Asymptotic analysis of unstable solutions of stochastic DE 9783030412906, 9783030412913

Table of contents :
Preface......Page 6
Contents......Page 10
About the Authors......Page 13
Notation......Page 14
1 Introduction to Unstable Processes and Their Asymptotic Behavior......Page 15
1.1.1 Description and Motivation of the Model......Page 17
1.1.2 Asymptotic Growth and Normalizing Multipliers for the Solutions......Page 18
1.1.3 Bilayer Environment and Transition Density of the Limit Homogeneous Markov Process......Page 21
1.1.4 Comparison with the Smooth Disturbing Process......Page 23
1.1.5 Spatial Averaging of the Vibrational Type Coefficient......Page 24
1.2 Equation with the Non-unit Diffusion Coefficient......Page 25
2.1 Preliminaries......Page 29
2.2 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Brownian Motion in a Bilayer Environment......Page 34
2.3 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Process of Skew Brownian Motion Type......Page 50
2.4 Examples......Page 57
3.1 Criteria of Instability and Ergodicity for the Solutions......Page 65
3.2 Convergence of Normalized Stochastically Unstable Solutions to the Bessel Diffusion Process......Page 70
3.3 Influence of the Coefficients of the Equation on the Limit Behavior of the Solutions......Page 80
3.4 Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions......Page 82
3.5 Examples......Page 86
4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions......Page 90
4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer Environment......Page 94
4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process......Page 112
4.3 Results About Weak Convergence of the Mixed Functionals......Page 128
4.4 Examples......Page 133
5.1 Preliminaries......Page 141
5.2 Theorem Concerning the Weak Compactness......Page 144
5.3 Weak Convergence to the Solutions of Itô SDEs......Page 149
5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type......Page 152
5.5 Asymptotic Behavior of Integral Functionals of Martingale Type......Page 161
5.6 Weak Convergence of Mixed Functionals......Page 167
5.7 Examples......Page 173
5.8 Auxiliary Results......Page 180
6.1 Preliminaries......Page 188
6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs......Page 191
6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type......Page 195
6.4 Weak Convergence of Martingale Type Functional and of Mixed Functional......Page 200
6.5 Examples......Page 201
6.6 Auxiliary Results......Page 203
A.1.1 Basic Facts Regarding Stochastic Processes......Page 207
A.1.1.3 Wiener Process as an Example of a Gaussian Process......Page 209
A.1.2 Notion of Stochastic Basis with Filtration......Page 210
A.1.3 Notion of (Sub-, Super-) Martingale. Elementary Properties. Square-Integrable Martingales. Quadratic Variations and Quadratic Characteristics......Page 211
A.1.4 Markov Moments and Stopping Times......Page 213
A.1.5 Construction of Stochastic Integral w.r.t. a Wiener Process and a Square-Integrable Continuous Martingale......Page 214
A.1.6 Generalized Itô Formula......Page 217
A.1.7 Skorokhod's Representation Theorem and Prokhorov's Theorem......Page 219
A.2 Convergence of Stochastic Integrals and Some Properties of Solutions of SDEs......Page 220
A.3 Brownian Motion in a Bilayer Environment......Page 235
A.4 Functions Regularly Varying at Infinity......Page 244
References......Page 246

Citation preview

Bocconi & Springer Series 9 Mathematics, Statistics, Finance and Economics

Grigorij Kulinich Svitlana Kushnirenko Yuliya Mishura

Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations

Bocconi & Springer Series Mathematics, Statistics, Finance and Economics Volume 9

Editors-in-Chief Lorenzo Peccati, Dipartimento di Scienze delle Decisioni, Università Bocconi, Milano, Italy Sandro Salsa, Department of Mathematics, Politecnico di Milano, Milano, Italy Series Editors Beatrice Acciaio, Department of Statistics, London School of Economics and Political Science, London, UK Carlo A. Favero, Dipartimento di Finanza, Università Bocconi, Milano, Italy Eckhard Platen, Department of Mathematical Sciences, University of Technology, Sydney, NSW, Australia Wolfgang J. Runggaldier, Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Padova, Italy Erik Schlögl, School of Finance & Economics, University of Technology, Sydney, City Campus, NSW, Australia

The Bocconi & Springer Series aims to publish research monographs and advanced textbooks covering a wide variety of topics in the fields of mathematics, statistics, finance, economics and financial economics. Concerning textbooks, the focus is to provide an educational core at a typical Master’s degree level, publishing books and also offering extra material that can be used by teachers, students and researchers. The series is born in cooperation with Bocconi University Press, the publishing house of the famous academy, the first Italian university to grant a degree in economics, and which today enjoys international recognition in business, economics, and law. The series is managed by an international scientific Editorial Board. Each member of the Board is a top level researcher in his field, well-known at a local and global scale. Some of the Board Editors are also Springer authors and/or Bocconi high level representatives. They all have in common a unique passion for higher, specific education, and for books. Volumes of the series are indexed in Web of Science - Thomson Reuters. Manuscripts should be submitted electronically to Springer’s mathematics editorial department: [email protected] THE SERIES IS INDEXED IN SCOPUS

More information about this series at http://www.springer.com/series/8762

Grigorij Kulinich • Svitlana Kushnirenko • Yuliya Mishura

Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations

Grigorij Kulinich Department of General Mathematics Taras Shevchenko National University of Kyiv Kyiv, Ukraine

Svitlana Kushnirenko Department of General Mathematics Taras Shevchenko National University of Kyiv Kyiv, Ukraine

Yuliya Mishura Department of Probability Theory, Statistics and Actuarial Mathematics Taras Shevchenko National University of Kyiv Kyiv, Ukraine

ISSN 2039-1471 ISSN 2039-148X (electronic) Bocconi & Springer Series ISBN 978-3-030-41290-6 ISBN 978-3-030-41291-3 (eBook) https://doi.org/10.1007/978-3-030-41291-3 © Springer Nature Switzerland AG 2020 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is devoted to unstable solutions of stochastic differential equations (SDEs). Despite the huge interest in the theory of SDEs, with the help of which the phenomena of nature, technology, economics and finance are modeled, as far as we know this is the first book to present a systematic study of the instability and asymptotic behavior of the corresponding unstable stochastic systems. The book is the result of many years of work by its main co-author, G.L. Kulinich, who devoted a considerable part of his scientific research to the unstable solutions of SDEs. Two other co-authors, S.V. Kushnirenko and Yu.S. Mishura, were very pleased to translate the main ideas of this theory into rigorously stated and clearly proved results, as well as to investigate more general cases of asymptotic behavior of unstable solutions and give relevant examples. The study of the conditions of existence and the asymptotic behavior of unstable solutions, which is proposed herein, started in 1965 when A.V. Skorokhod offered such a problem to his PhD student G.L. Kulinich. At that time the concept of an unstable solution was very poorly investigated; most experts, rather than focusing on instability problems, were attempting to study the conditions of existence and the asymptotics of stable solutions. This can be explained by the fact that, both in the past and, indeed still today, the problem of the stability of the solution of an SDE, in one or another sense (stability in probability, asymptotic stability in probability, stability in the mean-square sense, exponential stability etc.) is the key problem, important for the various applications. Since stability is an asymptotic property, we are immediately faced with the asymptotic problems which play a leading role in the theory of SDEs, regardless of whether their solutions are stable or not. One of the possible asymptotic approaches is to study stability under random perturbations of the parameters including random perturbations of the noise. In this case, one often considers the situation when the initial condition and noise are asymptotically small, and the stability of the trivial solution is studied, in one or another sense. When we study the asymptotic behavior, as time t → +∞, of SDE solutions, we are mostly interested in the conditions of “stabilization” of the solution. Among these conditions we have the conditions under which the solution is stable in its direct sense, that is, converges v

vi

Preface

to a certain constant in some stochastic sense, and the conditions under which the solution is ergodic, that is, converges to a certain random variable. This approach is applied to study properties, for example, of mechanical systems subjected to random perturbations and evolution processes involved in finance, economics, medicine, biology, electronics and telecommunications. This and the accompanying problems are discussed in detail in the books [26, 62, 80]; see also references therein. We are not going to review all of the many books and articles on the theory of SDEs, but for an initial acquaintance with this theory and its applications, we suggest that the reader consults the following textbooks and monographs [3, 23, 27, 63, 70, 74]. In contrast to the above books and numerous other works, we are considering the case in which the solution of the SDE is unstable. The notion of instability will be discussed in detail in what follows; however, we prefer to give one definition immediately with the goal of explaining to the reader, albeit briefly, the range of issues under consideration. Note that there can be different definitions of instability, but the following definition is, in our opinion, very simple, clear and easy to check. Definition 1 A stochastic process ξ = {ξ(t), t ≥ 0} is called (i) Stochastically unstable if 1 lim t →+∞ t

t P {|ξ(s)| < N} ds = 0 0

for any constant N > 0. (ii) Unbounded in probability at infinity if lim P {|ξ(t)| < N} = 0

t →+∞

for any constant N > 0. For example, the Wiener process is stochastically unstable. Also, it is clear that a stochastically continuous stochastic process is stochastically unstable assuming that it is unbounded in probability. Within our theory, we will apply these concepts to the solutions of Itô’s SDEs, which are stochastically continuous processes, even continuous with probability 1. The book itself describes in detail the “unstable” limit theorems and asymptotic properties of solutions. The limit theorems contained in our book are not merely of purely mathematical value; rather, they also have quite practical value. Instability or violations of stability have been noted in many phenomena, and the authors of their descriptions have tried to apply certain mathematical and stochastic methods to deal with them. Here are a few examples: the unstable states of physical and biological systems, as well as the instability of chemical reactions, have been described in [1, 4, 5, 8, 10, 68, 76, 83]. From our side, let us consider some examples of the behavior of dynamical systems where the instability occurs. For example, a volcanic eruption has occurred, and we are interested in the problem of finding the most likely areas

Preface

vii

of accumulation of volcanic particles, or in the problem of how to adjust the flow of these particles so that it goes in the necessary direction. If the drift coefficient in the equation has a periodic structure, then the situation arises whereby so-called shaking off takes place and at the same time the structure of the environment is changed. As a very particular example that will, however, be clear to every reader, with intensive shaking of milk, butter and buttermilk are produced; in other words, a bilayer environment is formed. In general, a bilayer environment naturally appears when we consider a simple stochastic system, e.g., described by a Wiener process, perturbed by some variable intensity λ(x). The asymptotic behavior of such system depends on the asymptotics of λ(x) at ±∞ that immediately gives us a bilayer environment. Evidently, ergodic (stable) behavior of the stochastic process does not cover the above anomalies and bundles. Even such a brief description of the applications convinces us that the study of instability is completely logical. Note also that the bilayer environment in our framework is described by so called skew Brownian motion, see, e.g., [2, 21, 57]. It should be mentioned, in particular, that the bilayer environment described by skew Brownian motion, appears in finances. The paper [12] provides evidence of systematic mispricing of the Black–Scholes model when the log-returns of the underlying asset are skewed and leptokurtic. This is described mathematically in [11]. Our main goals achieved in this book are: we have explored Brownian motion in environments with anomalies and we have studied the motion of the Brownian particle in layered media. In the limit we have obtained a fairly wide class of continuous Markov processes. In particular, this class includes Markov processes with discontinuous transition densities. This class includes also processes that are not solutions of any Itô’s SDEs, and the Bessel diffusion process. In addition, in this monograph we study the weak convergence of the additive integral type functionals, under a suitable normalization for unstable solutions of SDEs, to certain functionals of their limit processes. The book is organized as follows. In Chap. 2 we consider the homogeneous one-dimensional Itô’s SDE and obtain necessary and sufficient conditions of weak convergence of the solution to Itô’s SDE to Brownian motion in a bilayer environment. For the limit process the explicit form of the transient density is found. In addition, we consider classes of equations for which necessary and sufficient conditions of weak convergence of the normalized solutions to the processes of the skew Brownian motion type are obtained. We note that from the results of Chap. 2 the principle of spatial averaging of the coefficients of an SDE follows: under certain conditions, in the equation instead of its coefficients we can put constant coefficients which are a certain spatial averaging of the given coefficients. At the same time, the asymptotic form of the distribution for the solution to the SDE does not change. In Chap. 3 we conduct an analysis of the asymptotic behavior of solutions of equations that are on the verge of equations whose solutions have an ergodic distribution and equations with stochastically unstable solutions. Sufficient conditions for weak convergence of the normalized solutions to the Bessel diffusion process

viii

Preface

are established for them. The results of Chap. 3 are refinement and generalization of the papers [38, 40, 46]. Chapter 4 focuses on the asymptotic behavior of the additive integral-type functionals under a suitable normalization for unstable solutions of SDEs. These results summarize the papers [34] and [36] and contain the results from [52–54]. New classes of limit distributions, which are certain functionals of the Brownian motion in a bilayer environment or certain functionals of the Bessel diffusion process are obtained. In particular, they contain the local time of the limit processes at the point 0 on the interval [0, t]. Similar limit functionals for the Wiener process were first considered in the monograph [81, Chapter 5]. In Chap. 5 we consider homogeneous one-dimensional SDEs with non-regular dependence on parameter. We study the asymptotic behavior of integral functionals of an ordinary Lebesgue integral type, integral functionals of martingale type and mixed functionals defined on the solutions of the equations. We obtain the limit processes in the form of integral functionals of an ordinary Lebesgue integral type, integral functionals of martingale type and mixed functionals with subordinate Wiener processes. These results summarize some results of the paper [40] and contain the results from [55] and [56]. Chapter 6 generalizes some results from the previous chapter to the case of the solutions to inhomogeneous equations with non-regular dependence on the parameter. Under certain conditions, we prove that the asymptotic behavior of the solutions and some functionals of the solutions to inhomogeneous Itô SDEs is the same as that for the solutions to homogeneous Itô SDEs. All chapters contain a range of examples that illustrate statements about the weak convergence of the solutions and various types of functionals of the solutions to Itô SDEs. To make the book self-contained, definitions and auxiliary results are presented in the Appendix. We include here both the well-known classical theorems concerning the weak convergence of stochastic processes and auxiliary calculations for the main theorems describing the limit behavior of unstable solutions. The book will be interesting and useful for specialists in stochastic analysis and SDEs, as well as for physicists, chemists, economists, sociologists and other researchers who deal with unstable systems and for practitioners who apply stochastic models to describe phenomena of instability. The basic concepts of this book are quite accessible to graduate students. We are thankful to everybody who contributed to the improvement of this book. In the earliest stages of his research, G.L. Kulinich was inspired and directed by his teacher A.V. Skorokhod, whose ideas influenced the overall content of the book. Our special thanks are due to Springer Milan and the Editorial Board of the Bocconi & Springer Series for their helpful comments and recommendations which helped to significantly improve the book’s presentation. Kyiv, Ukraine Kyiv, Ukraine Kyiv, Ukraine 2019

Grigorij Kulinich Svitlana Kushnirenko Yuliya Mishura

Contents

1

Introduction to Unstable Processes and Their Asymptotic Behavior .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Equation with the Unit Diffusion Coefficient . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Description and Motivation of the Model . . . . . . . . . . . . . . . . . . . . 1.1.2 Asymptotic Growth and Normalizing Multipliers for the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 Bilayer Environment and Transition Density of the Limit Homogeneous Markov Process . . . . . . . . . . . . . . . . . 1.1.4 Comparison with the Smooth Disturbing Process . . . . . . . . . . . 1.1.5 Spatial Averaging of the Vibrational Type Coefficient . . . . . . 1.1.6 Functionals of the Solution . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Equation with the Non-unit Diffusion Coefficient .. . . . . . . . . . . . . . . . . . .

2 Convergence of Unstable Solutions of SDEs to Homogeneous Markov Processes with Discontinuous Transition Density . . . . . . . . . . . . . . 2.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Brownian Motion in a Bilayer Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Process of Skew Brownian Motion Type .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Criteria of Instability and Ergodicity for the Solutions .. . . . . . . . . . . . . . 3.2 Convergence of Normalized Stochastically Unstable Solutions to the Bessel Diffusion Process .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Influence of the Coefficients of the Equation on the Limit Behavior of the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 3 3 4 7 9 10 11 11 15 15

20

36 43 51 51 56 66

ix

x

Contents

3.4 3.5

Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

68 72

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 77 4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer Environment.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 4.3 Results About Weak Convergence of the Mixed Functionals . . . . . . . . 115 4.4 Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 120 5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the Solutions of Itô SDEs with Non-regular Dependence on a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Theorem Concerning the Weak Compactness . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Weak Convergence to the Solutions of Itô SDEs .. . . . . . . . . . . . . . . . . . . . 5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Asymptotic Behavior of Integral Functionals of Martingale Type .. . 5.6 Weak Convergence of Mixed Functionals .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.7 Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.8 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to Inhomogeneous Itô SDEs with Non-regular Dependence on a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Weak Convergence of Martingale Type Functional and of Mixed Functional . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A Selected Facts and Auxiliary Results . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Selected Definitions and Facts for Stochastic Processes and Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1.1 Basic Facts Regarding Stochastic Processes . . . . . . . . . . . . . . . . . A.1.2 Notion of Stochastic Basis with Filtration . . . . . . . . . . . . . . . . . . .

129 129 132 137 140 149 155 161 168

177 177 180 184 189 190 192 197 197 197 200

Contents

A.2 A.3 A.4

xi

A.1.3 Notion of (Sub-, Super-) Martingale. Elementary Properties. Square-Integrable Martingales. Quadratic Variations and Quadratic Characteristics .. . . . . . . . . . . . . . . . . . . . A.1.4 Markov Moments and Stopping Times . . .. . . . . . . . . . . . . . . . . . . . A.1.5 Construction of Stochastic Integral w.r.t. a Wiener Process and a Square-Integrable Continuous Martingale .. . . A.1.6 Generalized Itô Formula . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1.7 Skorokhod’s Representation Theorem and Prokhorov’s Theorem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Convergence of Stochastic Integrals and Some Properties of Solutions of SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Brownian Motion in a Bilayer Environment . . . . . .. . . . . . . . . . . . . . . . . . . . Functions Regularly Varying at Infinity . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

201 203 204 207 209 210 225 234

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237

About the Authors

Prof. Grigorij Kulinich received his PhD in probability and statistics from Kyiv University in 1968 and completed his postdoctoral degree in probability and statistics (Habilitation) in 1981. His research work focuses mainly on asymptotic problems of stochastic differential equations with nonregular dependence on parameter, theory of stochastic differential equations, and theory of stochastic processes. He is the author of more than 150 published papers and 3 books. Dr. Svitlana Kushnirenko is an Associate Professor in the Department of General Mathematics, Taras Shevchenko National University of Kyiv, where she also completed her PhD in probability and statistics in 2006. Her research interests include theory of stochastic differential equations and stochastic analysis. She is the author of 20 papers. Prof. Yuliya Mishura received her PhD in probability and statistics from Kyiv University in 1978 and completed her postdoctoral degree in probability and statistics (Habilitation) in 1990. She is currently a professor at Taras Shevchenko National University of Kyiv. She is the author/coauthor of more than 270 research papers and 9 books. Her research interests include theory and statistics of stochastic processes, stochastic differential equations, fractional processes, stochastic analysis, and financial mathematics.

xiii

Acronyms

Abbreviations a.s. a.e. SDE w.r.t.

Almost surely Almost everywhere Stochastic differential equation With respect to

Notation R+ (Ω, F, F, P) ξ = ξ(ω), ω ∈ Ω Eξ W = {W (t), t ≥ 0} χA δ(·) x Lζ 0 (t) x ∧y P

−→ d

= ζ (t) σ {ζ (s), s ≤ t} N

V σ

−N

[0, +∞) Complete probability space with filtration F = {Ft }t ≥0 Random variable (a real F-measurable function) Expectation of a random variable ξ Wiener process Indicator function of a set A Dirac’s delta function Local time of the process ζ at the point x0 on the interval [0, t] min {x, y} Convergence in probability Equality in distribution Quadratic characteristic of the martingale ζ(t) Smallest σ -algebra such that all random variables ζ(s) for s ≤ t are measurable Variation of the function σ on an interval [−N, N]

xv

Chapter 1

Introduction to Unstable Processes and Their Asymptotic Behavior

The purpose of this chapter is to introduce the reader to the basic concepts related to unstable processes. To convince the reader that stochastically unstable processes are an important subject for consideration, let us continue with further definitions and visualizations. Definition 1.1 A solution ξ of Itô’s SDE is called ergodic with a distribution function F (x) if lim P {ξ(t) < x} = F (x)

t →+∞

for any x ∈ R. Note that ergodic solutions ξ are not stochastically unstable, since for them there exists the limit in Definition 1, which is not equal to zero for any N > 0 (see Lemma A.14 in Appendix). Let us compare the minor changes in the equations from Figs. 1.1 and 1.2 with the significantly different properties of the solutions of these equations. So, is not it important to be able to make certain changes in stochastically unstable systems (in other words, to control stochastic systems) in order to compensate anomalies and to obtain a system with specified properties? Now we turn to such (rather interesting and important) cases, when the solution of Itô’s SDE is unbounded in probability. In such a case, it is necessary to introduce a suitable normalization, in order to obtain the limit distribution of such a solution. To find non-degenerate limit distributions of unstable solutions ξ to SDEs, we introduce a non-random normalizing factor B(T ) > 0, B(T ) → +∞, as T → +∞, where T > 0 is a parameter.

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_1

1

1 Introduction to Unstable Processes and Their AsymptoticBehavior

−10

−5

0

X(t)

5

10

2

0

2

4

6

8

10

t

−10

−5

0

X(t)

5

10

Fig. 1.1 Realizations of stochastically unstable solution to SDE dX(t) = sign X(t) dt + dW (t) with initial condition X(0) = 1

0

2

4

6

8

10

t

Fig. 1.2 Realizations of ergodic solution to SDE dX(t) = −sign X(t) dt + dW (t) with initial condition X(0) = 1

1.1 Equation with the Unit Diffusion Coefficient

3

Definition 1.2 A solution ξ to Itô’s SDE has an exact order of growth B(t), as t → +∞, if there exists a non-random function B(t) → +∞, as t → +∞, and a constant c0 = 0 such that  P

 ξ(t) = c0 = 1. t →+∞ B(t) lim

It is clear that the solutions to SDEs, which have an exact order of growth B(t), are stochastically unstable. Next we consider the asymptotic behavior of T) distributions of the processes ξT (t) = ξ(t B(T ) , t ≥ 0, as T → +∞. In addition to the convergence of finite-dimensional distributions of the processes ξT , as T → +∞, we also study the weak convergence to some limit process ζ in the following sense. Definition 1.3 A family ξT = {ξT (t), t ≥ 0} of stochastic processes is said to converge weakly, as T → +∞, to a process ζ = {ζ (t), t ≥ 0} if, for any L > 0, the measures μT [0, L], generated by the processes ξT (·) on the interval [0, L] converge weakly to the measure μ[0, L] generated by the process ζ(·). Remark 1.1 Since, as it was mentioned above, the processes ξT are continuous with probability 1 as the solutions to Itô’s SDEs, Definition 1.3 is a definition of the weak convergence of the processes ξT to the continuous process ζ in a uniform topology of the space of continuous functions.

1.1 Equation with the Unit Diffusion Coefficient 1.1.1 Description and Motivation of the Model Let ξ be the solution of the stochastic differential equation t ξ(t) = x0 +

a (ξ(s)) ds + W (t),

t ≥ 0,

(1.1)

0

where W = {W (t), t ≥ 0} is a Wiener process, a = a(x) is a continuous, absolutely integrable on the whole axis function with  a(x) dx = λ.

(1.2)

R

The initial problem from the range of problems considered in this monograph is to study the limiting behavior, as t → +∞, of the distribution of the process ξ .

4

1 Introduction to Unstable Processes and Their AsymptoticBehavior

To characterize the situation in general, from the point of view of appearance a Wiener process, we can say that it is the case where the deterministic system t x(t) = x0 +

a (x(s)) ds 0

is intrinsically perturbed by the stochastic “irregular” process and this perturbation is an important and unavoidable thing. As it is well known, a Wiener process W with probability 1 has no derivative at any point and has unbounded variation on any interval [t1 , t2 ]. So, a deterministic system is considered, whose motion is described by a nonlinear differential equation x(t) ˙ = a (x(t)), and which is intrinsically perturbed by a “white noise” process; shortly speaking, the deterministic system is distorted by noise. But, referring to the variety of applications, we can take a different point of view. Namely, the problem (1.1) and (1.2) can be considered as the description of an external perturbation by the coefficient a of the physical environment described by the Wiener process. So, our goal is to study the corresponding state changes of this environment with time. In this case, we solve the problem of non-random control of the structure of a chaotic environment in which a small Brownian particle moves. In particular, under the condition λ = 0 we can assume that at the initial point x = 0 an energy source of a high power is implemented in a homogeneous environment. The main motivation for considering of our model (1.1) and (1.2) is the fact that it can be used in the mathematical description of anomalous phenomena. The book explores Brownian motion in environments with anomalies, that is, environments where certain high power sources can be the energy at certain points. This is, for example, underwater volcanic eruptions or nuclear explosions, tsunamis, tornadoes, and other turbulence, while in the classical case, the motion of a Brownian particle in “smooth” media is considered. Note that the mutual influence of tsunamis, tornadoes, hurricanes, technological disasters with climate has acquired a decisive significance nowadays. Our approach also covers more peaceful phenomena such as the motion of the Brownian particle in layered media, such as, for example, oil and water. Some other examples are provided in Preface.

1.1.2 Asymptotic Growth and Normalizing Multipliers for the Solutions When studying Eq. (1.1) under the condition (1.2), it turned out that the solution ξ is unbounded in probability, as t → +∞, the details are contained in [30] for λ = 0 and in [29] for λ = 0. Therefore, for establishing a non-degenerate boundary distribution of the solution ξ , it was necessary to introduce some nonrandom normalizing multiplier B(t) → +∞, as t → +∞. In this connection, it was

1.1 Equation with the Unit Diffusion Coefficient

5

necessary to develop new methods for investigating the behavior when T → +∞ T) of the distribution of the normalized random process ξT (t) = ξ(t B(T ) , t > 0, where T is parameter. Considering Eq. (1.1) under the condition (1.2), it was noticed that one √ can chose B(T ) = T , and the normalized solution ξT (t) satisfies the equation x0 ξT (t) = √ + T

t aT (ξT (s)) ds + WT (t),

t ≥ 0,

(1.3)

0

√ √ where aT (x) = T a(x T ), WT (t) = W√(t T ) is a family of Wiener processes. T By the way, note that Eq. (1.3) is an integral form of the stochastic differential Itô equation dξT (t) = aT (ξT (t)) dt + dWT (t),

t ≥ 0.

(1.4)

Let us √ also emphasize that the coefficient aT (x) of Eq. (1.3) contains the multiplier T that tends to infinity, as T → +∞, therefore the standard asymptotic methods are not suitable for the investigation of the limit behavior of the solutions ξT (t) of Eq. (1.3), as T → +∞. In particular, in the case where λ = 0, aT (x) is a “δ”-shaped family of functions, whose  “δ”-shaped property is realized at the point x = 0 and it has the weight λ = a(x) dx. In this connection, the cases R

where λ = 0 and λ = 0 in (1.2) are essentially different concerning the limit behavior of the distributions of the solutions ξT (t) to Eq. (1.3). To specify the above description of the situation with butter and buttermilk, we can say that in the case where λ = √ 0, and√a has a periodic structure, our Eq. (1.3) contains the coefficient aT (x) = T a(x T ) that corresponds to “intensive shaking” and to creation of a bilayer limit environment. Concerning the case λ = 0, a new asymptotic method t of the investigation of the “inconvenient” term aT (ξT (s)) ds was proposed in the 0

paper [30]. This method applies the representation t

t aT (ξT (s)) ds = ΦT (ξT (t)) − ΦT (x0 ) −

0

ΦT (ξT (s)) dWT (s),

0

where x ΦT (x) = 2 0

⎡ ⎣1 − σ0 e

u −2 aT (v) dv 0

⎤ ⎦ du,

σ0 = e

2

+∞  0

a(z) dz

.

(1.5)

6

1 Introduction to Unstable Processes and Their AsymptoticBehavior

As one can see, the right-hand side of the representation (1.5) contains a stochastic Itô integral. According to the Itô formula, the representation (1.5) holds with probability 1 for any t > 0. Moreover, to obtain (1.5), we use the equality 1 ΦT (x)aT (x) + ΦT (x) = aT (x), 2 that holds for any x ∈ R. Therefore, the investigation of the limit behavior of t the “inconvenient” term aT (ξT (s)) ds can be reduced to the investigation of two 0

things: first, the limit behavior of the functional consisting of the function ΦT (x), where the value ξT (t) is substituted, and, second, the asymptotic behavior of the family of martingales t ηT (t) =

ΦT (ξT (s)) dWT (s).

0

Recall that the functions ΦT (x) and ΦT (x) contain the coefficient aT (x) of Eq. (1.3) under the sign of an integral. This fact simplifies essentially the asymptotic study of the behavior of the solutions ξT (t). When studying the asymptotic problems connected to Eq. (1.1) under the condition (1.2) with λ = 0, this method allows to get the convergence ξT (t) − WT (t) → 0, as T → +∞, in probability, for any t > 0. Therefore, in the case where the spaceaverage value of the function a is zero, i.e., a(x) dx = 0, we have that R

1 √ t

t a (ξ(s)) ds → 0 0

in probability, as t → +∞. In this case a = a(x) does not affect the limit behavior of the distribution of the solution ξ of Eq. (1.1), as t → +∞. Later on, this result was obtained in the monograph [17], also via the analysis of the right-hand side of the representation (1.5), however, the convergence of the right-hand side to zero was established by different methods. The case where λ = 0 in (1.1) and (1.2) differs essentially from the case λ = 0. According to [29], it turned out that in this case the value 1 √ t

t a (ξ(s)) ds  0 0

1.1 Equation with the Unit Diffusion Coefficient

7

in probability, as t → +∞. In view of this fact, another new probabilistic method was proposed. This method is to deal with the transformation ζT (t) = fT (ξT (t)) of the solution ξT (t) of Eq. (1.3). Such a transformation “annihilates” the term t aT (ξT (s)) ds when choosing fT properly. It means that it is reasonable to 0

consider harmonic functions fT (x) that solve the equation 1 fT (x)aT (x) + fT (x) = 0 2 for any x ∈ R. In particular, we can take the functions fT (x) as follows: ⎧ ⎨

x fT (x) =

u

exp −2 ⎩ 0

⎫ ⎬ aT (v) dv du, ⎭

x∈R

(1.6)

0

and investigate the behavior of the distributions of the process ζT (t) = fT (ξT (t)), where ξT (t) is the solution of Eq. (1.3). According to the Itô formula, t ζT (t) = fT (x0 ) +

 σT (ζT (s)) dWT (s),

(1.7)

0

where  σT (x) = fT (ϕT (x)), and ϕT (x) are the inverse functions to the functions fT (x) for any T > 0. According to condition (1.2), there exist constants δ > 0 and C > 0 such that 0 < δ ≤ fT (x) ≤ C for any x ∈ R. Therefore, the investigation of the behavior of the distributions of the processes ζT (t) is reduced to the investigation t σT (ζT (s)) dWT (s). Further, we can apply of the behavior of the martingales  0

A.V. Skorokhod subsequence convergence theorem (see Theorem A.12 or [79, Chapter I, § 6]) which states that for some subsequence Tn → +∞ we have the convergence in probability of the subsequences ζTn (t) → ζ(t) and WTn (t) → W (t) for any t > 0.

1.1.3 Bilayer Environment and Transition Density of the Limit Homogeneous Markov Process Condition (1.2) provides an opportunity to establish the relationship t P {ζ (s) = 0} ds = 0 0

(1.8)

8

1 Introduction to Unstable Processes and Their AsymptoticBehavior

for any t > 0, and this in turn allows one to come to the limit in (1.7) and get that the limit process ζ (t) satisfies the equation t ζ (t) =

σ (ζ (s))dW (s),

(1.9)

0

 +∞ −∞   σ1 , x ≥ 0, σ1 = exp{−2 a(x) dx}, σ2 = exp{−2 a(x) dx}. σ2 , x < 0, 0 0 By virtue of equality (1.8) the integral in (1.9) is an Itô integral, which means that Eq. (1.9) is an Itô stochastic differential equation and ζ is a weak solution of this equation. Since for any T > 0 the process ζT is a homogeneous Markov process, see [79], the limit process ζ is a homogeneous Markov process as well. Applying the equality (1.8), the explicit form of the transition density ρ(t, x, y) of the process ζ was established in the paper [29]. This transition density has a discontinuity of the first kind, i.e., a jump, at the point y = 0, and this phenomena appears as the result of adhesion at zero of two Gaussian distributions: N(0, σ12 t), restricted to the positive semi-axis, and N(0, σ22 t), restricted to the negative semiaxis, correspondingly. Therefore, the process ζ describes the projection on the real axis of the trajectory of a Brownian particle, moving in a bilayer environment with the boundary at the point y = 0, subject to the continuous passage of the boundary by this particle. In this case, a certain refraction of the trajectory occurs, similar to the refraction of the trajectory of the light beam as it passes through the boundary of x two media. Moreover, due to the convergence fT (x) → xσ (x) and ϕT (x) → σ (x) for any x = 0, as T → +∞, and the inequality 0 < δ1 ≤ ϕT (x) ≤ C1 that holds for any x and T with some constants δ1 and C1 not depending on x and T , we get the weak convergence, as T → +∞, of the solution ξT of Eq. (1.3) to the homogeneous x Markov process  ξ (t) = l(ζ (t)). Here ζ (t) is the solution of Eq. (1.9), l(x) = σ (x) . Therefore, the transition density of the process  ξ (t) has the form where σ (x) =

   ρξ (t, x, y) = ρζ t, l −1 (x), l −1 (y) l −1 (y) ,

(1.10)

where l −1 (x) = xσ (x) is the inverse function to the function l(x). Let us emphasize that the process  ξ describes quite adequately the trajectory of motion of a Brownian particle in a homogeneous environment having a barrier at the point y = 0. In this case the symmetry which is inherent in the Wiener process is broken. More precisely, the symmetry of probability of hitting the set y > 0 and the set y < 0, starting from the point y = 0, is broken. For a Wiener 2 process such probabilities equal 12 , while for the process  ξ they equal σ2σ+σ and 1 σ1 , respectively (see [17, Chapter 3, § 15, Theorem 4]). Therefore, (1.1) and σ2 +σ1 (1.2) with λ = 0 can be applied to construct a mathematical model of an atomic explosion leading to multidirectional streams of radioactive particles. Note that in the book [24] the process  ξ was called a skew Brownian motion. Besides this, it was

1.1 Equation with the Unit Diffusion Coefficient

9

established by M.I. Portenko in the book [72] that the diffusion process  ξ admits Kolmogorov local characteristics in the generalized sense. This fact served him as the basis for introducing the notion of a generalized diffusion process that is a homogeneous Markov process, admitting Kolmogorov local characteristics in the generalized sense. For example, the process  ξ admits a generalized drift coefficient 1 of the form cδ(·), where δ(·) is Dirac’s delta function, c = σσ22 −σ +σ1 = tanh λ, and the diffusion coefficient equals 1. Therefore, if we consider the problem (1.1) under the condition (1.2) with λ = 0, then the process ξT converges weakly to the process  ξ , which is a generalized process in the sense described above, and for which one can formally write the stochastic differential equation d ξ (t) = tanhλ δ(·) dt + dW (t),

(1.11)

where δ(·) is Dirac’s delta function at the point zero. Compare this situation to the case when λ = 0. In this case the solutions ξT of Eq. (1.3) converge weakly to a Wiener process, i.e.,  ξ = W. Note that A. Friedman [15] established that in the problem (1.1) and (1.2) the condition λ = 0 is necessary for the convergence, as T → +∞, of the distributions √ T ) to a Wiener process W . Besides this, it follows of the process ξT (t) = ξ(t T from [42] that for the process  ξ the equality  ξ (t) = β(t) + W (t) holds, where β(t) is some additive functional depending on the solution ζ of Eq. (1.9).

1.1.4 Comparison with the Smooth Disturbing Process We stress again that in this way, the situation with the limit behavior, as T → +∞, √ T ) , where ξ is the of the finite-dimensional distributions of the process ξT (t) = ξ(t T solution of the problem (1.1) and (1.2), depends on the value of λ. In other words, it depends on the limit behavior of the coefficient a(x), as |x| → +∞. This fact is connected to the non-regular behavior of the disturbing process W . If the disturbing process is smooth, the situation is different. For example, consider the stationary process in the wide sense η = {η(t), t ≥ 0}, where Eη(t) = 0 and the random variables η(t1 ) and η(t2 ) are independent for |t2 − t1 | ≥ l > 0 and any ti . It was √T ) , proved in [32] that finite-dimensional distributions of the process ξT (t) = ξ(t T where ξ is the solution of the equation t ξ(t) = x0 +

t a (ξ(s)) ds +

0

η(s) ds, 0

t ≥ 0,

10

1 Introduction to Unstable Processes and Their AsymptoticBehavior

and condition (1.2) holds, converge, as T → +∞, to the finite-dimensional l distributions of the process σ0 W , where W is a Wiener process, σ0 = R(u) du −l

and R(u) is the covariance function of the process η. It means that the limit finitedimensional distributions of the process ξT (t), as T → +∞, do not depend on λ.

1.1.5 Spatial Averaging of the Vibrational Type Coefficient Undoubtedly, the question on the influence of the coefficient a of a vibrational type, e.g., a(x) = sin x, on the behavior of the solution ξT , as T → +∞, is very intriguing. For the first time such a problem for Eq. (1.1) was studied in [31] using the methods developed in the papers [29] and [30]. In particular, the results of this paper demonstrate that the existence of the spatial averaging of a of the form 1 x

x

⎧ ⎨

exp −2 ⎩ 0

⎫ ⎬

u a(v) dv 0

⎭

du → σ1 ,

1 x

x

⎧ u ⎫ ⎨  ⎬ exp 2 a(v) dv du → σ2 , ⎩ ⎭

0

0

(1.12) as |x| → +∞, 0 < σi < +∞, supplies that the finite-dimensional distributions √T ) , t > 0, where ξ is the solution of (1.1), converge, as of the process ξT (t) = ξ(t T T → +∞, to the finite-dimensional distributions of the process σ0 W , where W is 1 a Wiener process, σ0 = (σ1 σ2 )− 2 . In the case where a(x) = sin x, we have that −1  +∞ 1 σ0 = , or, in other words, σ10 is the so-called Bessel’s constant. It is n=0 (n!)2 clear that here we have σ0 < 12 , and the coefficient a “eats” in the limit some part of the coefficient of the Wiener process W , or, in other words, coefficient a changes in the limit the structure of the environment, the mathematical model of which is described by Eq. (1.1). We note that the principle of spatial averaging of the coefficients of the stochastic differential equations was first substantiated in the paper [31]. This principle means that the existence of the spatial averaging (1.12) in Eq. (1.1) allows to substitute the averaged constants, instead of the coefficients of the equation, and get the equation 1 η(t) = √ W (t). σ1 σ2 √ ) ∼ η(t √ ) , as t → Herewith the limit distributions of the solutions coincide, i.e., ξ(t t t +∞. Let us mention that the problem (1.1) and (1.2) with λ = 0 was studied in [75], using purely analytical methods. More precisely, the limiting behavior of the characteristic operators of the Markov processes was investigated.

1.2 Equation with the Non-unit Diffusion Coefficient

11

1.1.6 Functionals of the Solution We ask the reader to pay attention to the fact that by investigating the problem (1.1) and (1.2), we deal with the study of the limiting behavior at +∞ of the distribution t of the functional √1t a (ξ(s)) ds depending on the solution ξ of Eq. (1.1). The 0

significance of the investigation of the limit behavior, at +∞, of integral functionals of the solutions of SDE is very well known (see, e.g., [81]). The limit behavior of the functionals of integral type of unstable solutions ξ of Eq. (1.1) was first reviewed in [34] and [36], whereby a method was used which is based on the representation of the type (1.5). Furthermore, the problem of the asymptotic behavior of the finite-dimensional distributions of the solution of Eq. (1.3) is very important in the case where the external disturbance is non-regular, but is a Wiener process only in the limit. For example, consider the equations of the form t ξT (t) = x0 +

aT (ξT (s)) ds + ηT (t),

(1.13)

0

where ηT (t) is a family of a.s. continuous, square-integrable martingales with P

quadratic characteristic ηT (t) → t, as T → +∞. It means that a Wiener process in Eq. (1.13) appears only in the limit. The asymptotic behavior of the finite-dimensional distributions of the solutions ξT of Eq. (1.13) and of the respective integral functionals, in the case where ηT (t) = WT (t), was first obtained in the paper [39]. In this case it is necessary to impose additional conditions that connect with the rate of degeneracy of the coefficients aT = aT (x) at some points, for example, if |aT (xk )| → +∞, as T → +∞, and with the rate of convergence ηT (t) − t→0, under the same condition T → +∞. Not less important is the discrete analogue of the limit theorems for unstable solutions. The first results in this direction were obtained in the paper [37].

1.2 Equation with the Non-unit Diffusion Coefficient For a deeper understanding of the essence of the established facts concerning the influence of the coefficients on the limit behavior of the distribution of the solution of Eq. (1.1), it is reasonable to consider Eq. (1.1) as a mathematical model of the physical phenomenon of diffusion, or as the model of the trajectory of the motion of a small Brownian particle, the mass of which we can neglect and which is in a liquid or in gas and moves under the influence of micro- and macroscopic factors of the environment. However, the coefficient which determines its motion, so-called diffusion coefficient, can differ from being simply 1. In such a more general case,

12

1 Introduction to Unstable Processes and Their AsymptoticBehavior

when the motion is guided by some nontrivial diffusion coefficient, it is known that the projection on the axis of the trajectory of the motion of the small Brownian particle in the environment, homogeneous in time, is well described by the SDE t ξ(t) = x0 +

a (ξ(s)) ds + 0

t

where

t σ (ξ(s)) dW (s),

(1.14)

0

a (ξ(s)) ds is an ordinary Lebesgue integral,

0

t

σ (ξ(s)) dW (s) is a

0

stochastic Itô integral, and W , as usual, is a Wiener process defined on some probability space (Ω, F, P), Ft = σ {W (s), s ≤ t}. Recall that W describes the diffusion in the homogeneous environment. This means that a more general class of stochastic differential equations appears on the scene. Obviously, (1.14) with σ (x) ≡ 1 coincides with Eq. (1.1). Equation (1.14) can be written in the differential Itô form dξ(t) = a (ξ(t)) dt + σ (ξ(t)) dW (t),

ξ(0) = x0 .

(1.15)

Discretizing, we can say that the drift ξ(t) = ξ(t + t) − ξ(t) of the Brownian particle on a comparatively small time interval t has the form ξ(t)  a (ξ(t)) t + σ (ξ(t)) W (t), where W (t) = W (t + t) − W (t) is the drift of the Brownian particle in a motionless homogeneous environment. So, a(x) is a macroscopic characteristic of the environment, in other words, the velocity of its movement calculated at the point x, while σ 2 (x) is a microscopic characteristic of the environment, in other words, the intensity of thermal chaotic motion of the environment’s molecules at the point x. Therefore coefficient a in Eq. (1.15) is called drift coefficient, while σ is called diffusion coefficient of the environment. Standard conditions (I1 )

there exists a constant L > 0 such that for any x ∈ R |a(x)|2 + |σ (x)|2 ≤ L(1 + |x|2 );

(I2 )

for any N > 0 there exist constants LN > 0, δN > 0 such that 0 < δN ≤ σ (x) ≤ LN for |x| ≤ N

imply that Eq. (1.15) has a weak solution (ξ, W ), and this solution is weakly unique. Moreover, ξ is a homogeneous process having the strong Markov property [80]. Under the conditions (I1 ), (I2 ) and (I3 )

for any N > 0 there exists a constant LN > 0 such that the variation N

V σ (x) ≤ LN , and σ (x) ≥ σ0 > 0 for any x ∈ R,

−N

1.2 Equation with the Non-unit Diffusion Coefficient

13

Eq. (1.15) has a strong solution ξ , and this solution is strongly unique [18]. Various situations concerning existence and uniqueness of the strong and weak solutions are described in the Introduction to the book [9]. Concerning the present book, sufficient conditions for the weak convergence, as T → +∞, were obtained for integral functionals of unstable solutions ξ of Eq. (1.15) and real-valued function g = g(x). Namely, they are established for βT(1) (t)

1 = B1 (T )

tT

βT(2) (t)

g (ξ (s)) ds, 0

1 = B2 (T )

tT g (ξ (s)) dW (s) , t ≥ 0, 0

where Bi (T ) are normalizing constants, and Bi (T ) → +∞, as T → +∞, (ξ, W ) is a weak solution of Eq. (1.15). New classes of limit distributions are obtained that are distributions of the following functionals ⎡

⎤ ζ(t ) t ⎢ ⎥ β (1)(t) = 2 ⎣ b(u) du − b(ζ (s)) dζ (s)⎦ , 0

 β (2) (t) = ξ β (1) (t),

0

where, b(x) = b1 for x ≥ 0 and b(x) = b2 for x < 0, ζ(t) is the strong solution of the Itô equation dζ (t) = σ (ζ(t)) dW (t). Here σ (x) = σ1 for x ≥ 0 and σ (x) = σ2 for x < 0, 0 < σi , i = 1, 2, ξ is a standard Gaussian random variable, independent of ζ(t). Previously, such limit  (t), where W  is a Wiener distributions were obtained in the book [81] for ζ (t) = W process, in the process of studying additive functionals of random walk. Now we mention, for the convenience of the readers, some related results, which either complement the theorems proved in the book, or can serve as illustrations and applications. Namely, in the paper [35], a consistent non-asymptotically normal drift parameter estimator for the Itô SDE is constructed. The law of the iterated logarithm is established for the one-dimensional diffusion process in [43]. The asymptotic behavior of the solution of the Cauchy problem for parabolic equations of the second order with non-regular dependence of the coefficients on a parameter is investigated in [49] in a nonclassical case, where the pointwise convergence of the coefficients of the equation is not required. An analysis of the asymptotic behavior of a harmonic oscillator with external perturbation by random processes of the “white and shot noises” types is carried out in [44]. As a kind of the pioneer result, an analog of the well-known arcsine law for a Wiener process is obtained in [7] for a Brownian motion in a bilayer environment. In the paper [46], the asymptotic behavior of the one-dimensional Itô SDE with non-regular dependence of the coefficients on a parameter is investigated. In this case, the convergence (in a certain sense) of the drift coefficient to functions of the

14

1 Introduction to Unstable Processes and Their AsymptoticBehavior

form ck |x − xk |−α , where α > 0, is assumed. In [66], a method for studying the convergence rate of the normalized unstable SDEs’ solutions to the limit process is proposed. In [51], the results of the paper [37] are generalized in the sense that the weak convergence of a sequence of Markov chains to a diffusion type process is obtained. Similar problems for unstable solutions of systems of SDEs are considered in the paper [15]. In [50], an exact order of growth and the law of the iterated logarithm for a part of components of the solutions for SDEs in Rn are obtained. In the paper [38], the weak convergence for a part of normalized unstable components of the solutions for SDEs in Rn to the solutions of some SDE with discontinuous coefficients or to the Bessel diffusion process is considered. The paper [47] investigates the asymptotic behavior of distributions for normalized unstable solutions of some systems of SDEs with Poisson jumps. The weak convergence for the solutions of SDEs to the discontinuous processes in the Skorokhod topology is considered in [45]. The asymptotic behavior of functionals of the solutions to inhomogeneous Itô stochastic differential equations with non-regular dependence on a parameter is studied in [48]. For modern issues of existence, uniqueness of solutions, their properties and asymptotic analysis, all this in the context of singularity and instability, we refer the readers to the papers [6, 13, 58, 58–64, 71]. Instability in the dynamical system involving fractional Brownian motion is considered in [16].

Chapter 2

Convergence of Unstable Solutions of SDEs to Homogeneous Markov Processes with Discontinuous Transition Density

In this chapter we consider one-dimensional homogeneous stochastic differential equations with stochastically unstable solutions. Conditions on the coefficients of the equations leading to instability of the solutions are established in Sect. 2.1. Necessary and sufficient conditions for the weak convergence of the stochastically unstable solutions to a Brownian motion in two-layer environment are formulated and proved in Sect. 2.2. Necessary and sufficient conditions for the weak convergence of stochastically unstable solutions of SDEs to a process of skew Brownian motion type are obtained in Sect. 2.3. Section 2.4 contains several examples that illustrate statements about the weak convergence of the stochastically unstable solutions. Auxiliary results are collected in Appendix A.

2.1 Preliminaries Let the real-valued measurable functions a = a(x) and σ = σ (x) : R → R be given. Assume that a and σ satisfy the following conditions: (i) a and σ are of linear growth, i.e., the following inequality holds: for some L > 0 and any x ∈ R |a(x)| + |σ (x)| ≤ L(1 + |x|); (ii) for any N > 0 there exists δN > 0 such that σ (x) ≥ δN > 0 for |x| ≤ N. Also, let (Ω, F, F, P) be a complete probability space with filtration F = {Ft }t ≥0 satisfying the standard assumptions, and let W = {W (t), t ≥ 0} be a onedimensional Wiener process which is adapted to the filtration F.

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_2

15

16

2

Convergence of Unstable Solutions to Homogeneous Markov Process

Consider the stochastic differential equation (SDE) dξ(t) = a (ξ(t)) dt + σ (ξ(t)) dW (t),

t ≥0

(2.1)

with F0 -measurable initial condition ξ(0). The function a is called the drift coefficient, and the function σ is called the diffusion coefficient of the SDE (2.1). Note that everywhere in the book where we refer to Eq. (2.1), we assume that the conditions (i)–(ii) are satisfied. Definition 2.1 A strong solution of Eq. (2.1) is a progressively measurable process {ξ(t), t ≥ 0} satisfying the integral equation t ξ(t) = ξ(0) +

t a (ξ(s)) ds +

0

σ (ξ(s)) dW (s) 0

a.s. for any t ≥ 0. Here

t

a (ξ(s)) ds is a Lebesgue integral, and

0

t

σ (ξ(s)) dW (s) is a stochastic

0

Itô integral. Definition 2.2 A weak solution to Eq. (2.1) is a triple consisting of the following components: – a stochastic basis (Ω , F , {F t }t ≥0, P ); – a Wiener process W on this basis; d

– an adapted process ξ = {ξ (t), t ≥ 0} on this basis such that ξ (0) = ξ(0) and     dξ (t) = a ξ (t) dt + σ ξ (t) dW (t),

t ≥ 0.

If for any two solutions ξ1 and ξ2 of Eq. (2.1), their finite-dimensional distributions coincide, then the solution ξ of Eq. (2.1) is called weakly unique. It means that we have uniqueness of the solution in distribution. If with probability 1 the paths for any two solutions ξ1 and ξ2 of Eq. (2.1) coincide, then the solution of Eq. (2.1) is called strongly unique, that is, we have a pathwise uniqueness. It is known [80, Chapter 1, § 2.1] that a weakly unique solution ξ of Eq. (2.1) is a homogeneous strong Markov process. Remark 2.1 The existence of a weak solution of Eq. (2.1) was first proved in [79], Chapter 3, §3 under the conditions of continuity and linear growth of the coefficients. Later in the paper [28] the existence of a weak solution and its weak uniqueness were proved under the condition σ (x) ≥ δ > 0 and the condition of boundedness of coefficients. The existence of a weak solution and its weak uniqueness follow from the paper [28] under conditions (i)–(ii).

