semi-norms, f m,n supx∈R 1 x f x . ˙ The white noise Wt (ω) is an S - valued random variable such that the characteristic functional is given in the following form:   √ ˙ 1 2 ∀ Φ(u) := E[exp{ −1(W, u)}] = exp − u 2 , u(x) ∈ S . 2 L

The existence of such a probability measure on the dual space S is a simple con- sequence 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 √ √ [ { − (F ˙ , )}] = [ { − ( ˙ , F )}] = {− 1 F 2} E exp 1 W u E exp 1 W u exp 2 u = {− 1 2}, ∈ L , exp 2 u u 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 +·))dBt , 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)dBs we have 1.3 Some Noncausal Problems 9  

Z ∗ Z(u) := dBt u(t + s)dBs .

We find that in this line of argument, the possibility of defining the direct product ˙ 2 (W) is reduced to the possibility of defining the convolution product with respect to the stochastic integral dB, 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 (1.9)

X(0) = x0, X(1) = x1, where x0, x1 are real constant, and A(t,ω),B(t,ω)and Zt , 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 d ( ) d + ( ) dt p t dt 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:   1 1 X(t,ω)= f (t,ω)+ L(t, s,ω)X(s)ds + K (t, s)X(s)dZ(s), 0 0

L(t, s,ω) = A(s,ω))K (t, s) f (t,ω) = 1 B(s,ω)K (t, s)ds where and 0 , and 1 ( , ) ( ) ( ) 0 K t s X s dZ 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)}.Thisisavery 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 Xt is still causal (i.e. adapted) to the natural filtration generated by the underlying fundamental process Zt . Hence the stochastic integral term K (t, s)X(s) dZs 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 Zt , Xt are the random fields, namely the stochastic processes with multi- dimensional parameters t ∈ J =[0, 1]d , we can think of the SIE for the random