2.1 Preliminaries

17

An overview of the basic definitions and facts related to different types of existence and uniqueness of the solution to SDEs is contained in the monograph [9], see also [65]. Remark 2.2 On the one hand, it is clear that a strong solution of Eq. (2.1) is of course also a weak solution, and the strong uniqueness of the solution implies weak uniqueness. On the other hand, to study the asymptotic behavior, as t → +∞, of the distribution of the solution, it is sufficient to assume the existence of a weak solution ξ to Eq. (2.1). In this connection, we will not distinguish between weak and strong solutions in the problems concerning the distribution of solutions. It should also be emphasized that the asymptotic behavior, as t → +∞, of solutions to SDE (2.1) is closely related to the behavior, as |x| → +∞, of the function ⎫ ⎧ x ⎬ ⎨ u a(v) dv f (x) = exp −2 du. (2.2) ⎩ σ 2 (v) ⎭ 0

0

Evidently, f is a positive function satisfying the equation 1 f (x)a(x) + f (x)σ 2 (x) = 0 2

(2.3)

a.e. with respect to the Lebesgue measure. The role of this function has been studied in detail in [17, § 16]. Since we consider stochastically unstable solutions, our goal is to formulate the conditions for the stochastic instability of the solutions to SDEs. Lemma 2.1 Let ξ be a solution to Eq. (2.1) and let, additionally to (i)–(ii), the following assumption hold for some constant C > 0 and for all x ∈ R: 0 < f (x) σ (x) ≤ C.

(2.4)

Then for any constant N > 0 1 lim t →+∞ t

t P {|ξ(s)| < N} ds = 0, 0

i.e., the solution of Eq. (2.1) is stochastically unstable. Proof Consider the function x ΦN (x) = 2 0

⎛ u  f (u) ⎝ 0

⎞ χ|v|≤N dv ⎠ du, f (v)σ 2 (v)

18

2

Convergence of Unstable Solutions to Homogeneous Markov Process

where χA is the indicator of the set A. Function ΦN (x) has the continuous first derivative ΦN (x)



x

= 2f (x)

χ|v|≤N dv. f (v)σ 2 (v)

0

Also, its second derivative ΦN (x)

x



= 2f (x)

χ|v|≤N χ|x|≤N dv + 2f (x) f (v)σ 2 (v) f (x)σ 2 (x)

0

exists a.e. with respect to the Lebesgue measure and is locally integrable. Taking into account the relation (2.3), we obtain 1 ΦN (x) a(x) + ΦN (x) σ 2 (x) = χ|x|≤N 2

(2.5)

a.e. with respect to the Lebesgue measure. Applying the Itô formula from Lemma A.3 to the process ΦN (ξ(t)), where ξ is a solution to Eq. (2.1), and equality (2.5), we conclude that t  1 ΦN (ξ(s)) a(ξ(s)) + ΦN (ξ(s)) σ 2 (ξ(s)) ds ΦN (ξ(t)) − ΦN (ξ(0)) = 2 0

t +

ΦN (ξ(s)) σ (ξ(s)) dW (s).

0

Therefore, we have t

t χ|ξ(s)|≤N ds = ΦN (ξ(t)) − ΦN (ξ(0)) −

0

ΦN (ξ(s)) σ (ξ(s)) dW (s).

0

Recall that according to (2.4) we have 0 < f (x) σ (x) ≤ C. Also, according to linear growth of a (assumption (i)) and to assumption (ii), both functions f (v) and σ 2 (v) are separated from zero on the interval [−N, N] by some positive constant u χ|v|≤N that depends on N. As a result, both functions f (v)σ 2 (v) dv and ΦN (x) σ (x) are 0

bounded, and so ⎞ ⎛ t  E ⎝ ΦN (ξ(s)) σ (ξ(s)) dW (s)⎠ = 0. 0

2.1 Preliminaries

19

Furthermore, x ΦN (x) − ΦN (y) = 2

⎛ u  ⎝ f (u)

y

⎞ χ|v|≤N dv ⎠ du, f (v)σ 2 (v)

0

and taking into account the boundedness of the positive integral

u 0

χ|v|≤N dv f (v)σ 2 (v)

by

some constant depending on N, we can deduce that ! x ! ! ! ! ! ! |ΦN (x) − ΦN (y)| ≤ CN ! f (u)du!! = CN |f (x) − f (y)|. ! ! y

So, we can bound from above the desirable probability as follows: 1 t

t P {|ξ(s)| < N} ds =

1 E [ΦN (ξ(t)) − ΦN (ξ(0))] t

0

1 ≤ CN E |f (ξ(t)) − f (ξ(0))| ≤ CN t

"

1 E |f (ξ(t)) − f (ξ(0))|2 t2

#1 2

.

Applying the Itô formula to the process f (ξ(t)), we conclude that t  f (ξ(t)) − f (ξ(0)) =

1 f (ξ(s)) a(ξ(s)) + f (ξ(s)) σ 2 (ξ(s)) ds 2

0

t +

t



f (ξ(s)) σ (ξ(s)) dW (s) = 0

f (ξ(s)) σ (ξ(s)) dW (s).

0

Taking into account the property of stochastic integrals and the assumptions of the lemma, we have t E |f (ξ(t)) − f (ξ(0))| = 2

%2 $ E f (ξ(s)) σ (ξ(s)) ds ≤ Ct.

0

Therefore, we conclude that 1 lim t →+∞ t

&

t P {|ξ(s)| < N} ds ≤ lim CN t →+∞

C = 0, t

0

and the solution ξ to Eq. (2.1) is stochastically unstable.

 

20

2

Convergence of Unstable Solutions to Homogeneous Markov Process

2.2 Necessary and Sufficient Conditions for the Weak Convergence of Solutions of SDEs to a Brownian Motion in a Bilayer Environment Let T > 0 be a parameter. Consider the processes ζT (t) =

f (ξ(tT )) , √ T

WT (t) =

W (tT ) √ , T

(2.6)

where the function f (x) is defined by (2.2), and the processes ξ(t) and W (t) are related via Eq. (2.1). Evidently, for any fixed T > 0 the process WT = {WT (t), t ≥ 0} is a Wiener process. Let ϕ(x) be the inverse function to f (x), and   √    √   σT (x) = f ϕ x T σ ϕ x T ,

x QT (x) =

 σT−2 (u) du.

(2.7)

0

Note that the function QT exists under the conditions (i)–(ii). Also, let ζ be a solution of the SDE t ζ (t) =

σ (ζ (s)) dW (s),

(2.8)

0

where  σ (x) =

σ1 , x ≥ 0, σ2 , x < 0,

0 < σi < +∞.

Now, let c1 > 0 and c2 > 0 be two constants. Denote  ξ (t) = l(ζ (t)),

(2.9)

where l(x) =

x , c(x)

 c(x) =

c1 , x ≥ 0, c2 , x < 0.

Remark 2.3 The process (2.9) with c(x) = σ (x) is called the skew Brownian motion (see [24, Section 4.2, Problem 1]). Therefore the process (2.9) will be called the process of the skew Brownian motion type. See Appendix A.3 for more details.

2.2

Convergence to a Brownian Motion in a Bilayer Environment

21

Remark 2.4 The solution ζ to Eq. (2.8) is a homogeneous strong Markov process with the transition density

ρ(t, x, y) =

⎧ ( ⎪ − ⎪ 1 ⎪ √ ⎪ e ⎪ σ1 2πt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2σ1 · √1 σ1 +σ2

(y−x)2 2σ 2 t 1

σ2 2πt

e

−

−

σ1 −σ2 σ1 +σ2 e

(σ1 y−σ2 x)2 2σ12 σ22 t

⎪ − ⎪ ⎪ 2σ2 ⎪ √1 · e ⎪ σ1 +σ2 σ1 2πt ⎪ ⎪ ( (y−x)2 ⎪ ⎪ − ⎪ ⎪ √1 2σ22 t ⎪ − ⎪ ⎩ σ2 2πt e

−

,

σ2 −σ1 σ1 +σ2 e

) ,

x ≥ 0, y > 0,

x ≥ 0, y < 0,

,

(σ2 y−σ1 x)2 2σ 2 σ 2 t 1 2

(y+x)2 2σ 2 t 1

−

(2.10)

x ≤ 0, y > 0, )

(y+x)2 2σ22 t

,

x ≤ 0, y < 0.

See Appendix A.3 for a proof of this fact. Note that the transition density of the solution ζ to Eq. (2.8) was first obtained in [29]. In what follows constants C and L do not depend on x and T . Now, we are in a position to prove the following statement. Lemma 2.2 Let ξ be a solution to Eq. (2.1) and let, additionally to (i)–(ii) and (2.4), the following assumption hold for some constant C > 0 and for all x ∈ R: 1 f (x)

x

du ≤ C. f (u)σ 2 (u)

(2.11)

0

 > 0 such that for any ε > 0 and any t ≥ 0 the Then there exist some constant C following inequality holds: t

√  t. P {|ζT (s)| < ε} ds ≤ ε C

0

Proof Indeed, applying the Itô formula to the process Φ1 (ζT (t)), where ⎞ ⎛ x u Φ1 (x) = 2 ⎝  σT−2 (v)χ(−ε,ε) (v) dv ⎠ du, 0

0

we obtain similarly to Lemma 2.1 that t P {|ζT (s)| < ε} ds = E [Φ1 (ζT (t)) − Φ1 (ζT (0))] 0

ε ≤2 −ε

dv E |ζT (t) − ζT (0)|  σT2 (v)

(2.12)

22

2

Convergence of Unstable Solutions to Homogeneous Markov Process

⎡ 1 ⎢ = 2ε ⎣ √ f (ϕ(ε T ))

+

√ ϕ(−ε  T)

1

√ f (ϕ(−ε T ))

√ ϕ(ε  T)

1 f (u)σ 2 (u)

du

0

⎤

1 ⎥ du⎦ E |ζT (t) − ζT (0)| f (u)σ 2 (u)

0

≤ 4εCE |ζT (t) − ζT (0)| . From the last equality, using the boundedness of the function  σT (x) and the inequality ⎛ t ⎞2  t 2 σT (ζT (s)) dWT (s)⎠ = E σT2 (ζT (s)) ds ≤ C 2 t, E |ζT (t) − ζT (0)| = E ⎝  0

0

 = 4C 2 . we have (2.12) for C

 

Theorem 2.1 Let ξ be a solution of Eq. (2.1) and let, additionally to (i)–(ii), there exist a constant C > 0 such that, for all x ∈ R, the inequalities (2.4) and (2.11) hold. Then the stochastic process ζT = {ζT (t), t ≥ 0} from (2.6) converges weakly, as T → +∞, to the process ζ satisfying (2.8) if and only if ⎧ x ⎨ 12 , x → +∞, 1 du σ1 → (2.13) ⎩ 12 , x → −∞. f (x) f (u)σ 2 (u) σ2

0

Proof Sufficiency Let convergence (2.13) hold. Note that the functions f (x) and f (x) are continuous, the function f (x) is locally integrable and we have equality (2.3) a.e. with respect to the Lebesgue measure. Therefore, we can apply the Itô formula from Lemma A.3 to the process ξ(tT ) and obtain t T  1 f (ξ(tT )) − f (ξ(0)) = f (ξ(s)) a(ξ(s)) + f (ξ(s)) σ 2 (ξ(s)) ds 2 0

t T +

t T



f (ξ(s)) σ (ξ(s)) dW (s) = 0

f (ξ(s)) σ (ξ(s)) dW (s).

0

Therefore, we have 1 ζT (t) = ζT (0) + √ T

t 0

f (ξ(sT )) σ (ξ(sT )) dW (sT ).

2.2

Convergence to a Brownian Motion in a Bilayer Environment

23

 √  Taking into account the equality ξ(tT ) = ϕ (f (ξ(tT ))) = ϕ ζT (t) T , we proceed as follows: t ζT (t) = ζT (0) +

 σT (ζT (s)) dWT (s).

(2.14)

0

Since the functions f (x)σ (x) and, consequently,  σT (x) are bounded, we have the following relations: lim

sup P {|ζT (t)| > C} = 0,

lim

C→+∞ T →+∞ 0≤t ≤L

lim lim sup

sup

h→0 T →+∞ |t1 −t2 |≤h, ti ε} = 0

for any L > 0 and ε > 0. It is clear that such relations hold for the process WT as well. According to Theorem A.12, given an arbitrary sequence Tn → +∞, we * * * can choose  a subsequence  Tn → +∞, a probability space (Ω, F, P), and a *Tn (t) of stochastic processes, defined on this space and such sequence * ζTn (t), W that their finite-dimensional distributions coincide with those of the processes * *   P P *Tn (t) −→ * (t), as Tn → +∞, ζTn (t), WTn (t) and, moreover, * ζTn (t) −→ * ζ (t), W W * (t) are some stochastic processes. for all t ≥ 0, where * ζ (t), W   It follows from the weak equivalence of the processes ζTn (t), WTn (t) and the   *Tn (t) , combined with Eq. (2.14), Lemma A.11 (see Corolprocesses * ζTn (t), W lary A.2) and Lemma A.13, that * ζTn (t) = * ζTn (0) +

t

  *Tn (s).  σTn * ζTn (s) d W

(2.15)

0

Now we are in a position to establish that t

  ζTn (s) ds −  σT2n *

0

t

  * P ζ (s) ds −→ 0, σ2 *

(2.16)

0

as Tn → +∞, where the function σ (x) is defined by (2.8). Consider the function x ΦTn (x) = 2 0

⎡ ⎣u −

u 0

⎤ σ 2 (v)  σT2n (v)

dv ⎦ du.

24

2

Convergence of Unstable Solutions to Homogeneous Markov Process

It satisfies the following relations: ⎡ ΦT n (x) = 2 ⎣x −

x 0

⎤ σ 2 (v) dv ⎦ ,  σT2n (v)

ΦT n (x) = 2

 σT2n (x) − σ 2 (x)  σT2n (x)

a.e. with respect to the Lebesgue measure. The functions ΦTn (x) and ΦT n (x) are continuous in x for every Tn , function ΦT n (x) is locally integrable. Therefore, we can apply the Itô formula from   Lemma A.3 to the process ΦTn * ζTn (t) and obtain     1 ΦTn * ζTn (t) − ΦTn * ζTn (0) = 2

t

  2   ζTn (s)  ζTn (s) ds ΦT n * σTn *

0

t +

    *Tn (s) ΦT n * σTn * ζTn (s)  ζTn (s) d W

0

t  t         2 * 2 * *Tn (s). =  σTn ζTn (s) − σ ζTn (s) ds + ΦT n * ζTn (s)  ζTn (s) d W σTn * 0

0

Therefore, we have t  σT2n

  * ζTn (s) ds −

0

t

  σ2 * ζTn (s) ds

0

    ζTn (t) − ΦTn * ζTn (0) − = ΦTn *

t

    *Tn (s). ζTn (s)  ζTn (s) d W ΦT n * σTn *

0

(2.17) Since ! ! ! ! ! !Φ (x)! = 2 !!x − σ 2 (x)   x√  Tn ! f ϕ x Tn !

√ ϕ ( x Tn )

0

! ! ! dv ! !, 2 f (v) σ (v) ! !

according to the condition (2.13), we have for some constant C > 0 and for any constant N > 0 ! ! !Φ (x)! ≤ C |x| , Tn

lim

! ! sup !ΦT n (x)! = 0.

Tn →+∞ |x|≤N

2.2

Convergence to a Brownian Motion in a Bilayer Environment

25

Consider the set + , + , ! ! ! ! - . B = sup !* and let PN = P sup !* ζTn (s)! ≤ N ζTn (s)! > N = P B c . 0≤s≤t

0≤s≤t

For any ε > 0 and N > 0 we have the following inequalities: .    ! -! ζTn (t) − ΦTn * ζTn (0) ! > ε P !ΦTn * -!  !   . = P !ΦTn * ζTn (t) − ΦTn * ζTn (0) ! (χB + χB c ) > ε /!    ! ε0 ζTn (t) − ΦTn * ζTn (0) ! χB > ≤ P !ΦTn * 2 /!  !   ε0 ζTn (t) − ΦTn * ζTn (0) ! χB c > +P !ΦTn * 2 , + /! 0 ! !    ! ε ! ! ! !* * * ≤ P ΦTn ζTn (t) − ΦTn ζTn (0) χB > + P sup ζTn (s) > N 2 0≤s≤t    !  2 ! ≤ PN + E !ΦTn * ζTn (t) − ΦTn * ζTn (0) ! χB ε ! ! 4N sup !ΦT n (x)! , ≤ PN + ε |x|≤N ⎫ ! ⎧! t ! ⎬ ⎨!! !     *Tn (s)! > ε ζTn (s)  ζTn (s) d W σTn * P !! ΦT n * ! ⎭ ⎩! ! 0 ⎧! t ! ⎫ ! ⎨!! ⎬ !     ε *Tn (s)! > ≤ P !! ΦT n * ζTn (s)  ζTn (s) χB d W σTn * ! 2⎭ ⎩! ! 0 ⎧! t ⎫ ! ! ⎨!! ⎬ !     ε *Tn (s)! > ζTn (s)  ζTn (s) χB c d W +P !! ΦT n * σTn * ! 2 ⎭ = I1 + I2 . ⎩! ! 0

We apply Chebyshev’s inequality and obtain ⎡ t ⎤2 " #2      2 *Tn (s)⎦ I1 ≤ ζTn (s)  ζTn (s) χB d W E ⎣ ΦT n * σTn * ε 0

4 = 2 ε

t 0

!2 !   $  %2 4 E ΦT n * σTn * ζTn (s)  ζTn (s) χB ds ≤ 2 Ct sup !ΦT n (x)! . ε |x|≤N

26

2

Convergence of Unstable Solutions to Homogeneous Markov Process

It is clear that ⎫ ⎧! t ! ! ⎬ ⎨!! !     *Tn (s)! > ε ≤ PN . ζTn (s)  ζTn (s) χB c d W σTn * I2 = P !! ΦT n * ! 2⎭ ⎩! ! 0

Therefore, from the previous inequalities passing to the limit as Tn → +∞ then as N → +∞ we obtain that right-hand side in (2.17) tends to zero in probability. Consequently, we have the convergence t  σT2n

  * ζTn (s) ds −

0

t

*   P ζTn (s) ds −→ 0, σ2 *

(2.18)

0

as Tn → +∞. Taking into account (2.12), we obtain t

. P * ζ (s) = 0 ds = 0.

0

Therefore, t

*   P ζTn (s) ds −→ σ2 *

0

t

  ζ (s) ds, σ2 *

0

as Tn → +∞. Taking into account (2.18), we have (2.16). Now, consider the process from relation (2.15): t γTn (t) =

  *Tn (s). ζTn (s) d W  σTn *

0

For every fixed Tn , γTn (t) is a martingale with respect to the σ -algebra * . P σ γTn (s), s ≤ t and γTn (t) −→ ζ (t), as Tn → +∞. For the quadratic characteristic of this martingale we have t  σT2n

γTn (t) = 0

*   P * ζTn (s) ds −→

t 0

  σ2 * ζ (s) ds.

2.2

Convergence to a Brownian Motion in a Bilayer Environment

27

* According to-Lemma A.9, . the limit process ζ (t) is a martingale with respect to the σ -algebra σ * ζ (s), s ≤ t whose quadratic characteristic is t

* ζ (t) =

  ζ (s) ds. σ2 *

0

Thus, according to the Doob theorem (see Theorem A.4), there exists a Wiener  (t) such that process W * ζ (t) =

t

   (s). σ * ζ (s) d W

(2.19)

0 Since the sequence Tn → +∞ is arbitrary and since the solution * ζ (t) is strongly unique (see Theorem A.11), then the finite-dimensional distributions of the process ζT (t) converge, as T → +∞, to the corresponding finite-dimensional distributions of the solution ζ (t) to Eq. (2.8). According to Theorem A.13, for the weak convergence of the processes ζT (t) to ζ (t) it is sufficient to prove that for every N > 0 and for any ε > 0 + ,

lim lim sup P

h→0 T →+∞

sup

|t1 −t2 |≤h, ti ≤L

|ζT (t2 ) − ζT (t1 )| > ε = 0.

(2.20)

In order to prove (2.20), we use the inequalities sup

|t1 −t2 |≤h, ti ≤L

|ζT (t2 ) − ζT (t1 )| ≤ 2 sup

sup

|ζT (t) − ζT (kh)| (2.21)

≤ 4 sup

sup

|ζT (t) − ζT (kh)| .

kh≤L kh≤t ≤(k+2)h

kh≤L kh≤t ≤(k+1)h

Therefore, relation (2.20) follows from the inequalities + , P

sup

|t1 −t2 |≤h; ti ≤L

|ζT (t2 ) − ζT (t1 )| > ε ,

+ ≤ P 4 sup

sup

kh≤L; kh≤t ≤(k+1)h

≤

1 kh≤L

+

|ζT (t) − ζT (kh)| > ε

ε |ζT (t) − ζT (kh)| > P sup 4 kh≤t ≤(k+1)h

,

⎛ (k+1)h ⎞4  1 1 " 4 #4 E⎝  σT (ζT (s)) dWT (s)⎠ ≤ C h2 . ≤ ε kh≤L

kh

kh≤L

28

2

Convergence of Unstable Solutions to Homogeneous Markov Process

The sufficiency is proved. Necessity Let the assumptions of Theorem 2.1 hold and let the process ζT converge weakly, as T → +∞, to the solution ζ of Eq. (2.8). Note that the process ζT satisfies Eq. (2.14) and  σT (x) ≤ C. Then for the process (ζT (t), WT (t), ζT  (t)) , where ζT  (t) is the quadratic characteristic of martingale ζT , the assumptions of Lemma A.12 (Skorokhod’s representation theorem or the Skorokhod’s convergent subsequence principle) hold. According to this principle, given an arbitrary sequence Tn → +∞, we can choose a subsequence Tn → +∞, a probability space * * (Ω, F, * P), and a stochastic process   *Tn (t), * * ζTn (t), W ζTn (t) defined on this space such whose finite-dimensional distributions coincide with those of the process 

 ζTn (t), WTn (t), ζTn (t) ,

and, moreover, *

*

*

P P P *Tn (t) −→ * (t), * * ζTn (t) −→ * ζ (t), W W ζTn (t) −→ β(t),

* (t), β(t) are some stochastic processes. as Tn → +∞, where * ζ (t), W According to Lemma A.13, relation (2.15) holds for the processes * ζTn (t) and * WTn (t), as well. -Therefore, * ζTn.(t) is a martingale for every fixed Tn with respect to the σ -algebra σ * ζTn (s), s ≤ t . For the quadratic characteristic of this martingale we have for some β(t) the convergence * ζTn (t) =

t

*   P  σT2n * ζTn (s) ds −→ β(t),

0

as Tn → +∞. Note that * ζTn (0) → 0, as Tn → +∞, with probability 1. According- to Lemma.A.9, we obtain that * ζ is a martingale with respect to the σ -algebra σ * ζ (s), s ≤ t with the quadratic characteristic * ζ (t) = β(t). Since the * finite-dimensional distributions of the processes ζ (t) and ζ(t) coincide, according to Eq. (2.8) and the boundedness of the coefficient σ , we obtain that t ζ  (t) =

σ 2 (ζ (s)) ds. 0

2.2

Convergence to a Brownian Motion in a Bilayer Environment

29

Consequently t β(t) =

  ζ (s) ds. σ2 *

0

Therefore, t  σT2n

*   P * ζTn (s) ds −→

t

0

  ζ (s) ds, σ2 *

0

as Tn → +∞. Taking into account (2.12), we obtain the following convergence t  σT2n

  * ζTn (s) ds −

0

t

*   P ζTn (s) ds −→ 0 σ2 *

0

for all t ≥ 0. Using the Itô formula, we have t  σT2n

  * ζTn (s) ds −

0

t

  ζTn (s) ds σ2 *

0

    = ΦTn * ζTn (t) − ΦTn * ζTn (0) −

t

  ΦT n * ζTn (s), ζTn (s) d*

0

where ΦTn (x) = 2

x 

 u − σ 2 (u)QTn (u) du.

0

Therefore     ζTn (t) − ΦTn * ζTn (0) − ΦTn *

t 0

as Tn → +∞, for every t ≥ 0.

*   P ζTn (s) d* ΦT n * ζTn (s) −→ 0,

(2.22)

30

2

Convergence of Unstable Solutions to Homogeneous Markov Process

The functions QTn (x) are increasing in x for every Tn . Using the assumption (2.11) and the fact that ϕ(x) is the inverse function to f (x), we obtain ! ! ! ! !! 1 !QT (x)! = ! √ n ! Tn ! ! ! ! 1 ! = |x| !   √  ! f ϕ x Tn !

√ ϕ(x  Tn )

0

√ ϕ(x  Tn )

0

! ! ! du ! ! f (u)σ 2 (u) ! !

! ! ! du ! ! ≤ C |x| . f (u) σ 2 (u) ! !

Therefore, there exist a subsequence T*n → +∞ of the sequence Tn and an *n → +∞, at each point increasing function Q(x) such that QT*n (x) → Q(x), as T of continuity of the function Q(x). *n → +∞, in each term of the left-hand part in (2.22). Let us pass to the limit, as T In order to do this, we introduce the function  Φ(x) =2

x 

 u − σ 2 (u) Q(u) du,

0

*n 0 that and obtain for any T t

ΦT* n

  * ζT*n (s) d* ζT*n (s) −

0

t

   * ζ (s) d* ζ (s) Φ

0

⎛ t ⎞  t     * * = ⎝ ΦT* ζT*n (s) − ΦT* ζT*n (s) d* ζ (s) d* ζ (s)⎠ n0

n0

0

0

⎞ ⎛ t  t     *  * ζ (s) d* ζ (s)) − Φ ζ (s) d* ζ (s)⎠ + ⎝ ΦT* n0

0

0

⎛ t ⎞  t     * * + ⎝ ΦT* ζT*n (s) d* ζT*n (s) d* ζT*n (s) − ΦT* ζT*n (s)⎠ n

0

n0

0

= I1 + I2 + I3 .

2.2

Convergence to a Brownian Motion in a Bilayer Environment

31

It is clear that the function ΦT*n (x) is continuous in x for each T*n 0 . According 0

* P

*n → +∞. to Lemma A.7, I1 −→ 0, as T Let τN be the moment of the first exit of the process * ζT*n (t) from interval (−N, N). According to Lemma A.10, we have + P {|I3 | > ε} ≤ P

! ! sup !* ζT*n (s)! > N

,

0≤s≤t

⎛ t ∧τ ⎞2  N      4 * ζTn (s) − ΦT n * ζT*n (s) d* + 2E⎝ ΦT* ζT*n (s)⎠ n 0 ε 0

+ ≤P

! ! ζT*n (s)! > N sup !*

0≤s≤t

,

4 + 2C ε

N ! !2 ! ! (x)! dx. !ΦT*n (x) − ΦT* n 0

−N

Similarly, + P {|I2 | > ε} ≤ P

! ! ζ (s)! > N sup !*

,

0≤s≤t

4 + 2C ε

N ! !2 !  (x)!! dx !ΦT*n (x) − Φ

−N * P

0

* P

*n → +∞, for any ε > 0, L > 0 and N > 0. Therefore, I2 −→ 0, I3 −→ 0, as T *n 0 → +∞. T Consequently, t

ΦT* n

*   P * ζT*n (s) d* ζT*n (s) −→

0

t

   * ζ (s) d* ζ (s) , Φ

0

*n → +∞. Furthermore, for any N > 0 as T ! !  !≤C sup !ΦT*n (x) − Φ(x)

|x|≤N

N

! ! !Q * (u) − Q(u)! du → 0 , Tn

−N

*n → +∞. Thus, as T   *   P  * ΦT*n * ζT*n (t) −→ Φ ζ (t) ,

32

2

Convergence of Unstable Solutions to Homogeneous Markov Process

*n → +∞. Taking into account (2.22), we have the equality as T    * Φ ζ (t) −

t

   * ζ (s) d* ζ (s) = 0 Φ

(2.23)

0

with probability 1 for all t ≥ 0.  (x) = b for all x, where b is some Using (2.23) and Lemma A.12, we have that Φ  (0) = 0, we obtain Φ  (x) = 0 for all x. Therefore, constant. Using condition Φ Q(x) =

x 2

σ (x)

for all x. Taking into account the relation QT*n (x) → Q(x), as T*n → +∞, we get that for x = 0  √  ϕ x T*n

1   2  f ϕ x T*n



du 1 → 2 , f (u) σ 2 (u) σ (x)

0

*n → +∞. as T Since the sequence T*n → +∞ is arbitrary, we have relation (2.13). Theorem 2.1 is proved.   Corollary 2.1 Since the processes ζT and WT satisfy the assumptions of Lemma A.13 and Theorem A.12, in order to study the weak convergence of this processes, as T → +∞, without loss of generality, we can assume that for an arbitrary sequence Tn → +∞ there exist some processes ζ and W such that P

P

ζTn (t) −→ ζ (t), WTn (t) −→ W (t), as Tn → +∞. The considerations of Corollary 2.1 are used for simplification of the proof of some theorems, for example, the proofs of Theorems 4.6 and 4.8 can be simplified. We proceed with some applications of Theorem 2.1. The next remark presents important examples of coefficients a and σ , satisfying conditions (2.4), (2.11), and (2.13). Remark 2.5 The conditions of Theorem 2.1 are satisfied in the following cases: (1) if there exist a derivative σ (x) of the function σ (x) and the integrals +∞ λ1 := 0

1 σ (x) a(x) − dx, λ2 := σ 2 (x) 2 σ (x)

−∞ 0

1 σ (x) a(x) − dx σ 2 (x) 2 σ (x)

2.2

Convergence to a Brownian Motion in a Bilayer Environment

33

with σ1 = σ (0)e−2λ1 and σ2 = σ (0)e−2λ2 , where   R

1 σ (x) a(x) − dx = λ1 − λ2 = λ; σ 2 (x) 2 σ (x)

(2) if the following spatial averaging holds: 1 x

⎧ ⎨

x

u

exp −2 ⎩ 0

1 x

x

⎫ ⎬

a(v) dv du → σ 2 (v) ⎭



λ1 , λ2 ,

0

⎫ ⎧ u  ⎨  a(v) ⎬ du λ3 , dv → exp 2 ⎩ λ4 , σ 2 (v) ⎭ σ 2 (u)

0

0

x → +∞, x → −∞, x → +∞, x → −∞,

0 < λi < +∞ with σ1 =



λ1 λ3

and σ2 =



λ2 λ4 .

Note that case (2) may include, in particular, vibrational coefficients, for example a(x) = sin x, σ (x) = 1, etc. Now, our goal is to study the asymptotic behavior, as t → +∞, of the distribution of the solution ξ to Eq. (2.1). Note that if we know the behavior of the distribution of f (ξ(t)) for regularly varying (at infinity) functions f (x) with index α > 0 (see Definition 4.1), we can find the behavior of the distribution of the solution ξ itself. Lemma 2.3 Let the conditions of Theorem 2.1 be satisfied and let the function f (x) be such that  f (kx) x α , x > 0, = b(x) = lim (2.24) k→+∞ f (k) −c0 |x|α , x < 0, where α > 0 and c0 >√0. Also, let the normalizing factor B = B(T ) > 0 be a solution to the equation T = f (B(T )). Then the finite-dimensional distributions of the process ξT (t) =

ξ(tT ) B(T )

converge, as T → +∞, to the corresponding finite-dimensional distributions of a homogeneous Markov process with the transition density ρ1 (t, x, y) = ρ (t, b(x), b(y)) b (y),

(2.25)

34

2

Convergence of Unstable Solutions to Homogeneous Markov Process

where the function ρ(t, x, y) is defined by (2.10). T) Proof Consider the normalized random process ξT (t) = ξ(t B(T ) , t > 0, where B(T ) √ is the solution to the equation T = f (B(T )). Note that for a monotonically increasing function f (x) there exists the inverse √ function ϕ(x) and the normalizing factor has the form B(T ) = ϕ T . So,   √  √  f (ξ(t √ T )) · T (t) T ϕ ϕ ζ T ξ(tT ) ϕ (f (ξ(tT ))) √  = T√  √  . ξT (t) = √  = = ϕ T ϕ T ϕ T ϕ T

ζTn (see Taking into account the weak equivalence of the processes ζTn and * Theorem 2.1), let us establish that the processes * ξTn (t) =

 √  ϕ * ζTn (t) Tn √  ϕ Tn

and ξTn (t) are weakly equivalent for all Tn . Indeed, , +  √  /  2 0 2  . ϕ ζTn (t) Tn √  P ξTn (t) < x = P < x = P ϕ ζTn (t) Tn < xϕ Tn ϕ Tn +  √  ,  2 0 f xϕ Tn , Tn = P ζTn (t) < = P ζTn (t) Tn < f xϕ √ Tn /

2

and in a similar way we have +  √  , . f xϕ Tn * ξTn (t) < x = * . P * P * ζTn (t) < √ Tn Consequently, . . P ξTn (t) < x = * P * ξTn (t) < x . Similarly we can prove that for any t1 , t2 , . . . , tl and x1 , x2 , . . . , xl . P ξTn (t1 ) < x1 , ξTn (t2 ) < x2 , . . . , ξTn (tl ) < xl . =* P * ξTn (t2 ) < x2 , . . . , * ξTn (tl ) < xl . ξTn (t1 ) < x1 , *

2.2

Convergence to a Brownian Motion in a Bilayer Environment

35 *

P From the proof of Theorem 2.1 we can get that * ζTn (t) −→ * ζ (t), as Tn → +∞. It is clear that for regularly varying at infinity function ϕ(x) we have the relation

⎧ 1 ⎨ x α , x > 0, ϕ(kx)  1 = b1 (x) = lim ⎩ − 1 |x| α , x < 0, k→+∞ ϕ(k) c0

where b1 (x) is the inverse function to the function b(x). Using the properties from [25] of regularly varying at infinity functions (see Lemma A.17), we can get that !  √  ! ! ϕ |x| T ! ! ! α ! √  − |x| !! → 0, sup ! 0 0, we have  √  √ !! 1 !!  |ξT (t1 ) − ξT (t2 )| = √ !ϕ ζT (t1 ) T − ϕ ζT (t2 ) T ! T ≤ C |ζT (t1 ) − ζT (t2 )| , where ϕ(x) is the inverse function to f (x). This inequality yields (2.20) for the process ξT for every N > 0 with any ε > 0. Necessity Let the assumptions of Theorem 2.2 be fulfilled and let the process 1 ξT (t) = ξ(tT )T − 2 converge weakly, as T → +∞, to the solution  ξ (t) of Eq. (2.9). √ 1 Note that ξT (t) = ϕT (ζT (t)) , where ϕT (x) = T − 2 ϕ(x T ), and the process ζT (t) is the solution to Eq. (2.14). According to Theorem A.12, given an arbitrary sequence Tn → +∞, we can choose a subsequence Tn → +∞, a probability space * * (Ω, F, * P), and a stochastic process * ζTn (t), defined on this space, which is weakly *

P ζTn (t) −→ * ζ (t), as Tn → +∞. equivalent to the process ζTn (t), herewith *

38

2

Convergence of Unstable Solutions to Homogeneous Markov Process

In addition, the function ϕT (x) is monotonically increasing in x for each T and |ϕT (x)| ≤ C|x|. Since ϕT (x) = f (ϕ1T (x)) , we have that 0 < δ ≤ ϕT (x) ≤ C. Therefore, there exists a subsequence Tn → +∞ of the sequence Tn and a monotonically increasing continuous function * ϕ (x) such that ϕTn (x) → * ϕ (x) at every point x, as Tn → +∞. Thus,  *    P * ϕ * ζ (t) ξTn (t) = ϕTn * ζTn (t) −→ * at every point t > 0, as Tn → +∞. It follows from the proof of Theorem 2.1, that the process * ζ (t) is the solution to Eq. (2.19) with σ (x) = 1 . Note that the processes ξTn (t) and * ξTn (t) are weakly b(x) equivalent and the process ξTn (t) converges weakly, as T → +∞, to the process n   l(ζ(t)), defined by (2.9). Therefore * ϕ * ζ (t) = l (ζ (t)), where ζ(t) is the solution to equation t ζ (t) =

 (s). σ (ζ (s)) d W

0

     ** ζ (t) , σ (ζ (t)) = σ * ζ (t) and Eq. (2.19), Using the equalities ζ (t) = l −1 ϕ we obtain

l

−1

   ϕ * * ζ (t) =

t

    σ * ζ (s) b * ζ (s) d* ζ (s)

0

with probability 1 for all t > 0. According to Lemma A.12, we have the equalities ϕ (x)) l −1 (* = σ (x) b(x) = k0 x for all x, where k0 is some constant. From here we have that b(x) =

k0 , σ (x)

* ϕ (x) = x

k0 . c(x)

Thus, fTn (x) → * ϕ −1 (x) = x

c(x) , k0

as Tn → +∞. Since  √  f x Tn f √ =x fTn (x) = Tn

 √  x Tn , √ x Tn

2.3

Convergence to a Process of Skew Brownian Motion Type

39

and the sequence Tn → +∞ is arbitrary, we have the convergence f (x) c (x) − → 0, x k0 as |x| → +∞. Theorem is proved with k =

 

1 k0 .

Theorem 2.3 Let ξ be a solution to Eq. (2.1) and let ! ! x ! ! ! ! a(u) ! ≤ C. du 0 < δ ≤ σ (x) ≤ C, !! ! 2 ! σ (u) ! 0

1

The stochastic process ξT (t) = ξ(tT )T − 2 converges weakly, as T → +∞, to the process  ξ (t), defined by relation (2.9), if and only if f (x) 1 − kc(x) → 0, x f (x)

x

1 du − →0 f (u) σ 2 (u) k 2 σ 2 (x)

0

as |x| → +∞, where k is some positive constant. Proof Sufficiency Sufficiency of the conditions of this theorem follows from Theorem 2.2. Necessity Let the assumptions of Theorem 2.2 be fulfilled. It is clear that 0 < δ ≤ f (x) ≤ C and 0 < δ ≤ f (x) σ (x) ≤ C for some constants δ > 0 and C > 0. Therefore, the families of monotonically increasing functions fT (x) and QT (x) are compact. Consequently, given an arbitrary sequence Tn → +∞ we can choose a subsequence Tn → +∞ and monotonically increasing functions f(x) and Q(x) such that fTn (x) → f(x),

QTn (x) → Q(x),

as Tn → +∞, at the points of continuity of the limit functions. The derivatives f (x) and Q (x), which exist a.e. with respect to the Lebesgue measure, satisfy the inequalities 0 < δ ≤ f (x) ≤ C, 0 < δ ≤ Q (x) ≤ C. In addition, in this case the conditions of Theorem 2.1 are fulfilled. Therefore, without loss of generality, we *Tn (t) in Theorem 2.1, will assume that for the corresponding processes * ζTn (t) and W we have the equality (2.15) and the convergence *

P * ζTn (t) −→ * ζ (t),

*

P *Tn (t) −→ * (t), W W

40

2

Convergence of Unstable Solutions to Homogeneous Markov Process

as Tn → +∞. Next, consider the processes t    ζTn (s) . ηTn (t) = ζTn (s) d* Q *

(2.27)

0

Since Q (x) exists everywhere except the set Λ of zero Lebesgue measure, the integral on the right-hand side in (2.27) is correctly defined. According to Lemma A.11, we have that t

. * P * ζTn (s) ∈ Λ ds = 0

0

for all t ≥ 0. Let us prove that the quadratic characteristic ηTn (t) of the family of martingales ηTn (t) converges in probability to t, as Tn → +∞. In order to do this, we introduce the functions x ΦTn (x) = 2

$

% Q(u) − QTn (u) du,

0

and, using the Itô formula, obtain that ηTn (t) − t = ΦTn (* ζTn (t)) − ΦTn (* ζTn (0)) −

t

ΦT n (* ζTn (s)) d* ζTn (s). (2.28)

0

Further, for any N > 0 ! ! !ΦT (x)! χ|x|≤N ≤ 2 n

N

! ! !Q(u) − QT (u)! du → 0, n

−N

as Tn → +∞, and according to Lemma A.10 t E 0

$

%2 ζTn (s))  σTn (* ζTn (s)) χ|*ζTn (s)|≤N ΦT n (*

N ds ≤ CN −N

! ! !Q(u) − QTn (u)!2 du.

2.3

Convergence to a Process of Skew Brownian Motion Type

41

From the above we have the obvious inequalities  ! .   -! * ζTn (s) − ΦTn * ζTn (0) ! > ε P !ΦTn * + ≤* P

! ! sup !* ζTn (s)! > N

,

0≤s≤t

4 + ε

N

! ! !G(u) − GT (u)! du, n

−N

⎧! t ! ⎫ , + ! ⎨!! ⎬ ! ! !   * P !! ΦTn * ζTn (s) d* ζTn (s)! > N ζTn (s)!! > ε ≤ * P sup !* ⎩! ⎭ 0≤s≤t ! 0

4 + 2E ε

t

$

 %2   σTn * ΦT n * ζTn (s)  ζTn (s) χ|* ζTn (s)|≤N ds.

0 * P

It follows from (2.28) that ηTn  (t) −→ t, as Tn → +∞. Consequently, the  (t), that is the limit in probability of the sequence ηTn (t), is a Wiener process W process, and, according to (2.27), we have the equality  (t) = W

t 

  ζ (s) Q * ζ (s) d*

(2.29)

0

with probability 1 for all t > 0.   Consider the process * ξTn (t) = ϕTn * ζTn (t) , where ϕTn (x) is the inverse function ξTn (t) converges in probability, as Tn → +∞, to the to fTn (x). It is clear that * ϕ (x) is the inverse function to f(x). The process ξTn (t) is process  ϕ * ζ (t) , where  weakly equivalent to the process* ξTn (t) for every Tn , and, according to the conditions of Theorem 2.3, we have that  ϕ * ζ (t) = l (ζ (t)), where ζ (t) is the solution of the equation t ζ (t) =

 (s). σ (ζ (s)) d W

0

     ϕ * Note that ζ (t) = l −1  ζ (t) and σ (ζ (t)) = σ * ζ (t) . So, taking into account equality (2.29), we have

l

−1

    ϕ * ζ (t) =

t 0

    ζ (s) σ * ζ (s) d* ζ (s) Q *

42

2

Convergence of Unstable Solutions to Homogeneous Markov Process

with probability 1 for all t > 0. Therefore, according to Lemma A.12, we obtain 2 l −1 ( ϕ (x)) = σ (x) Q (x) = k0 x for all x, where k0 is some constant. Thus, " # k0 2 , Q (x) = σ (x)

c(x) f(x) = x . k0

In addition, the following convergence holds:  √   √  f x Tn f x Tn c(x) fTn (x) = √ =x →x , √ k0 Tn x Tn and x QTn (x) =

$ 0

f

√ ϕ(x  Tn )

1 = √ Tn

=x

1   √  f ϕ x Tn

dv   √    √ % 2 ϕ v Tn σ ϕ v Tn dv f (v) σ 2 (v) 0

√ ϕ(x  Tn )

k02 dv → x , f (v) σ 2 (v) σ 2 (x)

0

as Tn → +∞. Since the subsequence Tn → +∞ is arbitrary, the proof of the necessity with k = k10 is complete.   Consider an important partial case of Theorem 2.3. Theorem 2.4 Let ξ be a solution of Eq. (2.1), and let ! x ! ! ! ! ! a(u) ! 0 < δ ≤ σ (x) ≤ C, ! du!! ≤ C. 2 ! σ (u) ! 0

√T ) converges weakly, as T → +∞, to Then the stochastic process ξT (t) = ξ(t T the process σ0 W (t), where W is a Wiener process, if and only if

1 x

f (x) → σ1 , x

x 0

1

as |x| → +∞, with σ0 = ( σ1 σ2 )− 2 .

du → σ2 , f (u) σ 2 (u)

2.4 Examples

43

It is easy to see that the latter assertion follows from Theorem 2.3 with c1 = c2 and σc11 = σc22 = σ0 (or from Theorem 2.2). We note that sufficient conditions of Theorem 2.4 are obtained in [31], and necessary conditions of Theorem 2.4 derive from the paper [41]. Remark 2.7 It is clear that the process ξT (t) = Lemma 2.3, is a solution of the equation

ξ(t T ) B(T ) ,

which is considered in

dξT (t) = aT (ξT (t)) dt + σT (ξT (t)) dWT (t), where √ T σT (x) = σ (x B(T )) . B(T )

T aT (x) = a (xB(T )) , B(T )

In particular, the following cases are possible: √

T → +∞; (1) B(T )

√ T (2) → 0; B(T )

(3) B(T ) ∼

√ T , as T → +∞.

Therefore, aT (x) and σT2 (x) can be sequences of “δ”-type as well as can have degeneration of another nature (see Example 2.6).

2.4 Examples Consider the following examples of the coefficients a(x) and σ (x) in Eq. (2.1). Example 2.1 Let a(x) = xe−x and σ (x) = 1. +∞ −∞    2 2 2 Since xe−x dx = 0, then xe−x dx = xe−x dx and 2

R

0

⎧ ⎨

f (x) = exp −2 ⎩

x

0

⎫ ⎬

⎧ ⎨

⎫ +∞ ⎬ a(v) dv →  a(v) dv , σ1 = exp −2 ⎭ ⎩ ⎭

0

0

as |x| → +∞. Therefore f (x) → σ1 , x

1 x

x 0

as |x| → +∞.

1 dv → , f (v)  σ1

44

2

Convergence of Unstable Solutions to Homogeneous Markov Process

Here  σ2 =  σ1−1 and σ0 = ( σ1 σ2 )− 2 = 1. Consequently, according to Theorem 2.4, the stochastic process ξT (t) = 1 ξ(tT )T − 2 converges weakly, as T → +∞, to the Wiener process W . 1

Example 2.2 Let a(x) = sin βx, σ (x) = 1, β = 0. Then ⎧ ⎫ ⎨ x ⎬ − 2 2 cos βx f (x) = exp −2 a(v) dv = e β e β . ⎩ ⎭ 0

Therefore 1 x

x

f (u) du =

0

1 x

u −2 a(v)dv

x e

0

du = e

− β2

1 x

x

0

2

eβ

cos βu

du → e

− β2

1 2π

2π

2

eβ

0

cos z

dz =  σ1 ,

0

as |x| → +∞, and 1 x

x

1 du = f (u) x

0

u 2 a(v)dv

x e

0

1 du → e 2π 2 β

0

2π e

− β2 cos z

dz =  σ2 ,

0

as |x| → +∞. Taking into account the equalities 2π

2

e β cos z dz =

0

2π

2

e− β cos z dz and

π

0

2

e β cos z dz =

2π

2

e β cos z dz,

π

0

we get that ⎡ σ0 = ( σ1 σ2 )− 2 = ⎣ 1

1 2π

⎤−1

2π e

2 β

cos z

dz⎦

0

⎡ =⎣

1 π

⎤−1

π e

2 β

cos z

dz⎦

.

0

According to Theorem 2.4, the stochastic process ξ(tT )T − 2 converges weakly, as T → +∞, to the process σ0 W (t). Note that the period of the periodic drift coefficient asymptotically affects the diffusion coefficient: if the period of oscillation decreases, the coefficient of diffusion increases asymptotically, while the drift coefficient is averaged and equals zero in the equation for the limit process. 1

Example 2.3 Let a(x) = 0, σ (x) =

1 2+cos βx .

2.4 Examples

45

In this case x

1 x

−2

e

u 0

a(v) dv σ 2 (v)

du = 1,

for x = 0,

0

and 1 x

x

2

u

e

0

a(v) dv σ 2 (v)

1 1 du = 2 σ (u) x

0

[2 + cos βu]2 du 0

4 =4+ x

x

1 cos βu du + x

0

11 2x

=4+

x

x cos2 βu du 0

x [1 + cos 2βu] du + o(1) = 4 +

9 1 + o(1) → , 2 2

0

as |x| → +∞. Therefore, according to Theorem 2.4 with σ0 = − 12

√ 2 3 ,

the stochastic √

process ξ(tT )T converges weakly, as T → +∞, to the process 32 W (t). Consequently, the magnitude of the period of the periodic diffusion coefficient does not asymptotically affect the diffusion coefficient in the equation of the limit process. Example 2.4 a(x) = xe−x , σ (x) = In this case 2

1 x

x

−2

e

u 0

a(v) dv σ 2 (v)

1 2+cos βx .

⎧ ⎨

+∞

du → c0 = exp −2 ⎩

0

⎫ ⎬

a(v) dv , σ 2 (v) ⎭

0

as |x| → +∞, and, using the previous example, we have 1 x

x

2

e

u 0

a(v) dv σ 2 (v)

1 1 du = c0 σ 2 (u)

0

x [2 + cos βu]2 du + o(1) →

9 , 2c0

0

as |x| → +∞. According to Remark 2.6 with c1 = c2 = c0 and

σ1 c1

=

σ2 c2

=

√ 2 3 ,

the stochastic process ξ(tT )T − 2 converges weakly, as T → +∞, to the process √ 2 3 W (t). 1

46

2

Convergence of Unstable Solutions to Homogeneous Markov Process

Example 2.5 (1) Let a(x) =

1 , 1+x 2

σ (x) = 1.

Since +∞ −∞ π π a(x)dx = , a(x)dx = − , 2 2 0

0 1

then, according to Theorem 2.3, the stochastic process ξT (t) = ξ(tT )T − 2 converges weakly, as T → +∞, to the process  ξ (t) = (ζ (t)), where ζ(t) is a solution to the Itô equation dζ (t) = σ (ζ(t)) dW (t), where  σ (x) =

e−π , x ≥ 0, eπ , x < 0,

(x) =

x . σ (x)

Using the explicit form (2.10) of the transition density ρζ (t, x, y) of the process ζ(t), we obtain an explicit form of the transition density of the process  ξ (t): ρξ (t, x, y) = ρζ (t, −1 (x), −1 (y))(−1 (y))

=

 ⎧ 2 (y−x)2 1 − (y+x) ⎪ 1 2t √ ⎪ e− 2t + σσ22 −σ , x ≥ 0, y > 0; · e ⎪ +σ1 2πt ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 2σ1 √ 1 e− (y−x) 2t , x ≥ 0, y < 0; σ1 +σ2

2πt

2 ⎪ 2σ2 √ 1 − (y−x) ⎪ 2t e , x ≤ 0, y > 0; ⎪ σ +σ ⎪ 1 2  2πt ⎪ ⎪ 2 2 (y−x) ⎪ ⎪ − (y+x) 1 ⎩ √1 2t e− 2t − σσ22 −σ , x ≤ 0, y < 0, +σ1 · e

2πt

where σ1 = e−π , σ2 = eπ . It is easy to get that for the limit process  ξ (t) there exist Kolmogorov’s local characteristics in the generalized sense: the diffusion coefficient σ (x) = 1 and the 1 drift coefficient is cδ(·), where δ(·) is Dirac’s delta function, c = σσ22 −σ +σ1 = tanh π. Furthermore, we can formally write (see [42]) the Itô SDE for the process  ξ (t) as d ξ (t) = tanhπ δ(·) dt + dW (t). Note that in this case the diffusion coefficient in the differential equation for the √ √ process ξT (t) equals to 1, and the drift coefficient aT (x) = T a(x T ) is a “δ”type sequence. After passing to the limit formally in the equation for the process ξT (t) we obtain the drift coefficient πδ(·) instead of tanh πδ(·) for the limit process  ξ (t). This fact emphasizes the specificity of passing to the limit in the Itô SDEs. See Appendix A.3 for more details.

2.4 Examples

47

(2) Let a(x) =

a(x) , 1 + x2

 a(x) =

a1 , x ≥ 0, a2 , x < 0,

σ (x) = 1.

In this case all the conclusions from (1) hold with σ1 = e−a1 π for x ≥ 0, σ2 = for x < 0. The limit process  ξ (t) satisfies the Itô SDE

e−a2 π

d ξ (t) = c0 δ(·)dt + dW (t), 1 where c0 = σσ22 −σ +σ1 . In particular, in the case where 1

(2.1) a2 = −a1 we have c0 = 0 and the stochastic process ξT (t) = ξ(tT )T − 2 converges weakly, as T → +∞, to the Wiener process W . We have here  a(x)dx = 0; R

(2.2) a2 = a1 = a0 we have c0 = tanha0 π and the stochastic process ξT (t) converges weakly, as T → +∞, to the solution of the equation d ξ (t) = tanha0 π δ(·)dt + dW (t). We have here

 R

a(x)dx = a0 π.

Example 2.6 x (1) Let a(x) = − (1+x 2 )3 , σ (x) =

1 . 1+x 2

Note that x

a(v) dv = − σ 2 (v)

0

x 0

v 1 dv = − ln(1 + x 2 ) 2 1+v 2

and ⎧ ⎨

f (x) = exp −2 ⎩

x 0

⎫ ⎬

a(v) dv = 1 + x 2 , σ 2 (v) ⎭

48

2

Convergence of Unstable Solutions to Homogeneous Markov Process

then f (x)σ (x) = 1. The conditions of Theorem 2.1 are fulfilled with σ1 = σ2 = 1 1 and the stochastic process f (ξ(tT ))T − 2 converges weakly, as T → +∞, to the Wiener process W (t). Moreover, x f (x) =

f (x) dx = x +

x3 3

and

lim

k→+∞

f (kx) = x 3. f (k)

0

According to Lemma 2.3, the finite-dimensional distributions of the process √ 1 ξT (t) = ξ(tT )B −1 (T ), where B(T ) = c0 T 6 , c0 = 3 3, converge,√as T → +∞, to the corresponding finite-dimensional distributions of the process 3 W (t) with the transition density 3y 2 − (y 3 −x 3 )2 2t . e ρ(t, x, y) = √ 2πt Note that the process ξT (t) for every fixed T > 0 satisfies the following Itô SDE (see Remark 2.7) dξT (t) = aT (ξT (t)) dt + σT (ξT (t)) dWT (t), where WT (t) = aT (x) =

W√(t T ) T

ξT (0) = 0,

(2.30)

is a Wiener process for every fixed T > 0,

T xB(T ) T Tx a(xB(T )) = −   3 = −  1 3 B(T ) B(T ) 1 + x 2 B 2 (T ) 1 + x 2 c02 T 3 5

T 6x =− · aT (x) with  aT (x) =  c0

c0 T

1 6

1 + x 2 c02 T

1 3

3 .

Note that for any continuous function ϕ(x) with compact support 3



 ϕ(x) aT (x) dx = R

ϕ R

as T → +∞, where m0 =



dz

R

(1+z2 )3

4

z c0 T

1 6

dz  3 → ϕ(0)m0, 1 + z2

is the weight of the “δ”-type sequence  aT (x)

at the point x = 0. The diffusion coefficient in Eq. (2.30) has the form √ T T 1 σ (xB(T )) = σT (x) = 1 B(T ) 6 c0 T 1 + x 2 c02 T √

1

1 3

=

T6 · σT (x) c02

2.4 Examples

49

with  σT (x) =

c0 T

1 6

1 + x 2 c02 T

1 3

.

It is a “δ”-type sequence at the point x = 0 with weight m0 = In this case ⎧ ⎨

x fT (x) =

u

exp −2 ⎩ 0

0

 R

dz 1+z2

= π.

⎫ c0 x aT (v) ⎬ dv du = 1 + x 3 . 2 σT (v) ⎭ T3

(2) Let the diffusion coefficient in Eq. (2.30) equal σT (x) =

T

1 6

1 + x 2T

1 3

.

It is a “δ”-type sequence at the point x = 0 with weight π. The drift coefficient aT (x) + will be found , from the equality fT (x)σT (x) = 1, where fT (x) = x exp −2 aT2 (v) dv . 0

σT (v)

As a result of differentiation of the equality x −2 0

1 aT (v) dv = ln 2 σ σT (v) T (x)

we get that 1 σT (x)σT (x). 2

aT (x) = Thus,

⎞

⎛ aT (x) =

xT

2 3

1 6

T 1 2 1 + x 2T

1 2

1 3

1 3

1 2xT ⎟ ⎜ · ⎝−T 6   ⎠ 1 2 1 + x2T 3

T

1 6

= − aT (x) with  aT (x) =   = −xT   , 1 3 1 2 1 + x 2T 3 1 + x 2T 3

50

2

Convergence of Unstable Solutions to Homogeneous Markov Process

which is a “δ”-type sequence at the point x = 0 with weight



dz . (1+z2 )2 In this case ζT (t) = fT (ξT (t)) = WT (t) converges weakly, as T → +∞, to the 1 Wiener process W (t). Here fT (x) = 13 x 3 T 3 +o(1), where o(1) → 0, as T → +∞, for all x ∈ R. R

(3) Let the diffusion coefficient in Eq. (2.30) equal 7 8 8 σT (x) = 9

T

1 6

1 + x2T

1 3

.

Note that σT2 (x) is a “δ”-type sequence at the point x = 0 with weight π. The drift coefficient aT (x) will be found from the equality fT (x)σT (x) = 1 similarly to (2). Thus, ⎛ aT (x) =

⎞

1 12

5 12

⎜ ⎟ 2xT 1 1 T ⎜− ⎟ σT (x)σT (x) =  · 1 3  ⎝   ⎠ 2 2 1 2 1 2 1 + x2T 3 1 + x 2T 3 xT

1 2

1 3

T

1 6

= − aT (x) with  aT (x) =   = −xT   , 1 2 1 2 1 + x 2T 3 1 + x 2T 3 which is a “δ”-type sequence at the point x = 0 with weight m0 =



dz . (1+z2 )2 Consequently, ζT (t) = WT (t) converges weakly, as T → +∞, to the Wiener 1 2 process W . In this case fT (x) = x2 T 12 sign x + o(1). R

Chapter 3

Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

In this chapter, we consider one-dimensional homogeneous stochastic differential equations whose coefficients place these equations on the border between equations whose solutions have ergodic distribution, and equations with stochastically unstable solutions. To simplify calculations and to visualize better the influence of the drift coefficient of the equation on the asymptotic behavior of solution, we consider Eq. (2.1) with σ (x) ≡ 1. Statements about the instability and ergodicity for the solutions are formulated and proved in Sect. 3.1. Weak convergence of normalized stochastically unstable solutions to the Bessel diffusion process we consider in Sect. 3.2. Section 3.3 includes more general results about the influence of the coefficients of the equation on the limit behavior of the solutions. Influence of the diffusion coefficient on the limit behavior of the stochastically unstable solutions we study in Sect. 3.4. Section 3.5 contains several examples.

3.1 Criteria of Instability and Ergodicity for the Solutions Let us consider Eq. (2.1) with σ (x) ≡ 1, namely an equation of the form dξ(t) = a(ξ(t))dt + dW (t),

t > 0, ξ(0) = x0 ,

(3.1)

with real measurable drift coefficient satisfying additional assumption: |x a(x)| ≤ L for a certain constant L and for all x ∈ R. In particular, it can be a (x) ∼

c , |x|

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_3

51

52

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

as |x| → +∞. The behavior of the solutions ξ to the class of equations of type (3.1), in which a (x) ∼

c , |x|α

as |x| → +∞, for α > 1, was investigated in Chap. 2. Indeed, in !this case ! !x !  ! ! |a(u)| du < +∞ and for all x ∈ R we obtain that ! a(u) du! ≤ C. ! ! 0 R Consequently, the conditions (2.4) and (2.11) hold. For −1 < α < 1 the solutions of Eq. (3.1) have an exact order of behavior, as t → +∞, i.e., ξ(t) → +∞ with probability 1, as t → +∞, and there exists a non-random function B(t) → +∞, as t → +∞, such that  ξ (t) = 1 = 1. t →+∞ B (t)

 P

lim

For more details see the book [17, § 17]. It is well known (see [82, Theorem 4]) that the SDE (3.1) possesses a unique strong pathwise solution and this solution is a homogeneous strong Markov process. In this chapter we use the following notations: x f (x) =

⎧ ⎨

⎫ ⎬

u

exp −2 ⎩ 0

a(v)dv du ⎭ 0

and 1 ψ(x, c) = ln |x|

x a(v)dv − c.

(3.2)

0

In what follows the constants C > 0 and L > 0 do not depend on x and t. Theorem 3.1 Let ξ be a solution to Eq. (3.1) and let lim

|x|→+∞

ψ(x, c0 ) = 0.

(3.3)

Then we have the following cases, depending on the value of c0 . 1. For 2c0 < −1, the solution ξ is ergodic and ⎡  ⎣ P {ξ(t) < x} → R

as t → +∞.

⎤−1 x  dv dv ⎦ , f (v) f (v) −∞

3.1 Criteria of Instability and Ergodicity for the Solutions

53

2. For 2c0 > −1, the solution ξ is stochastically unstable, in other words 1 lim t →+∞ t

t P {|ξ(s)| < N} ds = 0 0

for any constant N > 0. The case c0 = − 12 will be considered in Theorem 3.2. Proof Statement 1 For x = 0 we have the representation f (x) = |x|−2c0 exp {−2 ln |x| ψ(x, c0 )} .

(3.4)

In the case 2c0 < −1, we can choose ε > 0 such that 2c0 + 2ε < −1. According to relation (3.3), there exists a constant L > 0 such that |ψ(x, c0 )| < ε for |x| > L. Using representation (3.4), we obtain the inequality f (x) ≥ |x|−2c0 |x|−2ε for |x| > L. Therefore, we have that f (−∞) = −∞, f (+∞) = +∞ and  R

dx < +∞. f (x)

(3.5)

Consider the process η(t) = f (ξ(t)). Using the Itô formula, we conclude that dη(t) =  σ (η(t))dW (t), where  σ (x) = f (ϕ(x)) and the function ϕ(x) is the inverse function to f (x). It is clear that the function ϕ(x) is monotonically increasing and ϕ(−∞) = −∞, ϕ(+∞) = +∞. Using the substitution rule for the integral  R

with u = ϕ(x), du =

dx f (x)

dx =  σ 2 (x)

 R

dx [f (ϕ(x))]2

we obtain the equality  R

dx =  σ 2 (x)

 R

du . f (u)

54

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

Taking into account the boundedness of the function a(x), the last equality and (3.5) we have (see Theorem A.10) that the process η(t) is ergodic and ⎡ P {η(t) < x} → ⎣

⎤−1 ϕ(x)  dv ⎦ dv , f (v) f (v)



−∞

R

as t → +∞. Statement 1 follows from the last relation. Statement 2 Consider the function ⎫ ⎧ # ⎬ x ⎨ u " χ |v| 0 such that −2c0 + 2ε < 1. Using representation (3.4), we obtain the inequality f (x) ≤ |x|−2c0 +2ε for |x| > L. Therefore, f (x) →0 x

and

Φ(x) → 0, x2

as |x| → +∞. ! ! So, for arbitrary ε > 0 there exists a constant Lε such that !x −2 Φ(x)! < ε for |x| > Lε . Consequently, Cε 1 1 E |Φ(ξ(t))| ≤ + ε Eξ 2 (t), t t t

3.1 Criteria of Instability and Ergodicity for the Solutions

55

where Cε = sup |Φ(x)| . Furthermore, |x| 0. Proof Since the inequality 2c0 < −1 is used only in the proof of relation (3.5) in Theorem 3.1, the further considerations in the proof of statement 1 in Theorem 3.1 can be fully used also here. The inequality 2c0 > −1 is used only in the proof of the convergence x −1 f (x) → 0, as |x| → +∞ in Theorem 3.1. The further considerations in the proof of statement 2 in Theorem 3.1 can be fully used also here.  

56

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

Remark 3.1 It follows from Theorem 3.2 that the case 2c0 = −1 is critical. Actually, for 2c0 = −1 we have equations with ergodic solutions and equations with stochastically unstable solutions depending on the rate of convergence (3.3).

3.2 Convergence of Normalized Stochastically Unstable Solutions to the Bessel Diffusion Process Next, consider the case 2c0 > −1 and investigate the asymptotic behavior of stochastically unstable solutions. Theorem 3.3 Let ξ be a solution to Eq. (3.1), and let there exist the constants c1 and c2 such that ⎡ ⎣1 |x|→+∞ x

x

lim

⎤



va(v)dv − c(x)⎦ = 0,

c(x) =

c1 , x ≥ 0, c2 , x < 0.

(3.9)

0

Then we have three cases. 1

(1) If c1 = c2 = c0 , 2c0 > −1, then the stochastic process |ξ(tT )| T − 2 converges weakly, as T → +∞, to the process r(t), which is the solution of Itô’s SDE t r (t) = (2c + 1)t + 2 2

r(s)dW (s)

(3.10)

0

for c = c0 ; 1 (2) if 2c1 > 1 and 2c2 < 1, then the stochastic process ξ(tT )T − 2 converges weakly, as T → +∞, to the solution r(t) to Eq. (3.10) for c = c1 ; 1 (3) if 2c1 < 1 and 2c2 > 1, then the stochastic process −ξ(tT )T − 2 converges weakly, as T → +∞, to the solution r(t) to Eq. (3.10) for c = c2 . Definition 3.1 A nonnegative homogeneous Markov process ζ with transition density of the form ρ(t, x, y) =

 2   xy  x + y2 1 2ν−1 , exp − I y ν−1 t (xy)ν−1 2t t

where Iν is the modified Bessel function, is called a Bessel diffusion process of index ν > 0. This process ζ is a solution of Itô’s SDE t ζ (t) = ζ (0) + νt + 2 2

2

ζ (s)dW (s) . 0

3.2

Convergence of Normalized Unstable Solutions to the Bessel Diffusion. . .

57

Remark 3.2 The process r (t), which is the solution to Eq. (3.10), is the Bessel diffusion process of index ν = 2c + 1 (see Definition 3.1). Note that for −1 < 2c < 1, r (t) is a process with reflection at the origin, and for 2c ≥ 1 the origin is not attainable by the process r (t). See Example IV–8.3 in [23] and [24, 77] for further information on Bessel diffusions. To prove Theorem 3.3 we need the following statement. Lemma 3.1 If condition (3.9) holds, then lim

|x|→+∞

ψ (x, c(x)) = 0.

(3.11)

Proof Let us first consider the limit for x → +∞. For arbitrary ε > 0 there exists a constant Cε > 0 such that, for x > Cε , we have the inequality ! ! ! ! x ! !1 ! ! va(v) dv − c1 ! < ε. ! ! !x ! ! Cε Then for x > Cε 1 I (x) := ln x

x

1 a(v) dv − c1 = ln x

0

Cε

1 a(v) dv + ln x

0

1 c1 = o(1) + ln x

x

1 = o(1) + ln x

a(v) dv − c1 Cε

x

1 1 dv + v ln x

Cε

x

va(v) − c1 dv − c1 v

Cε

x

va(v) − c1 dv v

Cε

⎡ = o(1) +

1 ⎢1 ⎣ ln x x

x

x (za(z) − c1 ) dz +

Cε

⎡ 1 ⎢1 = o(1) + ⎣ ln x x

x Cε

⎛ 1 ⎜ ⎝ v2

Cε

⎤ 1 ⎥ (za(z) − c1 ) dz⎦ + ln x

v

⎞

⎤

⎟ ⎥ (za(z) − c1 ) dz⎠ dv ⎦

Cε

x Cε

1 v2

v (za(z) − c1 ) dz dv. Cε

58

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

So, for x > Cε , |I (x)| ≤ o(1) +

1 [ε (ln x − ln Cε )] = o(1) + ε. ln x

Consequently, lim sup |I (x)| < ε x→+∞

for any ε > 0, whence ⎡

x

1 lim ⎣ x→+∞ ln x

⎤ a(v) dv − c1 ⎦ = 0.

0

Similarly, we obtain the proof of (3.11) for x → −∞.

 

Proof Statement 1 of Theorem 3.3 Let us introduce the parameter T > 0 and denote for 0≤t ≤L rT (t) =

|ξ(tT )| , √ T

T (t) = W

WT (t) =

W (tT ) √ , T

t sign ξ(sT ) dWT (s), 0

1 βT (t, c0 ) = T

t T (ξ(s)a (ξ(s)) − c0 ) ds. 0

The process WT (t), for every fixed T > 0, is a Wiener process. Note that t P {ξ(s) = 0} ds = 0 0

T (t), for every fixed for every t ≥ 0. According to Theorem A.4, the process W T > 0, is also a Wiener process. Using the Itô formula, we obtain rT2 (t) =

x02 + (2c0 + 1)t + 2 T

t 0

T (s) + 2βT (t, c0 ). rT (s)d W

(3.12)

3.2

Convergence of Normalized Unstable Solutions to the Bessel Diffusion. . .

59

Let us prove that lim E sup |βT (t, c0 )| = 0.

T →+∞

(3.13)

0≤t ≤L

In order to do this, consider the function ⎛ ⎞ x u Φ(x) = 2 ⎝ (va(v) − c0 ) dv ⎠ du. 0

0

According to the Itô formula, we have the equality βT (t, c0 ) = IT(1) (t) − IT(2) (t) − IT(3) (t),

(3.14)

where (1) IT (t)

Φ (ξ(tT )) − Φ (ξ(0)) , = T

(2) IT (t)

1 = T

t T

Φ (ξ(s)) a (ξ(s)) ds,

0

IT(3) (t)

1 = T

t T

Φ (ξ(s)) dW (s).

0

It is clear that Φ(x) → 0, x2 Φ (x)a(x) =

Φ (x) → 0, x

Φ (x) (xa(x)) → 0, x

(3.15)

as |x| → +∞. Taking into account equality (3.12) we conclude that ! ! t ! ! ! ! 2 !  E sup rT (t) ≤ C + 2E sup ! rT (s)d WT (s)!! 0≤t ≤L 0≤t ≤L ! ! 0

!2 ⎞ 12 ! t ⎛ L ⎞ 12 ! !  ! ! ⎜ 2 T (s)! ⎟ ⎝ ⎠ . ≤ C + 2 ⎝E sup !! rT (s)d W ! ⎠ ≤ C + 2 4 ErT (s)ds 0≤t ≤L ! ! ⎛

0

0

60

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

The latter inequalities imply that E sup rT2 (t) ≤ CL . 0≤t ≤L

(3.16)

Using !(3.15) we !conclude that for arbitrary ε > 0 there exists a constant Cε > 0 such that !x −2 Φ(x)! < ε for |x| > Cε . Therefore, ! ! ! Φ (ξ(tT )) ! C ! ≤ + ε E sup r 2 (t) ≤ C + εCL . ! E sup ! T ! T T T 0≤t ≤L 0≤t ≤L Consequently, ! ! ! (1) ! lim sup E sup !IT (t)! ≤ εCL . T →+∞

0≤t ≤L

Since the ε > 0 is arbitrary, we conclude that ! ! ! (1) ! lim E sup !IT (t)! = 0.

T →+∞

0≤t ≤L

(3.17)

! ! Next, let us take arbitrary ε > 0 and a constant Cε > 0 such that !Φ (x)a(x)! < ε for |x| > Cε . Note that LT ! ! ! (2) !  1 P {|ξ(s)| < Cε } ds + εL. E sup !IT (t)! ≤ Cε T 0≤t ≤L 0

According to Lemma 3.1, we have that statement 2 of Theorem 3.1 holds. Consequently, ! ! ! ! lim sup E sup !IT(2) (t)! ≤ εL. T →+∞

0≤t ≤L

Therefore, ! ! ! (2) ! E sup !IT (t)! → 0, 0≤t ≤L

(3.18)

as T → +∞. Similarly to the proof of relation (3.18), we can apply the inequalities LT ! ! %2 $ 4 ! (3) !2 E Φ (ξ(s)) ds E sup !IT (t)! ≤ 2 T 0≤t ≤L 0

3.2

Convergence of Normalized Unstable Solutions to the Bessel Diffusion. . .

ε 1 ≤C T2

61

LT P {|ξ(s)| < Cε } ds 0

L +4ε

2

ErT2 (s)χ|ξ(sT )|≥Cε ds

ε 1 ≤C T2

0

LT P {|ξ(s)| < Cε } ds + Cε2 , 0

where ! ! ε = sup !Φ (x)! χ|x| 0 ! -! . * P !* rTn (t2 ) − * rTn (t1 )! > ε ! . -! rTn (t1 )! > ε, * rTn (t1 ) > ε ≤* P !* rTn (t2 ) − * rTn (t2 ) + * ! /! 0 ! ! 2 ≤* P !* rT2n (t1 )! > ε2 . rTn (t2 ) − * Taking into account (3.20), we get lim lim sup

sup

h→0 Tn →+∞ |t1 −t2 |≤h, ti ≤L

! -! . * P !* rTn (t2 ) − * rTn (t1 )! > ε = 0.

(3.21)

62

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

Therefore, according to Lemma A.7, we can pass to the limit in the stochastic integral in (3.20). Thus, t 2

* r (t) = (2c0 + 1)t + 2

* (s). * r(s)d W

0

Since the subsequence Tn is arbitrary and the solution of the latter equation is unique, the finite-dimensional distributions of the process rT (t) tend, as T → +∞, to the corresponding finite-dimensional distributions of the process r(t). Applying inequalities (2.21), we get that for arbitrary ε > 0 ⎧ ⎨

! t ! ⎫ ! 2 ! ⎬ ! ! ! rT (s) d W T (s)! > ε P sup ! ⎩|t1 −t2 |≤h, ti ≤L !! ⎭ ! t1

⎫ ! t ! ! ! ⎬ ! ! ! rT (s)d W T (s)! > ε sup ≤ P 4 sup ! ⎭ ⎩ kh ≤ P sup ! 4⎭ ⎩kh≤t ≤(k+1)h !! ! kh 0.

|rT (t2 ) − rT (t1 )| > ε

=0

3.2

Convergence of Normalized Unstable Solutions to the Bessel Diffusion. . .

63

According to Theorem A.13 the stochastic process rT (t) converges weakly, as T → +∞, to the solution r(t) of Eq. (3.10) with c = c0 . Statement 2 Let us take ε > 0 such that 2c1 − 2ε > 1 and 2c2 + 2ε < 1. Using Lemma 3.1 and equality (3.4) with c0 = c1 for x > L and c0 = c2 for x < −L, we obtain the following inequalities f (x) ≤ x −2c1 x 2ε for x > L and f (x) ≥ |x|−2c2 |x|−2ε

for x < −L.

So, the function f (x) is bounded from above and f (x) → −∞, as x → −∞. Thus (see Lemma A.5),  P

 lim ξ (t) = +∞ = 1.

(3.22)

t →+∞

Now let us use the analog of equality (3.12), that is

rT2 (t)

x2 1 = 0 + T T

t T

t [2c (ξ(s)) + 1] ds + 2

0

T (s) + 2βT (t, c (ξ(s))) , rT (s)d W

0

where 1 βT (t, c (ξ(t))) = T

t T [ξ(s)a (ξ(s)) − c (ξ(s))] ds. 0

According to the Itô formula for βT (t, c (ξ(t))), we have the analog of equality (3.14), where x Φ(x) = 2 0

⎛ ⎝

u

⎞ (va(v) − c(v)) dv ⎠ du.

0

Completely analogous to the proof of (3.13), we obtain lim E sup |βT (t, c (ξ(t)))| = 0.

T →+∞

0≤t ≤L

64

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

The rest of the proof can be done in the same way as in the proof of statement 1. In 1 1 doing so use (3.22) and the convergence for t ≥ 0: ξ(tT )T − 2 − |ξ(tT )| T − 2 → 0, as T → +∞, with probability one, that follows from (3.22). The proof of statement 3 is literally the same as that of statement 2 with the only difference that the function f (x) is bounded from below and f (x) → +∞, as x → +∞. In this case (see Lemma A.6)  P

 lim ξ (t) = −∞ = 1.

t →+∞

(3.23)  

Remark 3.3 Let the relation xa (x) − c (x) → 0, as |x| → +∞, hold. Then we have (3.9). In fact, it follows from the equality 1 x

x

1 va (v) dv − c (x) = x

0

x [va (v) − c (v)] dv. 0

Corollary 3.1 For 2c − 1 ≥ 0 there exists a unique strong solution r (t) to SDE d r (t) =

c dt + dW (t) . r (t)

(3.24)

Indeed, the proof of Theorem 3.3 implies the existence of a unique process r (t), which is a strong solution to Eq. (3.10). According to Remark 3.3, for 2c −1 ≥ 0 the origin is not attainable by the process r (t). Therefore, we can apply the Itô formula  √ to the process Φ r 2 (t) , where Φ (x) = |x|, and obtain that the process r (t) satisfies Eq. (3.24). Next, we show that, under additional conditions on the convergence rate in (3.9), we obtain equalities instead of inequalities for c1 and c2 in Theorem 3.3. That is, the following theorem holds. Theorem 3.4 Let the assumptions of Theorem 3.3 be fulfilled. 1. If c1 = c2 = − 12 and " # 1 ln |x| ψ x, − = +∞, |x|→+∞ 2 lim

then the stochastic process T − 2 ξ(tT ) converges weakly, as T → +∞, to the process r(t) ≡ 0. 1

3.2

Convergence of Normalized Unstable Solutions to the Bessel Diffusion. . .

65

2. If (1) 2c1 > 1, 2c2 = 1, and there exist ε(x) > 0 and constants C > 0 and L > 0 such that +∞

ε(x) dx = +∞ x

and

" # 1 ln ε(x) + 2 ln |x| ψ x, 0 such that +∞

ε1 (x) dx < +∞ x

and

" # 1 ln ε1 (x) + 2 ln |x| ψ x, > −C 2

0 1

for x > L, then the stochastic process T − 2 ξ(tT ) converges weakly, as T → +∞, to the solution r(t) of Eq. (3.10) with c = c1 . 3. If (1) 2c1 = 1, 2c2 > 1 and there exist ε(x) > 0 and constants C > 0 and L > 0 such that +∞

ε(x) dx = +∞ x

and

" # 1 ln ε(x) + 2 ln |x| ψ x, L, or (2) 2c1 = 1, 2c2 = 1 and, in addition to the assumptions (3.26), there exists a function ε1 (x) > 0 such that −∞

ε1 (x) dx < +∞ x

and

# " 1 ln ε1 (x) + 2 ln |x| ψ x, > −C 2

0 1

for x < −L, then the stochastic process −T − 2 ξ(tT ) converges weakly, as T → +∞, to the solution r(t) of Eq. (3.10) with c = c2 . Proof Assumption (1) of Theorem 3.4 implies the Statement 2 of Theorem 3.2. Furthermore, the proof of assertion (1) of Theorem 3.3 implies that assumption (1) 1 of Theorem 3.4 is sufficient for the weak convergence of the process T − 2 ξ(tT ), as

66

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

T → +∞, to the solution r(t) of Eq. (3.10) with 2c + 1 = 0. From the uniqueness of the solution of Eq. (3.10) we have that r(t) ≡ 0. Assumption (2) of Theorem 3.4 implies the relations f (x) < C and f (x) → −∞, as x → −∞, that is, relation (3.22) holds. The proof of assertion (2) of Theorem 3.3 implies that statement (2) of Theorem 3.4 is sufficient for the weak 1 convergence of the process T − 2 ξ(tT ), as T → +∞, to the solution r(t) of Eq. (3.10) with c = c1 . In the proof of statement (3) of Theorem 3.4 we use relation (3.23), which follows from the fact that in this case f (x) → +∞, as x → +∞, and the function f (x) is bounded from below. The theorem is proved.  

3.3 Influence of the Coefficients of the Equation on the Limit Behavior of the Solutions The results obtained in the process of studying of Eq. (3.1) allow us to investigate the asymptotic behavior of the solutions to a class of equations of the form dη(t) = a (η(t)) dt + σ (η(t)) dW (t), η(0) = x0 ,

(3.27)

where the function σ (x) > 0 is continuously differentiable, and g(x) → −∞, as x x → −∞, g(x) → +∞, as x → +∞, where g(x) = σdy (y) . 0

In fact, let us consider the process ξ(t) = g (η(t)) . According to the Itô formula, we have the equality dξ(t) =  a (ξ(t)) dt + dW (t),

(3.28)

where  a (x) =

1 a (l(x)) − σ (l(x)) , σ (l(x)) 2

here the function l(x) is the inverse function to g(x). So, we obtain that ξ is the solution of Eq. (3.1). 1 > 0 for all x ∈ R. Consequently, the function g(x) is In this case g (x) = σ (x) strictly monotonously increasing. The equality g (l(x)) = x, that holds for all x ∈ R, implies the convergences l(x) =→ −∞, as x → −∞, and l(x) =→ +∞, as 1 x → +∞. Since l (x) = g (l(x)) > 0 the inverse function is strictly monotonously increasing. Let us formulate the analogs of Theorems 3.1 and 3.3.

3.3 Influence of the Coefficients of the Equation on the Limit Behavior of the. . .

67

Theorem 3.5 Let η be a solution of Eq. (3.27) and let 1 lim |x|→+∞ ln |x|

l(x)

1 σ (z) a(z) − dz = c0 . σ 2 (z) 2 σ (z)

0

Then we have two cases. 1. If 2c0 < −1, then the stochastic process η is ergodic and ⎡  lim P {η(t) < x} = ⎣

t →+∞

⎤−1 g(x)  dz ⎦ dz , f (z) f (z)

(3.29)

−∞

R

where ⎧ ⎨

f (x) = exp −2 ⎩

l(x)

1 σ (v) a(v) − σ 2 (v) 2 σ (v)

⎫ ⎬ dv . ⎭

0

a (x)| ≤ C, then the stochastic process η is stochastically 2. If 2c0 > −1 and |x unstable, in other words 1 lim t →+∞ t

t P {|η(s)| < N} ds = 0

(3.30)

0

for any constant N > 0. Proof In this case, all the conditions of Theorem 3.1 are satisfied for Eq. (3.28) with the given c0 . Therefore, if 2c0 < −1, then we have relation (3.29) for the process η, and if 2c0 > −1, then relation (3.30) holds.   Theorem 3.6 Let η be a solution to Eq. (3.27) and let |x a (x)| ≤ C. If ⎫ ⎧  ⎬ ⎨ 1 x a (y) 1 σ (y) lim dy − g (y) c = 0, − (x) ⎭ |x|→+∞ ⎩ g (x) σ 2 (y) 2 σ (y) 0

where  c (x) =

c1 , x ≥ 0, c2 , x < 0,

then the stochastic process ξ (t) = g (η (t)) satisfies statements (1)–(3) of Theorem 3.3.

68

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

Proof It is easy to get that 1 x

x

1 v a (v) dv = x

0

 l(x) a (y) 1 σ (y) dy, g (y) 2 − σ (y) 2 σ (y) 0

c (x) = c (g (x)) . Therefore, the conditions of Theorem 3.3 are satisfied for the coefficients of Eq. (3.28). So, Theorem 3.3 implies Theorem 3.6.   Similarly, we can obtain analogs of Theorems 3.2 and 3.4 for the process g (η (t)), where η (t) is the solution to Eq. (3.27). Note that the analogs of Theorems 3.3 and 3.4 assert the weak convergence of the 1 stochastic process T − 2 |g (η(tT ))|, as T → +∞, to the process r(t). If additionally we have the convergence 1 x

x

dv → σ0 , σ0 > 0, σ (v)

0

as |x| → +∞, then we obtain the weak convergence of the stochastic process 1 |η(tT )| T − 2 to the process σ0−1 r(t).

3.4 Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions Now, we consider the asymptotic behavior, as t → +∞, of the distributions of the solutions ξ of Eq. (2.1), in which the drift coefficients a (x) ≡ 0. More precisely, we consider equations of the form dξ (t) = σ (ξ (t)) dW (t) ,

ξ (0) = x0 ,

(3.31)

where σ (x) ∼ c |x|α , as |x| → +∞, 0 < α < 12 , and σ (x) > 0 is a continuously differentiable function. Note that for α > 12 the solution ξ to Eq. (3.31) is an ergodic process (see Theorem A.10), and ⎡ lim P {ξ (t) < x} = ⎣

t →+∞



R

⎤−1 dv ⎦ σ 2 (v)

x

−∞

dv . σ 2 (v)

3.4 Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions

69

Theorem 3.7 Let ξ be a solution to Eq. (3.31). If for all x σ (x) = c0 |x|α + β (x) , where c0 > 0, 0 < α
0. 1−α 1−α

Consequently, all the conditions of statement 1 of Theorem 3.3 are fulfilled 1 for Eq. (3.33). Therefore, the process rT (t) = T − 2 |ξ(tT )| converges weakly, as α T → +∞, to the solution r(t) of Eq. (3.10) with c = − 2(1−α) . According to relation (3.32), for arbitrary ε > 0 there exists a constant Cε such that for |x| > Cε we have the inequality ! ! ! ! !|g(x)|c0 (1 − α)|x|α−1 − 1! < ε. Therefore, ! ! 1 ! ! |rT (t) − ξT (t)| ≤ √ sup ! |g(x)| − |x|1−α [c0 (1 − α)|]−1 ! + εξT (t), T |x|≤cε

(3.34)

70

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

where ξT (t) =

√ −1 |ξ(tT )|1−α . T c0 (1 − α)

Using the boundedness of the function x a (x), and taking into account (3.33) as well as the inequality rT (t) =

|g(ξ(tT ))| c0 (1 − α) |ξ(tT )|1−α

ξT (t) ≥ δ ξT (t),

we conclude that EξT (t) ≤ C + C1 t.

(3.35)

Since ε > 0 is arbitrary, it follows from (3.34) and (3.35) that rT (t) − ξT (t) → 0 in probability, as T → +∞. So, the finite-dimensional distributions of the process ξT (t) converge, as T → +∞, to the corresponding distributions of the solution r(t) of Eq. (3.10) with c = α − 2(1−α) . Therefore, the finite-dimensional distributions of the process 1

T − 2(1−α) |ξ(tT )| converge, as T → +∞, to the corresponding distributions of the process 1

[c0 (1 − α)r(t)] 1−α .   Remark 3.4 Note that the case α = is critical. In particular, if we add to Eq. (3.31) the drift coefficient a(x) = − 12 sign x, then the solution ξ to such an equation has an ergodic distribution. This fact is an immediate consequence of the following theorem. 1 2

Theorem 3.8 Let η(t) be a solution to Eq. (3.27). If 2xa(x) = −1, σ (x) = lim |x|→+∞ σ 2 (x)



c1 |x|α1 + β1 (x), c2 |x|α2 + β2 (x),

x ≥ 0, x < 0,

where ci > 0, 0 < αi ≤ 1, βi (x) = o(|x|αi ), as |x| → +∞, then lim P {η(t) < x} = B

t →+∞

−1

x

−∞

du f (u)σ 2 (u)

,

3.4 Influence of the Diffusion Coefficient on the Limit Behavior of the Solutions

71

where ⎫ ⎧  ⎬ ⎨ x a(u) du du . f (x) = exp −2 , B = 2 ⎩ σ (u) ⎭ f (u)σ 2 (u) R

0

Proof According to the Itô formula for the process ξ(t) = f (η(t)), where f (x) = x f (u)du, we obtain the equation 0

t  σ (ξ(s))dW (s),

ξ(t) = f (η(0)) + 0

where  σ (x) = f (ϕ(x)) σ (ϕ(x)) and the function ϕ(x) is the inverse function to f (x). Since 2xa(x) → −1, σ 2 (x) as |x| → +∞, for arbitrary ε > 0 there exists a constant Cε such that for |x| > Cε we have the inequalities −1 − ε
0. For x > Cε ⎧ ⎪ ⎨

f (x) = exp −2 ⎪ ⎩

Cε

a(u) du − 2 σ 2 (u)

0

x Cε

⎫ ⎪ ⎬

⎫ ⎧ ⎪ ⎪ x ⎨ a(u) −2ua(u) ⎬ *ε exp > K du du ⎪ ⎪ σ 2 (u) ⎪ uσ 2 (u) ⎭ ⎭ ⎩ Cε

  " #1−ε *ε exp (1 − ε) ln x = K *ε x >K = Kε x 1−ε . Cε Cε The similar situation is for x < 0. So, we obtain the inequality f (x) > Kε |x|1−ε . Consequently, f (+∞) = +∞,

f (−∞) = −∞

and  R

du < +∞.  σ 2 (u)

72

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

According to Theorem A.10, we obtain ⎡ P {η(t) < x} = P {ξ(t) < f (x)} → ⎣



R

⎤−1 f (x)  du ⎦ du , 2  σ (u)  σ 2 (u) −∞

as t → +∞. Hence the proof of Theorem 3.8 follows.

 

3.5 Examples Consider the following examples of the drift coefficient a(x) in Eq. (3.1). Example 3.1 Let a(x) = c(x)

x x sin x + , c(x) = 1 + x2 1 + x2



c1 , x > 0, c2 , x < 0,

σ (x) = 1.

Since for all x we have |xa(x)| ≤ C and 1 x

x

1 va(v)dv = x

0

1 = c(x) + x

x 0

x 0

v2 1 c(v) dv + 1 + v2 x



v2 1 c(v) − 1 dv + 2 x 1+v

x

x 0

v 2 sin v dv 1 + v2

1 sin v dv − x

0

x 0

sin v dv → c(x), 1 + v2

as |x| → +∞, then ⎡ 1 lim ⎣ |x|→+∞ x

x

⎤ va(v)dv − c(x)⎦ = 0.

0

According to Theorem 3.3 we obtain: (1) if c1 = c2 = c0 and 2c0 > −1, then the stochastic process |ξ(tT )| T −1 converges weakly, as T → +∞, to the Bessel diffusion process r(t), which is the solution to Eq. (3.10) for c = c0 ; 1 (2) if 2c1 > 1, 2c2 < 1, then the stochastic process ξ(tT )T − 2 converges weakly, as T → +∞, to the Bessel diffusion process r(t), which is the solution to Eq. (3.10) for c = c1 ;

3.5 Examples

73

(3) if 2c1 < 1, 2c2 > 1, then the stochastic process −ξ(tT )T − 2 converges weakly, as T → +∞, to the Bessel diffusion process r(t), which is the solution to Eq. (3.10) for c = c2 . 1

If c1 = c2 = c0 and 2c0 < −1, then, according to Theorem 3.1, the stochastic process ξ(t) is ergodic and ⎡ P {ξ(t) < x} → ⎣



R

⎤−1 x  dv ⎦ dv , f (v) f (v) −∞

x −2 a(v)dv

f (x)

as t → +∞, where =e 0 . Consequently, the ergodicity and instability of the solution depend on the magnitude c0 . Example 3.2 Let a(x) = −

1 x x , σ (x) = 1. −2 2 2 + x2 (2 + x 2 ) ln(2 + x 2 )

Since x

1 1 a(v)dv = − ln(2 + x 2 ) + ln 2 − ln ln(2 + x 2 ) + ln ln 2, 4 4

0

we have ⎧ ⎫ ⎨ x ⎬ 2  2 f (x) = exp −2 a(v) dv = 2 + x 2 ln(2 + x 2 ) · c1 , ⎩ ⎭ 0

where c1 = e− 2 ln 2−2 ln 1

ln 2

.

Taking into account that  R

dv √ %2 = 2 $ 2 2 + v ln(2 + v 2 )

+∞ 0

dv √ %2 = $ 2 2 + v ln(2 + v 2 )

we obtain the inequality  R

dx < +∞. f (x)

+∞ 2

dz < +∞, √ √ z z − 2 (ln z)2

74

3 Asymptotic Analysis of Equations with Ergodic and Stochastically Unstable Solutions

Consequently, according to Theorem 3.2, the solution ξ to Eq. (3.1) is ergodic, and ⎡ P {ξ(t) < x} → ⎣



R

⎤−1 x  dv ⎦ dv , f (v) f (v) −∞

as t → +∞. Example 3.3 Let a(x) = − x

1 x 2x + , σ (x) = 1. 2 1 + x2 (2 + x 2 ) ln(2 + x 2 )

1 1 a(v)dv = − ln(2 + x 2 ) + ln 2 + ln ln(2 + x 2 ) − ln ln 2. 4 4

0

Then ⎧ ⎨

f (x) = exp −2 ⎩

⎫ ⎬

x a(v) dv

⎭

=

2

 −2 2 + x 2 ln(2 + x 2 ) · c2 ,

0

where 1

c2 = e− 2 ln 2+2 ln

ln 2

.

Thus, f (x) = x

√ 1 2 + x2 ·$ %2 → 0, x ln(2 + x 2 )

as |x| → +∞. Consequently, according to Theorem 3.2, the solution ξ to Eq. (3.1) is stochastically unstable, that is, for an arbitrary constant N > 0 1 lim t →+∞ t

t P {|ξ(s)| < N} ds = 0. 0

In addition, in this case c1 = c2 = − 12 and "

1 lim ln |x| ψ x, − |x|→+∞ 2

#

√  √  4 4 |x| 2 2 2 = +∞. ln(2 + x ) + ln √ = lim ln √ 4 2 |x|→+∞ ln 2 2+x

3.5 Examples

75

According to Theorem 3.4, the stochastic process ξ(tT )T − 2 converges weakly, as T → +∞ to the process r(t) ≡ 0. 1

Consider the following examples of the diffusion coefficient σ (x) in Eq. (3.31). √ 6 Example 3.4 Let σ (x) = 1 + x 2 . 1 Thus, σ (x) ∼ |x| 3 , as |x| → +∞. Therefore, according to Theorem 3.7 with 1 c0 = 1 and α = 13 , the finite-dimensional distributions of the process |ξ(tT )| T − 3  3 2 converge, as T → +∞, to the corresponding distributions of the process 23 r(t) , where r(t) is the solution of Eq. (3.10) for c = − 14 . Example 3.5 Let 1 . σ (x) = √ 6 1 + x2 1

Thus, σ (x) ∼ |x|− 3 , as |x| → +∞. Therefore, according to Theorem 3.7 with c0 = 1 and α = − 13 , the finite-dimensional distributions of the process 2

|ξ(tT )| T − 3 converge, as T → +∞, to the corresponding distributions of the 3  4 process 43 r(t) , where r(t) is the solution to Eq. (3.10) for c = 18 .

Chapter 4

Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

A very important class of functionals from solutions ξ to SDEs is represented by stochastic integrals t

t g (ξ (s)) ds and

0

g (ξ (s)) dξ (s) , 0

where g = g (x) is a non-random function. It is supposed that ξ is stochastically unstable solution of some Itô’s SDE, and the integrals exist for every t > 0 in the respective sense. Also, we assume that the integrals are unbounded in probability, as t → +∞. In this chapter we study the behavior of the distributions, as t → +∞, of these functionals after some normalization. For example, according to the Itô formula, we have with probability 1 the equality 1 1 − cos W (t) = 2

t

t cos W (s) ds +

0

sin W (s) dW (s),

(4.1)

0

for all t ≥ 0, where W is a standard Wiener process. It is clear that W is stochastically unstable solution of SDE (2.1) for a(x) ≡ 0, σ (x) ≡ 1, and ξ(0) = 0. We know that (see Example 4.2) the stochastic process (2) βT (t)

1 = √ T

t T sin W (s) dW (s),

t ≥0

0

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_4

77

78

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

converges weakly, as T → +∞, to the process √1 W ∗ (t), where W ∗ is a Wiener 2 t process. So the functional sin W (s) dW (s), t > 0 is unbounded in probability, the 0

equality (4.1) imply that the functional

t

cos W (s) ds is unbounded in probability

0

as well. Since √1 cos W (T t) → 0 with probability 1, as T → +∞, for all t ≥ 0, T the stochastic processes (1) βT (t)

1 = √ T

t T cos W (s) ds 0

√ converge weakly, as T → +∞, to the process − 2W ∗ (t), where W ∗ is a Wiener process. It should be emphasized that these convergences are similar, in some sense, to the central limit theorem for the sum of the dependent random variables. We note that during the study of asymptotic behavior, as t → +∞, of the distributions of unstable solutions ξ of Eq. (2.1), which is equivalent to the integral equation t ξ(t) = ξ(0) +

t a (ξ(s)) ds +

0

σ (ξ(s)) dW (s),

(4.2)

0

we investigate the asymptotic behavior, as t → +∞, of the distributions of the integral functionals, that are unbounded in probability. Therefore, in the study of the asymptotic behavior, as t → +∞, of the distributions of stochastically unbounded integral functionals two problems arise: the problem of finding appropriate non-random normalizing multipliers B(t) → +∞, as t → +∞, that ensure convergence of the distributions of normalized functionals to the distributions of some non-degenerate distributions, and the problem of describing the class of limit distributions. Note that t

t g (ξ (s)) dξ (s) =

0

t g (ξ (s)) a (ξ (s)) ds +

0

g (ξ (s)) σ (ξ (s)) dW (s) , 0

so in this chapter we study the asymptotic behavior of the distributions, as T → +∞, of the following functionals: (1) βT (t)

1 = B1 (T )

t T g (ξ (s)) ds, 0

(2) βT (t)

1 = B2 (T )

t T g (ξ (s)) dW (s) , t ≥ 0, 0

4

Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

79

where Bi (T ) are non-random normalizing multipliers and Bi (T ) → +∞, as (1) T → +∞, g(x) is non-random measurable locally integrable (for βT (t)) or (2) locally square integrable (for βT (t)) function, the stochastic processes ξ and W are related via Eq. (2.1) or via Eq. (3.1). The chapter consists of three sections. In Sect. 4.1 the weak convergence is studied, as T → +∞, for the functionals βT(1) (t) and βT(2) (t) of unstable solutions of Eq. (2.1). In Theorem 4.1 the sufficient conditions are established for the weak (1) convergence, as T → +∞, of the functional βT (t) to the process ⎤ ζ (t ) t ⎥ ⎢ β (1) (t) = 2 ⎣ |u|α b (u) du − |ζ (s)|α b (ζ (s)) dζ (s)⎦ , ⎡

0

0

where ζ is the solution to Eq. (2.8), α ≥ 0 is the order of regularity at infinity of the function ψ (see Definition 4.1 in Sect. 4.1),  b (x) =

b1 , x ≥ 0, b2 , x < 0,

b1 and b2 are some constants. In Theorem 4.2 the sufficient conditions are (2) established for the weak convergence, as T → +∞, of the functional βT (t) to the  (1)  (2) ∗ (1) process β (t) = W β (t) , where the process β (t) has form (4.4), W ∗ (t) is a Wiener process, the processes W ∗ (t) and β (1) (t) are independent. It should be emphasized that the limit processes β (1) (t) and β (2) (t) are some functionals from the process ζ of Brownian motion in a bilayer environment with known explicit form of distributions. In particular, for α = 0 and b(x) = b sign x the limit process has the form β (1) (t) = bL0ζ (t), where L0ζ (t) is the local time at zero of the process ζ in the interval [0, t]. Note that similar limit processes of the form β (1) (t) and β (2) (t), where ζ = W is a Wiener process, were first obtained in the book [81]. In Theorems 4.3 and 4.4, which are, in some sense, analogues of the central limit theorem, the results about weak convergence, as T → +∞, of the processes (2) (1) βT (t) (Theorem 4.3) and βT (t) (Theorem 4.4) to a Wiener process are obtained. In Theorem 4.5 the sufficient conditions for the weak convergence, as T → +∞, of  (1)  (1)  (t) are established, where the Wiener the processes βT (t) to the process W ∗ β ∗ (1)  process W (t) and the process β (t) are independent. In Sect. 4.2, using similar methods we study the asymptotic behavior, as T → (1) (2) +∞, of the functionals βT (t) and βT (t) of the solutions of Eq. (3.1). The obtained results are completely similar to the results of Sect. 4.1 with the only difference that the corresponding limit processes β (1) (t) and β (2) (t) are the functionals of the diffusion Bessel process r(t). Note that in this case we know the explicit form of the distributions of the process r(t).

80

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

In Sect. 4.3 the functionals of the form t I (t) = F (ξ(t)) +

g (ξ(s)) dW (s) 0

are considered, where F (x) is a continuous functions, x ∈ R, g(x) is a real valued and measurable function, which is locally square integrable, the processes ξ and W are related via Eq. (2.1) for σ (x) ≡ 1, ξ(0) = x0 or via Eq. (3.1). Here the integral t functional g (ξ(s)) dW (s) is unbounded in probability. Sufficient conditions for 0 (t T ) the weak convergence, as T → +∞, of the processes IT (t) = IB(T ) , t > 0 are formulated, where B(T ) is normalizing multiplier, B(T ) → +∞, as T → +∞, to the functional

t b (ζ(s)) |ζ(s)|α dζ(s),

I0 (t) = a (ζ (t)) ζ (t) |ζ (t)| + α

0

where ζ is the solution of Eq. (2.8) (Theorem 4.11) or to the functional I0∗ (t) = a (ζ (t)) |ζ (t)|

1+α1 2

  + W ∗ β (1) (t) ,

where ⎡

⎤ ζ(t ) t ⎢ ⎥ β (1) (t) = 2 ⎣ b(x) |x|α1 dx − b (ζ (s)) |ζ(s)|α1 dζ(s)⎦ , 0

0

ζ is the solution of Eq. (2.8), W ∗ is a Wiener process, and the processes W ∗ and ζ are independent (Theorem 4.12). The results of weak convergence, as T → +∞, of the processes IT to the functionals I0 and I0∗ of the diffusion Bessel process r are formulated in Theorems 4.13 and 4.14. If we replace ζ with r in functionals I0 (t) and I0∗ (t) from Theorems 4.11 and 4.12, we obtain the limit functionals I0 (t) and I0∗ (t) in Theorems 4.13 and 4.14, respectively. Section 4.4 contains several examples that illustrate the results of previous sections.

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

81

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer Environment Now we study the behavior of the distributions, as t → +∞, of the integral functionals of the solutions ξ to Eq. (2.1), whose coefficients satisfy the conditions of Theorem 2.1. In what follows we use the notation: x J (x) = 0

g (ϕ (u)) [f (ϕ (u))

σ (ϕ (u))]2

du,

where ⎧ ⎨

f (x) = exp −2 ⎩

x

⎫ ⎬

a(v) dv , σ 2 (v) ⎭

0

ϕ(x) is the inverse function to f (x) from (2.1). Definition 4.1 Let Ψ denote the class of functions ψ (r) > 0, r ≥ 0, that are nondecreasing and regularly varying at infinity of order α ≥ 0, so that ψ (rT ) → rα, ψ (T ) as T → +∞ for all r > 0. Theorem 4.1 Let ξ be the solution to Eq. (2.1). Let the assumptions of Theorem 2.1 hold and ⎧ x ⎨ 12 , x → +∞, 1 du σ1 → ⎩ 12 , x → −∞. f (x) f (u)σ 2 (u) σ2

0

Let the real-valued measurable function g be locally integrable and there exist a function ψ = ψ(r) ∈ Ψ and constants bi , i = 1, 2 for which 1 lim |x|→+∞ x

x q 2 (v) dv = 0, 0

where q (x) =

1 J (x) − b (x) , ψ (|x|)

(4.3)

82

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

and  b (x) =

b1 , x ≥ 0, b2 , x < 0.

Then the stochastic processes (1) βT (t)

1 √  = √ Tψ T

t T g (ξ (s)) ds 0

converge weakly, as T → +∞, to the process ⎤ ζ (t ) t ⎥ ⎢ β (1) (t) = 2 ⎣ |u|α b (u) du − |ζ (s)|α b (ζ (s)) dζ (s)⎦ , ⎡

0

(4.4)

0

where ζ is the solution to Eq. (2.8) and α ≥ 0 is the order of regularity of the function ψ at infinity. Proof Consider the function f(x)

F (x) = 2

J (u) du. 0

Note that the functions F (x) and the derivative F (x) are continuous, the second derivative F (x) is locally integrable and 1 F (x) a (x) + F (x) σ 2 (x) = g (x) 2 a.e. with respect to the Lebesgue measure. Therefore, we can apply the Itô formula (Lemma A.3) to the process F (ξ(tT )) and obtain 1 (1) √  [F (ξ (tT )) − F (ξ (0))] βT (t) = √ Tψ T

1 √  −√ Tψ T

t T 0

F (ξ (s)) σ (ξ (s)) dW (s) .

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

83

Let us rewrite this equality in the form (1)

βT (t) = √

     √  √  1 √  F ϕ ζT (t) T − F ϕ ζT (0) T Tψ T

1 − √  ψ T

t

  √    √  F ϕ ζT (s) T σ ϕ ζT (s) T dWT (s) .

0

According to the proof of Theorem 2.1, given an arbitrary sequence Tn → +∞, * * * we can choose a subsequence Tn →   +∞, a probability space (Ω, F, P), and *Tn (t) defined on this space such that its finitea stochastic process * ζTn (t), W   dimensional distributions coincide with those of the process ζTn (t), WTn (t) and, moreover, *

P * ζ (t), ζTn (t) −→ *

*

P *Tn (t) −→ * (t), W W

* as Tn → +∞, for all t ≥ 0, where * ζ (t),  W (t) are some stochastic processes. Note that the processes ζTn (t) , WTn (t) are weakly equivalent to the processes   * *Tn (t) . Therefore, according to Corollary A.3, the processes β (1) (t) are ζTn (t) , W Tn *(1) (t), where weakly equivalent to the processes β Tn

*(1) (t) = √ β Tn −

ψ

1 √

     2  2  1 √  F ϕ * ζTn (0) Tn ζTn (t) Tn − F ϕ * Tn ψ Tn t

Tn

 0

* ζ Tn (t)

=2

t

b (u) |u|α du − 2

* ζTn (0)

  2    2  *Tn (s) F ϕ * ζTn (s) Tn σ ϕ * ζTn (s) Tn d W

4 1 !α !  (k) ζTn (s)! d* b * ζTn (s) !* ζTn (s) + 2 STn (t) , k=1

0

(1) STn (t)

(

)  √  ψ |u| Tn √  − |u|α du, b (u) ψ Tn

* ζ Tn (t )

= * ζTn (0)

(2) STn (t)

* ζ Tn (t )

= * ζTn (0)

 2  ψ |u| √Tn  √  du, q u Tn ψ Tn

(4.5)

84

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

(3) STn (t)

t =−

  b * ζTn (s)

(

0

) !√  ! ! !α ψ !* ζTn (s)! Tn √  ζTn (s) , − !* ζTn (s)! d* ψ Tn

and (4) STn (t)

!√  !  2  ψ !* ζTn (s)! Tn * √  d* ζTn (s) . q ζTn (s) Tn ψ Tn

t =− 0 *

P We have that * ζTn (t) −→ * ζ (t), as Tn → +∞. Moreover, according to the proof of Theorem 2.1, + , ! ! ! !* lim lim sup P sup ζTn (t) > N = 0, N→+∞ Tn →+∞

0≤t ≤L

+ lim lim sup P

sup

h→0 Tn →+∞

|t1 −t2 |≤h, ti ≤L

, ! ! !* ζTn (t1 )! > ε = 0 ζTn (t2 ) − *

for arbitrary ε > 0, L > 0. The convergence * ζTn (t) → * ζ (t), as Tn → +∞, is a weak convergence w.r.t. the measure * P. Note that * ζTn (0) → 0, as Tn → +∞, with probability 1, and obtain * ζ Tn (t )

* ζ (t )  b (u) |u| du −→ b (u) |u|α du. * P

α

* ζTn (0)

(4.6)

0

According to Lemma A.7, t

!α *  ! P b * ζTn (s) !* ζTn (s) −→ ζTn (s)! d*

0

t

!α  ! b * ζ (s) !* ζ (s)! d* ζ (s) ,

(4.7)

0

as Tn → +∞. Now, we are in a position to prove that ! ! P ! (k) ! * sup !STn (t)! −→ 0, k = 1, 4.

0≤t ≤L

Let + PN = P

, ! ! sup !* ζTn (t)! > N .

0≤t ≤L

(4.8)

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

85

In order to prove relations (4.8), we produce the following inequalities: , N ! ! ! ! 2 ! (1) ! !b (u)! P sup !STn (t)! > ε ≤ PN + ε 0≤t ≤L +

−N

+ P

! ! ! (2) ! sup !STn (t)! > ε

,

0≤t ≤L

2 ≤ PN + ε

N −N

⎛ 2 1 ⎜ ≤ PN + CN · ⎝2N √ ε Tn + P

! !  √  ! ! ψ |u| T n ! ! α √  − |u| ! du, ! ! ! ψ Tn

(4.9)

 2  ψ |u| √Tn  √  du q u Tn ψ Tn

√ N Tn √ −N Tn

⎞ 12

(4.10)

⎟ q 2 (u) du⎠ ,

! ! ! (3) ! sup !STn (t)! > ε

, ≤ PN

0≤t ≤L

( ! )42 !√  " #2 L 3 ! !α   ψ !*   ζTn (s)! Tn 2 !* ! * √  ζTn (s) ds − ζTn (s) + 4E b ζTn (s) * σT2n * ε ψ Tn 0

≤ PN +

" #2 L ! -! . 2 N P !* ζTn (s)! < δ ds C ε 0

+

" #2 2 4CE ε

L 3 0

(

)42 !√  ! ! !α   ψ !* ζTn (s)! Tn ! !* ! √  − ζTn (s) b * ζTn (s) χ0 ε ≤ PN

0≤t ≤L

!√  ! " #2 L  2  ψ !*  ζTn (s)! Tn 2  2 2 * √   σTn * ζTn (s) χ|* + 4E q ζTn (s) Tn ζTn (s)|≤N ds ε ψ Tn 0

" #2 L  2    2 σT2n * ζTn (s) χ|* 4CN E q 2 * ≤ PN + ζTn (s) Tn  ζTn (s)|≤N ds ε 0

(4.12) for any ε > 0, L < +∞, N < +∞. The properties of regularly varying at infinity functions ψ (r), r ≥ 0 (Lemma A.17), Lemma 2.1 and the arbitrary choice of N, together with (4.9)– (4.11) imply relations (4.8) for k = 1, 3.

86

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Let us prove that L E

 2  2   q2 * ζTn (s) Tn  σTn * ζTn (s) · χ|*ζTn (s)|≤N ds → 0,

(4.13)

0

as Tn → +∞. Using the Itô formula we have t E

 2  2   ζTn (s) · χ|*ζTn (s)|≤N ds q2 * ζTn (s) Tn  σTn *

0

 %   $ ζTn (t) − QTn * ζTn (0) , = E QTn * where x QTn (x) = 2 0

⎛ ⎝

u

⎞  2  q 2 v Tn χ|v|≤N dv ⎠ du.

0

Since ! ! !QT (x)! ≤ √2 n Tn

√ N Tn

q 2 (v) dv |x| , √ −N Tn

taking into account the conditions of the theorem we have that lim

Tn →+∞

$    % ζTn (t) − QTn * ζTn (0) = 0 E QTn *

and, consequently, the relation (4.13). So, taking into account (4.12) we get (4.8) for k = 4. Using relations (4.6)–(4.8) we conclude that (4.5) implies the following convergence: *

P *(1) (t) −→ *(1) (t) , β β Tn

as Tn → +∞, where ⎡ *(1) (t) = 2 ⎢ β ⎣

* ζ (t ) 

t b (u) |u|α du −

0

and * ζ (t) is the solution of Eq. (2.19).

0

⎤ ! !α   ⎥ !* ζ (s) d* ζ (s)⎦ , ζ (s)! b *

(4.14)

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

87 (1)

Thus, we obtain the convergence of the distribution of the processes βTn (t), as Tn → +∞, to the corresponding distribution of the process β (1) (t). Since the sequence Tn → +∞ is arbitrary and since the solution to Eq. (2.19) is unique, then we have the weak convergence of the finite-dimensional distributions of the process (1) βT (t), as T → +∞, to the corresponding distribution of the process β (1) (t). Next, let us show that for any L > 0 + , ! ! ! (1) ! (1) lim lim sup P sup (4.15) !βT (t2 ) − βT (t1 )! > ε = 0. h→0 T →+∞

|t1 −t2 |≤h, ti ≤L

(t), k It is clear that the relation (4.15) holds for ST(k) n tion (4.5), as Tn → +∞. Using the inequality ! ! ⎧ ! ! *ζTn (t2 ) ⎪ ⎨ ! ! ! ! sup b(u) |u|α du! > P ! ⎪ ! ! ⎩|t1 −t2 |≤h, ti ≤L ! ! * ζ (t )

= 1, 4 in the representa⎫ ⎪ ⎬ ε

⎪ ⎭

≤ PN

1

Tn

+

! ! ε !* +P CN sup ζTn (t2 ) − * ζTn (t1 )! > 2 |t1 −t2 |≤h, ti ≤L

,

and the convergence (2.20) we have that the relation (4.15) holds for the process * ζ Tn (t )

b(u) |u|α du. * ζTn (0)

Next, similarly to the proof of relation (2.20), we obtain ⎫ ⎧ ! t ! ! 2 ! ⎬ ⎨ ! ! !α  ! ! b * ! !* ! * > ε ≤ PN ζ P sup ζ d ζ (s) (s) (s) T T T n n n ! ⎭ ⎩|t1 −t2 |≤h, ti ≤L !! ! t1

! t ⎫ ! ! 2 ! ⎬ !α ! ! !  ε !* ! χ* ! b * ! > * P sup d ζ ≤ PN ζ ζ (s) (s) (s) T T T n n n |ζTn (s)|≤N ! ⎩|t1 −t2 |≤h, ti ≤L !! 2⎭ ! ⎧ ⎨

t1

⎤4 ⎡ (k+1)h  1 " 8 #4 !α !    *Tn (s)⎦ + E⎣ b * σTn * ζTn (s)! χ|*ζTn (s)|≤N  ζTn (s) d W ζTn (s) !* ε kh≤L

kh

≤ PN + CN

1 kh≤L

h2 .

88

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Consequently, the relation (4.15) holds for all terms of the right-hand side of equality (4.5). Therefore, the relation (4.15) holds for the process βT(1) (t). Since n

the sequence Tn is arbitrary, the relation (4.15) holds for the process βT (t), as (1) well. According to Theorem A.13, the stochastic processes βT converge weakly, as T → +∞, to the process β (1).   (1)

We emphasize that for α = 0 and b(x) = b sign x the limit process has the form β (1)(t) = bL0ζ (t), where L0ζ (t) is the local time at zero of the process ζ on the interval [0, t]. For α = 1 and b(x) = b sign x the limit process has the t   form β (1)(t) = b σ 2 * ζ (s) ds, where σ (x) is defined in (2.8). In particular, for 0

σ 2 (x) = σ02 the limit process is β (1) (t) = bσ02 t. Theorem 4.2 Let (ξ, W ) be a solution of Eq. (2.1). Let the assumptions of Theorem 2.1 hold and ⎧ x ⎨ 12 , x → +∞, 1 du σ1 → ⎩ 12 , x → −∞. f (x) f (u)σ 2 (u) σ2

0

Let the real-valued measurable function g = g (x) be locally square integrable and such that there exist functions ψi ∈ Ψ of orders αi ≥ 0, respectively, and constants bi , i = 1, 2, and E |f (ξ (0))|δ+α2 < +∞, for some δ > 1,

1 lim |x|→+∞ x

x q 2 (v) dv = 0,

(4.16)

0

where 1 q (x) = ψ1 (|x|)

x 0

g 2 (ϕ (u)) [f (ϕ (u)) σ (ϕ (u))]2

du − b (x)

with  b (x) =

b1 , x ≥ 0, b2 , x < 0,

and ! ! ! ! x ! 1 ! |g (ϕ (v))| ! dv !! ≤ C, ! ψ (|x|) 2 [f (ϕ (v)) σ (ϕ (v))] ! 2 ! 0

(4.17)

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

√  ψ2 T lim √  1 = 0. T →+∞ 1  2 T 4 ψ1 T

89

(4.18)

Then the stochastic processes βT(2) (t)

1 =  √  1 1 2 T 4 ψ1 T

t T g (ξ (s)) dW (s) 0

  converge weakly, as T → +∞, to the process β (2) (t) = W ∗ β (1) (t) , where the process β (1) has the form (4.4) for α = α1 , W ∗ is a Wiener process, and the processes W ∗ and β (1) are independent. *Tn (t) be the processes introduced in the proof of TheoProof Let * ζTn (t) and W (2) rem 4.1. According to Corollary A.3, the processes βTn (t) are weakly equivalent to the processes *(2) (t) β Tn

√ t   2  Tn *Tn (s) g ϕ * ζTn (s) Tn dW = 1  √  1 Tn4 ψ1 Tn 2 0

√ t   4 2  Tn *Tn (s) . =  √  1 dW g ϕ * ζTn (s) Tn 2 ψ1 Tn 0 Let τTn (t) be the minimal solution to the equation   *(1) τTn (t) = t, β Tn where *(1)(t) = β Tn

√ t   2  Tn g2 ϕ * √ ζTn (s) Tn ds. ψ1 ( Tn ) 0

Note that for every Tn and t,-the random variable τTn (t) is a Markov moment . with respect to the σ -algebras σ * ζTn (s), s ≤ t . According to Lemma A.8, +∞   2  g2 ϕ * ζTn (s) Tn ds = +∞ 0

90

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

*(1)(t) attains any with probability 1. Thus, given an arbitrary Tn , the process β Tn  (2)  * positive level with probability 1. Hence βTn τTn (t) is a sequence of Wiener processes (see Theorem A.8). Denote these processes by WT∗n (t).   *(2)(t) = W ∗ τ −1 (t) , where τ −1 (t) = β *(1)(t), that is, Now, β Tn Tn Tn Tn Tn   *(2) (t) = WT∗ β *(1) (t) . β Tn Tn n *

P *(1) (t) −→ *(1)(t), as Tn → +∞, It follows from the proof of Theorem 4.1 that β β Tn *(1)(t) are measurable *(1) (t) is given by (4.14) for α = α1 . The processes β where β . with respect to the σ -algebra σ * ζ (s), s ≤ t . Using the inequality

0 /!  !   ! * *(1)(t) − WT∗ β *(1)(t) !! > ε P !WT∗n β Tn n / 0 / 0 (1) *(1)(t) > N + * ≤* P β P β (t) > N Tn ! 0 /! ! *(1) *(1) (t)!! > δ + * P (t) − β +* P !β Tn

+ sup

|t1 −t2 |≤δ, ti≤N

! ∗ ! !W (t2 ) − W ∗ (t1 )! > ε Tn Tn

,

and the convergence + lim sup * P

δ→0 Tn

sup

|t1 −t2 |≤δ, ti≤N

! ∗ ! !W (t2 ) − W ∗ (t1 )! > ε Tn Tn

, =0

for any ε > 0, N > 0 (see [19, Chapter IX, § 3]), we obtain that     * P *(1) (t) − WT∗ β *(1) (t) −→ WT∗n β 0, Tn n as Tn → +∞. Therefore,  *  P *(2)(t) − WT∗ β *(1) (t) −→ β 0, Tn n

(4.19)

as Tn → +∞. The proof of Theorem 2.1 and the representation (2.19) imply that the process t ηTn (t) = 0

d* ζT (s)  n  σ * ζTn (s)

 (t) for every t > 0. converges, as Tn → +∞, in probability to a Wiener process W

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

91

Using properties of stochastic integrals (see Theorem A.6), together with the fact that τTn (t) is a Markov moment, and with the inequality ! ! ! ! ! σTn (x) ! ≤ C, ! σ (x) ! we get ! ! !EηT (t)W ∗ (t)! n Tn

! ! ! ! t τ Tn (t )   √ ! !    * 2  σTn ζTn (s) T ! n *Tn (s)!! *Tn (s) *   dW d W g ϕ (s) T = !E ζ T n n 1  √  1 * ! ! ! ! 0 σ ζTn (s) Tn4 ψ1 Tn 2 0 ! ! ! ! t ∧τTn (t )   √ ! 4 2  !! ζTn (s)    σTn * Tn !   g ϕ * =  √  1 !E ζTn (s) Tn ds ! ! ! ζTn (s) σ * ψ1 Tn 2 ! ! 0 √ √ t !   2 !! Cψ2 ( Tn ) Tn ! ≤ 1  √  1 E ζTn (s) Tn ! ds. √ !g ϕ * Tn4 ψ1 Tn 2 ψ2 ( Tn ) 0 (4.20) For the functional under the sign of the expectation in the last term of (4.20), we can write a representation similar to representation (4.5). Using such a representation and taking into account the properties of regular at infinity functions, together with the conditions of Theorem 4.2, we obtain that for sufficiently large Tn √ t !   2 !! Tn ! E √ ζTn (s) Tn ! ds !g ϕ * ψ2 ( Tn ) 0

!! δ+α ! ! !δ+α ! !δ+α !! ! 2 t 2 2 1 + C ζTn (t) − * ζTn (0)! + Cα2 E !!* ζTn (t)! 2 + !* ζTn (0)! 2 ! ≤ C ≤ CN E !* i , i = 1, 2. Consequently, for certain constants C lim

Tn →+∞

EηTn (t) WT∗n (t) =

 (t) WT∗ (t) = 0. lim EW n

Tn →+∞

(4.21)

 and W ∗ are Wiener processes and they are asymptotically uncorrelated Since W Tn at any point, so, due to independency of their increments, they are asymptotically uncorrelated at any pair of points. So, their mutual quadratic characteristic is zero. Therefore, according to the Levy theorem (see [20, Chapter 1, § 3, Theorem 3]) . we conclude that, as Tn → +∞, WT∗n does not depend asymptotically on W (1)  * Further, the process β generates the same filtration as the process W , whence

92

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

*(1). It is we conclude that WT∗n , as Tn → +∞, does not depend asymptotically on β easy to show  that, as Tn → +∞, the finite-dimensional distributions of the process *(1) coincide with the corresponding finite-dimensional distributions of the WT∗n β  (1) * , where W ∗ is a Wiener process, and the processes W ∗ and β *(1) process W ∗ β are independent. Therefore, according to the convergence (4.19), the finite-dimensional distri(2) butions of the process βTn converge, as Tn → +∞, to the corresponding  (1)  * . finite-dimensional distributions of the process W ∗ β Furthermore, for every L > 0 we have the inequality + P

sup

|t1 −t2 |≤h, ti ≤L

+ ≤P

sup

|t1 −t2 |≤h, ti ≤L

+ ≤P

sup

0≤t ≤L

>N

+ +P

sup

|t1 −t2 |≤h, ti ≤L

+ +P

,

!    ! ! ∗ *(1) *(1) (t1 ) !! > ε !WTn βTn (t2 ) − WT∗n β Tn

, *(1)(t) β Tn

! ! ! *(2) *(2) (t1 )!! > ε !βTn (t2 ) − β Tn

sup

|t1 −t2 |≤δ, 0≤ti ≤N

,

! ! ! *(1) *(1)(t1 )!! > δ !βTn (t2 ) − β Tn

! ! ∗ !W (t2 ) − W ∗ (t1 )! > ε Tn Tn

,

,

for any ε > 0, N > 0. *(2) a From this inequality and the convergence (4.15) we have for the process β Tn convergence analogous to convergence (4.15). Thus, according to Theorem A.13,  (1)  *(2) converges weakly, as Tn → +∞, to the process W ∗ β * (·) . the process β Tn Since the subsequence Tn → +∞ is arbitrary and since the solution * ζ to Eq. (2.19) is unique, the proof of Theorem 4.2 is complete.   Theorem 4.3 Let (ξ, W ) be a solution of Eq. (2.1). Let the assumptions of Theorem 2.1 hold and ⎧ x ⎨ 12 , x → +∞, 1 du σ1 → ⎩ 12 , x → −∞. f (x) f (u)σ 2 (u) σ2

0

Let the real-valued measurable function g = g (x) be locally square integrable and such that there exist constants bi , i = 1, 2, for which 1 lim |x|→+∞ x

x q 2 (v) dv = 0, 0

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

93

where 1 q (x) = |x|

x

g 2 (ϕ (u)) [f (ϕ (u)) σ (ϕ (u))]2

0

du − b (x) .

Here  b (x) =

b1 , x ≥ 0, b2 , x < 0

and satisfy the following relation: ! ! !b (x)! σ 2 (x) = σ 2 0

for all x ∈ R and for some constant σ0 . Then the stochastic processes (2) βT (t)

1 = √ T

t T g (ξ (s)) dW (s) 0

converge weakly, as T → +∞, to the process σ0 W ∗ (t), where W ∗ is a Wiener process. Proof It is clear that the function g satisfies condition (4.16) of Theorem 4.2 for ψ1 (r) ∈ Ψ with α1 = 1. Note that we do not need now the conditions (4.17) and (4.18) because they used in the proof of Theorem 4.2 only to prove the *(1) (t). But now the limit process independence of the processes WT∗n (t) and β (1) * (t) is degenerate and it is not necessary to establish any independence. β ! ! According to the Itô formula, in the case α = 1 and !b (x)! σ 2 (x) = σ02 , *(1) (t) = σ 2 t. Using relation (4.19) and the fact that the finitewe obtain that β 0   dimensional distributions of the process WT∗n σ02 t coincide with the corresponding   finite-dimensional distributions of the process W ∗ σ02 t , where W ∗ (t) is a Wiener process, we have the convergence of the finite-dimensional distributions of the *(2) (t) to the corresponding finite-dimensional distributions of the process process β Tn ∗ σ0 W (t). Now we conclude about the weak convergence for the same reasons as those in the proof of Theorem 4.2.   Note that Theorem 4.1 does not hold for a functional of the form

t

sin W (s)ds.

0

For the functions g(x) with oscillatory properties the following theorem holds.

94

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Theorem 4.4 Let ξ be the solution to Eq. (2.1). Let the assumptions of Theorem 2.1 hold and ⎧ x ⎨ 12 , x → +∞, 1 du σ1 → ⎩ 12 , x → −∞. f (x) f (u)σ 2 (u) σ 0

2

Let the real-valued measurable function g(x) be locally bounded and such that there exist some constant C > 0 that bounds J (x) for all x: |J (x)| ≤ C. Assume that there exist constants σ0 > 0 and c0 ∈ R such that 1 lim |x|→+∞ |x|

x (J (u) − c0 ) du = 0, 0

and 1 |x|

x [J (u) − c0 ] 2 du → 0

⎧ 2 ⎪ ⎨ σ02 , σ1

σ2 ⎪ ⎩ − 02 , σ2

x → +∞, x → −∞.

Then the stochastic processes βT(1)(t)

1 = √ T

t T g (ξ(s)) ds 0

converge weakly, as T → +∞, to the process β (1)(t) = 2σ0 W ∗ (t), where W ∗ is a Wiener process. Proof Using the Itô formula, we obtain βT(1)(t)

1 1 = √ [F (ξ(tT )) − F (ξ(0))] − √ T T

t T

F (ξ(s)) σ (ξ(s)) dW (s),

0

where f(x)

F (x) = 2

(J (u) − c0 ) du. 0

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

95

Note that f −1 (x)F (x) → 0, as |f (x)| → +∞. Therefore, for every ε > 0, there ! ! exists Nε > 0 such that for all |f (x)| > Nε one has !f −1 (x)F (x)! < ε. Using the inequalities ! ! ! F (ξ(tT )) ! $ % ! ! χ|f (ξ(t T ))|≤N + χ|f (ξ(t T ))|>N sup ! √ ε ε ! T 0≤t ≤L ! ! ! ! ! F (ξ(tT )) ! ! )) !! ! χ|f (ξ(t T ))|>N + sup ! F (ξ(tT ≤ sup !! √ √ ε ! ! ! χ|f (ξ(t T ))|≤Nε T T 0≤t ≤L 0≤t ≤L 1 1 ≤ ε sup |ζT (t)| + sup |F (ϕ(x))| √ ≤ ε sup |ζT (t)| + Cε √ , T T 0≤t ≤L |x|≤Nε 0≤t ≤L we obtain ! ! ! F (ξ(tT )) ! ! ≤ CL ε lim sup E sup !! √ ! T T →+∞ 0≤t ≤L for any ε > 0. Consequently, ! ! ! F (ξ(tT )) ! ! ! → 0, E sup ! √ ! T 0≤t ≤L as T → +∞. For arbitrary L > 0 we have ! ! ! ! t T ! (1) ! P 1 ! sup !βT (t) + √ F (ξ(s)) σ (ξ(s)) dW (s)!! −→ 0, T 0≤t ≤L ! !

(4.22)

0

as T → +∞.  2 of Theorem 4.3 hold for the function F (x) σ (x) with ! The! conditions !b (x)! σ 2 (x) = 4σ 2 . That is why the stochastic process 0 1 √ T

t T

F (ξ(s)) σ (ξ(s)) dW (s)

0

converges weakly, as T → +∞, to the process 2σ0 W ∗ (t), where W ∗ (t) is a Wiener process. Taking into account (4.22), we obtain the proof of Theorem 4.4.  

96

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Theorem 4.5 Let ξ be a solution to Eq. (2.1) with initial condition ξ(0) = x0 . Let the assumptions of Theorem 2.1 hold and 1 f (x)

x 0

⎧ ⎨ 12 , x → +∞, du σ1 → ⎩ 12 , x → −∞. f (u)σ 2 (u) σ 2

Let the real-valued measurable function g (x) be locally bounded and such that there exist constants bˆi , i = 1, 2 and c0 ∈ R, together with functions ψi (r) ∈ Ψ , i = 1, 2 of order αi ≥ 0, respectively, for which (i1 ) 1 lim √ |x|→+∞ |x| ψ1 (|x|)

x [J (u) − c0 ] du = 0, 0

and (i2 ) ⎡ lim

|x|→+∞

⎣

1 ψ1 (|x|)

x

⎤ ˆ ⎦ = 0, [J (u) − c0 ]2 du − b(x)

0

where ˆ b(x) = (i3 )



bˆ1 , x ≥ 0, bˆ2 , x < 0.

Also, let for some constant C > 0 the following inequality hold for all x ∈ R: ! ! x ! ! ! ! 1 1 ! ≤ C, ! | |J du (u) − c 0 ! ! ψ2 (|x|) ! f (ϕ (u)) σ (ϕ (u)) ! 0

and (i4 ) √  ψ2 T lim  √  1 = 0. T →+∞ 1 2 T 4 ψ1 T

4.1 Weak Convergence to the Functionals of a Brownian Motion in a Bilayer. . .

97

Then the stochastic processes 1 =  √  1 1 2 T 4 ψ1 T

(1) βT (t)

t T g (ξ(s)) ds 0

 (1)   (t) , where W ∗ is a Wiener converge weakly, as T → +∞, to the process W ∗ β process, ⎤ ζ (t ) t ⎥ α α (1) (t) = 8 ⎢ β ⎣ |u| 1 bˆ (u) du − |ζ (s)| 1 bˆ (ζ (s)) dζ (s)⎦ , ⎡

0

(4.23)

0

(1) are and ζ is the solution to Eq. (2.8), the Wiener process W ∗ and the process β independent. Proof Using the Itô formula, we obtain (1)

βT (t) =

1  √  1 [F (ξ(tT )) − F (ξ(0))] 1 2 T 4 ψ1 T

1 −  √  1 1 2 T 4 ψ1 T

t T

F (ξ(s)) σ (ξ(s)) dW (s),

0

where f(x)

F (x) = 2

(J (u) − c0 ) du 0

and 1 F (x) = 0. √ |x|→+∞ |f (x)| ψ1 (|f (x)|) lim

Consequently, 1 P  √  1 |F (ξ(tT ))| −→ 0, 2 0≤t ≤L T ψ1 T sup

1 4

as T → +∞.

98

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Using the condition (i1 ), similarly to (4.22) we get for arbitrary L > 0 ! ! ! P ! sup !βT(1)(t) + βT(2) (t)! −→ 0,

(4.24)

0≤t ≤L

as T → +∞, where βT(2) (t)

1 =  √  1 1 2 T 4 ψ1 T

t T

F (ξ(s)) σ (ξ(s)) dW (s).

0

$ According % to conditions (i2 )–(i4 ), it is easy to obtain that the function ˆ F (x)σ (x) satisfy the assumptions of Theorem 4.2 for b (x) = 4b(x). Indeed, taking into account the equalities F (x) = 2 [J (f (x)) − c0 ] f (x) and F (ϕ(x)) = 2 [J (x) − c0 ] f (ϕ(x)) we have that the function 1 q (x) = ψ1 (|x|)

x $ 0

F (ϕ (u)) σ (ϕ (u))

%2

[f (ϕ (u)) σ (ϕ (u))]2

du − b (x)

from Theorem 4.2 has the form 4 q (x) = ψ1 (|x|)

x [J (u) − c0 ]2 du − b (x) . 0

Applying the condition (i2 ), we have that q (x) → 0, as |x| → +∞, x and consequently x1 q 2 (u) du → 0, as |x| → +∞. Therefore, the function 0 $ %2 ˆ F (x)σ (x) satisfies the assumptions of Theorem 4.1 for b (x) = 4b(x). Taking into account the assumption (i3 ) we obtain that ! ! x ! ! ! ! ! ! ! F (ϕ (u)) σ (ϕ (u))! 1 ! ! du ! ! 2 ψ2 (|x|) ! [f (ϕ (u)) σ (ϕ (u))] ! 0

! x ! ! ! ! ! 1 2 ! |J (u) − c0 | du!! ≤ C = ψ2 (|x|) !! f (ϕ (u)) σ (ϕ (u)) ! 0

for all x ∈ R. $ %2 Applying the assumption (i4 ) we conclude that the function F (x)σ (x) satisˆ According to Theorem 4.2, fies the conditions of Theorem 4.2 for b (x) = 4b(x). (2) the stochastic processes βT (t) converge weakly, as T → +∞, to the process

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

99

 (1)   (t) , where the process β (1) (t) has the form (4.23), W ∗ (t) is a Wiener W∗ β (1)(t) are independent. Consequently, the process, and the processes W ∗ (t) and β proof follows from relation (4.24).   Remark 4.1 Let in Eq. (2.1) a(x) = 0 and σ (x) = 1. If the function g (x) is absolutely integrable on R, +∞ −∞ g(x)dx = g(x)dx = c0 0

and

0

! x ! ! ! ! ! ! g(u) du − c0 ! ∼ 1 , ! ! |x|α ! ! 0

as |x| → +∞, with α ≥ 12 , then the conditions of Theorem 4.5 are fulfilled. For α > 12 the normalization in the functional βT(1)(t) is T −1/4 and for α = √ √ − 12 T ln T . equals to

1 2

it

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process (1)

(2)

Next, we consider the behavior of the functionals βT (t) and βT (t) of the solutions ξ to Eq. (3.1). Theorem 4.6 Let ξ be a solution of Eq. (3.1) and the conditions of Theorem 3.3 be fulfilled. Let for some constant C > 0 the following inequality hold: for all x ⎛ u ⎞ x    dv ⎠ du ≤ C 1 + x 2 , f (u) ⎝ (4.25) f (v) 0

f (x)

0

x −2 a(v) dv

where = e 0 . Let g be real-valued measurable locally integrable function, ψ ∈ Ψ with α > 0 and f (x) q (x) := ψ (|x|)

x 0

as |x| → +∞, for some constant b.

g (u) du − b sign x → 0, f (u)

100

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Then the stochastic processes (1) βT (t)

t T

1 √  = √ Tψ T

g (ξ (s)) ds 0

converge weakly, as T → +∞, to the process ⎡ r α+1 (t) − β (1) (t) = 2b ⎣ α+1

t

⎤  (s)⎦ , r α (s) d W

(4.26)

0

where r (t) ≥ 0 is the solution of the Itô equation t r (t) = (2c + 1) t + 2 2

 (s), r(s) d W

0

c = c0 in case (1), c = c1 in case (2), and c = c2 in case (3) of Theorem 3.3. Proof We proceed in a similar way to the proof of Theorem 4.1. Let us consider the function ⎛ ⎞ x u g (v) F (x) = 2 ⎝f (u) dv ⎠ du. f (v) 0

0

Since 1 F (x) a(x) + F (x) = g(x) 2 a.e. with respect to the Lebesgue measure, then, applying the Itô formula (see Lemma A.3) to the process F (ξ(t)), one has 1 (1) √  [F (ξ (tT ) − F (x0 ))] βT (t) = √ Tψ T

1 √  −√ Tψ T

t T 0

F (ξ(s)) dW (s) .

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

101

After some transformations, we have the representation F (x0 ) √  + 2b = −√ Tψ T

(1) βT (t)

t −2b

T (s) + 2 rTα (s) d W

rT (t )

uα du 0

4 1

(k)

ST (t) ,

(4.27)

k=1

0

where  √  ⎤ ψ u T ⎣ √  − uα ⎦ du, ψ T ⎡

rT (t )

(1)

ST (t) = b 0

(2) ST (t)

1 √  =√ Tψ T t

(3)

ST (t) = −b 0

t

(4)

ST (t) = − 0

ξ(t T )

q (u) ψ (|u|) du, 0

 ⎤ √  ψ rT (s) T ⎣ T (s) , √  − rTα (s)⎦ d W ψ T ⎡

 √  ψ rT (s) T √  q (ξ (sT )) dWT (s) . ψ T

It follows from the proof of Theorem 3.3 that the processes rT (t) converge weakly, as T → +∞, to the process r(t), which is the solution to Eq. (3.10) for c = c0 in case (1), c = c1 in case (2), and c = c2 in case (3) of Theorem 3.3. In addition, for arbitrary constants L > 0 and ε > 0 we have + lim lim sup P

N→+∞ T →+∞

lim lim sup

h→0 T →+∞

, sup rT (t) > N

0≤t ≤L

= 0,

P {|rT (t2 ) − rT (t1 )| > ε} = 0. sup |t1 − t2 | ≤ h ti ≤ L

(4.28)

Now we are in a position to establish that ST(k) (t), k = 1, 4, uniformly converge to zero in probability. In particular, it means that they satisfy analog of relations (4.28)

102

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

as well. Let + PN = P

, sup rT (t) > N

0≤t ≤L

and TN > 0 are introduced in Lemma A.16. For arbitrary ε > 0 and L > 0, we have the inequalities uniformly in T ≥ TN : , ! ! ! /! 0 2 ! (1) ! ! (1) ! P sup !ST (t)! > ε ≤ PN + E sup !ST (t)! χrT (t )≤N ε 0≤t ≤L 0≤t ≤L +

!  √  ! ! N ! ψ |u| T ! ! 2 √  − uα !! du, ≤ PN + |b| !! ε ! ψ ! T 0 ⎧ ⎪ ⎪ ⎨

! ξ(tT ) ⎫  √  !! ! √ ⎪ T ! ⎪ !  ! ! ⎬  √  ψ |u| T ! ! ! (2) ! ! ! √  du! > ε P sup !ST (t)! > ε ≤ P sup ! q u T ⎪ ⎪ 0≤t ≤L ⎪ ⎪ ! ψ T ⎩0≤t ≤L !! 0 ⎭ ! ! ξ(tT )  √  !! ! √ ! ! T   ψ |u| T √ ! ! 2 √  du!! χrT (t )≤N ≤ PN + E sup !! q u T ε 0≤t ≤L ! ! ψ T ! ! 0 +

,

2 1 ≤ PN + CN √ ε T + P

! ! ! ! sup !ST(3) (t)! > ε

√ N T

|q(u)| du, (4.29) √

−N T

, ≤ PN

0≤t ≤L

⎡  ⎤2 √  " #2 L ψ rT (s) T 2 √  − rTα (s)⎦ χrT (s)≤N ds, +4 b2 E ⎣ ε ψ T 0

+ P

! ! ! ! sup !ST(4) (t)! > ε

0≤t ≤L

, ≤ PN

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

103

⎡  √  ⎤2 " #2 L ψ rT (s) T 2 √  ⎦ χrT (s)≤N ds +4 E q 2 (ξ(sT )) ⎣ ε ψ T 0

" #2 L 2 2 ≤ PN + 4 CN E q 2 (ξ(sT )) χrT (s)≤N ds. ε 0

Taking into account the convergence (see Lemma A.17) as T → +∞, |u| = 0 and the boundedness of

 √  ψ |u| T √  ψ T

 √  ψ |u| T √  ψ T

− |u|α → 0,

− |u|α for |u| ≤ N,

together with the convergence q(x) → 0, as |x| → +∞, we can pass to the limit, as T → +∞, and then as N → +∞, in the inequalities for ST(k) (t) with k = 1, 2. So, we obtain ! ! ! P ! sup !ST(k) (t)! −→ 0,

0≤t ≤L

(4.30)

as T → +∞, for k = 1, 2. (k) Now we establish a similar convergence for ST (t) with k = 3, 4. It is known that the following convergence holds (see Lemma A.17): !  √  ! ! ψ |x| T ! ! ! α √  − |x| !! → 0, sup !! 0 0 there exists a constant C E |ΦT (ξ(LT ))| ≤

ε C + ε (C + C1 L) . T

Thus, E |ΦT (ξ(LT ))| → 0, as T → +∞. Therefore, L q 2 (ξ(sT )) χrT (s)≤N ds → 0,

E 0

as T → +∞. It means that relation (4.30) holds for ST(4) (t), as well. T (t) as It is clear that the relations (4.28) hold for ST(k) (t), k = 1, 4 and for W well. It means that we can apply Skorokhod’s  convergent subsequence principle T (t), S (k) (t), k = 1, 4 . Therefore, taking into account to the process rT (t), W T Theorem A.12 and Corollary A.3, we have that for any subsequence Tn → +∞ P

rTn (t) −→ r(t), for every t > 0.

P Tn (t) −→  (t), W W

(k)

P

STn (t) −→ S (k) (t), k = 1, 4

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

105

! ! ! P ! According to the relation (4.30), we have that sup !ST(k) (t) ! −→ 0, as Tn → n 0≤t ≤L

+∞ for k = 1, 4 and any L > 0. Taking into account Theorem 3.3, we have that  (t) are related via the equation the processes r(t) and W t r (t) = (2c + 1) t + 2 2

 (s), r(s) d W

(4.32)

0

where c = c0 in case (1), c = c1 in case (2), and c = c2 in case (3) of Theorem 3.3. Using Lemma A.7, we can go to the limit, as subsequence Tn → +∞ in the P

(1)

equality (4.27), and obtain that βTn (t) −→ β (1)(t) for every t ≥ 0, where ⎡ r α+1 β (1)(t) = 2b ⎣ − α+1

t

⎤  (s)⎦ , r α (s) d W

(4.33)

0

 (t) are related via Eq. (4.32). and the processes r(t) and W It follows from the strong uniqueness of the solution r(t) to Eq. (4.32) that the finite-dimensional distributions of the limit process β (1) (t) are unique, as well. Since the sequence Tn → +∞ is arbitrary, the finite-dimensional distributions of the processes βT(1)(t) tend, as T → +∞, to the corresponding finite-dimensional distributions of the process β (1)(t), defined by equality (4.33). In order to establish the weak convergence of the processes βT(1) (t) to the process β (1)(t), it is sufficient to prove the property (4.15) for these processes. It follows from the proof of Theorem 3.3 that the relation (4.15) holds for the process rT (t). According to the convergences (4.30), the property (4.15) holds for the processes ST(k) (t), k = 1, 4, as well. In addition, using the properties of stochastic Itô integrals, we get for any L > 0 and ε > 0 the following inequalities: ⎧ ⎨

! r (t ) ! ⎫ r T (t1 ) ! T 2 ! ⎬ ! ! ! P sup uα du − uα du!! > ε ! ⎩|t1 −t2 |≤h, ti ≤L ! ⎭ ! 0

0

, ε |rT (t2 ) − rT (t1 )| > , sup ≤ PN + P N 2 |t1 −t2 |≤h, ti ≤L +

α

⎧ ⎨

! t ! ⎫ ! 2 ! ⎬ ! ! ! r α (s) d W ! > ε ≤ PN  P (s) sup T T ! ⎩|t1 −t2 |≤h, ti ≤L !! ⎭ ! t1

106

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

! t ⎫ ! ! ! ! ! ε⎬ ! r α (s) χr (s)≤N d W !  > +P 4 sup sup (s) T T T ! 2⎭ ⎩ kh≤L kh≤t ≤(k+1)h !! ! ⎧ ⎨

kh

⎤4 ⎡ t " #4 1  8 α T (s)⎦ ⎣ rT (s) χrT (s)≤N d W ≤ PN + E sup ε kh≤t ≤(k+1)h kh 1. Under additional conditions on the function a(x) in Eq. (3.1) for the function q(x) we can also suppose “explosions” for |x| >> 1. In fact, we have the following theorem. Theorem 4.7 Let ξ be a solution to Eq. (3.1) and the conditions of Theorem 3.3 be fulfilled. Let g be a real-valued measurable locally integrable function and let ψ ∈ Ψ with α > 0. Let also one of the following conditions hold (a) for all x, one has

f (x)

=e

x −2 a(v) dv 0

1 |x|→+∞ x

x

lim

0

or

≤ C, where C is some constant, and q 2 (u) du = 0; f (u)

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

107

(b) for all x, 0 < δ ≤ f (x) and f (x) lim |x|→+∞ x

x q 2 (u) du = 0, 0

where f (x) q(x) = ψ (|x|)

x

g(u) du − b sign x, f (u)

0

and δ and b are some constants. Then the statement of Theorem 4.6 holds for the stochastic process 1 (1) √  βT (t) = √ Tψ T

t T g (ξ(s)) ds. 0

Proof Let us obtain from (4.27) the expression of the form (4.30) for ST(k) (t), (1) (3) k = 1, 4, as T → +∞. The proof of this convergence for ST (t) and ST (t) is completely analogous to the proof of the same convergence in Theorem 4.6. Next, according to the conditions of Theorem 4.7, it is easy to establish convergence 1 x

x q 2 (u) du → 0, 0

as |x| → +∞. The inequality (4.29) can be extended if we use the Cauchy–Schwarz inequality. Therefore, + P

, ! ! 2 1 ! (2) ! sup !ST (t)! > ε ≤ PN + CN √ ε T 0≤t ≤L ⎛ 2 ⎜ 1 ≤ PN + CN (2N)1/2 ⎝ √ ε T (2)

The last inequality implies (4.30) for ST (t).

√ N T √ −N T

√ N T

|q(u)| du √

−N T

⎞ 12 ⎟ q 2 (u) du⎠ .

108

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

It is easy to obtain (4.31) from the assumptions of the theorem. Therefore, in the same way as in the proof of Theorem 4.6, we obtain (4.30) for ST(4) (t) as well. Weak convergence of the processes rT (t), as T → +∞, to the processes r(t) is established in Theorem 3.3. Thus, to prove Theorem 4.7 it is enough to take advantage of Remark 4.2.   Next, we consider certain statements for solutions ξ to Eq. (3.1), which are analogs of Theorems 4.2–4.4. Theorem 4.8 Let ξ be a solution of Eq. (3.1) and the conditions of Theorem 3.3 be fulfilled with c1 = c2 = c0 and 2c0 +1 > 0. Let there exist a real-valued measurable locally integrable function g 2 (x) such that g(−x) = −g(x) and functions ψi ∈ Ψ of order αi > 0, respectively, i = 1, 2, such that ⎡

(1)

f (x) lim ⎣ |x|→+∞ ψ1 (|x|)

x

⎤ g 2 (v) dv − b sign x ⎦ = 0; f (v)

0

(2)

! ! ! ! x ! f (x) |g(v)| !! ! ≤ C; dv ! ψ (|x|) f (v) !! ! 2 0

(3)

√  ψ2 T lim =0   √  12 T →+∞ 1 T 4 ψ1 T

for some constants C > 0 and b. Then the stochastic processes (2) βT (t)

1 =  √  1 1 2 T 4 ψ1 T

t T g (ξ(s)) dW (s), 0

where the processes ξ and W are related via   Eq. (3.1), converge weakly, as T → +∞, to the process β (2)(t) = W ∗ β (1) (t) , where W ∗ is a Wiener process, β (1) is the process from (4.33) with α = α1 , and the processes W ∗ and β (1) are independent. Proof It is easy to see that βT(2)(t)

√ t  4 √  T T (s), = &  g r (s) T dW T √  ψ1 T 0

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

109

where |ξ(tT )| rT (t) = √ , T

Since

t

T (t) = W

t WT (t) =

sign ξ(sT ) dWT (s), 0

W (tT ) √ . T

T (t) is an almost surely P {ξ(sT ) = 0} ds = 0 for all t > 0, W

0

continuous martingale for every fixed T > 0, and its quadratic characteristic is equal T (t) = t. According to Doob’s theorem (see [17, Chapter 1, § 1, Theorem 1]), to W T (t) is, for every fixed T > 0, a Wiener process with respect to the σ -algebra W σ {WT (s), s ≤ t}.   T (t) satisfies Skorokhod’s principle for a Moreover, the process rT (t), W convergent subsequence (Theorem A.12). Hence, without loss of generality, we may assume that (see Lemma A.13), for an arbitrary subsequence Tn → +∞, the P P Tn (t) −→  (t) holds, as Tn → +∞, convergence rTn (t) −→ r(t) as well as W W  (t) are for every t > 0. According to Theorem 3.3, the limit processes r(t) and W related via Eq. (3.10) with c = c0 . Moreover, for all L > 0 and ε > 0 + lim lim sup P

h→0 Tn →+∞

sup

|t1 −t2 |≤h; ti ≤L

! ! !rT (t2 ) − rT (t1 )! > ε n n

, = 0.

(4.34)

Tn (t), as well. Let τTn (t) be the It is clear that a similar convergence holds for W  (1)  minimal solution to the equation βTn τTn (t) = t, where βT(1) (t) n

√ t  2  Tn √  g 2 rTn (s) Tn ds. = ψ1 Tn 0

Note that τTn (t) is a Markov moment with respect to the σ -algebra +∞  2 . √  σ WTn (s), s ≤ t . According to Lemma A.8 we have that g rTn (s) Tn ds = 0 (1)

+∞ with probability 1. Thus, given an arbitrary Tn , the process βTn (t) attains any positive level with probability 1. Similarly to the proof of the convergence (4.19) we obtain   P (2) (4.35) βTn (t) − WT∗n β (1) (t) −→ 0, as Tn → +∞, where WT∗n (t) is a sequence of Wiener processes, β (1) (t) is the process from (4.33) with α = α1 .

110

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Using properties of stochastic integrals, we get ! ! ! !   !EW ∗ (t)W Tn (t)! = !!Eβ (2) τTn (t) W Tn (t)!! Tn Tn ! ! ! ! τ Tn (t ) √ ! ! 4   2 Tn ! Tn (s) · W Tn (t)!! g rTn (s) Tn d W = !E  √  1 ! ! ! ! ψ1 Tn 2 0 ! ! ! ! t ∧τTn (t ) √ ! ! 4   2 Tn ! ! =  √  1 !E g rTn (s) Tn ds ! ! 2 ! ψ1 Tn ! ! 0 √ t !  4 2 !! Tn ! ≤  √  1 E !g rTn (s) Tn ! ds ψ1 Tn 2 0 √  √ t !  2 !! ψ2 Tn Tn ! √  E !g rTn (s) Tn ! ds. = 1  √  1 · Tn4 ψ1 Tn 2 ψ2 Tn 0

(4.36)

Now we show that conditions (2) and (3) of Theorem 4.8 imply that the righthand side of the preceding inequality tends to zero, as Tn → +∞. Consider the function ⎞ ⎛ u x  |g(v)| dv ⎠ du. F (x) = 2 f (u) ⎝ f (v) 0

0

This function possesses a continuous derivative F (x) and an almost everywhere (with respect to the Lebesgue measure) locally integrable second derivative F (x). According to Lemma A.3, we can apply the Itô formula to the process F (ξ(t)) and obtain ) ( √ t !  2 !! Tn F (ξ(tTn )) − F (x0 ) ! √  E √  !g rTn (s) Tn ! ds = E . √ ψ2 Tn Tn ψ2 Tn o

According to condition (2) of Theorem 4.8, we have |F (x)| ≤ Cψ2 (|x|) · |x| for all x for some constant C > 0.

(4.37)

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

111

Therefore, ! !  √  ! F (ξ(tT )) − F (x ) ! ψ2 rTn (t) Tn n 0 ! ! √  ! ≤ C ErTn (t) √  E! √ ≤ C1 ErTαn2 +1 (t). ! ! Tn ψ2 Tn ψ2 Tn Since the drift coefficient in Eq. (3.1) is such that |x a(x)| ≤ C and ξ(0) = x0 , we obtain ErTαn2 +1 (t) ≤ C2 + C3 t

α2 +1 2

for some constants C2 and C3 . Thus, equality (4.37) implies that √ t !  2 !! α +1 Tn ! 2 t 22 . 1 + C E √  !g rTn (s) Tn ! ds = C ψ2 Tn o

In turn, condition (3) of Theorem 4.8 and inequality (4.36) imply that ! ! !EW ∗ (t)W Tn (t)! → 0 Tn for all t > 0, as Tn → +∞.  and W ∗ are Wiener processes and they are asymptotically Since the processes W Tn  . The uncorrelated, we conclude that WT∗n does not depend asymptotically on W further proof of the theorem is similar to the end of the proof of Theorem 4.2.   Theorem 4.9 Let ξ be a solution to Eq. (3.1) and the conditions of Theorem 3.3 be fulfilled for c1 = c2 = c0 and 2c0 + 1 > 0. Let the real-valued measurable function g 2 (x) be locally integrable and such that f (x) |x|

x

g 2 (v) dv − b sign x → 0, f (v)

0

as |x| → +∞, for a certain constant b. Then the stochastic processes (2) βT (t)

1 = √ T

t T g (ξ(s)) dW (s) 0

converge weakly, as T → +∞, to the process Wiener process.

√ b (2c0 + 1)W ∗ (t), where W ∗ is a

112

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Proof It is clear that (2) βT (t)

t =

g (ξ(sT )) dWT (s) 0

with probability 1 for every t > 0. Let τT (t) be the minimal solution to the equation (1) βT (τT (t)) = t, where βT(1)(t)

t =

g 2 (ξ(sT )) ds. 0

Since τT (t) is a Markov moment with respect to the σ -algebra σ {W (sT ), s ≤ t} for every fixed T > 0 and, according to Lemma A.8, the equality +∞  2  g 2 rTn (s) Tn ds = +∞ 0 (2)

holds with probability 1, then for every fixed T > 0 the process βT (τT (t)) is a Wiener process. Denote this process by WT∗ (t). Thus,   βT(2) (τT (t)) = WT∗ βT(1)(t) . It is easy to see that the function g 2 (x) satisfies the conditions of Theorem 4.6 for (1) α = 1. Therefore, according to Theorem 4.6, the process βT (t) converges weakly, as T → +∞, to the process ⎡ r 2 (t) β (1) (t) = 2b ⎣ − 2

t

⎤  (s)⎦ = b (2c0 + 1) t. r(s) d W

0

It follows from the condition of Theorem 4.9 that b ≥ 0. Since the limit process for βT(1)(t) is degenerate, then convergence of βT(1) (t) to β (1) (t), as T → +∞, for (1) every t > 0 in probability, follows from the weak convergence of βT (t) to β (1)(t), as T → +∞. Consequently, similarly to the proof of the relation (4.19), we have   P (2) βT (t) − WT∗ β (1) (t) −→ 0

(4.38)

4.2 Weak Convergence to the Functionals of the Bessel Diffusion Process

113

as T → +∞. Since the process   WT∗ β (1)(t) √ b(2c0 + 1) is a Wiener process for b > 0, according to (4.38), we have the convergence of the finite-dimensional distributions of the process βT(2) to the corresponding √ distributions of the process b (2c0 + 1)W ∗ , where W ∗ is a Wiener process. The proof of the relation (4.15) for the process βT(2) coincides completely with (2) the proof of this convergence for the process βT in Theorem 4.2. To complete the proof, it remains to use Theorem A.13.   Theorem 4.10 Let ξ be a solution of Eq. (3.1) and the conditions of Theorem 3.3 be fulfilled for c1 = c2 = c0 and 2c0 + 1 > 0. Assume that a real-valued measurable function g(x) is locally integrable and there are two constants a and b for which

(1)

1 |x|

x

⎞ ⎛ u  g(v) dv ⎠ du → 0, f (u) ⎝ f (v) a

0

and

(2)

f (x) |x|

x

⎤2 ⎡ u  g(v) dv ⎦ du − b sign x → 0, f (u) ⎣ f (v) a

0

as |x| → +∞. Then the stochastic processes (1) βT (t)

1 = √ T

t T g (ξ(s)) ds 0

converge weakly, as T → +∞, to the process 2 2 b (2c0 + 1)W ∗ (t), where W ∗ is a Wiener process. Proof Consider the function x F (x) = 2 0

⎞ ⎛ u  g(v) dv ⎠ du. f (u) ⎝ f (v) a

114

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

This continuous function possesses a continuous derivative F (x) and an almost everywhere (with respect to the Lebesgue measure) locally integrable second derivative F (x). According to Lemma A.3, we can apply the Itô formula to the process F (ξ(t)). Using the equality 1 F (x)a(x) + F (x) = g(x) 2 a.e. with respect to the Lebesgue measure, we obtain that (1) βT (t)

1 F (ξ(tT )) − F (x0 ) −√ = √ T T

t T

F (ξ(s)) dW (s)

(4.39)

0

with probability 1 for all t ≥ 0 and for every T . It follows from condition (1) of Theorem 4.10 that F x(x) → 0, as |x| → +∞. In addition, taking into account the inequality (3.16), we obtain ξ 2 (tT ) ≤ CL . T 0≤t ≤L

E sup

It is easy to establish that (see the proof of Theorem 3.3) ! ! ! F (ξ(tT )) ! P ! ! −→ 0, sup ! √ ! T 0≤t ≤L as T → +∞. So, for every t ≥ 0 ! ! ! ! t T ! (1) ! P 1 ! F (ξ(s)) dW (s)!! −→ 0, sup !βT (t) + √ T 0≤t ≤L ! ! 0

as T → +∞. Since 1 √ T

t T 0

F (ξ(s)) dW (s) =

t

⎛ F (ξ(sT )) dWT (s) = WT∗ ⎝

0

t

$

%2

F (ξ(sT )) ds ⎠ ,

0

where WT∗ is a Wiener process for every T (see the proof of Theorem 4.7), then ! ⎞! ⎛ t ! !  ! P ! $ % 2 sup !!βT (t) + WT∗ ⎝ F (ξ(sT )) ds ⎠!! −→ 0, 0≤t ≤L ! ! 0

⎞

4.3 Results About Weak Convergence of the Mixed Functionals

115

$ %2 as T → +∞. According to condition (2), the function F (x) satisfies assumptions of Theorem 4.6 with α = 1. That is why the processes βT(1)(t)

t =

$

%2 F (ξ(sT )) ds

0

converge weakly, as T → +∞, to the degenerate process β (1) (t) = 4b (2c0 + 1) t. The rest of the proof is similar to the corresponding proof of Theorem 4.9.

 

4.3 Results About Weak Convergence of the Mixed Functionals Next, we consider the asymptotic behavior, as t → +∞, of the distributions of functionals of the following form: t I (t) = F (ξ(t)) +

g (ξ(s)) dW (s),

(4.40)

0

where F = F (x) is a continuous functions, g = g(x) is a real-valued measurable function, x ∈ R, g is locally square integrable, the processes ξ and W are related via Eq. (2.1) with σ (x) ≡ 1 and ξ(0) = x0 . Let us now consider the behavior of these functionals depending on the solutions of two classes of equations. Definition 4.2 Let K1 be the class of equations of the form (2.1) with σ (x) ≡ 1, ξ(0) = x0 , and such a that ! x ! ! ! ! ! ! sup ! a(v) dv !! ≤ C x∈R ! ! 0

for a certain constant C > 0, and convergence (2.13) holds for x f (x) =

⎧ ⎨

exp −2 ⎩ 0

⎫ ⎬

u a(v) dv 0

⎭

du.

116

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Also, let K2 be the class of equations of the form (2.1) with σ (x) ≡ 1, ξ(0) = x0 , and such a that sup |x a(x)| ≤ C, and x∈R

1 lim |x|→+∞ x

x

1 v a(v) dv = c0 > − . 2

0

According to Lemma 2.1 and Theorem 3.1, the solutions ξ of equations belonging to classes Ki , i = 1, 2 are stochastically unstable. It is clear that the (2) functional βT (t) is a functional of the form (4.40). It follows from (4.5) that the (1) functional βT (t) is a functional of the form (4.40), as well. In addition, using the Itô formula, we have equality t

t g (ξ(s)) dξ(s) =

0

t g (ξ(s)) a (ξ(s)) ds +

0

g (ξ(s)) dW (s) 0

t = Φ (ξ(t)) − Φ (x0 ) +

% $ g (ξ(s)) − Φ (ξ(s)) dW (s),

(4.41)

0

where x Φ (x) = 2 0

Thus, the functional

t

⎞ ⎛ u  g(v) a(v) dv ⎠ du. f (u) ⎝ f (v) 0

g (ξ(s)) dξ(s) is a functional of the form (4.40). There-

0 (1)

(2)

fore, using the results obtained earlier about the functionals βT (t) and βT (t), it is easy to get similar results for properly normalized functionals from (4.40). Here we present only the formulation of the following theorems. Theorem 4.11 Let ξ be a solution of Eq. (2.1) belonging to the class K1 . Let two functions F and g define the functional I (t) by equality (4.40) and let the function ψ ∈ Ψ be a regularly varying function of order α ≥ 0. Assume that   F (x) a1 , x ≥ 0, − a(x) = 0, a(x) = (1) lim |x|→+∞ f (x) ψ (|f (x)|) a2 , x < 0. (2)

! ! ! ! g(x) ! ! ! f (x) ψ (|f (x)|) ! χ(|x|≤N) ≤ CN for any N > 0 and 

lim

|x|→+∞

g(x) − b(x) = 0, f (x) ψ (|f (x)|)

for some constants CN > 0 and ai , bi , i = 1, 2.

 b(x) =

b1 , x ≥ 0, b2 , x < 0

4.3 Results About Weak Convergence of the Mixed Functionals

117

Then the stochastic processes I (tT ) √  IT (t) = √ Tψ T converge weakly, as T → +∞, to the process t I0 (t) = a (ζ (t)) ζ (t) |ζ (t)| +

b (ζ(s)) |ζ(s)|α dζ(s),

α

0

where ζ is the solution of Eq. (2.8). Remark 4.3 (1) If α = 0, a(x) = a0 sign x, and b(x) = −a0 sign x, then ⎡ I0 (t) = a0 ⎣|ζ (t)| −

⎤

t

sign ζ (s) dζ(s)⎦ = a0 L0ζ (t),

0

where L0ζ (t) is the local time at zero of the process ζ on the interval [0, t]. (2) If α = 1, a(x) = a0 sign x, and b(x) = −2a0 sign x, then the Itô formula for the process ζ 2 (t) yields t I0 (t) = a0

σ 2 (ζ (s)) ds, 0

and for σ1 = σ2 = σ0 we have I0 (t) = a0 σ02 t. Theorem 4.12 Let ξ be a solution of Eq. (2.1) belonging to the class K1 . Let two functions F and g define the functional I (t) by equality (4.40) and let the functions ψi ∈ Ψ be regularly varying functions of orders αi ≥ 0, i = 1, 2. Assume that  (1)

lim

|x|→+∞

F (x) √ − a(x) = 0, a(x) = |f (x)| ψ1 (|f (x)|) ⎡

(2)

1 lim ⎣ |x|→+∞ ψ1 (|f (x)|)

x



b1 , x ≥ 0, b2 , x < 0,

a1 , x > 0, a2 , x < 0.

⎤ g 2 (|v|) dv − b(x)⎦ = 0, f (v)

0

b(x) =



b1 ≥ 0, b2 ≤ 0.

118

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

√  ψ2 T lim  √  1 = 0 T →+∞ 1 2 T 4 ψ1 T

! ! ! ! x ! |g(v)| !! 1 ! dv ! ≤ C, ! ψ2 (|f (x)|) f (v) ! !

(3)

0

for some constants C > 0 and ai , bi , i = 1, 2. Then the stochastic processes IT (t) =

I (tT )  √  1 1 2 T 4 ψ1 T

converge weakly, as T → +∞, to the process I0∗ (t) = a (ζ (t)) |ζ (t)|

1+α1 2

  + W ∗ β (1) (t) ,

where ⎡

⎤ ζ(t ) t ⎢ ⎥ β (1) (t) = 2 ⎣ b(x) |x|α1 dx − b (ζ (s)) |ζ(s)|α1 dζ(s)⎦ , 0

0

ζ is a solution of Eq. (2.8), W ∗ is a Wiener process, and the processes W ∗ and ζ are independent. Remark 4.4 (1) If α1 = 0, b(x) = b0 sign x, then I0∗ (t) = a (ζ (t))

2

  |ζ (t)| + W ∗ 2b0 L0ζ (t) ,

where L0ζ (t) is the local time at zero of the process ζ on the interval [0, t], W ∗ is a Wiener process, and the processes W ∗ and ζ are independent. (2) If α1 = 1, b(x) = b0 sign x, then ⎛ I0∗ (t) = a (ζ (t)) |ζ (t)| + W ∗ ⎝b0

t

⎞ σ 2 (ζ(s)) ds ⎠ ,

0

where the processes W ∗ and ζ are independent. In particular, if σ1 = σ2 = σ0 , then   I0∗ (t) = a (ζ (t)) |ζ (t)| + W ∗ b0 σ02 t . (3) If b(x) ≡ 0, then I0∗ (t) = a (ζ (t)) |ζ (t)|

1+α1 2

, α1 ≥ 0.

4.3 Results About Weak Convergence of the Mixed Functionals

119

Theorem 4.13 Let ξ be a solution of Eq. (3.1) belonging to the class K2 . Let two functions F = F (x) and g = g(x) define the functional I (t) by equality (4.40) and let ψ = ψ(r) ∈ Ψ be a regularly varying function of order α > 0. Assume that (1)

(2)

the function

F (x) = a0 ; |x|→+∞ |x| ψ (|x|) lim



g(x) is bounded and ψ (|x|)

lim

|x|→+∞

g(x) − b0 sign x = 0, ψ (|x|)

where a0 and b0 are some constants. Then the stochastic processes I (tT ) √  IT (t) = √ T ψ1 T converge weakly, as T → +∞, to the process t I0 (t) = a0 r

α+1

(t) + b0

 (s), r α (s) d W

0

 are related via Eq. (4.32) where the stochastic process r and the Wiener process W with c = c0 . Remark 4.5 If α = 1, then # " b0 2 b0 r (t) − I0 (t) = a0 + (2c0 + 1) t. 2 2 In particular, in the case a0 = − b20 we have that I0 (t) = a0 (2c0 + 1) t. Theorem 4.14 Let ξ be a solution of Eq. (3.1) belonging to the class K2 . Let two functions F = F (x) and g = g(x) define the functional I (t) by equality (4.40). Assume that there are two regularly varying functions ψi = ψi (r) ∈ Ψ of orders αi > 0, i = 1, 2, and an odd locally square integrable function  g = g (x) such that (1) ⎡

(2)

lim

|x|→+∞

f (x) lim ⎣ |x|→+∞ ψ1 (|x|)

F (x) √ = a0 ; |x| ψ1 (|x|) x 0

⎤  g 2 (u) du − b0 sign x ⎦ = 0; f (u)

120

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

⎡

f (x) lim ⎣ |x|→+∞ ψ1 (|x|)

(3)

x

⎤ g (u)]2 ⎦ [g(u) −  du = 0; f (u)

0

(4)

√  ψ2 T lim √  1 = 0 T →+∞ 1  2 4 T ψ1 T

! ! ! ! x ! f (x) ! |g(u)| ! du!! ≤ C, ! ψ (|x|) f (u) ! 2 ! 0

for some constants C > 0 and a0 , b0 . Then the stochastic processes IT (t) =

I (tT )  √  1 2 T ψ1 T 1 4

converge weakly, as T → +∞, to the process I0∗ (t) = a0 [r(t)]

1+α1 2

  + W ∗ β (1) (t) ,

where ⎡

r α1 +1 (t) − β (1)(t) = 2b0 ⎣ α1 + 1

t

⎤  (s)⎦ . r α1 (s) d W

0

Here r ≥ 0 is a solution of Eq. (4.32) with c = c0 , W ∗ is a Wiener process, and the  are independent. processes W ∗ and W Remark 4.6 Condition (4) of Theorem 4.14 is used only when we prove that W ∗  are independent. If α1 = 1 and a0 = 0, then we can omit condition (4) and and W √ get I0∗ (t) = b0 (2c0 + 1) W ∗ (t).

4.4 Examples Example 4.1 Let the coefficients a(x) and σ (x) of Eq. (2.1) satisfy the sufficient conditions of Theorem 2.1. Let the real-valued measurable function g(x) be locally integrable. $ %−1 (1) If the function g(x) f (x)σ 2 (x) is absolutely integrable on R and  λ := R

g(x) f (x)σ 2 (x)

dx = 0,

4.4 Examples

121

then the stochastic processes βT(1) (t)

t T

1 = √ T

g (ξ(s)) ds 0

converge weakly, as T → +∞, to the process β (1) (t) ≡ 0. Indeed, in this case the conditions of Theorem 4.1 are fulfilled with ψ(r) ≡ 1, b1 = b2 = b0 , where +∞ b0 =

g(x) dx. f (x)σ 2 (x)

0

Therefore, according to Theorem 4.1 the stochastic processes βT(1)(t) converge weakly, as T → +∞, to the process β (1) (t) of the form (4.4) with α = 0 and b(x) = b0 . So, in this case, the limit process β (1) (t) ≡ 0. t Note, that for λ = 0 the behavior, as t → +∞, of the functional g (ξ(s)) ds depends on the rate of convergence α(x) → 0, as |x| → +∞, where x α(x) =

g(u) f (u) σ 2 (u)

0

du − b0 .

0

For example, if 1 lim √ |x|→+∞ |f (x)|

x



f (u) α(u) du = 0 and

f (x)α 2 (u) du < +∞,

R

0

then the stochastic processes (1) βT (t)

1 = √ 4 T

t T g (ξ(s)) ds 0

  converge weakly, as T → +∞, to the process W ∗ β (1)(t) , where W ∗ is a Wiener process, β (1) is a process of the form (4.4) with α = 0, +∞ b1 = 4 f (u) α 2 (u) du,

−∞ b2 = 4 f (u) α 2 (u) du,

0

and the processes W ∗ and β (1) are independent.

0

122

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

Indeed, it follows from the representation t

t g (ξ(s)) ds = F (ξ(t)) − F (ξ(0)) −

0

F (ξ(s)) σ (ξ(s)) dW (s),

0

x where F (x) = 2 f (u) α(u) du and from the proof of Theorem 4.5. Note, that the 0

function F (x)σ (x) satisfies the conditions of Theorem 4.2. (2) If λ = 0, then the conditions of Theorem 4.1 are fulfilled with ψ(r) ≡ 1, α = 0, +∞ −∞  −1  −1 2 b1 = g(u) f (u) σ (u) du, and b2 = g(u) f (u) σ 2 (u) du. 0

0

Therefore, according to Theorem 4.1, the stochastic processes (1) βT (t)

1 = √ T

t T g (ξ(s)) ds 0

converge weakly, as T → +∞, to the process ⎡

⎤ ζ(t ) t ⎢ ⎥ β (1) (t) = 2 ⎣ b(u) du − b(ζ(s)) dζ(s)⎦ , 0

0

where b(x) = b1 for x > 0 and b(x) = b2 for x < 0, ζ(t) is the solution to Eq. (2.8). In particular, if b2 = −b1, then ⎡ β

(1)

(t) = 2b1 ⎣|ζ (t)| −

t

⎤ sign ζ (s) dζ(s)⎦ = 2b1 L0ζ (t),

0

where L0ζ (t) is the local time at zero of the process ζ on the interval [0, t]. Example 4.2 Let in Eq. (2.1) the coefficients a(x) = 0 and σ (x) = 1 for all x. (1) If g(x) = sin x, then the conditions of Theorem 4.4 are fulfilled with c0 = 1 and σ02 = 12 . Therefore, the stochastic processes 1 √ T

t T sin W (s) ds 0

4.4 Examples

123

√ converge weakly, as T → +∞, to the process 2W ∗ (t), where W ∗ is a Wiener process. In addition, in this case the conditions of Theorem 4.3 are fulfilled with σ02 = 1 2 . Therefore, the stochastic processes 1 √ T

t T sin W (s) dW (s) 0

converge weakly, as T → +∞, to the process W ∗. (2) If c(x) sin x g(x) = √ , 1 + x2

√1 W ∗ (t) 2

 c(x) =

with a Wiener process

c1 , x ≥ 0, c2 , x < 0,

then the conditions of Theorem 4.2 are fulfilled with ψ1 (r) ≡ 1, ψ2 (r) ≡ ln (1 + r) , b1 = c12

+∞  0

sin2 x dx, b2 = c22 1 + x2

−∞  0

sin2 x dx. 1 + x2

Therefore, according to Theorem 4.2, the stochastic processes βT(2)(t)

1 = √ 4 T

t T 0

c (W (s)) sin W (s) 2 dW (s) 1 + W 2 (s)

  converge weakly, as T → +∞, to the process W ∗ β (1) (t) , where W ∗ (t) is a Wiener process, ⎡ W (t ) ⎤  t b(u) du − b(W (s)) dW (s)⎦ , β (1) (t) = 2 ⎣ 0

0

b(x) = b1 for x > 0 and b(x) = b2 for x < 0. Furthermore, the Wiener processes W ∗ (t) and W (t) are independent. In particular, if c12 = c22 , then β (1)(t) = 2b1 L0W (t). (3) If the function g(x) is such that x 0

⎧  ⎨ sin(x − n)n2 π, |x| ∈ n, n +   g(u) du − sign x = ⎩ 0, |x| ∈ / n, n + n12 ,

1 n2

 ,

124

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

then for this function the conditions of Theorem 4.1 are fulfilled with ψ(r) ≡ 1 and b(x) = sign x, and α = 0. Note, that q(x)  0, as |x| → +∞ and 1 x

x q 2 (u) du → 0, 0

as |x| → +∞. Then the stochastic processes

√1 T

weakly, as T → +∞, to the process ⎡ β (1)(t) = 2 ⎣|W (t)| −

t

t T

g (W (s)) ds converge

0

⎤ sign W (s) dW (s)⎦ = 2L0W (t).

o

Example 4.3 Let the coefficients of Eq. (2.1) be a(x) ≡ 0 and σ (x) ≡ 1. Let the initial condition of Eq. (2.1) be ξ(0) = 0. The solution of Eq. (2.8) has the form ζ(t) = W (t). In this case f (x) = x, ϕ(x) = x, and x J (x) = 0

x

g (ϕ (u)) [f (ϕ (u)) σ (ϕ (u))]2

du =

g(u) du. 0

Let the equality J (x) = √sin x 2 be fulfilled, that is, 1+x

" g(x) =

sin x √ 1 + x2

#

cos x x sin x  = √ − 2 1 + x2 1+x

and f(x)

F (x) = 2

x

J (u) du = 2 0

0

sin u √ du. 1 + u2

The conditions of the Theorem 4.5 are satisfied with ψ1 (|x|) = 1, (α1 = 0), c0 = 0, ˆ b(x) = b0 sign x, +∞ b0 = 0

  ψ2 (|x|) = ln 1 + |x|2 , (α2 = 0).

sin u du, √ 1 + u2

4.4 Examples

125

According to Theorem 4.5 the stochastic processes (1) βT (t)

t T

1 = √ 4 T

g (ξ(s)) ds 0

 (1)   (t) , where W ∗ is a Wiener converge weakly, as T → +∞, to the process W ∗ β process, ⎤ ⎡ W (t )  t (1) (t) = 8b02 ⎣ β sign u du − sign (W (s)) dW (s)⎦ = 8b02L0W (t), 0

0

(1) are independent. the Wiener process W ∗ and the process β c0 x 1+x 2

Example 4.4 Let in Eq. (3.1) a(x) =

for all x and 2c0 + 1 > 0.

(1) If g(x) = sin x and c0 = 1, then the conditions of Theorem 4.10 are fulfilled for a = 0 and b = 16 . Therefore, the stochastic processes t T

1 √ T

sin ξ(s) ds 0

√ converge weakly, as T → +∞, to the process 2W ∗ (t), where W ∗ is a Wiener process. In addition, the conditions of Theorem 4.9 are fulfilled for b = 16 . Therefore, according to Theorem 4.9 the stochastic processes 1 √ T

t T sin ξ(s) dW (s), 0

where the processes ξ(t) and W (t) are related via Eq. (2.1), converge weakly, as T → +∞, to the process √1 W ∗ (t), where W ∗ is a Wiener process. 2

sin x (2) If g(x) = √ and c0 = − 14 , then the conditions of Theorem 4.8 are fulfilled 8 1+x 2 √ 3 for ψ1 (r) = r ln r, ψ2 (r) = r 4 , b = 12 . Therefore, the stochastic processes

1 2 √ T 3/8 ln T

t T 2 8 0

sin ξ(s) 1 + ξ 2 (s)

dW (s),

126

4 Asymptotic Behavior of Integral Functionals of Stochastically Unstable Solutions

where the processes ξ(t) and W (t) via Eq. (2.1), converge weakly,  are related  as T → +∞, to the process W ∗ β (1)(t) , where W ∗ is a Wiener process,

β

(1)

2 (t) = r 3/2 (t) − 3

t 2

 (s), r(s) d W

1 r (t) = t + 2

t

2

0

 (s), r(s) d W

0

 and W ∗ are independent. and the Wiener processes W Example 4.5 Let in Eq. (2.1) a(x) =

x 1+x 4

and σ (x) ≡ 1. In this case, according to

Theorem 2.1, the solution to Eq. (2.8) is ζ (t) = σ0 W (t), where σ0 = e− 2 . π

(1) Let g(x) → g0 , as |x| → +∞. Using the equality (4.41), we obtain that the t conditions of Theorem 4.11 are fulfilled for the functional g (ξ(s)) dξ(s) with 0

ψ (|x|) ≡ 1, a(x) = 2α(x), b(x) = g0 σ0−1 − 2α(x), where  α(x) =

α1 , x ≥ 0, α1 = α2 , x < 0,

+∞

g(v)a(v) dv, α2 = f (v)

0

−∞

g(v)a(v) dv. f (v)

0

Therefore, the stochastic processes 1 √ T

t T g (ξ(s)) dξ(s) 0

converge weakly, as T → +∞, to the process ⎡ I0 (t) = 2σ0 ⎣α (W (t)) W (t) −

t

⎤ α (W (s)) dW (s)⎦ + g0 W (t).

0

In particular, (a) if α0 = α1 = −α2 , then I0 (t) = 2α0 σ0 L0W (t) + g0 W (t); (b) if α0 = α1 = α2 , then I0 (t) = g0 W (t). (2) Let t I (t) = |ξ(t)| −

sign ξ(s) dξ(s) = L0ξ (t) 0

be the local time at zero of the process ξ on the interval [0, t].

4.4 Examples

127

Using the equality (4.41), we obtain that the conditions of Theorem 4.11 are fulfilled for the functional I (t) with ψ (|x|) ≡ 1, a (x) = sign x, b (x) = − sign x. Therefore, the stochastic processes √1 L0ξ (tT ) converge weakly, as T → +∞, to the process I0 (t) = σ0 L0W (t).

T

x Example 4.6 Let in Eq. (3.1) a(x) = 1+x 2 . The conditions of Theorem 4.14 are t  fulfilled for the functional sin ξ(s) dξ(s) with ψ1 (|x|) = |x|, a0 = 0, α1 = 1, 0

 g (x) = sin x, b0 = 16 , c0 = 1. So, according to Remark 4.6, the stochastic processes 1 √ T

t T sin (ξ(s)) dξ(s) 0

converge weakly, as T → +∞, to the process I0∗ (t) = Wiener process.

√1 W ∗ (t), 2

where W ∗ is a

Chapter 5

Asymptotic Behavior of Homogeneous Additive Functionals Defined on the Solutions of Itô SDEs with Non-regular Dependence on a Parameter

In this chapter, we consider homogeneous one-dimensional stochastic differential equations with non-regular dependence on a parameter. The asymptotic behavior of t the mixed functionals of the form IT (t) = FT (ξT (t)) + gT (ξT (s)) dξT (s), t ≥ 0 0

is studied as T → +∞. Here ξT is a strong solution to the stochastic differential equation dξT (t) = aT (ξT (t)) dt + dWT (t), T > 0 is a parameter, aT = aT (x) is a set of measurable functions, |aT (x)| ≤ LT for all x ∈ R, WT = WT (t) are standard Wiener processes, FT = FT (x), x ∈ R are continuous functions, and gT = gT (x), x ∈ R are locally bounded real-valued functions. Section 5.1 contains some preliminary remarks. We prove a theorem about the weak compactness of the family of some processes in Sect. 5.2. Section 5.3 includes a theorem concerning the weak convergence of some stochastic processes to the solutions of Itô SDEs. In Sect. 5.4 we consider the asymptotic behavior of integral functionals of Lebesgue integral type. Section 5.5 is devoted to asymptotic behavior of the integral functionals of martingale type. The explicit form of the limiting processes for IT (t) is established in Sect. 5.6 under very non-regular dependence of gT and aT on the parameter T . This section summarizes the main results and their proofs. Section 5.7 contains several examples. Auxiliary results are collected in Sect. 5.8.

5.1 Preliminaries Consider the following Itô stochastic differential equation: dξT (t) = aT (ξT (t)) dt + dWT (t), t ≥ 0, ξT (0) = x0 ,

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_5

(5.1)

129

130

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

where T > 0 is a parameter, aT = aT (x), x ∈ R are real-valued measurable functions, such that for some constants LT > 0 and for all x ∈ R |aT (x)| ≤ LT , and WT = {WT (t), t ≥ 0}, T > 0 is a family of standard Wiener processes defined on a complete probability space (Ω, F, P). It is known from Theorem 4 in [82] that for any T > 0 and x0 ∈ R Eq. (5.1) possesses a unique strong pathwise solution ξT = {ξT (t), t ≥ 0} and this solution is a homogeneous strong Markov process. We suppose that the drift coefficient aT (x) in Eq. (5.1) can have a very nonregular dependence on the parameter. For example, the drift coefficient can be a “δ”-type sequence at some points xk , as T → +∞. Otherwise, it can be equal to √ √ T sin (x − xk ) T or it can have degeneracies of some other types. It is known from [17, § 16] that the asymptotic behavior of the solution ξT of Eq. (5.1) is closely related to the asymptotic behavior of harmonic functions, i.e., functions satisfying the following ordinary differential equation a.e. with respect to the Lebesgue measure: fT (x)aT (x) +

1 f (x) = 0. 2 T

It is obvious that the functions fT (x) have the form (1)

x

fT (x) = cT

⎧ ⎨

exp −2 ⎩ 0

⎫ ⎬

u aT (v) dv

⎭

(2)

du + cT ,

(5.2)

0

where cT(1) and cT(2) are some families of constants. The latter functions possess the continuous derivatives fT (x) and their second derivatives fT (x) exist a.e. with respect to the Lebesgue measure and are locally integrable. Note that cT(1) play the role of normalizing constants and cT(2) of centering constants, respectively, in the limit theorems (see [81, §6]). Further, for simplicity, (1) (2) we assume that cT ≡ 1, cT ≡ 0. In this chapter our assumption concerning the coefficient aT (x) of Eq. (5.1) is that there exists a family of functions GT = GT (x), x ∈ R with continuous derivatives GT (x) and locally integrable second derivatives GT (x) a.e. with respect to the Lebesgue measure, such that for all T > 0 and x ∈ R the following inequalities hold true:  2  2   (A1 ) GT (x)aT (x) + 12 GT (x) + GT (x) ≤ C 1 + (GT (x))2 , |GT (x0 )| ≤ C. Suppose additionally that the unique strong solution of Eq. (5.1) satisfies the following assumptions: + , (A2 )

lim lim sup P

N→+∞ T →+∞

sup |ξT (t)| > N

0≤t ≤L

= 0 for any constant L > 0.

5.1 Preliminaries

131

(A3 ) There exist constants δ > 0, C > 0, and m1 ≥ 0 such that the following inequality holds: ! ! ! u ! 1 !x ! ! ! 1+δ ! ! ! ! % $ 1 ! ! ! dv du ! fT (u) !! ! ≤ C 1 + |GT (x)|m1 ! 1+δ ! ! ! |fT (v)| ! !0 ! 0 + for all x ∈ R and T > 0, where

fT (x)

,

x

= exp −2 aT (v) dv . 0

(A4 ) There exist a bounded function ψ (|x|), x ∈ R and a constant m2 ≥ 0 such that ψ (|x|) → 0 as |x| → 0, and for all x ∈ R, T > 0 and any measurable bounded set B the following inequality holds: x

⎛ fT (u) ⎝

0

u 0

⎞ $ % χB (GT (v)) ⎠ dv du ≤ ψ (λ(B)) 1 + |GT (x)|m2 , fT (v)

where χB (v) is the indicator function of a set B and λ(B) is the Lebesgue measure of B. Let K (GT ) be the class of equations of the form (5.1) whose coefficients aT (x) and solution satisfy conditions (A1 )–(A4 ). It is easy to understand that the class (1) (2) K (GT ) does not depend on the constants cT and cT in the representation (5.2). It is clear that if there exist constants δ > 0 and C > 0 such that 0 < δ ≤ fT (x) ≤ C for all x ∈ R, T > 0, then such equations (5.1) belong to the class K (GT ) for GT (x) = fT (x). We denote this subclass as K1 . Note that the class K (GT ) contains in particular the equations for which at some points xk we have a convergence fT (xk ) → +∞ or a convergence fT (xk ) → 0, as T → +∞. For c0 T x example, consider Eq. (5.1) with aT (x) = 1+x 2 T . It is easy to obtain that fT (x) = 1 and if c0 > − 12 , such equations belong to the class K (GT ) with GT (x) = (1+x 2 T )c0 x 2 (here at point x = 0 we have fT (x) → 0 for c0 > 0, fT (x) → +∞ for − 12 < c0 < 0, as T → +∞, and fT (x) ≡ 1 for c0 = 0). For the class of equations K (GT ) we study the asymptotic behavior, as T → +∞, of the distributions of the following functionals:

(1) βT (t)

t =

gT (ξT (s)) ds, 0

(2) βT (t)

t =

gT (ξT (s)) dWT (s), 0

t IT (t) = FT (ξT (t)) +

t gT (ξT (s)) dWT (s), βT (t) =

0

gT (ξT (s)) dξT (s), 0

132

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

where the processes ξT and WT are related via Eq. (5.1), gT is a family of measurable, locally bounded real-valued functions, FT is a family of continuous real-valued functions. Assume that for certain locally bounded functions qT (x) and any constant N > 0 the following conditions hold: (A5 )

ψT(1) (x) := fT (x)

x 0

qT (v) fT (v)

dv → 0, a.e., as T → +∞,

herewith |ψT(1) (x)|χ|x|≤N ≤ CN for arbitrary N > 0; for measurable and locally bounded functions gT (x) and g0 (x) x (2) ψT (x) := fT (x) fgT (v) dv − g0 (GT (x)) GT (x) → 0, a.e., as T → +∞, (v)

(A6 )

herewith

0 T (2) |ψT (x)|χ|x|≤N

≤ CN ;

(A7! ) there exist some constants C > 0 and αi ≥ 0, i = 1, 2 such that ! ! ! x  ! ! q (v) !fT (x) fT (v) dv ! ≤ C [1 + |x|α1 ] , |GT (x)| ≥ C|x|α2 ! ! T 0 for all x ∈ R.

5.2 Theorem Concerning the Weak Compactness In what follows we denote by C, L, N, CN any constants which do not depend on T and x. Theorem 5.1 Let ξT be a solution to Eq. (5.1) and there exists a family of functions GT (x), which satisfies assumption (A1 ). Then the family of the processes ζT = {ζT (t) = GT (ξT (t)), t ≥ 0} is weakly compact.

Proof The functions GT (x) have continuous derivatives GT (x) for all T > 0, the second derivatives GT (x) exist a.e. with respect to the Lebesgue measure and are locally integrable. Therefore (Lemma A.3) we can apply the Itô formula to the process ζT (t) = GT (ξT (t)), and with probability 1, for all t ≥ 0, we obtain t ζT (t) = GT (x0 ) +

t LT (ξT (s)) ds +

0

G T (ξT (s)) dWT (s),

0

where LT (x) = G T (x) aT (x) +

1 G (x). 2 T

(5.3)

5.2 Theorem Concerning the Weak Compactness

Let χN (t) = χ+

,. sup |ζT (s)|≤N

133

It is clear that for s ≤ t we have χN (t)χN (s) =

0≤s≤t

χN (t) with probability 1. Thus, according to (5.3), the following equality holds with probability 1: ζT (t)χN (t) = ζT (0)χN (t) t +χN (t) LT (ξT (s))χN (s) ds + χN (t) G T (ξT (s))χN (s) dWT (s). t 0

(5.4)

0

Hence, using condition (A1 ) and the properties of stochastic integrals, we obtain that ⎡ ⎛ t ⎞2  ⎢ EζT2 (t)χN (t) ≤ 3 ⎣EζT2 (0)χN (t) + E ⎝ LT (ξT (s))χN (s) ds ⎠ 0

⎛ + E⎝

t

⎞2 ⎤ ⎥ G T (ξT (s))χN (s) dWT (s)⎠ ⎦

0

⎡ ≤ 3 ⎣EζT2 (0)χN (t) + t

t

t EL2T (ξT (s))χN (s) ds +

0

⎡ ≤ 3 ⎣C + t

t

$

E G T (ξT (s))

%2

⎤ χN (s) ds ⎦

0





C 1 + EζT2 (s)χN (s) ds + C 0

t 



⎤

1 + EζT2 (s)χN (s) ds ⎦

0

t ≤C

(1)

+C

EζT2 (s)χN (s) ds,

(2)

(5.5)

0

where C (1) = 3C(1+t +t 2 ), C (2) = 3C(1+t), C > 0 is a constant from condition (A1 ), 0 ≤ t ≤ L. Using the Gronwall inequality, we conclude that there exists a constant KL , not depending on T , and such that for 0 ≤ t ≤ L EζT2 (t)χN (t) ≤ KL . Let N ↑ +∞, then ζT2 (t)χN (t) ↑ ζT2 (t), and we get the inequality sup EζT2 (t) ≤ KL .

0≤t ≤L

(5.6)

134

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Similarly to (5.5), using (5.3) and the inequality ⎡ E sup ⎣ 0≤t ≤L

t

⎤2 G T (ξT (s)) dWT (s)⎦

L ≤4

0

$ %2 E G T (ξT (s)) ds,

0

we conclude that ⎡ E sup |ζT (t)|2 ≤ 3 ⎣G2T (x0 ) + L 0≤t≤L

L

⎤ L    C 1 + EζT2 (s) ds + C 1 + EζT2 (s) ds ⎦ 

0

*(1) + C *(2) ≤C L L

0

L E [ζT (s)]2 ds. 0

Therefore, considering (5.6), we obtain the inequality *L E sup |ζT (t)|2 ≤ K 0≤t ≤L

(5.7)

*L do not depend on T . for all L > 0, where the constants K Using the inequalities for martingales and for stochastic integrals (see. [17, Part I, § 3, Theorem 6]), we obtain that ! t !2m ! ! ! ! E sup !! G T (ξT (s))χN (s) dWT (s)!! 0≤t ≤L ! ! 0

" ≤ " ≤

2m 2m − 1

2m 2m − 1

#2m

! L !2m ! ! ! ! ! E ! GT (ξT (s))χN (s) dWT (s)!! ! ! 0

#2m

L [m(2m − 1)]

m−1

m−1

L

$ %2m E G T (ξT (s)) χN (s) ds,

0

for any natural number m. Therefore, similarly to (5.7), we have the inequality E sup |ζT (t)|2m ≤ KL,m . 0≤t ≤L

(5.8)

5.2 Theorem Concerning the Weak Compactness

135

Furthermore, for all α > 0 there exists m ∈ N such that α ≤ 2m and, for a random variable η, we have E|η|α ≤ 1 + E|η|2m . The last inequality together with (5.8) implies that E sup |ζT (t)|α ≤ KL,α

(5.9)

0≤t ≤L

for all α > 0 and L > 0, where the constants KL,α do not depend on T . We have that for t1 < t2 ≤ L E [ζT (t2 ) − ζT (t1 )]4 ⎡ ⎛ ⎞4 ⎛ t ⎞4 ⎤ t2 2 ⎥ ⎢ ≤ 8 ⎣E ⎝ LT (ξT (s)) ds ⎠ + E ⎝ G T (ξT (s)) dWT (s)⎠ ⎦ t1

⎡ ≤ 8 ⎣(t2 − t1 )3

t1

t2

t2 E |LT (ξT (s))|4 ds + 36(t2 − t1 )

t1

$

E G T (ξT (s))

%4

⎤ ds ⎦ .

t1

Therefore, considering condition (A1 ) and inequality (5.9), we get E [ζT (t2 ) − ζT (t1 )]4 ≤ CL |t2 − t1 |2 ,

(5.10)

where the constants CL do not depend on T . According to (5.6) and (5.10), we have the relations of weak compactness: lim lim sup sup P {|ζT (t)| > N} = 0,

N→+∞ T →+∞ 0≤t ≤L

lim lim sup

sup

h→0 T →+∞ |t1 −t2 |≤h, ti ≤L

P {|ζT (t2 ) − ζT (t1 )| > ε} = 0

(5.11)

for any L > 0, ε > 0. It means that we can apply Skorokhod’s convergent subsequence principle (see Theorem A.12) for the process ζT (t) for all 0 ≤ t ≤ L. According to this principle, given an arbitrary sequence Tn → +∞, we can choose a subsequence Tn → +∞, * * a probability space (Ω, F, * P), and a stochastic processes * ζTn (t) and ζ(t), defined on this space, such that their finite-dimensional distributions coincide with those of the *

P processes ζTn (t), and, moreover, * ζTn (t) → ζ (t), as Tn → +∞, for all 0 ≤ t ≤ L. The processes * ζTn (t) and ζ (t) can be assumed to be separable.

136

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Using (5.10), we have %4 $ ζTn (t2 ) − * ζTn (t1 ) ≤ CL |t2 − t1 |2 E * for all 0 ≤ t1 ≤ t2 ≤ L. By Fatou’s lemma, E [ζ (t2 ) − ζ (t1 )]4 ≤ CL |t2 − t1 |2 . Thus, the processes * ζTn (t) and ζ (t) are continuous with probability 1. We have that the finite-dimensional distributions of the processes * ζTn (t) converge, as Tn → +∞, to the correspondent finite-dimensional distributions of the process ζ(t). For the weak convergence of the processes ζTn (t) it is sufficient (see [19, Chapter IX, § 2]) to prove that + lim lim sup P

, sup

h→0 Tn →+∞

|t1 −t2 |≤h, ti ≤L

|ζTn (t2 ) − ζTn (t1 )| > ε

=0

(5.12)

for any L > 0, ε > 0. In order to do this, we use the Hölder and Burkholder–Gundy inequalities and get ,

+ P

sup

|t1 −t2 |≤h, ti ≤L

|ζTn (t2 ) − ζTn (t1 )| > ε

+

, ε P sup |ζTn (t) − ζTn (kh)| > ≤ 4 kh≤t ≤(k+1)h kh≤L ⎧ ⎞4 ⎛ t " #4 1 ⎪  ⎨ 4 ⎝ LTn (ξTn (s)) ds ⎠ E sup 8 ≤ ⎪ ε kh≤t ≤(k+1)h ⎩ kh≤L 1

kh

⎞4 ⎫ ⎪ t ⎬ " 4 #4 1 ⎝ G T (ξTn (s)) dWTn (s)⎠ ≤ +E sup KL (h4 + h2 ), n ⎪ ε kh≤t ≤(k+1)h ⎭ kh≤L ⎛

kh

(5.13) where KL do not depend on Tn . Obviously, (5.13) implies (5.12). The proof of Theorem 5.1 is complete.  

5.3 Weak Convergence to the Solutions of Itô SDEs

137

5.3 Weak Convergence to the Solutions of Itô SDEs In this section we obtain sufficient conditions for the weak convergence of some stochastic processes to the solutions of Itô SDEs. Theorem 5.2 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and GT (x0 ) → y0 , as T → +∞. Assume that there exist measurable locally bounded functions a0 (x) and σ0 (x) such that: (1) the functions

qT(1)(x) = GT (x) aT (x) +

1 G (x) − a0 (GT (x)) , 2 T

 2 qT(2)(x) = GT (x) − σ02 (GT (x)) , satisfy assumption (A5 ); (2) the Itô equation t ζ (t) = y0 +

t a0 (ζ (s)) ds +

0



 (s) σ0 (ζ(s)) d W

(5.14)

0

  . has a unique weak solution ζ, W Then the stochastic processes ζT = GT (ξT (·)) converge weakly, as T → +∞, to the solution ζ of Eq. (5.14). Proof Rewrite Eq. (5.3) as t ζT (t) = GT (x0 ) +

(1)

a0 (ζT (s)) ds + αT (t) + ηT (t),

(5.15)

0

where (1) αT (t)

t =

(1)

qT (ξT (s)) ds,

(1)



qT (x) = GT (x)aT (x) +

0

t ηT (t) = 0

G T (ξT (s)) dWT (s).

1 G (x) − a0 (GT (x)) , 2 T

138

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . . (1)

The functions qT (x) satisfy the conditions of Lemma 5.2. Thus, for any L > 0 ! ! ! (1) ! P sup !αT (t)! → 0,

0≤t ≤L

(5.16)

as T → +∞. It is clear that ηT (t) is a family of continuous martingales with quadratic characteristics t  t 2 (2) ηT (t) = GT (ξT (s)) ds = σ02 (ζT (s)) ds + αT (t), 0

(5.17)

0

where (2) αT (t)

t =

(2)

qT (ξT (s)) ds,

 2 (2) qT (x) = GT (x) − σ02 (GT (x)) .

0

The functions qT(2)(x) satisfy the conditions of Lemma 5.2. Thus, for any L > 0 ! ! ! (2) ! P sup !αT (t)! → 0,

0≤t ≤L

(5.18)

as T → +∞. We have that relations (5.10) and (5.11) hold for the processes ζT (t) and ηT (t). According to convergence (5.16) and (5.18), these relations hold for the processes αT(k) (t), k = 1, 2 as well. It means that we can apply Skorokhod’s convergent subsequence principle (see Theorem A.12) for the process   (1) (2) ζT (t), ηT (t), αT (t), αT (t) . According to this principle, given an arbitrary sequence Tn → +∞, we can * * choose a subsequence Tn → +∞, a probability space (Ω, F, * P), and a stochastic process   (1) (2) * ηTn (t),* αTn (t),* αTn (t) ζTn (t), * defined on this space and such that its finite-dimensional distributions coincide with those of the process   (2) (t), α (t) ζTn (t), ηTn (t), αT(1) T n n

5.3 Weak Convergence to the Solutions of Itô SDEs

139

and, moreover, *

*

*

*

P P P P (1) (2) * ζ (t), * ηTn (t) → * η(t), * αTn (t) → * α (1) (t), * αTn (t) → * α (2) (t) ζTn (t) → *

for all 0 ≤ t ≤ L, where * ζ (t), * η(t), * α (1) (t), * α (2) (t) are some stochastic processes. Evidently, relations (5.16) and (5.18) imply that * α (k) (t) ≡ 0, k = 1, 2 a.s. * According to (5.10), the processes ζ (t) and * η(t) are continuous. Moreover, applying Lemma 5.5 together with equalities (5.15) and (5.17), we obtain that * ζTn (t) = GTn (x0 ) +

t 0

and * ηTn (t) =

t 0

a0 (* ζTn (s)) ds + * αTn (t) + * ηTn (t) (1)

(5.19) σ02 (* ζTn (s)) ds + * αT(2) (t), n

! ! * * P P P ! (k) ! * αTn (t)! → 0, k = 1, 2 as Tn → where * ζTn (t) → * ζ (t), * ηTn (t) → * η(t) and sup !* 0≤t ≤L

+∞. In addition, an analog of the convergence (5.12) holds for the processes * ζTn (t) and * ηTn (t). Therefore, ! ! * P sup !* ζ (t)! → 0, ζTn (t) − *

! ! * P ηTn (t) − * sup !* η(t)! → 0,

0≤t ≤L

0≤t ≤L

as Tn → +∞. According to Lemma 5.3 we can pass to the limit in (5.19) and obtain * ζ (t) = y0 +

t

a0 (* ζ (s)) ds + * η(t),

(5.20)

0

where * η(t) is a continuous martingale with the quadratic characteristics t * η(t) =

σ02 (* ζ (s)) ds.

0

Now, it is well known that the latter representation provides the existence of a  (t) such that Wiener process W t * η(t) = 0

 (s). σ0 (* ζ (s)) d W

(5.21)

140

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

   (t) satisfies Eq. (5.14) and the processes * Thus, the process * ζ (t), W ζTn (t) converge weakly, as Tn → +∞, to the process * ζ (t). Since the subsequence Tn → +∞ is arbitrary and since the solution of Eq. (5.14) is weakly unique, the proof of Theorem 5.2 is complete.  

5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type In this section we obtain the sufficient conditions for the weak convergence of some integral functionals of the Lebesgue integral type. Theorem 5.3 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and let the assumptions of Theorem 5.2 hold. Assume that for measurable and locally bounded functions gT there exists a measurable and locally bounded function g0 such that the functions qT (x) = gT (x) − g0 (GT (x)) satisfy assumption (A5 ). Then the stochastic processes (1) βT (t)

t =

gT (ξT (s)) ds 0

converge weakly, as T → +∞, to the process t β (1)(t) =

g0 (ζ (s)) ds, 0

where ζ is the solution of Eq. (5.14). Proof It is clear that, for all t > 0, with probability 1 βT(1) (t)

t =

g0 (ζT (s)) ds + αT (t), 0

where t αT (t) =

qT (ξT (s)) ds, 0

qT (x) = gT (x) − g0 (GT (x)) .

5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

141

The functions qT (x) satisfy the conditions of Lemma 5.2. Thus, for any L > 0 P

sup |αT (t)| → 0,

0≤t ≤L

as T → +∞. Similarly to (5.19), we obtain the equality *(1) (t) = β Tn

t

g0 (* ζTn (s)) ds + * αTn (t),

(5.22)

0

where *

P * ζTn (t) → * ζ (t)

! * ! P αTn (t)! → 0, sup !*

and

0≤t ≤L

as Tn → +∞. The process * ζ (t) is a solution to Eq. (5.20), whereas by Lemma 5.5 (1) the finite-dimensional distributions of the stochastic process βTn (t) coincide with *(1)(t). those of the process β Tn Using Lemma 5.3 and equality (5.22) we conclude that ! ! ! ! t ! (1) ! * P ! * * sup !βTn (t) − g0 (ζ (s)) ds !! → 0 0≤t ≤L ! ! 0

as Tn → +∞. Thus, the process βT(1) (t) converges weakly as Tn → +∞ to the n t  process β (1)(t) = g0 (ζ (s)) ds, where ζ is a solution of Eq. (5.14). Since the 0

subsequence Tn → +∞ is arbitrary and since a solution ζ of Eq. (5.14) is weakly unique, the proof of Theorem 5.3 is complete.   Theorem 5.4 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the assumptions of Theorem 5.2 hold. Assume that, for measurable and locally bounded functions gT , there exists a measurable locally bounded function g0 such that assumption (A6 ) holds. Then the stochastic processes (1) βT (t)

t =

gT (ξT (s)) ds 0

converge weakly, as T → +∞, to the process ⎛

⎞ ζ(t ) t  (s)⎟ *(1) (t) = 2 ⎜ β ⎝ g0 (x) dx − g0 (ζ (s)) σ0 (ζ(s)) d W ⎠, y0

0

142

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

 are related via Eq. (5.14). where ζ and the Wiener process W Proof Consider the function ⎞ ⎛ u  g (v) T dv ⎠ du. fT (u) ⎝ fT (v)

x ΦT (x) = 2 0

0

Applying the Itô formula to the process ΦT (ξT (t)), where ξT (t) is a solution to Eq. (5.1), we get that βT(1) (t) = ΦT (ξT (t)) − ΦT (x0 ) − =2 +2

ξT(t )

x0 ξT(t )

=2

t 0

ΦT (ξT (s)) dWT (s)

t g0 (GT (u)) GT (u) du − 2 g0 (ζT (s)) GT (ξT (s)) dWT (s)

0

t

 qT (u) du − 2  qT x0 0 ζT(t ) t

(ξT (s)) dWT (s)

g0 (u) du − 2 g0 (ζT (s)) dηT (s) + γT(1)(t) − γT(2)(t), 0

GT (x0 )

where (1) γT (t)

ξT (t )

=

(2) γT (t)

 qT (u) du, x0

=

 qT (ξT (s)) dWT (s), 0

⎛  qT (x) = 2 ⎝fT (x)

x 0

⎞ gT (v) dv − g0 (GT (x)) GT (x)⎠ . fT (v)

+ Denote PNT = P

t

, sup |ξT (t)| > N . It is clear that for any constants ε > 0,

0≤t ≤L

N > 0, and L > 0, we have the inequalities ! ! ! ξT (t ) ! , ! ! ! ! 2 ! ! ! (1) ! P sup !γT (t)! > ε ≤ PNT + E sup !  qT (u) du! χ{|ξT (t )|≤N} ! ! ε 0≤t ≤L 0≤t ≤L ! ! x0 +

≤ PNT

2 + ε

N | qT (u)| du −N

5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

143

and ! t !2 , ! ! ! ! ! ! 4 ! (2) ! ! P sup !γT (t)! > ε ≤ PNT + 2 E sup !  qT (ξT (s)) χ{|ξT (s))≤N|} dWT (s)!! ε 0≤t ≤L ! 0≤t ≤L ! +

0

≤ PNT

16 + 2E ε

L  qT2 (ξT (s)) χ{|ξT (s))≤N|} ds.

(5.23)

0

Using assumption (A2 ) we obtain that lim

lim sup PNT = 0.

N→+∞ T →+∞

Thus, according to the conditions of Theorem 5.4, we get the convergence ! ! ! P ! sup !γT(k) (t)! → 0, for k = 1,

0≤t ≤L

(5.24)

as T → +∞. Let us show that the last convergence holds with k = 2. Consider the function *T (x) = 2 Φ

x 0

⎞ ⎛ u  2  q (v)χ |v|≤N T dv ⎠ du. fT (u) ⎝ fT (v) 0

We can apply the Itô formula to the process ΦT (ξT (t)), where ξT (t) is a solution to Eq. (5.1). Furthermore, the equality *T (x)aT (x) + 1 Φ * (x) =  Φ qT2 (x)χ|x|≤N 2 T holds a.e. with respect to the Lebesgue measure. Using the latter equality and the properties of stochastic integrals, we conclude that L E 0

$ % *T (x0 ) . *T (ξT (L)) − Φ  qT2 (ξT (s)) χ|(ξT (s))|≤N ds = E Φ

144

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Using the Hölder inequality, we obtain the following bounds: ! ! ! u !1 ! u !1 !x ! ! q !  ! p !! !  ! ! ! ! ! ! p 1 ! !Φ ! ! *T (x) − Φ *T (x0 )! ≤ 2 !! fT (u) !  qT2 (v) χ|v|≤N dv !! du! ! |f (v)|q dv ! ! ! ! ! ! ! ! T !x0 ! 0 0 ! N ! p1 ! ! ! ! 2p ! ≤ 2 ! | qT (v)| dv !! ! ! −N

! ! ! u !1 ! !x ! !q ! ! ! ! 1 ! ! ! ! dv ! du, ! fT (u) ! ! ! |fT (v)|q !! ! ! !0 0

where p > 1 and q > 1 are arbitrary constants with p1 + q1 = 1. Taking into account assumption (A3 ) for a certain q = 1 + δ, we conclude that ⎛ N ⎞ p1  ! $ %! $ % !E Φ *T (ξT (L)) − Φ *T (x0 ) ! ≤ 2 ⎝ | qT (v)|2p dv ⎠ C 1 + E|ζT (L)|m , −N

where the constants C > 0 and m ≥ 0 do not depend on T . According to assumption (A5 ) and to Lebesgue’s dominated convergence theoN | rem, we have the convergence qT (v)|2p dv → 0, as T → +∞, for arbitrary −N

N > 0. Hence, taking into account the inequality E |ζT (L)|m ≤ CL we obtain the convergence $ % *T (ξT (L)) − Φ *T (x0 ) → 0, E Φ as T → +∞, and consequently L  qT2 (ξT (s)) χ|(ξT (s))|≤N ds → 0,

E 0

as T → +∞. The latter convergence and inequality (5.23) imply the convergence (5.24) for k = 2. The same arguments as we used establishing (5.19) yield that *(1)(t) = 2 β Tn

* ζ Tn (t )

GTn (x0 )

t g0 (u) du − 2

  (1) (2) g0 * ηTn (s) + * γTn (t) − * γTn (t), ζTn (s) d*

0

(5.25)

5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

145

where ! * ! P ζTn (t) − * sup !* ζ (t)! → 0,

0≤t ≤L

! * ! P ηTn (t) − * sup !* η(t)! → 0,

0≤t ≤L

! ! P ! (k) ! * γTn (t)! → 0, sup !*

k = 1, 2

0≤t ≤L

   (t) satisfies ζ (t), W as Tn → +∞ for all L > 0. According to (5.20), the process * Eq. (5.14), and * η(t) is defined in (5.21). By Lemma 5.5 the finite-dimensional distributions of the stochastic process (1) *(1) (t). Using Lemma 5.4, we can pass to βTn (t) coincide with those of the process β Tn the limit, as Tn → +∞, in (5.25) and obtain ! * ! P ! *(1) *(1)(t)!! → sup !β (t) − β 0, Tn

0≤t ≤L

(5.26)

as Tn → +∞, where *(1)

β

* ζ (t )  t   ζ (s) d* η(s) (t) = 2 g0 (u) du − 2 g0 * y0

0

⎡

⎤ * ζ (t )  t t       ⎥ ⎢ = 2 ⎣ g0 (u) du − g0 * ζ (s) d* ζ (s) + g0 * ζ (s) a0 * ζ (s) ds ⎦ , y0

0

0

and * ζ is a solution of Eq. (5.14). Therefore, we have that Theorem 5.4 holds for the , as Tn → +∞. Since the subsequence Tn → +∞ is arbitrary and since process βT(1) n a solution ζ of Eq. (5.14) is weakly unique, the proof of Theorem 5.4 is complete.   Theorem 5.5 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the assumptions of Theorem 5.2 hold. Let for the coefficient aT (x) of Eq. (5.1) assumption (A5 ) hold. Assume that, for measurable and locally bounded functions gT (x), there exist certain constants cT , mT and C > 0 such that |cT | ≤ C, 0 ≤ mT ≤ C and for arbitrary N > 0 ! x⎡ ⎤ ! ! ! u ! ! ! ⎣f (u) gT (v) dv − cT ⎦ du! → 0, T ! ! fT (v) ! ! 0

0

146

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

a.e., as T → +∞, ! ! ! ! x ! ! !f (x) gT (v) dv − cT ! χ|x|≤N ≤ CN ! ! T fT (v) ! ! 0

and for the functions ⎡  qT (x) = ⎣fT (u)

u 0

⎤2 gT (v) dv − cT ⎦ − m2T fT (v)

assumption (A5 ) holds. Then, (1) for the subsequences Tn → +∞ such that lim inf mTn > 0, the stochastic Tn →+∞ processes (1)(t) β Tn

1 = 2mTn

t

  gTn ξTn (s) ds

0

converge weakly, as Tn → +∞, to the Wiener process W ; (2) for the subsequences Tn → +∞ such that lim mTn = 0, the stochastic Tn →+∞ processes βT(1) (t) n

t =

  gTn ξTn (s) ds

0

converge weakly, as Tn → +∞, to zero. Proof We apply the Itô formula to the process ΦT (ξT (t)), where ξT is a solution to Eq. (5.1) and the function ΦT has the form x ΦT (x) = 2 0

⎞ ⎛ u  gT (v) ⎠ dv du. fT (u) ⎝ fT (v) 0

As a result, we get the representation βT(1)(t) = 2cT

t 0

aT (ξT (s)) ds + αT (t) + ηT(1)(t),

5.4 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

147

where βT(1) (t) =

t

gT (ξT (s)) ds, 0 ( ) ξT(t ) u gT (v) fT (u) f (v) dv − cT du, αT (t) = 2 x0 0 T t $  % (1) ηT (t) = − ΦT (ξT (s)) − 2cT dWT (s). 0

Let condition (A5 ) hold for the functions aT (x). Then, according to Lemma 5.2, for arbitrary L > 0, we conclude ! t ! ! ! ! ! P ! sup ! aT (ξT (s)) ds !! → 0 , 0≤t ≤L ! ! 0

as T → +∞. We have the obvious inequalities + P

, sup |αT (t)| > ε ≤ PNT

0≤t ≤L

≤ PNT

4 + ε

! ! ! ! ξT (t ) ! $ % !! 2 ! ΦT (u) − 2cT du! χ|ξT (t )|≤N + E sup ! ! ε 0≤t ≤L ! ! ! x0

! ! ! N ! u ! ! g (v) T !f (u) dv − cT !! du ! T fT (v) ! !

−N

0

+ for arbitrary N > 0, L > 0, and ε > 0, where PNT = P Taking into account that, according to assumption (A2 ), we get P

sup |αT (t)| → 0,

as T → +∞. So we have ! ! ! P ! sup !βT(1) (t) − ηT(1) (t)! → 0,

as T → +∞.

sup |ξT (t)| > N .

0≤t ≤L

lim sup PNT = 0,

N→+∞ T →+∞

0≤t ≤L

0≤t ≤L

lim

,

148

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . . (1)

It is clear that ηT (t) is a continuous martingale with the quadratic characteristics (1) ηT (t)

t =

4m2T t

+

qT (ξT (s)) ds, 0

where ⎡ qT (x) = 4 ⎣fT (x)

x 0

⎤2 gT (v) dv − cT ⎦ − 4m2T . fT (v)

Condition (A5 ) holds for the function qT (x), therefore, by Lemma 5.2 ! ! ! P ! sup !ηT(1) (t) − 4m2T t ! → 0,

0≤t ≤L

as T → +∞, for any L > 0.   (1) (1) Next we use the random time change, i.e., ηT (t) = WT∗ ηT (t) , where WT∗ is a Wiener process. Similarly to the proof of the relation (4.19) we have ! ! P  ! ! sup !βT(1)(t) − WT∗ 4m2T t ! → 0,

0≤t ≤L

(5.27)

as T → +∞. Let Tn → +∞ be a subsequence such that lim inf mTn > 0. The process Tn →+∞

  WT∗n 4m2Tn t 2mTn is a Wiener process for every Tn . Let us denote it as W ∗ (t). According to the convergence (5.27) and the equality ! ! t ! ! ! ! ! t !!  ! ! ! 1 ∗ 2 ∗ ! ! ! sup ! gTn (ξTn (s)) ds − WTn 4mTn t ! = 2mTn sup ! gTn (ξTn (s)) ds − W (t )!! 2m T 0≤t≤L ! 0≤t≤L ! ! ! n 0

0

we have the convergence ! ! ! ! t ! 1 ! sup !! gTn (ξTn (s)) ds − W ∗ (t)!! → 0, 0≤t ≤L ! 2mTn ! 0

5.5 Asymptotic Behavior of Integral Functionals of Martingale Type

149

as Tn → +∞, whence the proof of statement (1) follows. Statement (2) follows from (5.27).  

5.5 Asymptotic Behavior of Integral Functionals of Martingale Type In this section we obtain the sufficient conditions for the weak convergence of some integral functionals of martingale type. Theorem 5.6 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and let the assumptions of Theorem 5.2 hold. Assume that, for measurable and locally bounded functions gT , there exists a measurable locally bounded function g0 such that the function  2 qT (x) = gT (x) − g0 (GT (x)) GT (x) satisfies assumption (A5 ). Then the stochastic processes βT(2) (t) =

t gT (ξT (s)) dWT (s), 0

where ξT (t) and WT (t) are related via Eq. (5.1), converge weakly, as T → +∞, to the process t β

(2)

(t) =

 (s), g0 (ζ (s)) σ0 (ζ (s)) d W

0

   is a solution to Eq. (5.14). where ζ, W Proof It is clear that (2) βT (t)

t =

g0 (ζT (s)) dηT (s) + γT (t),

(5.28)

0

where t γT (t) =

qT (ξT (s)) dWT (s), 0



qT (x) = gT (x) − g0 (GT (x)) GT (x).

150

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

The process γT (t) for every T > 0 is continuous with probability 1 and a martingale with the quadratic characteristics t qT2 (ξT (s)) ds.

γT (t) = 0

Since for the functions qT2 (x) assumption (A5 ) holds, according to Lemma 5.2, we obtain that L

P

qT2 (ξT (s)) ds → 0, 0

as T → +∞, for any constant L > 0. For arbitrary constants ε > 0, δ > 0, and L > 0 the following inequality holds (see Theorem A.7): + P

, sup |γT (t)| > ε ≤ δ + P

0≤t ≤L

⎧ L ⎨ ⎩

⎫ ⎬

qT2 (ξT (s)) ds > ε2 δ . ⎭

0

So, we have the convergence P

sup |γT (t)| → 0,

0≤t ≤L

(5.29)

as T → +∞, for any constant L > 0. Similarly to the representation (5.19), for an arbitrary subsequence Tn we can get * * on certain probability space (Ω, F, * P), the equality *(2)(t) β Tn

t =

g0 (* ζTn (s)) d* ηTn (s) + * γTn (t),

0

where ! ! * P sup !* ζ (t)! → 0, ζTn (t) − *

0≤t ≤L

! ! * P sup !* η(t)! → 0, ηTn (t) − *

0≤t ≤L

! * ! P *Tn (t)! → 0, sup !γ

0≤t ≤L

5.5 Asymptotic Behavior of Integral Functionals of Martingale Type

151

   (t) satisfies Eq. (5.14), * as Tn → +∞, for any L > 0, the process * ζ (t), W η(t) is (2) (2) * defined in (5.21), and the processes βTn (t) and βTn (t) are stochastically equivalent. Similarly to the proof of the relation (5.26), we obtain ! ! * P ! *(2) *(2)(t)!! → sup !β (t) − β 0 Tn

0≤t ≤L

as Tn → +∞, where *(2)(t) = β

t

  ζ (s) d* η(s) = g0 *

0

t

  ζ (s) d* ζ (s) − g0 *

0

t

    ζ (s) a0 * ζ (s) ds. g0 *

0

*(2) (t) converges weakly, as Tn → +∞, to the process β *(2)(t). Thus, the process β Tn (2) * (t) and Since the subsequence Tn → +∞ is arbitrary and since the processes β Tn

βT(2) (t) are stochastically equivalent, the proof of Theorem 5.6 is complete. n

 

Theorem 5.7 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and let the assumptions of Theorem 5.2 hold. Assume that, for measurable and locally gT and g0 bounded functions gT there exist measurable locally bounded functions  such that for the functions (1)

gT (GT (x))]2 , qT (x) = [gT (x) −  (2) gT2 (GT (x)) − g02 (GT (x)) qT (x) =  (3) assumption (A5 ) holds, and the function qT (x) = | gT (GT (x))| satisfies assumptions (A5 ) and (A7 ). Then the stochastic processes

βT(2) (t)

t =

gT (ξT (s)) dWT (s), 0

where ξT (t) and WT (t) are  related via Eq. (5.1), converge weakly, as T → +∞, to the process β (2)(t) = W ∗ β (1)(t) , where t β

(1)

(t) =

g02 (ζ (s)) ds, 0

here ζ is a solution to Eq. (5.14), W ∗ = {W ∗ (t), t ≥ 0} is a Wiener process and the processes W ∗ (t) and β (1)(t) are independent.

152

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Proof It is clear that (2) βT (t)

t =

 gT (ζT (s)) dWT (s) + γT (t),

(5.30)

0

where t γT (t) =

qT (ξT (s)) dWT (s),

qT (x) = gT (x) −  gT (GT (x)) .

0

Since for the functions qT2 (x) assumption (A5 ) holds, we obtain relation (5.29) for the process γT (t). We can apply Skorokhod’s convergent subsequence principle (see Theo* * rem A.12) and obtain (5.19) on a probability space (Ω, F, * P) with additional representation *(2)(t) β Tn

t =

*Tn (s) + *  gTn (* ζTn (s)) d W γTn (t).

0

Here ! * ! P sup !* ζ (t)! → 0, ζTn (t) − *

0≤t ≤L

! * ! P *Tn (t) − W * (t)! → sup !W 0,

0≤t ≤L

! * ! P *Tn (t)! → 0, sup !γ

0≤t ≤L

* (t) ζ (t) is the solution to Eq. (5.14), and W as Tn → +∞ for any L > 0, the process * is a Wiener process. *(2)(t) and β (2)(t) are stochastically According to Lemma 5.5, the processes β Tn Tn equivalent. Taking into account Lemma A.8, we use the random time change in stochastic integrals (see, e.g., [74]) and obtain that for any t ≥ 0 with probability 1   *(2) (t) = WT∗ β *(1) (t) + * β γTn (t), Tn Tn n where WT∗n (t) is a family of a Wiener processes, *(1)(t) β Tn

t = 0

  ζTn (s) ds.  gT2n *

(5.31)

5.5 Asymptotic Behavior of Integral Functionals of Martingale Type

153

The functions  gT2n (GT (x)) − g02 (GT (x)) satisfy condition (A5 ). The proof of Theorem 5.3 implies that ! ! * ! *(1) *(1) ! P sup !β Tn (t) − β (t)! → 0,

0≤t ≤L

as Tn → +∞, where *(1)

β

t (t) =

  ζ (s) ds. g02 *

0

It is clear that for any L > 0, N > 0, ε > 0, and δ > 0 we have the inequality + * P

!    ! ! *(1)(t) − WT∗ β *(1) (t) !! > ε sup !WT∗n β Tn n

,

0≤t ≤L

0 / 0 / *(1)(L) > N *(1) (L) > N + * P β ≤* P β Tn ,

+ +* P

sup

|t1 −t2 |≤δ; ti ≤N

|WT∗n (t2 )

− WT∗n (t1 )|

> ε +* P

+

, ! ! ! *(1) (1) ! * sup !βTn (t) − β (t)! > δ .

0≤t ≤L

We have the analog of the convergence (5.12) for the Wiener process WT∗n (t). Therefore, we obtain the convergence !    ! * P ! *(1) (t) − WT∗ β *(1)(t) !! → 0, sup !WT∗n β Tn n

0≤t ≤L

as Tn → +∞. According to (5.31), we get !  ! * ! *(2) ! P ∗ *(1) sup !β Tn (t) − WTn β (t) ! → 0,

0≤t ≤L

(5.32)

as Tn → +∞. Using the properties of stochastic integrals, we obtain the inequality ! ! ! τTn (t ) ! ! ! ! !   ! !EW ∗ (t)W *Tn (t)! = !E *Tn (s)W *Tn (t)!! = ζTn (s) d W  gTn * Tn ! ! ! 0 ! ! ! ! ! = !E ! !

! ! t !    !! !  gTn * gTn * ζTn (s) ds ! ≤ E ! ζTn (s) ! ds. ! ! 0

t ∧τ Tn (t )

0

154

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . . (3)

The conditions on the function qT (x), combined with Remark 5.2, imply the convergence t E

! !  ! ζTn (s) ! ds→0, gTn *

0

as Tn → +∞, for any t > 0. Thus, *Tn (t)→0, EWT∗n (t)W as Tn → +∞. *Tn (t) are Wiener processes and they are asympSince the processes WT∗n (t) and W totically uncorrelated, we conclude that WT∗n (t) does not depend asymptotically on * (t). It is clear that the process β *(1)(t) is completely determined by the process W * ζ (s) for s ≤ t. Using the strong uniqueness of the solution (ξT (t), WT (t)) to Eq. (5.1) we  (t) are measurable with respect to the σ * conclude that ζ (t) and W  the processes  * (s), s ≤ t , where W * (t) is a Wiener process, which is the limit process algebra σ W *Tn (t). So, the process W ∗ (t) does not depend asymptotically on the process for W Tn (1) * (t). The finite-dimensional distributions of the process W ∗ (t) do not depend β  (1)  Tn * (t) , where W ∗ is a Wiener on Tn , thus, the limit process can be written as W ∗ β ∗ (1) * are independent. Taking into account (5.32), process, and the processes W and β we have that !  ! * ! *(2) ! P ∗ *(1) β sup !β (t) − W (t) ! → 0, Tn

0≤t ≤L

as Tn → +∞. *(2) (t) converge weakly, as Tn → +∞, to the process Therefore, the processes β Tn   *(1)(t) . It means that the statement of Theorem 5.7 is valid for the process W∗ β . Since the subsequence Tn → +∞ is arbitrary and since the finite-dimensional βT(2) n  (1)  * (t) are determined in a unique way, the proof distributions of the process W ∗ β of Theorem 5.7 is complete.   Theorem 5.8 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the assumptions of Theorem 5.2 hold. Assume that, for measurable and locally gT and g0 bounded functions gT there exist measurable locally bounded functions  (1) such that the function qT (x) = [gT (x) − g0 (GT (x))]2 satisfies assumption (A5 ), gT2 (GT (x)) and g0 (x) satisfy assumption (A6 ), and the the functions qT (x) =  (3) function qT (x) = | gT (GT (x))| satisfies assumptions (A5 ) and (A7 ).

5.6 Weak Convergence of Mixed Functionals

155

Then the stochastic processes (2) βT (t)

t =

gT (ξT (s)) dWT (s), 0

where ξT (t) and WT (t) are  related via Eq. (5.1), converge weakly, as T → +∞, to the process β (2)(t) = W ∗ β (1)(t) , where ⎡

⎤ ζ(t ) t ⎢  (s)⎥ β (1) (t) = 2 ⎣ g0 (x) dx + g0 (ζ (s)) σ0 (ζ(s)) d W ⎦, y0

(5.33)

0

   (t) is a solution to Eq. (5.14), W ∗ = {W ∗ (t), t ≥ 0} is a Wiener here ζ (t), W process, and the processes W ∗ and β (1) are independent. Proof The proof of Theorem 5.8 completely coincides with the proof of Theo  rem 5.7 with the only difference that the process β (1) has the form (5.33).

5.6 Weak Convergence of Mixed Functionals In this section we obtain sufficient conditions for the weak convergence of some mixed functionals. Theorem 5.9 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the assumptions of Theorem 5.2 hold. Assume that, for continuous functions FT and a locally bounded measurable functions gT , there exist a continuous function F0 and locally bounded measurable function g0 such that, for arbitrary N > 0 lim

sup |FT (x) − F0 (GT (x))| = 0.

T →+∞ |x|≤N

Also, let the function  2 qT (x) = gT (x) − g0 (GT (x)) GT (x) satisfy assumption (A5 ). Then the stochastic processes t IT (t) = FT (ξT (t)) +

gT (ξT (s)) dWT (s), 0

156

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

where ξT and WT are related via Eq. (5.1), converge weakly, as T → +∞, to the process t I0 (t) = F0 (ζ (t)) +

 (s), g0 (ζ (s)) σ0 (ζ(s)) d W

0





 (t) is a solution of Eq. (5.14). where ζ (t), W Proof It is clear that t IT (t) = F0 (ζT (t)) +

g0 (ζT (s)) dηT (s) + αT (t) + γT (t), 0

where t αT (t) = FT (ξT (t)) − F0 (ζT (t)) ,

ηT (t) =



GT (ξT (s)) dWT (s), 0

t γT (t) =

qT (ξT (s)) dWT (s),



qT (x) = gT (x) − g0 (GT (x)) GT (x).

0

+ Denote as before, PNT = P

, sup |ξT (t)| > N . Since for any constants ε >

0≤t ≤L

0, N > 0 and L > 0 the following inequality holds: ,

+ P

sup |FT (ξT (t)) − F0 (GT (ξT (t)))| > ε

0≤t ≤L

2 ≤ PNT + E sup |FT (ξT (t)) − F0 (GT (ξT (t)))| χ|ξT (t )|≤N ε 0≤t ≤L ≤ PNT +

2 sup |FT (x) − F0 (GT (x)| , ε |x|≤N

we can apply the conditions of Theorem 5.9 and condition (A2 ) to get that P

sup |αT (t)| → 0,

0≤t ≤L

5.6 Weak Convergence of Mixed Functionals

157

as T → +∞. The proof of the fact that for γT (t) the convergence (5.29) holds is literally the same as the respective part of the proof of Theorem 5.6. Then, we can apply Skorokhod’s convergent subsequence principle for the process (ζT (t), ηT (t), αT (t), γT (t)) and, similarly to the representation (5.19), obtain equal* * ity for an arbitrary subsequence Tn in a certain probability space (Ω, F, * P) I*Tn (t) = F0 (* ζTn (t)) +

t

g0 (* ζTn (s)) d* ηTn (s) + * αTn (t) + * γTn (t),

0

where for any L > 0 ! ! * P ζTn (t) − * sup !* ζ (t)! → 0,

! ! * P ηTn (t) − * sup !* η(t)! → 0,

0≤t ≤L

0≤t ≤L

! * ! P αTn (t)! → 0, sup !*

! * ! P *Tn (t)! → 0, sup !γ

0≤t ≤L

0≤t ≤L

as Tn → +∞. Since the function F0 (x) is continuous, it is uniformly continuous in a closed region |x| ≤ N for any N > 0. Denote + *NTn = P P

! ! sup !* ζTn (t)! > N

0≤t ≤L

,

+ and P*N = P

, ! ! sup !* ζ (t)! > N .

0≤t ≤L

For any constants ε > 0, N > 0, and L > 0 there exists δ > 0 such that the following inequalities hold: + P +3 +P

!   !  sup !F0 * ζTn (t) − F0 * ζ (t) ! > ε

0≤t ≤L

0≤t ≤L

0≤t ≤L

∩

+P

*N ≤ P*NTn + P

4 4 3 ! !  !   ! sup !F0 * ζTn (t) − F0 * ζ (t) ! > ε ∩ sup !* ζTn (t)! ≤ N 3

+3

,

! ! ζ (t)! ≤ N sup !*

4,

0≤t ≤L

≤ P*NTn + P*N

4 3 4 3 4, ! ! ! ! ! ! ! ! !* !* !* ! * sup ζTn (t) − ζ (t) > δ ∩ sup ζTn (t) ≤ N ∩ sup ζ (t) ≤ N

0≤t≤L

0≤t≤L

0≤t≤L

158

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

+ ≤ P*NTn + P*N + P

, ! ! ζTn (t) − * sup !* ζ (t)! > δ .

0≤t ≤L

Therefore, ! ! ! ! t ! ! *     P ITn (t) − F0 * ζ (t) − g0 * ζTn (s) d* ηTn (s)!! → 0, sup !!* 0≤t ≤L ! !

(5.34)

0

as Tn → +∞. Using Lemma 5.4 we can pass to the limit, as Tn → +∞, in the stochastic integral from (5.34) and obtain ! ! ! ! t ! ! *     P sup !!I*Tn (t) − F0 * ζ (t) − g0 * ζ (s) d* η(s)!! → 0, 0≤t ≤L ! ! 0

as Tn → +∞. It is an analog of convergence (5.26). From here we have that !  ! *   P ζ (t) ! → 0, sup !F0 ζTn (t) − F0 *

0≤t ≤L

as Tn → +∞. To complete the proof of Theorem 5.9, we repeat the same arguments as in the proof of Theorem 5.6.   Theorem 5.10 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and let the assumptions of Theorem 5.2 hold. Assume that, for continuous functions FT and a locally bounded measurable functions gT , there exist a continuous function F0 and locally bounded measurable functions  gT and g0 such that, for arbitrary N >0 lim

sup |FT (x) − F0 (GT (x))| = 0,

T →+∞ |x|≤N

and let for the function gT ,  gT and g0 the assumptions of Theorem 5.7 hold. Then the stochastic processes t IT (t) = FT (ξT (t)) +

gT (ξT (s)) dWT (s), 0

where ξT and WT are related via Eq. (5.1), converge weakly, as T → +∞, to the process    I0 (t) = F0 (ζ (t)) + W ∗ β (1) (t) ,

5.6 Weak Convergence of Mixed Functionals

159

where t β

(1)

g02 (ζ (s)) ds,

(t) = 0

here ζ is a solution to Eq. (5.14), W ∗ = {W ∗ (t), t ≥ 0} is a Wiener process, and the processes W ∗ and β (1) are independent. Proof It is clear that t IT (t) = F0 (ζT (t)) +

 gT (ζT (s)) dWT (s) + αT (t) + γT (t), 0

where t αT (t) = FT (ξT (t)) − F0 (ζT (t)) ,

γT (t) =

qT (ξT (s)) dWT (s), 0

qT (x) = gT (x) −  gT (GT (x)) . Since the function qT2 (x) satisfies the assumptions of Lemma 5.2, we have L

P

qT2 (ξT (s)) → 0,

sup

0≤t ≤L

0

as Tn → +∞, for any L > 0. Thus, for the martingale γT (t) we have the analog of the convergence (5.29). The proof of this fact is literally the same as in the proof of Theorem 5.6. Then, taking into account the proof of Theorem 5.9 we can apply Skorokhod’s convergent subsequence principle to the process (ζT (t), WT (t), αT (t), γT (t)) and obtain equality for an arbitrary subsequence Tn in a certain probability space * * (Ω, F, * P)

I*Tn (t) = F0 (* ζTn (t)) +

t 0

*Tn (s) + *  gTn (* ζTn (s)) d W αTn (t) + * γTn (t),

160

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

where for any L > 0 ! ! * P sup !* ζTn (t) − * ζ (t)! → 0,

! ! * P *Tn (t) − W * (t)! → sup !W 0,

0≤t ≤L

0≤t ≤L

! ! * P αTn (t)! → 0, sup !*

! ! * P *Tn (t)! → 0, sup !γ

0≤t ≤L

0≤t ≤L

* is a Wiener process. as Tn → +∞. Here * ζ is the solution to Eq. (5.14), W Taking into account Lemma A.8, we change the time in the stochastic integral t

*Tn (s)  gTn (* ζTn (s)) d W

0

and obtain that for any t ≥ 0 with probability 1 ⎛ * ITn (t) = F0 (* ζTn (t)) + WT∗n ⎝

t

⎞  gT2n (* ζTn (s)) ds ⎠ + * γTn (t), αTn (t) + *

0

where WT∗n (t) is the Wiener process for every Tn . To complete the proof of Theorem 5.10 we use the equivalence of the processes ITn (t) and * ITn (t) and repeat the same arguments as in the proof of Theorem 5.7.   Theorem 5.11 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the assumptions of Theorem 5.2 hold. Assume that, for continuous functions FT and a locally bounded measurable functions gT , there exist a continuous function F0 and a locally bounded measurable functions  gT and g0 such that, for arbitrary N >0 lim

sup |FT (x) − F0 (GT (x))| = 0.

T →+∞ |x|≤N

Also, let for the functions gT (x),  gT (x), and g0 (x) the assumptions of Theorem 5.8 hold. Then the stochastic processes t IT (t) = FT (ξT (t)) +

gT (ξT (s)) dWT (s), 0

where ξT and WT are related via Eq. (5.1), converge weakly, as T → +∞, to the process    I0 (t) = F0 (ζ (t)) + W ∗ β (1) (t) ,

5.7 Examples

161

where ⎡

⎤ ζ(t ) t ⎢  (s)⎥ β (1)(t) = 2 ⎣ g0 (x) dx + g0 (ζ (s))σ0 (ζ(s)) d W ⎦, y0

(5.35)

0

   is a solution of Eq. (5.14), and the processes W ∗ and W  are independent here ζ, W Wiener processes. Proof The proof of Theorem 5.11 completely coincides with the proof of Theorem 5.10 with the only difference that the process β (1) has form (5.35).  

5.7 Examples Denote by bT the family of constants such that bT > 1 and bT ↑ +∞, as T → +∞. Consider the following examples of the coefficients aT (x) of Eq. (5.1). Example 5.1 Let aT (x) =

bT bT 1 x + − , 2 2 2 2 2 1 + x2 1 + x bT 1 + (x − 1) bT

and let the initial condition in Eq. (5.1) be x0 = 0. Taking into account the equality fT (x) = exp {−2 arctan bT x} · exp {−2 [arctan bT (x − 1) + arctan bT ]} ·

2

1 + x 2,

we conclude that the solution ξT (t) of Eq. (5.1) belongs to the class K(GT ) for x GT (x) = fT (x) =

⎧ ⎨

exp −2 ⎩ 0

⎫ ⎬

u aT (v) dv

⎭

du.

0

In this case the2conditions of Theorem 5.2 are fulfilled with a0 (x) ≡ 0 and σ0 (x) = σ (ϕ(x)) 1 + ϕ 2 (x), where ⎧ π ⎨ e for x < 0, σ (x) = e−π for 0 ≤ x < 1, ⎩ −3π for x ≥ 1, e and ϕ(x) is the inverse function to l(x) =

x 0

√ σ (u) 1 + u2 du.

162

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Therefore, according to Theorem 5.2, the stochastic processes ζT = fT (ξT (t)) converge weakly, as T → +∞, to the solution ζ of the equation ζ(t) = t  (s). σ0 (ζ (s)) d W 0

Example 5.2 Let aT (x) ≡ 0 and x0 = 0. Consider the family of functions gT (x) = bT (sin [(x − x1 ) bT ] + cos [(x − x2 ) bT ]) . It is clear that the solution of Eq. (5.1) has the form ξT (t) = WT (t). In this case Eq. (5.1) belongs to the class K(GT ) for GT (x) = x, fT (x) = 1, a0 (x) = 0, σ0 (x) = 1. The conditions of Theorem 5.5 hold for cT = cos(x1 bT ) + sin(x2 bT ) and m2T = 1 + sin ((x2 − x1 ) bT ) . The stochastic processes (1) (t) = β Tn

1 2mTn

t

  gTn WTn (s) ds

0

for the subsequences Tn → +∞ such that lim inf mTn > 0, converge weakly, as Tn →+∞

Tn → +∞, to a Wiener process W . The proof of Theorem 5.5 implies that for the subsequences Tn → +∞ such that mTn → 0 the stochastic processes (1) βTn (t)

t =

  gTn WTn (s) ds

0

converge weakly, as n → +∞, to the process β(t) ≡ 0. In particular, the functional of the form βT(1) (t)

t =

gT (WT (s)) ds, 0

for x1 = x2 converges weakly, as T → +∞, to the process β(t) = 2W ∗ (t), (1) where W ∗ is a Wiener process. If x1 < x2 , then the stochastic processes βTn (t) for   1 2π Tn = x2 −x 3 + 2nπ converge weakly, as n → +∞, to the process β(t) ≡ 0. 1 Example 5.3 Let aT (x) ≡ 0 in Eq. (5.1). Then ξT (t) = x0 + WT (t) is the solution of Eq. (5.1) belonging to the class K (GT ) for GT (x) = x. In this case,

5.7 Examples

163

the conditions of Theorem 5.2 are fulfilled with a0 (x) = 0 and σ0 (x) = 1. The assumptions of Theorem 5.5 with cT = cos bT x1 and m2T = 12 hold for gT (x) = bT sin (bT (x − x1 )). According to Theorem 5.2, the stochastic processes  (t), where W  is ξT (t) converge weakly, as T → +∞, to the process ζ(t) = x0 + W a Wiener process. According to Theorem 5.5, the stochastic processes (1) βT (t)

t =

bT sin (bT ξT (s)) ds 0

converge weakly, as T → +∞, to the process

√  (t). 2W

Example 5.4 Let in Eq. (5.1) aT (x) ≡ 0. Consider : gT (x) =

bT . 1 + bT2 x 2

It is clear that gT2 (x) is a δ-shaped family at the point x = 0 with weight π. The assumptions of Theorem 5.8 hold with GT (x) = x and g0 (x) = π2 signx. Thus, Theorem 5.8 implies that the stochastic processes βT(2)(t)

t : = 0

bT dWT (s) 1 + bT2 WT2 (s)

  converge weakly to the process β (2)(t) = W ∗ β (1)(t) , as T → +∞, where ⎡ ⎢ β (1) (t) = 2 ⎣

ζ(t )

π signx dx − 2

x0

t

⎤ π ⎥ signζ(s) dW (s)⎦ , 2

0

ζ(t) = x0 + W (t), and the processes W ∗ and W are independent.  x0 (2) ∗ Thus, β (t) = W πLW (t) , where x

t

LW0 (t) = |x0 + W (t)| − |x0 | −

sign (x0 + W (s)) dW (s) 0

is the local time on the interval [0, t] of a Wiener process W at the point x0 . $ %−1 Example 5.5 Consider Eq. (5.1), where aT (x) = bT 1 + (bT x − 1)2 is a δshaped family at the point x = 0 with weight π.

164

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

We are going to show that Eq. (5.1) belongs to the class K (GT ) for ⎧ ⎨

x GT (x) = fT (x) =

⎫ ⎬

u

exp −2 ⎩ 0

aT (v) dv

⎭

du.

0

Indeed, +

,

x

. = exp −2arctg(bT v − 1)|x0 0 + 3 e − 2 π , x > 0, = exp {−2 [arctg(bT x − 1) + arctg1]} → σ0 (x) = π x < 0, e2, fT (x) = exp −2 aT (v) dv

Since fT (x)aT (x) +

1 2

as T → +∞.

fT (x) = 0, we have

 1 GT (x)aT (x) + GT (x) 2

2

  2  2  + GT (x) = fT (x) ≤ C ≤ C 1 + |GT (x)|2 .

We derive |GT (x)| ≥ C|x|α for all x ∈ R with C = δ0 and α = 1 from the inequalities 0 < δ0 ≤ G T (x) = fT (x) ≤ C0 . In addition, ! 3 !x u ! ! fT (u) !0 0

4 χB (GT (v)) fT (v)

dv

! ! ! du! ≤ !

C0 δ0

! ! !x u ! ! ! χB (GT (v)) dv du! ! !0 0 !

≤ C1 λ(B)|x| ≤ ψ (λ(B)) [1 + |x|m2 ] . Therefore, condition (A4 ) holds for the case of ψ (|x|) = C1 |x| and m2 = 1. Let us check the assumptions of Theorem 5.2: (1)



qT (x) = GT (x) aT (x) +



GT (x) − a⎧0 (GT (x)) ≡ 0,  2 ⎪ ⎨ G (x) − e−3π → 0, x > 0,  2 T (2) as T → +∞. qT (x) = GT (x) − σ02 (GT (x)) =  2 ⎪ ⎩ G (x) − eπ → 0, x < 0, T 1 2

and ! ! x ! !  (2) N ! ! ! C0 ! q (v) ! (2) ! T !≤ dv sup fT (x) !! (v) ! dv → 0, when T → +∞. !q T ! fT (v) δ0 |x|≤N ! ! 0

−N

Therefore, the conditions of Theorem 5.2 hold for a0 (x) ≡ 0 and σ0 (x) = e− 2 π , π if x > 0, σ0 (x) = e 2 , if x ≤ 0, y0 = x0 σ0 (x0 ). Thus, the stochastic process ζT (t) = 3

5.7 Examples

165

GT (ξT (t)) converges weakly, as T → +∞, to the solution ζ of the following Itô’s equation: t ζ (t) = x0 σ0 (x0 ) +

 (s). σ0 (ζ (s)) d W

0

The assumptions of Theorem 5.8 hold for the functions gT (x) = cos (bT x)   −1 if  gT (x) = gT G−1 T (x) , where GT (x) are the inverse functions to GT (x) and g0 (x) ≡ 12 . In this case β (1) (t) = 12 t. Hence, the stochastic processes (2) βT (t)

t =

cos (bT ξT (s)) dWT (s) 0

converge weakly, as T → +∞, to the process process.

√1 W ∗ (t), 2

where W ∗ is a Wiener

Example 5.6 Let in Eq. (5.1) aT (x) = −

1 bT2 x . 4 1 + bT2 x 2

In this case, Eq. (5.1) belongs to the class K (GT )√for GT (x) = x 2 . The assumptions of Theorem 5.2 hold for a0 (x) = 12 , σ0 (x) = 2 |x|, and y0 = x02 . Therefore, the stochastic processes ζT (t) = ξT2 (t) converge weakly, as T → +∞, to the solution ζ of equation ζ (t) =

x02

1 + t +2 2

t 2

 (s). ζ (s) d W

0

The assumptions of Theorem 5.8 hold for the functions √ 4 bT cos (bT x)  gT (x) = √ ln bT 8 1 + b 2 x 2 T

(5.36)

166

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

if  √  √ 4 bT cos bT |x|   gT (x) = √ . ln bT 8 1 + b 2 x 2 T

1 1 , g0 (x) = √ 4 4 |x|

In this case,  gT (x 2 ) = gT (x). According to Theorem 5.8, the stochastic processes (2) βT (t)

√ t 4 bT cos (bT ξT (s))  dWT (s) = √ 8 ln bT 2 ξ 2 (s) 1 + b T T 0

  converge weakly, as T → +∞, to the process W ∗ β (1)(t) , where

β

(1)

 3 2 3 ζ 4 (t) − |x0 | 2 − (t) = 3

t 2 4

 (s), ζ(s) d W

0

   is a solution of Eq. (5.36), and W ∗ is a Wiener process such that W ∗ and ζ, W β (1) are independent. Example 5.7 Let aT (x) = bT χ

0,

λ bT

 (x)

x GT (x) = fT (x) =

and λ > 0. If

⎧ ⎫ ⎨ u ⎬ exp −2 aT (v) dv du, ⎩ ⎭

0

0

then Eq. (5.1) belongs to the class K (GT ). The assumptions of Theorem 5.2 hold for a0 (x) = 0,  σ0 (x) =

e−2λ , x > 0, 1, x ≤ 0,

and y0 = x0 σ0 (x0 ). Thus, the stochastic processes ζT (t) = GT (ξT (t)) converge weakly, as T → +∞, to the solution ζ of the following Itô’s equation: t ζ (t) = x0 σ0 (x0 ) + 0

 (s). σ0 (ζ (s)) d W

(5.37)

5.7 Examples

167

The assumptions of Theorem 5.8 with ⎞1

⎛ ⎜ and  gT (x) = ⎝

π signx g0 (x) = 2 σ0 (x)

2

bT ⎟  2 ⎠ 1 + bT2 G−1 T (x)

hold for the functions 3 gT (x) =

bT 1 + bT2 x 2

41 2

,

where G−1 T (x) denotes the inverse function to GT (x). According to Theorem 5.8, the stochastic processes (2) βT (t)

t : = 0

bT dWT (s) 1 + bT2 ξT2 (s)

(5.38)

  converge weakly, as T → +∞, to the process W ∗ β (1)(t) , where ⎡ ⎢ β (1) (t) = π ⎣

ζ(t )

sign v dv − σ0 (v)

t

⎤  (s)⎥ signζ(s) d W ⎦,

0

x0 σ0 (x0 )

   is a solution of Eq. (5.37), and W ∗ is a Wiener process, W ∗ and β (1) are ζ, W independent. Remark 5.1 The classes K (GT ), related to Eq. (5.1), are not defined uniquely. In particular, if aT (x) in Eq. (5.1) are the same as in Example 5.7, then Eq. (5.1) belongs to the class K (GT ) with GT√(x) = x 2 . In addition, according to Theorem 5.2 with a0 (x) = 1, σ0 (x) = 2 |x|, and y0 = x02 , the stochastic process ζT (t) = ξT2 (t) converges weakly, as T → +∞, to the solution ζ of the following Itô’s equation: ζ (t) =

x02

+t +2

t 2 0

 (s). ζ (s) d W

(5.39)

168

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Here ζ (t) ≥ 0 with probability 1 for all t ≥ 0. Moreover, the assumptions of Theorem 5.8 with 3  gT (x) =

bT 1 + bT2 |x|

41 2

,

π g0 (x) = √ 4 |x|

hold for the functions gT (x), defined in Example 5.7. (2) Thus, by Theorem 5.8, the stochastic processes βT (t), defined by relation (5.38)  (1) ∗ converge weakly, as T → +∞, to the process W β , where β (1)(t) = π

2

  (t) , ζ (t) − |x0 | − W

   is a solution of Eq. (5.39), and W ∗ is a Wiener process, W ∗ and β (1) are ζ, W independent.

5.8 Auxiliary Results Lemma 5.1 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ). Then for any N > 0 and for any Borel set B ⊂ [−N; N] there exists a constant CL such that L P{GT (ξT (s)) ∈ B} ds ≤ CL ψ (λ(B)) , 0

where λ(B) is the Lebesgue measure of the set B, ψ (|x|) is a certain bounded function satisfying ψ (|x|) → 0 as |x| → 0. Proof Consider the function x ΦT (x) = 2 0

⎞ ⎛ u  χB (GT (v)) ⎠ dv du. fT (u) ⎝ fT (v) 0

The function ΦT (x) is continuous, the derivative ΦT (x) of this function is continuous and the second derivative ΦT (x) exists a.e. with respect to the Lebesgue measure and is locally bounded. Therefore, we can apply the Itô formula to the process ΦT (ξT (t)), where ξT (t) is a solution of Eq. (5.1). Furthermore, the equality ΦT (x)aT (x) +

1 Φ (x) = χB (GT (x)), 2 T

5.8 Auxiliary Results

169

holds a.e. with respect to the Lebesgue measure. Using the latter equality we conclude that t

t χB (ζT (s)) ds = ΦT (ξT (t)) − ΦT (x0 ) −

0

ΦT (ξT (s)) dWT (s),

(5.40)

0

with probability 1 for all t ≥ 0, where ζT (t) = GT (ξT (t)). Hence, using the properties of stochastic integrals, we obtain that t P{ζT (s) ∈ B} ds = E [ΦT (ξT (t)) − ΦT (x0 )] .

(5.41)

0

According to condition (A4 ) we have $ % |ΦT (x) − ΦT (x0 )| ≤ Cψ (λ(B)) 1 + |GT (x)|m . Hence, using inequality (5.4), we obtain that |E [ΦT (ξT (L)) − ΦT (x0 )]| ≤ CL ψ (λ(B)) for a certain constant CL . The latter inequality and equality (5.41) prove Lemma 5.1.   Lemma 5.2 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ). If for measurable locally bounded functions qT (x) condition (A5 ) holds true, then for any L>0 ! t ! ! ! ! ! P ! sup ! qT (ξT (s)) ds !! → 0, 0≤t ≤L ! ! 0

as T → +∞. Proof Consider the function x ΦT (x) = 2 0

⎞ ⎛ u  q (v) T dv ⎠ du. fT (u) ⎝ fT (v) 0

The same arguments as used to obtain equality (5.40) yield that t

t qT (ξT (s)) ds = ΦT (ξT (t)) − ΦT (x0 ) −

0

0

ΦT (ξT (s)) dWT (s).

(5.42)

170

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

It is clear that for any constants ε > 0, N > 0, L > 0 the following inequalities hold true: ! ! u + , ! ! N ! qT (v) !! 4 ! dv ! du, P sup |ΦT (ξT (t)) − ΦT (x0 )| > ε ≤ PNT + fT (u) ! ε 0≤t ≤L ! fT (v) ! −N

0

⎧ ⎨

⎫ ! ! t ! ! ⎬ ! ! P sup !! ΦT (ξT (s)) dWT (s)!! > ε ⎩0≤t ≤L ! ⎭ ! 0

≤ PNT

! t !2 ! ! ! ! 4 + 2 E sup !! ΦT (ξT (s))χ|ξT (s)|≤N dWT (s)!! ε 0≤t ≤L ! ! 0

≤ PNT

16 + 2E ε

L

$

%2 ΦT (ξT (s)) χ|ξT (s)|≤N ds,

0

+ where PNT = P

, sup |ξT (t)| > N .

0≤t ≤L

Using equality (5.42) and the properties of stochastic integrals, we conclude that L E

$

%2 $ % *T (x0 ) , *T (ξT (L)) − Φ ΦT (ξT (s)) χ|ξT (s)|≤N ds = E Φ

(5.43)

0

where *T (x) − Φ *T (x0 ) = 2 Φ

x

⎛ u  ⎝ fT (u)

x0

0

⎞ %2 1 $ Φ (v) χ|v|≤N dv ⎠ du. fT (v) T

Using the Hölder inequality, we obtain the following bounds: ! ! ! u !1 ! u !1 !x ! ! q ! ! p !! ! ! ! ! ! $ ! ! %2p 1 ! ! ! !Φ *T (x0 )! ≤ 2 !! fT (u) ! *T (x) − Φ ΦT (v) χ|v|≤N dv !! du! ! |f (v)|q dv ! ! ! ! ! ! ! ! T !x0 ! 0 0

! N ! p1 ! ! ! ! !2p ! ! ! ! ≤ 2! ΦT (v) dv !! ! ! −N

! ! ! u !1 ! !x ! !q ! ! ! ! 1 ! ! ! ! dv ! du, ! fT (u) ! ! ! |fT (v)|q !! ! ! !x0 0

5.8 Auxiliary Results

171

where p > 1 and q > 1 are arbitrary constant with p1 + q1 = 1. Taking into account assumption (A3 ) for certain q = 1 + δ, we conclude that ⎛

! $ %! !E Φ *T (ξT (L)) − Φ *T (x0 ) ! ≤ 2CL ⎝

N

⎞ p1

! ! !Φ (v)!2p dv ⎠ , T

(5.44)

−N

where the constants CL > 0 and p > 1 do not depend on T . According to assumption (A5 ) and to Lebesgue’s dominated convergence theorem, we have for arbitrary N > 0 that N

! ! !Φ (v)!2p dv → 0, T

−N

as T → +∞. Hence, taking into account the inequality (5.44) and the equality (5.43), we obtain the convergence L E

$

ΦT (ξT (s))

%2

χ|ξT (s)|≤N ds → 0,

0

as T → +∞. According to assumptions (A1 ) and (A5 ), using Lebesgue’s dominated convergence theorem, we obtain that P

sup |ΦT (ξT (t)) − ΦT (x0 )| → 0,

0≤t ≤L

! t ! ! ! ! ! P ! sup ! ΦT (ξT (s)) dWT (s)!! → 0, 0≤t ≤L ! ! 0

as T → +∞. Thus, the equality (5.42) implies the statement of Lemma 5.2.

 

Remark 5.2 Let the assumptions of Lemma 5.2 hold. Suppose additionally that the measurable locally bounded functions gT satisfy assumption (A7 ). Then for any L>0 ! ! t ! ! ! ! ! E sup ! qT (ξT (s)) ds !! →0, 0≤t ≤L ! ! 0

as T → +∞.

172

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

Proof Using equality (5.42) we have that ! t !2 ! ! ! ! sup !! qT (ξT (s)) ds !! ≤ 2 sup |ΦT (ξT (t)) − ΦT (x0 )|2 0≤t ≤L ! 0≤t ≤L ! 0

! t !2 ! ! ! ! ! +2 sup ! ΦT (ξT (s)) dWT (s)!! . 0≤t ≤L ! ! 0

Taking into account the first inequality from (A7 ) we conclude ! x ⎞ !2 ⎛ u ! !  ! ! q (v) T 2 ! |ΦT (x) − ΦT (x0 )| = 2 ! fT (u) ⎝ dv ⎠ du!! fT (v) ! ! x0

0

! x !2 ! !  2 ! ! $ %  1 + |x|α1 +1 . ≤ 2 !! C 1 + |u|α1 du!! ≤ C ! ! x0

Therefore, E sup |ΦT (ξT (t)) − ΦT (x0 )|2 ≤ CL . 0≤t ≤L

Using condition (A7 ) and the properties of stochastic integrals, we conclude that !2 ! t ! ! L ! ! %2 $ ! ! *L . E sup ! ΦT (ξT (s)) dWT (s)! ≤ 4E ΦT (ξT (s)) ds ≤ C 0≤t ≤L ! ! 0

0

Therefore, ! t !2 ! ! ! ! * *L . E sup !! qT (ξT (s)) ds !! ≤ C 0≤t ≤L ! ! 0

Consequently, for any L > 0 ! t ! ! ! ! ! E sup !! qT (ξT (s)) ds !! →0, 0≤t ≤L ! ! 0

as T → +∞.

 

5.8 Auxiliary Results

173

Lemma 5.3 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ). Also P

let ζT (t) = GT (ξT (t)) → ζ (t), as T → +∞. Then for any measurable locally bounded function g the following convergence holds true: ! t ! ! ! t ! ! P ! sup ! g(ζT (s)) ds − g(ζ (s)) ds !! → 0, 0≤t ≤L ! ! 0

0

as T → +∞, for any constant L > 0. Proof Let ϕN (x) = 1 for |x| ≤ N, ϕN (x) = N + 1 − |x| for |x| ∈ [N, N + 1] and ϕN (x) = 0 for |x| > N + 1. Then for all T > 0, L > 0 ⎫ ! t ! + , ! ! ⎬ ! ! ! ! P sup ! [g(ζT (s)) − g(ζT (s))ϕN (ζT (s))] ds ! > 0 ≤ P sup |ζT (t)| > N , ⎭ ⎩0≤t≤L ! 0≤t≤L ! ⎧ ⎨

0

⎫ ! t ! + , ! ! ⎬ ! ! P sup !! [g(ζ (s)) − g(ζ (s))ϕN (ζ (s))] ds !! > 0 ≤ P sup |ζ(t)| > N ⎭ ⎩0≤t ≤L ! 0≤t ≤L ! ⎧ ⎨

0

+ ≤ lim sup P T →+∞

, sup |ζT (t)| > N .

0≤t ≤L

According to Theorem 5.7 the convergence (5.10) holds for the process ζT . So, to complete the proof of Lemma 5.3, we need to establish that L

P

|g(ζT (s))ϕN (ζT (s)) − g(ζ (s))ϕN (ζ (s))| ds → 0

(5.45)

0

as T → +∞. First, assume that the function g is continuous. Then P

g(ζT (s))ϕN (ζT (s)) − g(ζ (s))ϕN (ζ(s)) → 0 as T → +∞ for all 0 ≤ s ≤ L and |g(x)ϕN (x)| ≤ CN for all x. Thus, according to Lebesgue’s dominated convergence theorem, we have the convergence (5.45). Second, let the function g be measurable and locally bounded. Then, using Luzin’s theorem we conclude that for any δ > 0 there exists a continuous function g δ (x), which coincides with g(x) for x ∈ / B δ , where B δ ⊂ [−N − 1, N + 1] and its

174

5 Asymptotic Behavior of Homogeneous Additive Functionals Defined on the. . .

  Lebesgue measure satisfies inequality λ B δ < δ. Thus, for every δ > 0 the relation (5.45) holds true for the function g δ (x). Since for any ε > 0 ⎫ ⎧ L ⎬ ⎨ ! ! !g(ζT (s))ϕN (ζT (s)) − g δ (ζT (s))ϕN (ζT (s))! ds > ε P ⎭ ⎩ 0

2 ≤ E ε

L

! ! !g(ζT (s))ϕN (ζT (s)) − g δ (ζT (s))ϕN (ζT (s))! χ{B δ } (ζT (s)) ds

0

≤

CN ε

L

. P ζT (s) ∈ B δ ds,

0

and ⎧ L ⎫ ⎨ ! ⎬ ! !g(ζ (s))ϕN (ζ (s)) − g δ (ζ (s))ϕN (ζ(s))! ds > ε P ⎩ ⎭ 0

CN ≤ ε

L

-

P ζ (s) ∈ B 0

δ

.

CN lim sup ds ≤ ε T →+∞

L

. P ζT (s) ∈ B δ ds,

0

we can additionally take into account Lemma 5.1, and conclude that the relation (5.45) holds for a measurable and locally bounded function g, as well.   Lemma 5.4 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ) and let t P P ζT (t) = GT (ξT (t)) → ζ (t), ηT (t) = GT (ξT (s)) dWT (s) → η(t), as T → +∞. 0

Then for a measurable locally bounded function g the following convergence holds true: ! ! t ! ! t ! P ! sup !! g(ζT (s)) dηT (s) − g(ζ(s)) dη(s)!! → 0 0≤t ≤L ! ! 0

as T → +∞ for any constant L > 0.

0

5.8 Auxiliary Results

175

Proof Similarly to the proof of Lemma 5.3, it is sufficient to obtain an analog of the convergence (5.45), i.e., to get that for any N > 0, L > 0 ! ! t ! ! t ! P ! ! sup ! g(ζT (s))ϕN (ζT (s)) dηT (s) − g(ζ(s))ϕN (ζ(s)) dη(s)!! → 0 0≤t ≤L ! ! 0

0

(5.46) as T → +∞, where ϕN (x) is defined in the proof of Lemma 5.3. The proof of the convergence (5.46) for a continuous function g(x) is similar to the proof of the corresponding theorem in [79, Chap.2, §6]. The explicit form of the quadratic characteristic ηT (t) of the martingale ηT (t) and condition (A1 ) imply the inequality L [ϕN (ζT (t))]2 dηT (t) ≤ CN L, 0

which is used for the proof of the convergence (5.46). The extension of such a convergence to the class of measurable locally bounded functions is based on Lemma 5.1 and is provided similarly to the proof of Lemma 5.3.   Lemma 5.5 Let ξT be a solution of Eq. (5.1) belonging to the class K (GT ), and let the stochastic process (ζT (t), ηT (t)) , with ζT (t) = GT (ξT (t)) and ηT (t) = t   GT (ξT (s)) dWT (s) be stochastically equivalent to the process * ζT (t), * ηT (t) . 0

Then the process t

t g(ζT (s)) ds +

0

q(ζT (s)) dηT (s), 0

where g, q are measurable locally bounded functions, is stochastically equivalent to the process t 0

g(* ζT (s)) ds +

t

q(* ζT (s)) d* ηT (s).

0

Proof The proof is the same as that of Corollary A.3 from the Appendix.

 

Chapter 6

Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to Inhomogeneous Itô SDEs with Non-regular Dependence on a Parameter

In this chapter, we consider the asymptotic behavior, as T → +∞, of some t functionals of the form IT (t) = FT (ξT (t)) + gT (ξT (s)) dWT (s), t ≥ 0. Here 0

ξT (t) is the solution to the time-inhomogeneous Itô stochastic differential equation dξT (t) = aT (t, ξT (t)) dt + dWT (t), t ≥ 0, ξT (0) = x0 , where T > 0 is a parameter, aT (t, x), x ∈ R are measurable functions, |aT (t, x)| ≤ LT for all x ∈ R and t ≥ 0, WT are standard Wiener processes, FT (x), x ∈ R are continuous functions, and gT (x), x ∈ R are measurable locally bounded functions. Section 6.1 contains some preliminary remarks, notations and basic definitions. The asymptotic behavior of the integral functionals of the Lebesgue integral type is investigated in Sect. 6.3. Section 6.4 contains some results about the weak convergence of the martingale type functionals and the mixed functionals. Section 6.5 includes several examples. Auxiliary results are collected in Sect. 6.6.

6.1 Preliminaries Consider the time-inhomogeneous Itô stochastic differential equation dξT (t) = aT (t, ξT (t)) dt + dWT (t),

t ≥ 0, ξT (0) = x0 ,

(6.1)

where T > 0 is a parameter, aT (t, x), x ∈ R are real-valued measurable functions such that |aT (t, x)| ≤ LT for all (t, x) and some family of constants LT > 0, and WT = {WT (t), t ≥ 0}, T > 0 is a family of standard Wiener processes defined on a complete probability space (Ω, , P). © Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3_6

177

178

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

It is known from Theorem 4 in [82] that for any T > 0 and x0 ∈ R Eq. (5.1) possesses a unique strong solution ξT = {ξT (t), t ≥ 0}. In this chapter, we study the weak convergence, as T → +∞, of the t processes IT (t) = FT (ξT (t)) + gT (ξT (s)) dWT (s), where ξT (t) is the solution 0

to the stochastic differential equation (6.1), FT (x) is a family of continuous realvalued functions, and gT (x) is a family of measurable locally bounded real-valued functions. All the results about asymptotic behavior are obtained under the condition which provides a certain proximity of the coefficients aT (t, x) to some measurable functions  aT (x). In a such situation, the limit processes, obtained when T → +∞, are some functionals of the limits of the solutions  ξT (t) to the homogeneous stochastic differential equations d ξT (t) =  aT ( ξT (t)) dt + dWT (t).

(6.2)

The present chapter generalizes similar results from the previous chapter, that were formulated for the unique strong solutions  ξT to the homogeneous stochastic differential equations (6.2), to the case of the solutions ξT (t) to inhomogeneous equations (6.1). Under the proposed conditions, we prove that the asymptotic behavior of the solutions and some functionals of the solutions to the inhomogeneous Itô stochastic differential equations (6.1) is the same as that for the solutions to the homogeneous Itô stochastic differential equations (6.2). We assume that the drift coefficient aT (t, x) in Eq. (5.1) can have non-regular dependence on the parameter. For example, the drift coefficient aT (t, x) can tend, as T → +∞, to infinity at some points xk and at some points tk as well, or it can have degeneracies of some other types. In what follows we denote by C, L, N, CN , LN any constants that do not depend on T , x nor t. To formulate and prove the main results, we introduce functions of the form ⎧ ⎫ x ⎨ u ⎬ aT (v) dv du, T > 0. fT (x) = exp −2  (6.3) ⎩ ⎭ 0

0

Throughout the chapter we use the following notations: βT(1) (t)

t =

gT (ξT (s)) ds,

βT(2)(t)

0

t =

gT (ξT (s)) dWT (s), 0

t IT (t) = FT (ξT (t)) +

gT (ξT (s)) dWT (s), 0

6.1 Preliminaries

179

where ξT and WT are related via Eq. (6.1), gT is a family of measurable, locally bounded real-valued functions, and FT is a family of continuous real-valued functions. To study the weak convergence, as T → +∞, of the processes IT (t) = t FT (ξT (t)) + gT (ξT (s)) dWT (s), where ξT is the solution to the stochastic 0

differential equation (6.1), we suppose additionally that the drift coefficients satisfy the following assumption: There exists a family of measurable, locally bounded functions  aT (x) such that for any L > 0 L (A0 )

sup |aT (t, x) −  aT (x)| dt = 0.

lim

T →+∞

x

0

Note that, due to condition (A0 ), some results about solutions  ξT to the homogeneous equations (6.2), which are obtained in Chap. 5, can be extended to solutions ξT to the inhomogeneous equations (6.1). Therefore, by analogy to Chap. 5, we consider Eq. (6.1) belonging to the class K (GT ). Definition 6.1 The class of equations of the form (6.1) will be denoted by K (GT ), if there exist families of functions  aT (x) and GT (x), x ∈ R, such that: (1)  aT (x) are measurable locally bounded real-valued functions, satisfying condition (A0 ); (2) GT (x) have continuous derivatives GT (x) and locally integrable second deriva tives GT (x) a.e. with respect to the Lebesgue measure such that, for all T > 0, x ∈ R and t ≥ 0, for some constant C > 0 the following inequalities hold:  2  2 $ % (A1 ) GT (x)aT (t, x) + 12 GT (x) + GT (x) ≤ C 1 + |GT (x)|2 , |GT (x0 )| ≤ C; (3) there exist constants C > 0 and α > 0 such that, for all x ∈ R, |GT (x)| ≥ C|x|α ; (4) there exist a bounded function ψ (|x|) and a constant m ≥ 0 such that ψ (|x|) → 0, as |x| → 0, and, for all x ∈ R and T > 0 and for any measurable bounded set B, the following inequality holds:

(A2 )

! ! ! ! x ! ! % $ χ (u)) (G B T !f (x) du!! ≤ ψ (λ(B)) 1 + |x|m , ! T fT (u) ! ! 0

180

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

where χB (x) is the indicator function of a set B, λ(B) is the Lebesgue measure of B, and fT (x) is the derivative of the function fT (x) defined by equality (6.3). Assume that, for certain locally bounded functions qT (x) and any constant N > 0, the following condition holds: ! ! ! ! x ! ! q (v) T ! = 0. dv lim sup !!fT (x) T →+∞ |x|≤N ! fT (v) !!

(A3 )

0

6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs Now we are in a position to use the result concerning the weak compactness of stochastic processes ζT = {ζT (t) = GT (ξT (t)), t ≥ 0} (Theorem 6.1) in further investigation of asymptotic behavior of the solutions (Theorem 6.2) and some functionals of the solutions (Theorems 6.3–6.7) to the inhomogeneous Itô stochastic differential equations (6.1). In the proofs of the theorems, which are performed similarly to the proofs of the corresponding theorems in Chap. 5, we emphasize the differences associated with the inhomogeneous equations. Theorem 6.1 Let ξT be a solution of Eq. (6.1) and let there exist a family of continuous functions GT (x), x ∈ R with continuous derivative GT (x) and the second derivative GT (x) to be assumed to exist a.e. with respect to the Lebesgue measure and to be locally integrable. Let the functions GT (x) satisfy assumption (A1 ), for all T > 0, t ≥ 0, x ∈ R. Then the family of the processes ζT = {ζT (t) = GT (ξT (t)), t ≥ 0} is weakly compact. Proof The proof of Theorem 6.1 differs from the proof of Theorem 5.1 only by the representation of the process ζT (t) = GT (ξT (t)). Now we have that t ζT (t) = GT (x0 ) +

t LT (ξT (s)) ds +

0

G T (ξT (s)) dWT (s),

(6.4)

0

where LT (x) = G T (x) aT (t, x) + comparing to formula (5.3).

1 G (x), 2 T  

Theorem 6.2 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ) and GT (x0 ) → y0 , as T → +∞. Assume that there exist measurable locally bounded functions a0 (x) and σ0 (x) such that:

6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs

181

(1) the functions

aT (x) + qT(1)(x) = GT (x)

1 G (x) − a0 (GT (x)) , 2 T

 2 (2) qT (x) = GT (x) − σ02 (GT (x)) , satisfy assumption (A3 );

   . (2) the Itô equation (5.14) has a unique weak solution ζ, W Then the stochastic processes ζT = GT (ξT (t)) converge weakly, as T → +∞, to the solution ζ of Eq. (5.14). Proof Rewrite the equality (6.4) as t ζT (t) = GT (x0 ) +

a0 (ζT (s)) ds + ηT (t) + αT(0) (t) + αT(1) (t),

(6.5)

0

where t

G T (ξT (s)) dWT (s),

ηT (t) = 0

(0) αT (t)

t =

G T (ξT (s))ΔaT (s) ds,

ΔaT (s) = aT (s, ξT (s)) −  aT (ξT (s)),

0

αT(1) (t)

t =

qT(1)(ξT (s)) ds,

qT(1)(x) = G T (x) aT (x) +

1 G (x) − a0 (GT (x)) . 2 T

0

The conditions (A0 ) and (A1 ), together with the inequality (5.9), imply that ! L ! ! ! ! (0) ! sup !αT (t)! ≤ !G T (ξT (s))! |ΔaT (s)| ds

0≤t ≤L

( 3

0

4) 1 L 2

≤ C 1 + sup |ζT (s)|2 0≤s≤L

as T → +∞, for any L > 0.

P

sup |aT (s, x) −  aT (x)| ds → 0, x

0

(6.6)

182

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . . (1)

The functions qT (x) satisfy the conditions of Lemma 6.2. Thus, for any L > 0 ! ! ! (1) ! P sup !αT (t)! → 0

0≤t ≤L

(6.7)

as T → +∞. It is clear that ηT (t) is a family of continuous martingales with the quadratic characteristics t ηT (t) =

$

%2 G T (ξT (s))

t ds =

0

σ02 (ζT (s)) ds + αT(2) (t),

(6.8)

0

where αT(2) (t)

t =

qT(2)(ξT (s)) ds,

 2 qT(2)(x) = G T (x) − σ02 (GT (x)) .

0 (2)

The functions qT (x) satisfy the conditions of Lemma 6.2. Thus, for any L > 0 ! ! ! P ! sup !αT(2) (t)! → 0,

0≤t ≤L

(6.9)

as T → +∞. According to Theorem 6.1, the family of the processes ζT (t) is weakly compact. It is easy to see that the compactness conditions (5.12) are fulfilled for the processes ηT (t). Using the convergences (6.6), (6.7), and (6.9), we have that relations (5.12) (k) hold for the processes αT (t), k = 0, 1, 2, as well. It means that we can apply Skorokhod’s convergent subsequence principle (see Theorem A.12) for the process 

 ζT (t), ηT (t), αT(k) (t), k = 0, 1, 2 .

According to this principle, given an arbitrary sequence Tn → +∞, we can ˜ and a stochastic ˜ P), ˜ , choose a subsequence Tn → +∞, a probability space (Ω, process   (t), k = 0, 1, 2 ζ˜Tn (t), η˜ Tn (t), α˜ T(k) n defined on this space such that its finite-dimensional distributions coincide with those of the process   (t), k = 0, 1, 2 ζTn (t), ηTn (t), αT(k) n

6.2 Weak Compactness and Weak Convergence of the Solutions of Itô SDEs

183

and, moreover, ˜

˜

˜

P P P (k) ˜ α˜ Tn (t) → α˜ (k) (t), k = 0, 1, 2, ζ˜Tn (t) → ζ˜ (t), η˜ Tn (t) → η(t),

as Tn → +∞, for all 0 ≤ t ≤ L, where ζ˜ (t), η(t), ˜ α˜ (k) (t), k = 0, 1, 2 are some stochastic processes. Evidently, relations (6.6)–(6.9) imply that α˜ (k) (t) ≡ 0, k = 0, 1, 2 a.s. According to (5.10), the processes ζ˜ (t) and η(t) ˜ are continuous with probability 1. Moreover, applying Lemma 5.5 together with equalities (6.5) and (6.8), we obtain that ζ˜Tn (t) = GTn (x0 ) +

t

a0 (ζ˜Tn (s)) ds + α˜ T(0) (t) + α˜ T(1) (t) + η˜ Tn (t), n n

(6.10)

0

t η˜ Tn (t) =

σ02 (ζ˜Tn (s)) ds + α˜ Tn (t), (2)

0

where ˜

P ζ˜Tn (t) → ζ˜ (t),

! ! ! (k) ! P˜ sup !α˜ Tn (t)! → 0, k = 0, 1, 2

˜ P

η˜ Tn (t) → η(t), ˜

0≤t ≤L

as Tn → +∞. An analog of the convergence (5.12) holds for the processes ζ˜Tn (t) and η˜ Tn (t). Therefore, according to the well-known result of Prokhorov (see Theorem A.13), we conclude that for any L > 0 ! ˜ ! ! P ! sup !ζ˜Tn (t) − ζ˜ (t)! → 0,

! P˜ ! sup !η˜ Tn (t) − η(t) ˜ !→0

0≤t ≤L

0≤t ≤L

(6.11)

as Tn → +∞. According to Lemma 5.3, we can pass to the limit in (6.10) and obtain the representation ζ˜ (t) = y0 +

t

a0 (ζ˜ (s)) ds + η(t), ˜

0

where η(t) ˜ is the almost surely continuous martingale with the quadratic characteristic t η(t) ˜ = 0

σ02 (ζ˜ (s)) ds.

184

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

Now, it is well known that the latter representation provides the existence of a  such that Wiener process W t η(t) ˜ =

 (s). σ0 (ζ˜ (s)) d W

0

   satisfies Eq. (5.14), and the processes ζ˜Tn (t) converge Thus, the process ζ˜ , W

weakly, as Tn → +∞, to the process ζ˜ . Since the sequence Tn → +∞ is arbitrary, and since the solution to Eq. (5.14) is weakly unique, the proof of Theorem 6.2 is complete.  

6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type In this section we obtain sufficient conditions for the weak convergence of some integral functionals of the Lebesgue integral type. Theorem 6.3 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ) and let assumptions of Theorem 6.2 hold. Assume that for measurable and locally bounded functions gT there exists a measurable and locally bounded function g0 such that the function qT (x) = gT (x) − g0 (GT (x)) satisfies assumption (A3 ). Then the stochastic processes βT(1)(t) = converge weakly, as T → +∞, to the process

t

gT (ξT (s)) ds

0

t β

(1)

(t) =

g0 (ζ (s)) ds, 0

where ζ is the solution of Eq. (5.14). The proof of Theorem 6.3 is literally the same as that of Theorem 5.3. Theorem 6.4 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ), and let the assumptions of Theorem 6.2 hold. Assume that, for measurable and locally bounded functions gT , there exists a measurable locally bounded function g0 such that ! ! ! ! x ! gT (v) !! !f (x) dv χ|x|≤N ≤ CN , ! T fT (v) !! ! 0

6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

185

! ! ! ! x ! ! g (v) T !=0 dv − g lim sup !!fT (x) (x)) G (x) (G 0 T T ! T →+∞ |x|≤N ! fT (v) !

(A4 )

0

for all N > 0. Then the stochastic processes βT(1)(t)

t =

gT (ξT (s)) ds 0

converge weakly, as T → +∞, to the process ⎛

⎞ ζ(t ) t ⎜  (s)⎟ β˜ (1) (t) = 2 ⎝ g0 (x) dx − g0 (ζ (s)) σ0 (ζ(s)) d W ⎠, y0

0

   is the solution to Eq. (5.14). where ζ, W Proof The proof of Theorem 6.4 differs from the proof of Theorem 5.3 only in that t we use the different representation of the functional βT(1)(t) = gT (ξT (s)) ds. In 0

this case we have (1)

βT (t) = 2

ζT(t )

t (1) (2) (0) g0 (u) du − 2 g0 (ζT (s)) dηT (s) + γT (t) − γT (t) − γT (t), 0

GT (x0 )

where γT(1)(t)

ξT (t )

=

 qT (u) du,

γT(2) (t)

x0

(0) γT (t)

t =

 qT (ξT (s)) dWT (s), 0

t =

ΦT (ξT (s)) [aT (s, ξT (s)) −  aT (ξT (s))] ds,

0

x ΦT (x) = 2 0

⎞ ⎛ u  gT (v) ⎠ dv du, fT (u) ⎝ fT (v) 0



 qT (x) = ΦT (x) − 2g0 (GT (x)) GT (x), and fT (x) is the derivative of the function fT (x) defined by the equality (6.3).

186

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

The latter representation differs from the corresponding representation in the proof of Theorem 5.3 by the term γT(0)(t). For any constants ε > 0, N > 0, and L > 0, we have the inequalities , + ! ! ! (0) ! P sup !γT (t)! > ε ≤ PNT 0≤t ≤L

2 + ε

L

! ! E !ΦT (ξT (s))! |aT (s, ξT (s)) −  aT (ξT (s))| χ|ξT (s)|≤N ds

0

2 ≤ PNT + CN ε

L sup [aT (s, x) −  aT (x)] ds, x

0

+ where PNT = P

, sup |ξT (t)| > N . Using condition (3) from Definition 6.1 and

0≤t ≤L

the inequality (5.9), we obtain the convergence lim

lim sup PNT = 0. Taking

N→+∞ T →+∞

into account the assumptions of Theorem 6.4, we conclude that ! ! ! P ! sup !γT(0)(t)! → 0 0≤t ≤L

(6.12)

for any L > 0, as T → +∞. The rest of the proof of Theorem 6.4 is the same as that of Theorem 5.3.   Theorem 6.5 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ), and let the assumptions of Theorem 6.2 hold. Suppose that the functions  aT satisfy assumption (A3 ). Assume that, for measurable and locally bounded functions gT , there exist two constants c0 and b0 such that for all N > 0 ! ! ! ! x ! gT (v) !! !f (x) dv χ|x|≤N ≤ CN , ! T fT (v) !! ! 0

! x⎡ ⎤ ! ! ! u ! ! g (v) T ! ⎣ ⎦ fT (u) lim sup ! dv − c0 du!! = 0, T →+∞ |x|≤N ! fT (v) ! 0

0

and the functions ⎡ qT (x) = ⎣fT (x)

x 0

satisfy assumption (A3 ).

⎤2 gT (v) dv − c0 ⎦ − b02 fT (v)

6.3 Asymptotic Behavior of Integral Functionals of the Lebesgue Integral Type

187

Then the stochastic processes (1) βT (t)

t =

gT (ξT (s)) ds 0

converge weakly, as T → +∞, to the process 2b0 W (t), where W is a Wiener process. Proof For the functional βT(1)(t) = all t ≥ 0, and with probability 1

t

gT (ξT (s)) ds we have the representation, for

0

t βT(1) (t) = 2c0  aT (ξT (s)) ds + γT (t) − ηT(1)(t) − γT(0)(t) + γT(3) (t), 0

where ⎡

ξT (t )

u

x0

0

⎣fT (u)

γT (t) = 2

ηT(1)(t)

t =

$

⎤ gT (v) dv − c0 ⎦ du, fT (v)

% ΦT (ξT (s)) − 2c0 dWT (s),

0

and (3) γT (t)

t = 2c0

aT (ξT (s))] ds, [aT (s, ξT (s)) −  0

where γT(0) (t) and ΦT (x) are defined in the proof of Theorem 6.4. The functions  aT satisfy condition (A3 ). Thus, using Lemma 6.2, we conclude that for any L > 0 ! t ! ! ! ! ! P sup !!  aT (ξT (s)) ds !! → 0, 0≤t ≤L ! ! 0

as T → +∞.

188

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

For any constants ε > 0, N > 0 and L > 0, we have the inequalities + P

, sup |γT (t)| > ε

0≤t ≤L

≤ PNT

≤ PNT

! ! ! ! ξT (t ) ! ! $ % 1 ! ! ΦT (u) − 2c0 du! χ|ξT (t )|≤N + E sup ! ! ε 0≤t ≤L ! ! ! x0

! x⎡ ⎤ ! !  ! u ! ! 2 ⎣fT (u) gT (v) dv − c0 ⎦ du! , + N sup !! ! ε |x|≤N ! fT (v) ! x0

0

where PNT is the same as that in the proof of Theorem 6.4. Using the latter inequality and the assumptions of Theorem 6.5, we conclude that P

sup |γT (t)| → 0,

0≤t ≤L

as T → +∞. (0) Since the term γT (t) is the same as that in the proof of Theorem 6.4, we get (6.12). The inequality L ! ! ! (3) ! aT (x)] ds sup !γT (t)! ≤ 2|c0 | sup [aT (s, x) − 

0≤t ≤L

x

0

implies that for any L > 0 ! ! ! (3) ! P sup !γT (t)! → 0,

0≤t ≤L

as T → +∞. Thus, we have that for any L > 0 ! ! ! P ! sup !βT(1) (t) + ηT(1) (t)! → 0,

0≤t ≤L

as T → +∞. It is clear that ηT(1) (t) is the almost surely continuous martingale with the quadratic characteristic (1) ηT (t)

t =

4b02t

+

qT (ξT (s)) ds, 0

6.4 Weak Convergence of Martingale Type Functional and of Mixed Functional

189

%2 $ where qT (x) = ΦT (x) − 2c0 − 4b02. The functions qT (x) satisfy condition (A3 ). Thus, using Lemma 6.2, we conclude that for any L > 0 ! ! ! (1) ! P sup !ηT (t) − 4b02t ! → 0, 0≤t ≤L

as T → +∞. Then, using the random  time change in stochastic integrals (see, e.g., [74]), we (1) ∗ obtain ηT (t) = WT ηT(1) (t) , where WT∗ (t) is a Wiener process. The same arguments as used to get (4.19) yield that !  ! P ! (1) ! sup !βT (t) − WT∗ 4b02t ! → 0 0≤t ≤L

(1)

as T → +∞. Thus, the processes βT (t) converge weakly, as T → +∞, to the process 2b0W (t).  

6.4 Weak Convergence of Martingale Type Functional and of Mixed Functional In this section we formulate sufficient conditions for the weak convergence of some integral functional of martingale type and of the mixed functional. Theorem 6.6 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ) and let the assumptions of Theorem 6.2 hold. Assume that, for measurable and locally bounded functions gT (x), there exists a measurable locally bounded function g0 (x) such that the function  2 qT (x) = gT (x) − g0 (GT (x)) GT (x) satisfies assumption (A3 ). Then the stochastic processes βT(2) (t)

t =

gT (ξT (s)) dWT (s), 0

where ξT and WT related via Eq. (5.1), converge weakly, as T → +∞, to the process t β

(2)

(t) =

 (s), g0 (ζ (s)) σ0 (ζ (s)) d W

0





 is the solution of Eq. (5.14). where ζ, W

190

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

The proof of Theorem 6.6 is literally the same as that of Theorem 5.6. Theorem 6.7 Let ξT and WT be related via Eq. (6.1) belonging to the class K (GT ) and let the assumptions of Theorem 6.2 hold. Assume that, for continuous functions FT and locally bounded measurable functions gT , there exist a continuous function F0 and a locally bounded measurable function g0 such that, for all N > 0 lim

sup |FT (x) − F0 (GT (x))| = 0.

T →+∞ |x|≤N

Also, let the functions gT and g0 satisfy the assumptions of Theorem 6.6. Then the stochastic processes t IT (t) = FT (ξT (t)) +

gT (ξT (s)) dWT (s) 0

converge weakly, as T → +∞, to the process t I0 (t) = F0 (ζ (t)) +

 (s), g0 (ζ (s)) σ0 (ζ(s)) d W

0





 is the solution of Eq. (5.14). where ζ, W The proof of Theorem 6.7 is literally the same as that of Theorem 5.9.

6.5 Examples We denote by bT a family of constants such that bT > 1 and bT ↑ +∞, as T → +∞. Example 6.1 Consider Eq. (6.1) with the drift coefficient having non-regular dependence on the parameter T : γ

aT (t, x) = bT cos(xbT ) +

tbT sin ((x − 1)bT ) , 0 ≤ γ < 1. 1 + t 2 bT2

The family of measurable locally bounded real-valued functions  aT (x) = γ bT cos(xbT ) satisfies condition (1) from Definition 6.1. Indeed, for any L > 0 L

L sup |aT (t, x) −  aT (x)| dt ≤ lim

lim

T →+∞

x

0

T →+∞ 0

tbT dt = 0. 1 + t 2 bT2

6.5 Examples

191

The rest of the conditions from Definition 6.1 are fulfilled, if we take the family of functions ⎧ ⎫ x ⎨ u ⎬ aT (v) dv du, T > 0. GT (x) = fT (x) = exp −2  ⎩ ⎭ 0

0

/ 0 γ b Since fT (x) = exp −2 bTT sin(xbT ) , there exist two constants c0 and δ0 such

that, for all x ∈ R, we have 0 < δ0 ≤ fT (x) ≤ c0 . Taking into account that x aT (x) + 12 GT (x) ≡ 0. GT (x) = fT (v) dv, we obtain GT (x) 0

Therefore, condition (2) holds, because  GT (x)aT (t, x) +  = GT (x)

1 2



2

GT (x)

 2 + GT (x) 2

t bT 1+t 2 bT2

sin ((x − 1)bT )

 2  2 + GT (x) ≤ 2 GT (x)

$ % ≤ 2c02 ≤ 2c02 1 + |GT (x)|2 ; ! ! ! x0 ! ! ! |GT (x0 )| = ! fT (v) dv ! ≤ c0 · |x0 | = C. !0 ! Condition (3) has the form ! ! !x ! ! ! |GT (x)| = ! fT (v) dv ! ≥ C|x|α with C = δ0 , α = 1; !0 ! and condition (4) ! 3 !x u ! ! fT (u) !0 0

4 χB (GT (v)) fT (v)

dv

! ! ! du! ≤ !

C0 δ0

! ! ! !x u ! ! χB (GT (v)) dv du! ! ! !0 0

≤ C1 λ(B)|x| ≤ ψ (λ(B)) [1 + |x|m ] is fulfilled with ψ (|x|) = C1 |x|, m = 1. Thus, Eq. (6.1) belongs to the class K (GT ). According to Theorem 6.1, the family of processes ζT (t) = GT (ξT (t)) is weakly compact. We can find the form of the limit process using Theorem 6.2 with a0 (x) ≡ 0, σ0 (x) ≡ 1. According to Theorem 6.2, the stochastic processes ζT (t) converge weakly, as T → +∞, to the  (t), where W  is a solution ζ of Eq. (5.14) and the limit process is ζ (t) = x0 + W Wiener process.

192

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

Example 6.2 Let the conditions of Example 6.1 hold. For the family of functions γ

gT (x) =

bT 1 + bT2 x 2

, 0≤γ 0, L > 0 and any Borel set B ⊂ [−N; N], there exists a constant CL such that L P{GT (ξT (s)) ∈ B} ds ≤ CL ψ (λ(B)) , 0

where λ(B) is the Lebesgue measure of the set B, ψ (|x|) is a certain bounded function satisfying the assumption ψ (|x|) → 0, as |x| → 0. Proof Consider the function x ΦT (x) = 2 0

⎞ ⎛ u  χ (v)) (G B T dv ⎠ du. fT (u) ⎝ fT (v) 0

The function ΦT (x) is continuous, the derivative ΦT (x) of this function is continuous and the second derivative ΦT (x) exists a.e. with respect to the Lebesgue measure and is locally bounded. Therefore, we can apply the Itô formula to the process ΦT (ξT (t)), where ξT is the solution of Eq. (6.1). Furthermore, ΦT (x) aT (x) +

1 Φ (x) = χB (GT (x)), 2 T

6.6 Auxiliary Results

193

a.e. with respect to the Lebesgue measure. Using the Itô formula and the latter equality, we conclude that L

L χB (ζT (s)) ds = ΦT (ξT (L)) − ΦT (x0 ) −

0

ΦT (ξT (s)) dWT (s) − αT (L)

0

with probability 1 for all t ≥ 0, where ζT (t) = GT (ξT (t)), t αT (t) =

ΦT (ξT (s)) [aT (s, ξT (s)) −  aT (ξT (s))] ds.

0

Hence, using the properties of stochastic integrals, we obtain that L P{ζT (s) ∈ B} ds = E [ΦT (ξT (L)) − ΦT (x0 )] − EαT (L).

(6.13)

0

According to condition (A2 ), inequalities |GT (x)| ≥ C|x|α , C > 0, α > 0 and (5.9), we have that |E [ΦT (ξT (L)) − ΦT (x0 )]| ≤ CL(1) ψ (λ(B)) (1)

for a certain constant CL . Condition (A0 ) implies that |EαT (L)| ≤ CL(2) ψ (λ(B)) for a certain constant CL(2) . Here the function ψ (λ(B)) is from condition (A2 ). The latter inequalities and equality (6.13) prove Lemma 6.1.   Lemma 6.2 Let ξT be a solution of Eq. (6.1) belonging to the class K (GT ). If, for measurable locally bounded functions qT , condition (A3 ) holds, then, for any L > 0, ! t ! ! ! ! ! P ! sup ! qT (ξT (s)) ds !! → 0, 0≤t ≤L ! ! 0

as T → +∞.

194

6 Asymptotic Behavior of Homogeneous Additive Functionals of the Solutions to. . .

Proof Consider the function x ΦT (x) = 2 0

⎞ ⎛ u  q (v) T dv ⎠ du. fT (u) ⎝ fT (v) 0

The function ΦT (x) and the derivative ΦT (x) of this function are continuous, and the second derivative ΦT (x) exists a.e. with respect to the Lebesgue measure and is locally bounded. Therefore, we can apply the Itô formula to the process ΦT (ξT (t)), where ξT is the solution to Eq. (6.1). Furthermore, ΦT (x) aT (x) +

1 Φ (x) = qT (x) 2 T

a.e. with respect to the Lebesgue measure. Using the latter equality, we conclude that with probability 1, for all t ≥ 0, t

t qT (ξT (s)) ds = ΦT (ξT (t)) − ΦT (x0 ) −

0

ΦT (ξT (s)) dWT (s) − αT (t),

0

(6.14) where t αT (t) =

ΦT (ξT (s)) [aT (s, ξT (s)) −  aT (ξT (s))] ds.

0

For any constants ε > 0, N > 0 and L > 0, we have +

, sup |αT (t)| > ε ≤ PNT

P

0≤t ≤L

! L ! x ! ! ! qT (v) !! 4 dv + sup fT (x) !! aT (x)] ds, [aT (s, x) −  ! sup ε |x|≤N x ! fT (v) ! 0

+ where PNT = P

0

, sup |ξT (t)| > N .

0≤t ≤L

6.6 Auxiliary Results

195

The same arguments as used in the proof of Lemma 5.2 and the assumptions of Lemma 6.2 yield that P

sup |αT (t)| → 0,

0≤t ≤L

P

sup |ΦT (ξT (t)) − ΦT (x0 )| → 0,

0≤t ≤L

! t ! ! ! ! ! P sup !! ΦT (ξT (s)) dWT (s)!! → 0, 0≤t ≤L ! ! 0

as T → +∞. Thus, the equality (6.14) implies the statement of Lemma 6.2.

 

Appendix A

Selected Facts and Auxiliary Results

Let us consider main definitions and some auxiliary results that are used throughout the book. Section A.1 contains well-known facts from the theory of stochastic processes, construction of stochastic integrals, the generalized Itô formula, some classical results for SDEs, Skorokhod’s representation theorem, and Prokhorov’s theorem. For the proofs, we recommend [20, 59, 65], and some respective references are given in the text. Section A.2 contains more specific results, including convergence results for stochastic integrals and solutions of SDEs. They are given with proofs. Section A.3 is devoted to the detailed description of the Brownian motion in the bilayer environment, and Sect. A.4 contains a brief description of the regularly varying functions.

A.1 Selected Definitions and Facts for Stochastic Processes and Stochastic Integration A.1.1 Basic Facts Regarding Stochastic Processes Let (Ω, F, P) be a probability space. Here Ω is a sample space, i.e., a collection of all possible outcomes or results of the experiment, and F is a σ -field; in other words, (Ω, F) is a measurable space, and P is a probability measure on F. Let (S, Σ) be another measurable space with σ -field Σ, and let us consider the functions defined on the space (Ω, F) and taking their values in (S, Σ). Recall the notion of random variable. Definition A.1 A random variable on the probability space (Ω, F) with the values ξ

in the measurable space (S, Σ) is a measurable map Ω → S, i.e., a map for which

© Springer Nature Switzerland AG 2020 G. Kulinich et al., Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations, Bocconi & Springer Series 9, https://doi.org/10.1007/978-3-030-41291-3

197

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the following condition holds: the pre-image ξ −1 (B) of any set B ∈ Σ belongs to F. Definition A.2 Stochastic process on the probability space (Ω, F, P), parameterized by the set T and taking values in the measurable space (S, Σ), is a set of random variables of the form Xt = {Xt (ω), t ∈ T, ω ∈ Ω}, where Xt (ω) : T × Ω → S. Thus, each parameter value t ∈ T is associated with the random variable Xt taking its value in S. Here are the other common designations of stochastic processes: X(t), ξ(t), ξt , X = {Xt , t ∈ T}. If S = R, then the process is called real or real valued. Additionally, we assume in this case that Σ = B(R), i.e., (S, Σ) = (R, B(R)), where B(S) is a Borel σ -field on S. Concerning the parameter set T, as a rule, it is interpreted as a time set. If the time parameter is continuous, then usually either T = [a, b], or [a, +∞), or R+ . If the time parameter is discrete, then usually either T = N = 1, 2, 3, . . ., or T = Z+ = N ∪ 0, or T = Z. We consider the real-valued parameter T ⊂ R+ , so that we can regard the parameter as time, as described above. A stochastic process X = {Xt (ω), t ∈ T, ω ∈ Ω} is a function of two variables, one of them being a time variable t ∈ T and the other one is a sample point (elementary event) ω ∈ Ω. As mentioned earlier, fixing t ∈ T, we get a random variable Xt (·). In contrast, fixing ω ∈ Ω and following the values that X· (ω) takes as the function of parameter t ∈ T, we get a trajectory (path, sample path) of stochastic process. The trajectory is a function of t ∈ T and, for any t, it takes its value in S. Changing the value of ω, we get a set of paths, or trajectories, of stochastic process. Assume that the process is real valued and T = [a, b] or R+ . If its trajectories are a.s. continuous functions, then X is called continuous stochastic process. If its trajectories are a.s. nondecreasing (nonincreasing) functions, then X is called nondecreasing (nonincreasing) stochastic process. The σ -algebra generated by a stochastic process X is the smallest σ -algebra containing all the sets of the form {ω ∈ Ω : X(t1 , ω) ∈ A1 , . . . , X(tk , ω) ∈ Ak }, Ai ∈ Σ, ti ∈ T, 1 ≤ i ≤ k, k ≥ 1.

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A.1.1.1 Wiener Process Definition A.3 A stochastic process X = {Xt , t ≥ 0} is called a process with independent increments, if for any set of points 0 ≤ t1 < t2 < . . . < tn , the random variables Xt1 , Xt2 − Xt1 , . . . , Xtn − Xtn−1 are mutually independent. Definition A.4 A real-valued stochastic process W = {Wt , t ≥ 0} is called a (standard) Wiener process if it satisfies the following three conditions: 1. W0 = 0. 2. The process W has independent increments. 3. Increments Wt − Ws for any 0 ≤ s < t have the Gaussian distribution with zero mean and variance t − s. In other words, Wt − Ws ∼ N(0, t − s). Remark A.1 The Wiener process is often called Brownian motion.

A.1.1.2 Gaussian Processes Let X = {Xt , t ∈ T} be a real-valued stochastic process. Definition A.5 Stochastic process X is Gaussian if all its finite-dimensional distributions are Gaussian, i.e., for any m ≥ 1 and any t1 , . . . , tm ∈ T random vector (Xt1 , . . . , Xtm ) is Gaussian. For any Gaussian process, there exists function {a(t), t ∈ T} and function of two variables {R(t, s), (t, s) ∈ T × T} such that for any m ≥ 1, any ti ∈ T, 1 ≤ i ≤ m and any λ1 , . . . , λm ∈ R ⎧ ⎫ ⎫ ⎧ m m m ⎨ 1 ⎬ ⎬ ⎨ 1 1 1 E exp i λj Xtj = exp i λj a(tj ) − R(tj , tk )λj λk . ⎩ ⎭ ⎭ ⎩ 2 j =1

j =1

j,k=1

Function R has the properties: 1. R(t, s) = R(s, t), (s, t) ∈ T × T; 2. for any m ≥ 1, any t1 , . . . , tm ∈ T and any b1 , . . . , bm ∈ R m 1

R(tj , tk )bj bk ≥ 0.

j,k=1

A.1.1.3 Wiener Process as an Example of a Gaussian Process Consider a Wiener process W = {Wt , t ≥ 0} satisfying Definition A.4. Recall that for any t ≥ 0 EWt = 0 and the covariance function equals Cov(Ws , Wt ) = E Ws Wt = s ∧ t.

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The function R(s, t) = s ∧ t is nonnegative definite, as any covariance function. With this in mind, consider another definition of a Wiener process. Definition A.6 Stochastic process W = {Wt , t ≥ 0} is a Wiener process if it satisfies three assumptions: 1. W is a Gaussian process; 2. E Wt = 0, for any t ≥ 0; 3. Cov(Ws , Wt ) = s ∧ t, s, t ≥ 0. Definitions A.4 and A.6 of the Wiener process are equivalent.

A.1.2 Notion of Stochastic Basis with Filtration Consider a probability space (Ω, F, P). Let a family {Ft , t ≥ 0} of σ -fields satisfy the following assumptions that are often called “standard assumptions” or “usual conditions.” (i) For any 0 ≤ s < t Fs ⊂ Ft ⊂ F. (ii) For any t ≥ 0 Ft =

;

Fs (continuity “from the right”).

s>t

(iii) F0 contains all the sets from F of zero P-measure. Definition A.7 The family {Ft , t ≥ 0} satisfying assumptions (i)–(iii) is called a flow of σ -fields, or a filtration. Remark A.2 The notion of filtration reflects the fact that information is increasing in time: the more time passed, the more events we could observe, and the richer is corresponding σ -field. Continuity “from the right” means that each σ -field Ft is sufficiently rich to contain all “future sprouts,” and condition (iii) means the completeness of all σ -fields. Sometimes the collection (Ω, F, {Ft }t ≥0 , P) is called a stochastic basis with filtration. Definition A.8 Stochastic process X = {Xt , t ≥ 0} is said to be adapted to the filtration {Ft }t ≥0 or simply X is F-adapted, if for any t ≥ 0 Xt is Ft -measurable. If we write {Xt , Ft , t ≥ 0}, then it means that X is F-adapted.

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Remark A.3 Adaptedness of stochastic process means that for any moment of time the values of the process “agree” with the information available at this moment of time. Remark A.4 Let X = {Xt , t ≥ 0} be a real stochastic process. We can define σ algebra FX t generated by the process X restricted to the interval [0, t]: it is the smallest σ -algebra containing the sets {ω ∈ Ω : X(t1 , ω) ∈ A1 , . . . , X(tk , ω) ∈ Ak }, Ai ⊂ R, Ai- ∈ .B(R), ti ≤ t, 1 ≤ i ≤ k. We denote it FX t = σ {Xs , s ≤ t} and say that FX is a natural filtration generated by process X. Any stochastic t t ≥0 process is adapted to its natural filtration. Moreover, if X is adapted to {Ft }t ≥0, then FX t ⊂ Ft for t ≥ 0.

A.1.3 Notion of (Sub-, Super-) Martingale. Elementary Properties. Square-Integrable Martingales. Quadratic Variations and Quadratic Characteristics   Let Ω, F, {Ft }t ∈R+ , P be a stochastic basis with filtration. . Definition A.9 A stochastic process Xt , t ∈ R+ is said to be a martingale w.r.t. a filtration {Ft }t ∈R+ if it satisfies the following three conditions: (i) For any t ∈ R+ the random variable Xt ∈ L1 (Ω, F, P) (it means that the process X is integrable on R+ ). (ii) For any t ∈ R+ Xt is Ft -measurable, so the process X is {Ft }t ∈R+ -adapted. (iii) For any s, t ∈ R+ such that s ≤ t it holds that E(Xt |Fs ) = Xs P-a.s. If we change in condition (iii) the sign = for ≥ and obtain E(Xt |Fs ) ≥ Xs P-a.s. for any s ≤ t, we get the definition of a submartingale; if E(Xt |Fs ) ≤ Xs P-a.s. for any s ≤ t, s, t ∈ T, then we have a supermartingale. A vector process is called (sub-, super-) martingale if the corresponding property has each of its components. Evidently, any martingale is a (sub-, super) martingale. If X is a submartingale, then −X is a supermartingale and vice versa. Lemma A.1 (1) Each (sub-, super-) martingale has the same property w.r.t. its natural filtration. (2) Property (iii) is equivalent to the following one: for any s ≤ t, s, t ∈ R+ E(Xt − Xs |Fs ) = 0 (≥ 0, ≤ 0 for (sub-, super) martingales). Example A.1 Let {Xt , Ft , t ≥ 0} be an integrable process with independent increments, EXt = at . Then E(Xt − Xs |Fs ) = EXt − EXs = at − as .

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Therefore, X is a martingale if at = a, i.e., is the same for any t ≥ 0, and X is a sub- (super-) martingale if at is increasing (decreasing) in t. In particular, Wiener process W is a martingale w.r.t. a natural filtration. Definition A.10 Process X is called square-integrable, if for any t ≥ 0 EXt2 < ∞. In particular, martingale X is called square-integrable martingale if for any t ≥ 0 EXt2 < ∞. Now, let T > 0 and πn ([0, T ]) = {0 = t0n < t1n < · · · tknn = T } be any sequence of partitions of [0, T ]. Definition A.11 Let X be a stochastic process. (i) If for any T > 0 and any sequence πn ([0, T ]) such that n | → 0, n → ∞ |πn ([0, T ])| = max |tkn − tk−1 1≤k≤kn

there exists an a.s. limit [X]T = lim

n→∞

kn 1 2 n ) , (Xtkn − Xtk−1 k=1

then this limit is called a quadratic variation and we say that X has a quadratic variation. Process [X] is nondecreasing. (ii) If for any T > 0 and any sequence πn ([0, T ]) such that |πn ([0, T ])| → 0, n → ∞ there exists an a.s. limit XT = lim

n→∞

kn   1 2 n ) |Ft n , E (Xtkn − Xtk−1 k−1 k=1

then this limit is called a quadratic characteristic and we say that X has a quadratic characteristic. Process X is nondecreasing. Theorem A.1 Let X be a continuous square-integrable martingale w.r.t. a filtration {Ft }t ∈R+ . Then it has both quadratic variation [X] and quadratic characteristics X, they coincide, [X] = X a.s., they are continuous processes, and X2 − X is a continuous martingale.

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203

A.1.4 Markov Moments and Stopping Times   Let Ω, F, {Ft }t ∈R+ , P be a stochastic basis with filtration. Definition A.12 (1) Random variable τ = τ (ω) : Ω → R+ ∪ {+∞} is called Markov moment if for any t ∈ R+ the event {ω : τ (ω) ≤ t} ∈ Ft . (2) Markov moment τ = τ (ω) is called a stopping time if τ < ∞ a.s. (3) The σ -algebra generated by the Markov moment τ is the class of events . Fτ = A ∈ F : A ∩ {τ ≤ t} ∈ Ft , t ∈ R+ . Theorem A.2 (1) In the case T = R+ , a random variable τ : Ω → [0, +∞] is a Markov moment if and only if, for any t ∈ R+ , {τ < t} ∈ Ft . (2) If τ is a Markov moment, then for any nondecreasing f : R+ ∪ {+∞} → R+ ∪ {+∞} such that f (t) ≥ t for any t ∈ R+ , f (τ ) is a Markov moment. (3) Let σ and τ be Markov moments. Then σ +τ , σ ∧τ , σ ∨τ are Markov moments. ∞  (4) Let {τk , k ≥ 1} be the Markov moments. Then τk , sup τk , inf τk , lim sup τk , k=1

k≥1

k≥1

k→∞

lim inf τk are Markov moments. k→∞

Theorem A.3 (1) Let τ be a Markov moment and let the collection of sets Fτ be defined according to Definition A.12, (3). Then Fτ is indeed a σ -algebra and τ is Fτ -measurable random variable. (2) If T = R+ , then A ∈ Fτ if and only if for any t ∈ T, A ∩ {τ < t} ∈ Ft . (3) Let σ ≤ τ be two Markov moments. Then Fσ ⊂ Fτ . (4) For any two Markov moments σ and τ , Fσ ∧τ = Fσ ∩ Fτ . < (5) For any sequence of stopping times {τn , n ≥ 1}, Finfn≥1 τn = n≥1 Fτn . (6) Let σ and τ be two Markov moments. Then the events {σ = τ }, {σ ≤ τ }, and {σ < - τ } belong to F.σ ∧τ . (7) Let Xt , Ft , t ∈ R+ be an adapted right-continuous stochastic process and τ be a Markov moment. Then Xτ is Fτ -measurable. Theorem A.4 ([17, Chapter 1, § 1, Theorem 1]) Let ξ be a process defined and continuous with probability 1 for t ≥ 0, ξ(0) = 0. Let the σ -algebras Ft be defined for all t ≥ 0 and Ft1 ⊂ Ft2 for t1 < t2 . If (1) ξ(t) is Ft -measurable for all t ≥ 0; (2) E [ξ(t + h) − ξ(t)] |Ft = 0 with probability 1 for all t ≥ 0 and h > 0; (3) E [ξ(t + h) − ξ(t)]2 |Ft = h with probability 1 for all t ≥ 0 and h > 0, then ξ is a Wiener process.

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A.1.5 Construction of Stochastic Integral w.r.t. a Wiener Process and a Square-Integrable Continuous Martingale Definition A.13 A real-valued stochastic process {X(t), t ∈ R+ } is called progressively measurable if for any t > 0 and Borel set B {(s, ω) ∈ [0, t] × Ω : X(s, ω) ∈ B} ∈ Ft ⊗ B([0, t]), where B([0, t]) is the Borel σ -algebra on [0, t]. Now, for a, b ∈ R+ , a < b, we introduce the class H2 ([a, b]) of real-valued processes {ξ(t), t ∈ [a, b]} such that • ξ is progressively measurable; b • ||ξ ||2H2 ([a,b]) := a Eξ(t)2 dt < ∞. Let us first consider simple processes of the form ξ(t) =

n 1

(A.1)

αk χ[ak , bk )(t),

k=1

where n ≥ 1 is an integer, a ≤ ak < bk ≤ b are some real numbers, and αk is an Fak -measurable square-integrable random variable. Clearly, ξ ∈ H2 ([a, b]). Define Itô integral, or stochastic integral, of ξ with respect to W as 

b a

ξ(t)dW (t) =

n 1

  αk W (bk ) − W (ak ) .

k=1

For notation simplicity, we will also denote 

b

I (ξ, W, [a, b]) = I (ξ, [a, b]) =

ξ(t)dW (t). a

Further, we establish several properties of the Itô integral. Theorem A.5 Let ξ, ζ be simple processes in H2 ([a, b]). Then the following properties are true. I (ξ + ζ, [a, b]) = I (ξ, [a, b]) + I (ζ, [a, b]). For any c ∈ R I (cξ, [a, b]) = cI (ξ, [a, b]). For any c ∈ (a, b) I (ξ, [a, b]) = I (ξ, [a, c]) + I (ξ, [c, b]). EI (ξ, [a, b]) = 0. Moreover, {I (ξ, [a, t]), t ∈ [a, b]} is a martingale. b EI (ξ, [a, b])2 = ||ξ ||2H2 ([a,b]) = a Eξ(t)2 dt.   b 6. E I (ξ, [a, b])I (ζ, [a, b]) | Fa = a E(ξ(t)ζ (t) | Fa )dt, in particular: 1. 2. 3. 4. 5.

  ξ, ζ H2 ([a,b]) := E I (ξ, [a, b])I (ζ, [a, b]) =



b

E(ξ(t)ζ(t))dt. a

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205

Lemma A.2 Let ξ ∈ H2 ([a, b]). Then, there exists a sequence {ξn , n ≥ 1} of simple processes such that ||ξ − ξn ||H2 ([a,b]) → 0, n → ∞. With this at hand, the extension is done in a standard manner. Namely, if {ξn , n ≥ 1} is a sequence of simple processes converging in H2 ([a, b]) to ξ ∈ H2 ([a, b]), then, due to the isometry property, the sequence {I (ξn , [a, b]), n ≥ 1} is a Cauchy sequence in L2 (Ω). Then it has a limit in L2 (Ω), which justifies the following definition. Definition A.14 For ξ ∈ H2 ([a, b]), Itô integral of ξ with respect to Wiener process is the limit 

b

I (ξ, [a, b]) =

ξ(t)dW (t) = lim I (ξn , [a, b]). n→∞

a

(A.2)

in L 2 (Ω), where {ξn , n ≥ 1} is a sequence of simple processes in H2 ([a, b]) such that ||ξ − ξn ||H2 ([a,b]) → 0, n → ∞. The properties of Itô integral defined by (A.3) are essentially the same as for simple functions. For completeness, we give them in full. Theorem A.6 Let ξ, ζ ∈ H2 ([a, b]). 1. 2. 3. 4. 5.

I (ξ + ζ, [a, b]) = I (ξ, [a, b]) + I (ζ, [a, b]) almost surely; For any c ∈ R I (cξ, [a, b]) = cI (ξ, [a, b]) almost surely; For any c ∈ (a, b) I (ξ, [a, b]) = I (ξ, [a, c]) + I (ξ, [c, b]) almost surely; EI (ξ, [a, b]) = 0. Moreover, {I (ξ, [a, t]), t ∈ [a, b]} is a martingale; b EI (ξ, [a, b])2 = ||ξ ||2H2 ([a,b]) = a Eξ(t)2 dt (Itô isometry). Moreover, the process  M(t) = I (ξ, [a, t])2 −

t

ξ(s)2 ds, t ∈ [a, b],

a

is a martingale; b   6. E I (ξ, [a, b])I (ζ, [a, b]) = ξ, ζ H2 ([a,b]) = a Eξ(t)ζ (t)dt. Theorem A.7 ([17, Chapter 1, § 3, Theorem 2]) Let f ∈ H2 ([a, b]). Then the t separable process f (s) dW (s), t ∈ [a, b] is continuous with probability 1, and a

for arbitrary constants C > 0 and N > 0 ⎫ ⎧ b ! t ! ⎫ ! ! ⎬ ⎨ ⎬ ! ! N P sup !! f (s) dW (s)!! > C ≤ P f 2 (t) dt > N + 2 . ⎭ ⎩ ⎩a≤t ≤b ! ⎭ C ! ⎧ ⎨

a

a

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Theorem A.8 ([17, Chapter 1, § 4, Theorem 3]) Let the process f (t) be defined for t ≥ 0 and for all L > 0 f ∈ H2 [0, L]. Assume that +∞ f 2 (t) dt = +∞ 0

with probability 1. Let s f 2 (u) du ≥ t}.

τt = inf{s : 0

Then the process τt ζt =

f (s) dW (s) 0

is a Wiener process. Remark A.5 The notion of stochastic integral w.r.t. a Wiener process can be, without great difficulties, extended to the notion of stochastic integral w.r.t. a continuous square-integrable martingale η, because the increments of any square-integrable martingale are orthogonal in the sense that for any 0 ≤ t1 < t2 ≤ t3 < t4 , E(ηt2 − ηt1 )(ηt4 − ηt3 ) = 0, and only this property (and not the independence of increments) is used in the construction of the stochastic integral. This construction follows the same steps, from simple functions to the functions from the class H2 ([a, b], η) of real-valued processes {ξ(t), t ∈ [a, b]} such that • ξ is progressively measurable w.r.t. the filtration generated by η; b • ||ξ ||2H ([a,b],η) := a Eξ(t)2 dηt < ∞. 2 Recall that η exists, according to Theorem A.1. Denote, for simple function ξ(t) =  n k=1 αk χ[ak , bk )(t) from H2 ([a, b], η), integral I (ξ, [a, b], η) =

n 1

  αk η(bk ) − η(ak ) .

k=1

As before, for any stochastic function from H2 ([a, b], η) there exists a sequence of simple functions from H2 ([a, b], η) such that ||ξ − ξn ||H2 ([a,b],η) → 0, n → ∞.

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Definition A.15 For ξ ∈ H2 ([a, b], η), Itô integral of ξ with respect to continuous square-integrable martingale η is the limit 

b

I (ξ, [a, b], η) =

ξ(t)dη(t) = lim I (ξn , [a, b], η) n→∞

a

(A.3)

in L 2 (Ω), where {ξn , n ≥ 1} is a sequence of simple processes in H2 ([a, b]) such that ||ξ − ξn ||H2 ([a,b],η) → 0, n → ∞. Now, Theorem A.6 can be reformulated in the following way. Theorem A.9 Let ξ, ζ ∈ H2 ([a, b], η). Then 1. 2. 3. 4. 5.

I (ξ + ζ, [a, b], η) = I (ξ, [a, b], η) + I (ζ, [a, b], η) almost surely; For any c ∈ R I (cξ, [a, b], η) = cI (ξ, [a, b], η) almost surely; For any c ∈ (a, b) I (ξ, [a, b], η) = I (ξ, [a, c], η)+I (ξ, [c, b], η) almost surely; EI (ξ, [a, b], η) = 0. Moreover, {I (ξ, [a, t], η), t ∈ [a, b]} is a martingale; b EI (ξ, [a, b])2 = ||ξ ||2H2 ([a,b],η) = a Eη(t)2 dηt . Moreover, the process 

t

M(t) = I (ξ, [a, t], η)2 −

ξ(s)2 dηs , t ∈ [a, b],

a

is a martingale; b   6. E I (ξ, [a, b], η)I (ζ, [a, b], η) = ξ, ζ H2 ([a,b],η) = a Eξ(t)ζ (t)dηt .

A.1.6 Generalized Itô Formula For the definition and some properties of solutions of the Itô equations see Sect. 2.1, in particular, Definition 2.1. Lemma A.3 Let ξ be a solution of the Itô equation t ξ (t) = ξ (0) +

t a (ξ (s)) ds +

0

σ (ξ (s)) dW (s) ,

(A.4)

0

where ξ (0) is a random variable not depending on W and let a (x) , σ (x), x ∈ R, be real-valued measurable functions satisfying the following assumptions: (a) there exists a constant L > 0 such that for all x ∈ R   a 2 (x) + σ 2 (x) ≤ L 1 + |x|2 , (b) for any constant N > 0 and for |x| ≤ N we have that σ (x) ≥ δN > 0.

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Assume that a function Φ (x) has continuous derivative Φ (x), and the second derivative Φ (x) is assumed to exist a.e. with respect to the Lebesgue measure and to be locally integrable. Then with probability 1, for all t ≥ 0, the following equality holds (Itô’s formula for the process Φ (ξ (t))): t  1 Φ (ξ (s)) a (ξ (s)) + Φ (ξ (s)) σ 2 (ξ (s)) ds Φ (ξ (t)) = Φ (ξ (0)) + 2 0

t +

Φ (ξ (s)) σ (ξ (s)) dW (s) .

0

This formula follows from the more general result of M.V. Krylov [28, Theorem 4]. Consider the homogeneous SDE dξ(t) = a (ξ(t)) dt + σ (ξ(t)) dW (t),

(A.5)

where a(x) : R → R and σ (x) : R → R are measurable functions. Assume that 1 are locally bounded. Let f (x) be defined in (A.9) the functions a(x), σ (x), and σ (x) and ϕ(x) = f −1 (x),

 σ (x) = f (ϕ(x)) σ (ϕ(x)) ,

⎫ ⎧ −∞ ⎬ ⎨ z a(y) dz = f (−∞), r1 = dy exp −2 ⎩ σ 2 (y) ⎭ 0

0

⎫ ⎧ +∞ ⎬ ⎨ z a(y) r2 = dy dz = f (+∞), exp −2 ⎩ σ 2 (y) ⎭ 0

0

0 > r1 ≥ −∞,

0 < r2 ≤ +∞.

Lemma A.4 ([80, Chapter I, § 3, Lemma 9]) Let ξ be a solution of Eq. (A.5) on the interval [0, τ ), where τ is some Markov moment. Then the process ζ(t) = f (ξ(t)) is a solution of the equation dζ (t) =  σ (ζ (t)) dW (t)

(A.6)

on [0, τ ) and ζ ∈ (r1 , r2 ) for s < τ . Theorem A.10 ([80, Chapter I, § 3, Theorem 16]) Let f (−∞) = −∞, f (+∞) = +∞, then the process ζ (t) = f (ξ(t)) is a solution of Eq. (A.6).

A Selected Facts and Auxiliary Results

If additionally distribution

 R

209

 σ −2 (z) dz < ∞, then the process ζ is ergodic with ergodic

x F (x) = k

 σ −2 (z) dz,

−∞

⎛ ⎞−1  k=⎝  σ −2 (z) dz⎠ . R

Theorem A.11 (S. Nakao [67]) Let a(x) and σ (x) be bounded Borel functions. Suppose σ (x) is of bounded variation on any compact interval. Further, suppose that there exists a constant δ > 0 such that σ (x) ≥ δ for x ∈ R. Then, the pathwise uniqueness holds for (A.5). Let Eq. (A.5) have unique solution for any initial conditions ξ(0) and σ (x) > 0 for all x. Lemma A.5 ([17, Chapter 1, § 16, Remark 1]) Let f (−∞) = −∞ and f (x) ≤ C for all x. Then  P

     lim ξ (t) = +∞ = P sup ξ(t) = +∞ = P inf ξ(t) > −∞ = 1.

t →+∞

t >0

t >0

Lemma A.6 ([17, Chapter 1, § 16, Lemma 2]) Let f (+∞) = +∞ and f (x) ≥ C for all x. Then       P sup ξ(t) < +∞ = P inf ξ(t) = −∞ = P lim ξ (t) = −∞ = 1. t →+∞

t >0

t >0

A.1.7 Skorokhod’s Representation Theorem and Prokhorov’s Theorem The following two lemmas contain the well-known result of A.V. Skorokhod, namely Skorokhod’s convergent subsequence principle (see [79, Chapter I, § 6]) and the well-known result of Y.V. Prokhorov [73]. Theorem A.12 (Skorokhod’s Representation Theorem) Let the d-dimensional random processes ξn (t), t ≥ 0, n ≥ 1 be defined on a certain probability space (Ω, F, P). Assume that for any L > 0 and ε > 0 lim lim sup sup P {|ξn (t)| > N} = 0,

N→+∞ n→+∞ 0≤t ≤L

lim lim sup

sup

h→0 n→+∞ |t1 −t2 |≤h, ti ≤L

P {|ξn (t1 ) − ξn (t2 )| > ε} = 0.

210

A

Selected Facts and Auxiliary Results

  * * Then we can choose a subsequence nk → +∞, a probability space Ω, F, * P , ξ (t) defined on this space such that the and stochastic processes * ξnk (t) and * finite-dimensional distributions of the processes * ξnk (t) coincide with the finite* P

dimensional distributions of the processes ξnk (t) and, moreover, * ξnk (t) −→ * ξ (i) (t), i = 1, d, as nk → +∞, for all t ≥ 0. (i)

In what follows the processes * ξnk and ξnk , satisfying assumptions of Theorem A.12, are called weakly equivalent. Theorem A.13 (Prokhorov’s Theorem) Let the stochastic processes ζn , n ≥ 1 and ζ be continuous with probability 1 on the interval [a, b]. Let the finitedimensional distributions of the processes ζn converge, as n → +∞, to the corresponding finite-dimensional distributions of the process ζ . The stochastic processes ζn weakly converge in the uniform topology of the space of continuous functions, as n → +∞, to the process ζ on the interval [a, b] if and only if for any ε > 0: + lim sup P

h→0 n

, sup

|t1 −t2 |≤h, a≤ti ≤b

|ζn (t1 ) − ζn (t2 )| > ε = 0.

A.2 Convergence of Stochastic Integrals and Some Properties of Solutions of SDEs Consider some conditions of the convergence of stochastic integrals w.r.t. squareintegrable continuous martingales. Lemma A.7 Let {ηn , n ≥ 1} be a sequence of square-integrable continuous martingales with respect to the σ -algebras σ {ηn (s), s ≤ t}, gn be real-valued meaL surable locally bounded functions, and the stochastic integrals gn (ξn (s))dηn (s) 0

are well defined on the probability space (Ω, F, P) for each n ≥ 0. Assume that gn converges, as n → +∞, to the function g0 , almost everywhere (a.e.) with respect to the Lebesgue measure. If (1)

lim lim sup P{ sup |ξn (t)| > N} = 0;

N→+∞ n→+∞

(2) lim lim sup (3)

0≤t ≤L

sup

{P|ξn (t2 ) − ξn (t1 )| > ε} = 0 for every ε > 0;

h→0 n→+∞ |t2 −t1 |≤h P ξn (t) −→ ξ0 (t), ηn (t)

P

−→ η0 (t), as n → +∞, where η0 is a square-integrable continuous martingale with respect to the σ -algebra σ {η0 (s), s ≤ t}, and for the quadratic characteristics ηn  we have the inequality | ηn  (t2 ) − ηn  (t1 )| ≤ C|t2 − t1 | + |λn (t2 ) − λn (t1 )|

A Selected Facts and Auxiliary Results

211

L

L

0

0

with E V λn → 0, as n → +∞, where V λn is the variation of the (possibly, random) functions λn on an interval [0, L]; L P{ξn (s) ∈ A} ds ≤ Cλ(A), where A an arbitrary bounded (4) lim sup n→+∞

0

measurable set, C = C(L), λ(A) is the Lebesgue measure of the set A. Then T

P

T

gn (ξn (s)) dηn (s) −→ 0

g0 (ξ0 (s)) dη0 (s),

(A.7)

0

as n → +∞. Proof For an arbitrary constants N > 0 and δ > 0 there exists a subset Aδ ⊂ [−N, N] with λ(Aδ ) < δ such that according to Luzin’s theorem (see [78]), there exists a continuous function * g (x), x ∈ [−N, N] that coincides with g0 (x) for x ∈ / (Aδ ). According to Egorov’s theorem (see [78]), the pointwise convergence almost everywhere on [−N, N] of the sequence of functions gn (x) to the function g(x) implies uniform convergence to the function * g (x) everywhere except on the set Aδ ⊂ [−N, N]. Let qN (x) be a continuous nonnegative function that equals to 1, for |x| ≤ N, and equals to 0, for |x| > N + 1. Then L

L gn (ξn (s)) dηn (s) =

gn (ξn (s))qN (ξn (s))χAcδ (ξn (s)) dηn (s) + αN + αN,δ ,

0

0

where L αN =

gn (ξn (s)) [1 − qN (ξn (s))] dηn (s), 0

L αN,δ =

gn (ξn (s))qN (ξn (s))χAδ (ξn (s)) dηn (s), 0

χA (x) is the indicator function of a set A, Acδ = [−N, N] \Aδ . Let sup |g(x)| ≤ CN .

|x|≤N

212

A

+

Since P {|αN | > ε} ≤ P

Selected Facts and Auxiliary Results

, sup |ξn (t)| > N

for any ε > 0, and

0≤t ≤L

L gn2 (ξn (s))qN2 (ξn (s))χAδ (ξn (s)) d ηn  (s)

E|αN,δ | ≤ E 2

0

L ≤

L

2 P {ξn (s) ∈ Aδ } ds + CN E V λn ,

2 CN C

0

0

then, according to the conditions of our lemma, we have L

L gn (ξn (s)) dηn (s) =

0

gn (ξn (s))qN (ξn (s))χAcδ (ξn (s)) dηn (s) + o(1), 0

where lim

lim

lim

N→+∞ δ→0 n→+∞

o(1) = 0

in probability. Note that L

L gn (ξn (s))qN (ξn (s))χAcδ (ξn (s)) dηn (s) =

0

* g (ξn (s))qN (ξn (s)) dηn (s) + βn(1) + βn(2) , 0

where L βn(1)

=

* g (ξn (s))qN (ξn (s))χAδ (ξn (s)) dηn (s), 0

and L βn(2)

=

g (ξn (s))] qN (ξn (s))χAcδ (ξn (s)) dηn (s). [gn (ξn (s)) − * 0

A Selected Facts and Auxiliary Results

213

Moreover,  E

βn(1)

L



=E

$ %2 * g (ξn (s))qN (ξn (s)) χAδ (ξn (s)) dηn (s)

0

L ≤

L

2 P {ξn (s) ∈ Aδ } ds + CN CE V λn ,

2 CN

0

0

and  E

βn(2)



L =E

g (ξn (s))]2 qN2 (ξn (s))χAcδ (ξn (s)) dηn (s) [gn (ξn (s)) − * 0

L g (x)|2 E ≤ sup |gn (x) − * x∈Acδ

qN2 (ξn (s))dηn (s). 0

Therefore, using uniform convergence, as n → +∞ of the sequence of functions gn to the function * g on the set Acδ , we obtain L

L gn (ξn (s)) dηn (s) =

0

* g (ξn (s))qN (ξn (s)) dηn (s) + o(1),

(A.8)

0

as n → +∞. Furthermore, ⎧! ! ⎫ ! ! + , ⎪ ⎪ L ⎨!L ⎬ ! ! ! P ! * g (ξn (s))qN (ξn (s)) dηn (s) − * g (ξn (s)) dηn (s)! > 0 ≤ P sup |ξn (t)| > N , ⎪ ! ⎪ ⎩!! ⎭ 0≤t ≤L ! 0 0

and ⎫ ! ⎧! L ! L ⎬ ⎨!! ! g (ξ0 (s))qN (ξ0 (s)) dη0 (s) − * g (ξ0 (s)) dη0 (s)!! > 0 P !! * ⎭ ⎩! ! 0

0

+ ≤P

, sup |ξ0 (t)| > N

0≤t ≤L

≤ lim sup P {|ξn (t)| > N} . n→+∞

214

A

Selected Facts and Auxiliary Results

Therefore, to prove the lemma, it is sufficient to show that for all N > 0 L

L

P

* g (ξn (s))qN (ξn (s)) dηn (s) −→

* g (ξ0 (s))qN (ξ0 (s)) dη0 (s),

0

0

as n → +∞. Let 0 = s0 < s1 < . . . < sm = L be an arbitrary partition of the interval [0, L]. Then L

L * g (ξn (s)) qN (ξn (s)) dηn (s) −

0

* g (ξ0 (s))qN (ξ0 (s)) dη0 (s) 0

=

m−1 1

* g (ξn (sk ))qN (ξn (sk ))[ηn (sk+1 ) − ηn (sk )]

k=0

−

m−1 1

* g (ξ0 (sk ))qN (ξ0 (sk ))[η0 (sk+1 ) − η0 (sk )]

k=0

+

sk+1  [* g (ξn (s))qN (ξn (s)) − * g (ξn (sk ))qN (ξn (sk ))] dηn (s)

m−1 1

k=0 sk

+

sk+1  [* g (ξ0 (s))qN (ξ0 (s)) − * g (ξ0 (sk ))qN (ξ0 (sk ))] dη0 (s)

m−1 1

k=0 sk

= In(1) + In(2) + In(3) + In(4) . P

P

According to the convergence ξn (t) −→ ξ0 (t) and ηn (t) −→ η0 (t), as n → (1)

(2)

P

+∞, we have In + In −→ 0, as n → +∞. It is clear that s m−1 k+1 ! ! ! (3) !2 1 g (ξn (sk ))qN (ξn (sk ))]2 dηn (s) E !In ! = g (ξn (s))qN (ξn (s)) − * [* k=0 sk

≤ CL

sup

|t2 −t1 |≤h

where h =

L

2 E |* g (ξn (t2 ))qN (ξn (t2 )) − * g (ξn (t1 ))qN (ξn (t1 ))|2 + 4CN E V λn ,

max [sk+1 − sk ].

0≤k≤m−1

0

A Selected Facts and Auxiliary Results

215

Since the function * g (x) qN (x) is continuous on the set |x| ≤ N +1, it is uniformly continuous. Therefore, for an arbitrary ε > 0 there exists δ1 > 0 such that sup

|t2 −t1 |≤h

E|* g (ξn (t2 ))qN (ξn (t2 )) − * g (ξn (t1 ))qN (ξn (t1 ))|2

2 ≤ ε2 + 4CN

sup

|t2 −t1 |≤h

P{|ξn (t2 ) − ξn (t1 )| > δ1 }.

Hence, from the previous inequalities we have that lim

In(3) = 0

lim

h→0 n→+∞

in probability. Similarly we obtain that lim

In(4) = 0

lim

h→0 n→+∞

in probability. Thus, we have the convergence L

t

P

* g (ξn (s))qN (ξn (s)) dηn (s) −→ 0

* g (ξ0 (s))qN (ξ0 (s)) dη0 (s), 0

as n → +∞ for any δ > 0 and N > 0. The proof follows from the representation (A.8), combined with the equality * g (x) = g0 (x), as x ∈ / Aδ , and condition (4) from the assumptions.   Corollary A.1 If the processes ξn and ηn satisfy the conditions (1) – (3) of Lemma A.7, and the function g0 is continuous, then we have convergence (A.7). In particular, for gn (x) = x, n ≥ 0, η0 (t) = W (t), where W is a Wiener process, such a convergence was obtained in [79, Chapter II, § 3]. Lemma A.8 Let ξ be a solution of the Itô equation (A.4), ξ (0) = x0 and f (−∞) = −∞, f (+∞) = +∞, where ⎫ ⎧ x ⎬ ⎨ u a(v) dv f (x) = exp −2 du. (A.9) ⎩ σ 2 (v) ⎭ 0

0

Also, let g be a locally square-integrable real-valued function such that g 2 (x) > 0 for x ∈ A with a positive Lebesgue measure λ(A) > 0. Then with probability 1 +∞ g 2 (ξ(s)) ds = +∞. 0

216

A

Selected Facts and Auxiliary Results

Proof We may assume without loss of generality that inf

−1 σ1 : ξ(t) = −1}, . . . , σn = min{t > τn−1 : ξ(t) = 0}, τn = min{t > σn : ξ(t) = −1}, . . . , and τn ζn =

g 2 (ξ(s)) ds. σn

Note that (see [17, Chapter 4, § 16, Lemma 1])     P lim sup ξ(t) = +∞ = P lim inf ξ(t) = −∞ = 1. t →+∞

t →+∞

Taking into account that the process ξ has a strong Markov property, it is easy to see that for all n ≥ 1, σn < τn < +∞ with probability 1. Moreover, ζn is a sequence of independent identically distributed random variables (see [17, Chapter 3, § 15, Corollary 1]). According to the strong law of large numbers, 11 ζi → Eζ1 , n → +∞ n n

i=1

almost surely. Since the expectation of ζ1 exists and is nonnegative, τ1 Eζ1 = E

g 2 (ξ(s)) ds > σ1

then

n 

inf

−1 0. Let ηn (t) −→ η(t) and the quadratic

A Selected Facts and Auxiliary Results

217

P

characteristic ηn  (t) −→ β(t), as n → +∞, for any t > 0, then the limit process η is a martingale with respect to the σ -algebra σ {η(s), s ≤ t} with the quadratic characteristic β(t). Proof Since ηn (t) for each n is a square-integrable martingale, for any 0 ≤ t1 < t2 < . . . < tk < t and arbitrary bounded function g(x1 , x2 , . . . xk ) we have Eηn (t)g (ηn (t1 ), . . . ηn (tk )) = Eηn (tk )g (ηn (t1 ), . . . ηn (tk )) , E [ηn (t) − ηn (tk )]2 g (ηn (t1 ), . . . ηn (tk )) = E (ηn  (t) − ηn  (tk )) g (ηn (t1 ), . . . ηn (tk )) . Taking into account the inequality E (ηn (t))1+δ/2 ≤ Cδ E |ηn (t)|2+δ (see [69]), we pass to the limit, as n → +∞, in the previous equality and obtain Eη(t)g (η(t1 ), ..., η(tk )) = Eη(tk )g (η(t1 ), ..., η(tk )) , E [η(t) − η(tk )]2 g (η(t1 ), ..., η(tk )) = E (β(t) − β(tk )) g (η(t1 ), ..., η(tk )) . Since the function g(x1 , ..., xk ) is arbitrary, we have that η(t) is a martingale with respect to σ {η(s), s ≤ t} with the quadratic characteristic η(t) = β(t).   Lemma A.10 Let the process ζ be the solution of the equation t ζ (t) = x0 +

σ (ζ (s)) dW (s) ,

(A.10)

0

  where σ is real-valued measurable function and such that σ 2 (x) ≤ L 1 + |x|2 for all x ∈ R and σ (x) ≥ δN > 0, for |x| ≤ N, for any N > 0; W is a Wiener processes defined on a complete probability space (Ω, , P). Let x0 ∈ (−N, N), τN = inf {t : ζ (t) ∈ / (−N, N)}, g be a real-valued measurable function for which N |g (x)| dx < +∞. −N

Then the inequality holds t∧τN

E 0

1 |g (ζ (s))| ds ≤ 2 C (x0 , L, t) δN

where C (x0 , L, t) does not depend on N.

N

−N

|g (x)| dx,

218

A

Selected Facts and Auxiliary Results

Proof Consider the function x Q (x) = 2 0

⎛ u ⎞  ⎝ |g (v)| σ −2 (v) dv ⎠ du. 0

According to the Itô formula (see Lemma A.3) we obtain Q (ζ (t ∧ τN )) − Q (x0 ) t∧τN

=

1 Q (ζ (s)) σ (ζ (s)) dW (s) + 2

0

t∧τN

Q (ζ (s)) σ 2 (ζ (s)) ds.

0

Since τN is a Markov moment, taking into account the properties of stochastic integrals [17, §4], we get from the previous equality that t∧τN

|g (ζ (s))| ds = E [Q (ζ (t ∧ τN )) − Q (x0 )]

E 0

1 ≤ 2 δN

1 ≤ 2 δN

1 2 δN

N −N

N |g (x)| dx E |ζ (t ∧ τN ) − x0 | −N

N

⎛ |g (x)| dx ⎝E

−N

t∧τN

⎞ 12 σ 2 (ζ (s)) ds ⎠

0

⎛

⎞ 12 t   |g (x)| dx · L ⎝ 1 + E |ζ (s)|2 ds ⎠ . 0

From this inequality and the well-known bound for E |ζ (s)|2 (see [17, §6]) the statement of Lemma A.10 follows.   Lemma A.11 Let the process ζ be the solution of Eq. (A.10) and let A be a measurable bounded set, A ⊂ (−N, N). Then t P {ζ (s) ∈ A} ds ≤ 0

1 C (x 0 , L, t) λ (A) , 2 δN

where λ (·) is the Lebesgue measure, and C (x0 , L, t) is the same constant as in Lemma A.10.

A Selected Facts and Auxiliary Results

219

Proof Let x Q (x) = 2 0

⎛ u ⎞  ⎝ χA (v) σ −2 (v) dv ⎠ du. 0

According to the Itô formula we have for t > 0 Q (ζ (t)) − Q (x0 ) t =

1 Q (ζ (s)) σ (ζ (s)) dW (s) + 2

0

t

Q (ζ (s)) σ 2 (ζ (s)) ds.

0

From the last equality, in the same way as in the proof of Lemma A.10, we obtain the proof of Lemma A.11.   Corollary A.2 Let the assumptions of Lemma A.11 hold and additionally for all x ∈ R and for some constant C0 let σ (x) ≤ C0 . Then t P {ζ (s) ∈ A} ds ≤ 0

√ C0 t λ (A) . 2 δN

In fact, this statement follows directly from the proof of Lemma A.11, combined with the inequalities ⎛ t ⎞ 12   1 √ 2 E |ζ (t) − x0 | ≤ E |ζ (t) − x0 |2 ≤ ⎝ Eσ 2 (ζ (s)) ds ⎠ ≤ C0 t. 0

Lemma A.12 Let the process ζ be the solution of Eq. (A.10). If for all t > 0 the following equality holds a.s.: t F (ζ (t)) − F (x0 ) =

g (ζ (s)) dζ (s) ,

(A.11)

0

where F is a continuous function, and the function g is locally square-integrable, then for any x = x0 F (x) − F (x0 ) = g (x) = b x − x0 a.e. with respect to the Lebesgue measure, where b is some constant.

220

A

Selected Facts and Auxiliary Results

Proof Let τ = inf {t : ζ (t) ∈ / (x0 − h1 , x0 + h2 )}, hi > 0. It is clear that τ is a Markov moment. As in Theorem 2, §15, [17], we find that Eτ = v (x0 ), where x v (x) = −2 x0 −h1

⎛ ⎜ ⎝

y x0 −h1

⎞ dz ⎟ ⎠ dy + 2 σ 2 (z)

x 0 +h2

x0 −h1

⎛

y

⎜ ⎝

x0 −h1

⎞ x − x0 + h1 dz ⎟ . ⎠ dy σ 2 (z) h1 + h2

The function v (x), for x ∈ (x0 − h1 , x0 + h2 ), satisfies the equation 1 v (x) σ 2 (x) = −1 2 with boundary conditions v (x0 − h1 ) = v (x0 + h2 ) = 0. Thus, τ < +∞ with probability 1. Taking into account (A.10) and (A.11), using the properties of stochastic integrals, we obtain that Eζ (τ ) = x0 ,

EF (ζ (τ )) = F (x0 ) .

Therefore P {ζ (τ ) = x0 − h1 } =

h2 h1 + h2

P {ζ (τ ) = x0 + h2 } =

h1 . h1 + h2

and

Since EF (ζ (τ )) = F (x0 − h1 )P {ζ (τ ) = x0 − h1 } + F (x0 + h2 )P {ζ (τ ) = x0 + h2 } , for any hi > 0, we have the equality F (x0 + h2 ) − F (x0 ) F (x0 ) − F (x0 − h1 ) = , h1 h2 from which it follows that F (x) − F (x0 ) = b (x − x0 ), where b is some constant. According to equality (A.11), we obtain that t [g (ζ (s)) − b] dζ (s) = 0 0

(A.12)

A Selected Facts and Auxiliary Results

with probability 1 for all t ≥ 0. Since

221

t

[g (ζ (s)) − b] dζ (s) is continuous with

0

probability 1 and a square-integrable martingale, its quadratic characteristics t [g (ζ (s)) − b]2 σ 2 (ζ (s)) ds 0

with probability 1, for all t ≥ 0, is equal to zero. Consequently, +∞ [g (x) − b]2 σ 2 (x) dx = 0, −∞

because otherwise the process τt [g (ζ (s)) − b] σ (ζ (s)) dW (s) , 0

where τ t is the smallest solution of the equation τt t=

[g (ζ (s)) − b]2 σ 2 (ζ (s)) ds, 0

would be (see [17, §4]) a Wiener process, but according to (A.12), τt [g (ζ (s)) − b] σ (ζ (s)) dW (s) ≡ 0 0

with probability 1. Thus, g (x) = b a.e. with respect to the Lebesgue measure, where b is some constant. Lemma A.12 is proved.   Lemma A.13 Let the process ξ be the solution of Eq. (A.4) and the process (ξ, W ) * ), where the process * be weakly equivalent to the process (* ξ, W ξ is continuous with probability 1. If, for any N > 0 and for any Borel set A ⊂ [−N, N], the following inequality holds: t P {ξ (t) ∈ A} ds ≤ CN λ (A) , 0

(A.13)

222

A

Selected Facts and Auxiliary Results

where λ (·) is the Lebesgue measure, then * ξ (t) = * ξ (0) +

t

  a * ξ (s) ds +

0

t

  * (s) . σ * ξ (s) d W

(A.14)

0

Proof First, assume that the coefficients a (x) and σ (x) of Eq. (A.4) are continuous functions. Then the processes a (ξ (·)) and σ (ξ (·)) are continuous with probability 1 on the interval [0, t]. Consider the partition of the interval [0, t]: 0 = tn0 < tn1 < . . . < tnn = t, λn = max Δtnk , where Δtnk = tnk+1 − tnk . n

Then Sn(1)

=

n−1 1

P

t

a (ξ (tnk )) Δtnk −→

k=0

a (ξ (s)) ds,

(A.15)

0

as λn → 0. In accordance with the properties of stochastic integrals, Sn(2)

=

n−1 1

P

t

σ (ξ (tnk )) [W (tnk+1 ) − W (tnk )] −→

k=0

σ (ξ (s)) dW (s) ,

(A.16)

0

as λn → 0. Therefore, P

αn := ξ (t) − ξ (0) − Sn(1) − Sn(2) −→ 0, as λn → 0. The distribution of the random variables αn coincides with the distribution of * αn , where * αn = * ξ (t) − * ξ (0) − * Sn(1) − * Sn(2) , * Sn(1) =

n−1 1   a * ξ (tnk ) Δtnk , k=0

* Sn(2) =

n−1 1

(A.17)

$ %  * (tnk+1 ) − W * (tnk ) . σ * ξ (tnk ) W

k=0 P

* P

From the convergence αn −→ 0 it follows that * αn −→ 0, as λn → 0. It is clear that the convergences similar to (A.15) and (A.16) hold. Thus, we pass to the limit, as λn → 0, in (A.17) and obtain (A.14).

A Selected Facts and Auxiliary Results

223

Now, let the coefficients a (x) and σ (x) be measurable locally bounded functions. For any N > 0 we have the equality t ξ (t) = ξ (0) +

a (ξ (s)) qN (ξ (s)) ds + 0

t σ (ξ (s)) qN (ξ (s)) dW (s) + βN (t) ,

(A.18)

0

where t

t a (ξ (s)) [1 − qN (ξ (s))] ds +

βN (t) = 0

σ (ξ (s)) [1 − qN (ξ (s))] dW (s) , 0

and qN (x) is the function defined in Lemma A.7. It is clear that we have the inequality ,

+ P {|βN (t)| > 0} ≤ P

sup |ξ (s)| > N . 0≤s≤t

Since the process ξ is continuous with probability 1, the right-hand side of this P

inequality tends to 0, as N → +∞. Thus, βN (t) −→ 0, as N → +∞, for every t > 0. Using Lusin‘s theorem [78], we conclude that, for any ε > 0, there exist (ε) (ε) continuous functions aN (x) and σN (x), x ∈ R, that coincide with a (x) qN (x) (ε) (ε) and σ (x) qN (x), respectively, for x ∈ A / AN , where   N ⊂ [−N − 1, N + 1], and (ε)

its Lebesgue measure satisfies the inequality λ AN < ε. Therefore, taking into account (A.18) and the condition of Lemma A.10, we have the convergence lim

lim lim

N→+∞ ε→0 λn →0

  ξ (t) − ξ (0) − Sn(1) − Sn(2) = 0

in probability P, where Sn(1) =

n−1 1 k=0

(ε)

aN (ξ (tnk )) Δtnk , Sn(2) =

n−1 1 k=0

(ε)

σN (ξ (tnk )) [W (tnk+1 ) − W (tnk )] .

224

A

Selected Facts and Auxiliary Results

Thus, lim

lim lim

N→+∞ ε→0 λn →0

  * ξ (t) − * ξ (0) − * Sn(1) − * Sn(2) = 0

ξ in probability * P, where * Sn are the corresponding integral sums of the processes * * . Continuation of the proof is the same as in the case of continuous functions and W a (x) and σ (x). The analog of the condition (A.13) holds for the process * ξ according to the weak equivalence of the processes ξ and * ξ.   (i)

Corollary A.3 Let g and q be real measurable locally bounded functions, and let the assumptions of Lemma A.13 hold. Then the process t

t g (ξ (s)) ds +

S(t) = 0

q (ξ (s)) dW (s) 0

is stochastically equivalent to the process * S(t) =

t

  g * ξ (s) ds +

0

t

  * (s) . q * ξ (s) d W

0

In fact, this statement follows directly from the proof of Lemma A.13. Lemma A.14 If the solution ξ of an Itô SDE is ergodic with a distribution function F , then it is not stochastically unstable. Proof Note that for an ergodic solution ξ for any N > 0 we have P {|ξ(t)| < N} = P {−N < ξ(t) < N} = P {ξ(t) < N} − P {ξ(t) ≥ −N} → F (N) − F (−N + 0), as t → +∞. Taking into account the equalities F (−∞) = 0 and F (+∞) = 1, we obtain that there exists N0 for which F (N0 ) − F (−N0 + 0) = a > 0. Therefore, lim P {|ξ(t)| < N0 } = a > 0.

t →+∞

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225

So, for any ε > 0 there exists Tε > 0 so that for every t > Tε we have the inequality |P {|ξ(t)| < N0 } − a| < ε. From the obvious inequalities ! ! ! t ! t ! ! !  ! ! 1! ! !1 ! ! ! P {|ξ(s)| < N0 } ds − a ! = ! (P {|ξ(s)| < N0 } − a) ds !! ≤ !t t ! ! ! ! 0

1 t

Tε

0

1 |P {|ξ(s)| < N0 } − a| ds + t

0

t |P {|ξ(s)| < N0 } − a| ds ≤

Lε t − Tε + ε t t

Tε

we obtain ! t ! !  ! !1 ! ! lim sup ! P {|ξ(s)| < N0 } ds − a !! ≤ ε. t →+∞ ! t ! 0

Since ε > 0 is arbitrary, we conclude that 1 lim t →+∞ t

t P {|ξ(s)| < N0 } ds = a > 0. 0

For N = N0 the convergence from the Definition 1 of the stochastic instability is not satisfied, that is, ξ is not stochastically unstable.  

A.3 Brownian Motion in a Bilayer Environment Consider an SDE of the form dζ (t) = σ¯ (ζ(t)) dW (t),

t > 0,

ζ (0) = 0,

(A.19)

where W is a Wiener process, σ¯ (x) = σ1 for x ≥ 0, and σ¯ (x) = σ2 for x < 0, both σi > 0, i = 1, 2. The specific feature of Eq. (A.19) is a discontinuity of the coefficient σ¯ (x) at the point x = 0. Because of this reason, classical existence-uniqueness conditions cannot be applied. Necessity of the investigation of Eq. (A.19) first emerged in the study of the limit behavior of normalized unstable

226

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Selected Facts and Auxiliary Results

solutions of Itô SDEs with continuous coefficients. This problem was studied in [29], where the following relation was established for the limit process ζ : L P {ζ (s) = 0} ds = 0

(A.20)

0

for any L > 0. Due to this result, it was proved that the process ζ is a weak solution of Eq. (A.19), and ζ is a Markov process as the limit of the sequence of Markov processes. Besides this, it is proved in the mentioned paper that equality (A.20) allows to apply the first Kolmogorov equation (see [19, Chapter 8, § 1, Theorem 1]) in order to get the explicit form of the transition density ρ(t, x, y) of the Markov process ζ . Finally, it is established that the transition density has the form

ρ(t, x, y) =

⎧ ( ⎪ − ⎪ ⎪ √1 ⎪ e ⎪ ⎪ ⎪ σ1 2πt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2σ1 · √1 σ1 +σ2

(y−x)2 2σ12 t

σ2 2πt

e

−

−

σ1 −σ2 σ1 +σ2 e

(σ1 y−σ2 x)2 2σ 2 σ 2 t 1 2

−

(y+x)2 2σ12 t

) ,

x ≥ 0, y < 0,

,

(σ y−σ x)2 ⎪ − 2 2 12 ⎪ ⎪ 2σ 1 2σ1 σ2 t 2 ⎪ √ , ⎪ σ1 +σ2 · σ1 2πt e ⎪ ⎪ ( ⎪ (y−x)2 ⎪ − − ⎪ ⎪ 2σ 2 t 1 1 ⎪ 2 − σσ21 −σ ⎪ ⎩ σ2 √2πt e +σ2 e

x ≥ 0, y > 0,

(y+x)2 2σ 2 t 2

(A.21)

x ≤ 0, y > 0,

) ,

x ≤ 0, y < 0.

So, the weak uniqueness of the solution ζ of Eq. (A.19) is established. Moreover, it follows from the explicit form of the density ρ(t, x, y) that the process ζ is a homogeneous in time Markov process whose transition density is continuous in x and discontinuous in y at the point y = 0. Continuity in x implies that the process ζ is a Feller Markov process (see [14, Chapter 2, § 1]), consequently, it has a strong Markov property (see [14, Chapter 3, § 3, Theorem 3.10]). Note that σ¯ (x) ≥ σ0 > 0 N

for any x ∈ R, and the variation V σ (x) = |σ1 − σ2 | for any N > 0. Therefore, −N

the solution of Eq. (A.19) is pathwise unique [67], and consequently, it is strong and strongly unique [84]. Note that in the paper [29] the solution η of Eq. (A.19) with σ¯ (x) = a > 0, x ≥ 0 and σ¯ (x) = 1, x < 0, is obtained as a limit of the normalized unstable solutions of some specific SDEs. This process satisfies condition (A.20). The transition density ρη (t, x, y) has the form (A.21) with σ1 = a and σ2 = 1. Obviously, due to the condition (A.20) and the relation σ¯ (x) = σ¯ (σ2 x) for σ2 > 0, the process ζ(t) := σ2 η(t) is a solution of Eq. (A.19) with σ1 = aσ2 . It is well known (see [19, Chapter VI, § 5]), that Eq. (A.19) is the mathematical description of the physical Brownian particle moving in a liquid or gaseous environment, in the presence of collisions of this particle with molecules of the environment, with the assumption that these molecules are in a chaotic temperature

A Selected Facts and Auxiliary Results

227

movement. The intensity of the collision equals σ¯ (x) at the point x. Let ζ = ζ(t) be a projection of the Brownian particle’ position on the coordinate axis at the moment t > 0. Under these assumptions, during some small interval t, we get the approximate equality ζ (t + t) − ζ (t)  σ¯ (ζ(t)) [W (t + t) − W (t)] .

(A.22)

Let S1 = {x : x ≥ 0}, S2 = {x : x < 0}, and the intensity of collision equal σ¯ (x) = σ1 , x ∈ S1 , and σ¯ (x) = σ2 , x ∈ S2 . Then ζ (t + t) − ζ (t)  σ1 [W (t + t) − W (t)] for ζ (t) ∈ S1 , while ζ (t + t) − ζ (t)  σ2 [W (t + t) − W (t)] for ζ (t) ∈ S2 . Therefore in this case the solution ζ of Eq. (A.19) is a process that describes the trajectory of the movement of the Brownian particle in the bilayer environment S = S1 ∪ S2 with the continuous-type border crossing at the point x = 0. Definition A.16 The solution ζ of Eq. (A.19) is called the process of Brownian motion in the bilayer environment S = S1 ∪ S2 . For the reader’s convenience, we give here the proof of formula (A.21). Lemma A.15 The transition density ρ(t, x, y) of the solution of Eq. (A.19) has the form (A.21). Proof Introduce the solution u(t, x) of the Cauchy problem for the parabolic equation −

∂u(t, x) 1 ∂ 2 u(t, x) = σ 2 (x) , ∂t 2 ∂x 2

considered on the set 0 ≤ t < s, x ∈ (−∞, +∞), with conditions of conjugation ! ! ∂u(t, x) !! ∂u(t, x) !! u(t, +0) = u(t, −0), = , ∂x !x=+0 ∂x !x=−0 and the initial condition u(s, x) = q(x). The function q(x) is assumed to be twice continuously differentiable and bounded, together with its derivatives q (x) and q (x).

228

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Selected Facts and Auxiliary Results

Introduce the function V (t, x) = u(s − t, x). Then V (x, 0) = q(x) and for t > 0 and x ∈ R we have the equation 1 ∂ 2 V (t, x) ∂V (t, x) = σ 2 (x) . ∂t 2 ∂x 2 This equation is convenient to solve by means of the Laplace transformation. So, we multiply both parts of the equation by e−pt , denoting +∞ V (p, x) = e−pt V (t, x)dt. ∗

0

After integrating with respect to t over the interval [0, +∞) we get the following differential equations: ∂ 2 V ∗ (p, x) 2p ∗ 2 − 2 V (p, x) = − 2 q(x), x > 0, 2 ∂x σ1 σ1 and 2 ∂ 2 V ∗ (p, x) 2p ∗ − 2 V (p, x) = − 2 q(x), x < 0. 2 ∂x σ2 σ2 The solution of this couple of equations has the form 1 V (p, x) = − √ e σ1 2p ∗

+

1 − √ e σ1 2p

√ 2p σ1 x

x e

−

√ 2p σ1 y

√ 2p σ1 x

x e

−

√ 2p σ1 y

q(y) dy

+∞

q(y) dy + C1 e

√ 2p σ1 x

+ C2 e

−

√ 2p σ1 x

−∞

for x > 0, and 1 V (p, x) = − √ e σ2 2p ∗

+

1 − √ e σ2 2p

√ 2p σ2 x

x e

−

√ 2p σ2 y

√ 2p σ2 x

x e

−

√ 2p σ2 y

+∞

q(y) dy + C3 e

−∞

for x < 0, where Ci are arbitrary constants.

q(y) dy

−

√ 2p σ2 x

+ C4 e

√ 2p σ2 x

A Selected Facts and Auxiliary Results

229

Since the function V ∗ (p, x) is bounded, we have that C1 = C3 = 0. Applying the conditions of conjugation: V ∗ (p, +0) = V ∗ (p, −0),

∂V ∗ (p, −0) ∂V ∗ (p, +0) = , ∂x ∂x

we can determine the constants 1 σ1 − σ2 C2 = √ σ1 + σ2 σ1 2p 1 − √ σ1 2p

0 e

−

√ 2p σ1 y

−∞

0 e

−

√ 2p σ1 y

q(y) dy

+∞

2σ1 1 q(y)dy + √ σ1 + σ2 σ2 2p

0 e

√ 2p σ2 y

q(y) dy,

−∞

and 1 σ1 − σ2 C4 = √ σ1 + σ2 σ2 2p 1 + √ σ2 2p

0 e

√ 2p σ2 y

+∞

0 e

√ 2p σ2 y

q(y)dy

−∞

1 2σ2 q(y) dy − √ σ1 + σ2 σ1 2p

0 e

−

√ 2p σ2 y

q(y) dy.

+∞

Next, we use the inverse Laplace transform and get ⎡ +∞ ) 2 2  +∞ − (y−x) − (y+x) 1 − σ σ 2 2 1 2 ⎣ V (t, x) = √ q(y)e 2σ1 t dy− q(y) e 2σ1 t dy σ1 + σ2 σ1 2πt 0

2σ1 1 + √ σ1 + σ2 σ2 2πt

0

0 q(y)e

−

(σ1 y−σ2 x)2 2σ12 σ22 t

dy

−∞

for x ≥ 0, while ⎡ 0 ) 2 2  0 − (y−x) − (y+x) 1 σ2 − σ1 2σ22 t 2σ22 t ⎣ V (t, x) = √ q(y)e dy− q(y) e dy σ2 + σ1 σ2 2πt −∞

2σ2 1 + √ σ1 + σ2 σ1 2πt

−∞

+∞ (σ y−σ x)2 − 2 2 12 q(y)e 2σ1 σ2 t dy 0

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Selected Facts and Auxiliary Results

for x ≤ 0. Therefore, +∞ u(t, x) = q(y)ρ(t, x, y)dy, −∞

where ρ(t, x, y) has the form (A.21). The functions u(t, x) and u (t, x) are continuous, the function u xx (t, x) has jump discontinuities at the points (t, 0). Applying equality (A.20) and smoothing, it is possible to get the Itô formula for the process u(t, ζ (t)). Applying this formula, we get for any 0 ≤ t1 < t2 < s the relations u(t2 , ζ (t2 )) = u(t1 , ζ (t1 )) t2  1 + u t (t, ζ (t)) + σ 2 (ζ (t)) u xx (t, ζ (t))] dt 2 t1

t2 +

u x (t, ζ(t)) σ (ζ (t)) dW (t) = u (t1 , ζ(t1 ))

t1

t2 +

u x (t, ζ(t)) σ (ζ (t)) dW (t).

t1

It follows immediately from the latter relation that Eu(t2 , ζ (t2 )) = Eu (t1 , ζ (t1 )) . Let t2 → s. Taking into account the boundedness and continuity of the function u(t, x), together with the continuity a.s. of the process ζ , we have Eq (ζ (s)) = Eu (t1 , ζ (t1 )) for any t1 < s. From this equality and from the explicit form of the function u(t, x) we obtain the explicit form (A.21) of the transition density of the solution ζ to Eq. (A.19).   It was mentioned in [33] that the normalized unstable solutions of SDEs with continuous coefficients tend to processes of the form  ξ (t) = l(ζ (t)), where l(x) =

x 0

du , c(u)

(A.23)

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231

c(x) = c1 for x ≥ 0 and c(x) = c2 for x < 0, ci > 0, i = 1, 2, ζ is the solution of Eq. (A.19). Since the function l(x) is increasing, the process  ξ is a continuous homogeneous strongly Markov process with the transition density    (A.24) ρξ (t, x, y) = ρζ t, l −1 (x), l −1 (y) l −1 (y) , where l −1 (x) is the inverse function to the function l(x), and ρ(t, x, y) is the transition density of the process ζ . For the details, see [17, Chapter 3, § 9]. x It is obvious that in this case l(x) = c(x) , l −1 (x) = xc(x), and ρ(t, x, y) is defined by the equality (A.21). Therefore, ⎡ ⎤ ⎧ (c1 y−c1 x)2 (c1 y+c1 x)2 ⎪ − − ⎪ c1 ⎣ 2σ12 t 2σ12 t ⎪ 2 ⎦ , x ≥ 0, y > 0, √ ⎪ e − σσ11 −σ ⎪ +σ2 e σ1 2πt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (σ c y−σ c x)2 ⎪ − 1 2 2 22 1 ⎪ ⎪ 2σ σ t ⎨ 2σ1 · √c2 e 1 2 , x ≥ 0, y < 0, σ1 +σ2 σ2 2πt ρξ (t, x, y) = 2 (σ c y−σ c x) ⎪ − 2 1 2 12 2 ⎪ ⎪ 2σ2 c1 2σ1 σ2 t ⎪ √ · e , x ≤ 0, y > 0, ⎪ σ +σ ⎪ 1 2 ⎡ σ1 2πt ⎤ ⎪ ⎪ 2 2 ⎪ (c y−c x) (c y+c x) ⎪ − 2 22 − 2 22 ⎪ ⎪ c2 σ2 −σ1 2σ2 t 2σ2 t ⎣ ⎦ , x ≤ 0, y < 0. ⎪ √ − e e ⎪ σ1 +σ2 ⎩ σ2 2πt (A.25) In particular, for c(x) = σ¯ (x) we have that

ρξ (t, x, y) =

 ⎧ (y−x)2 1 ⎪ 2 − √ ⎪ e− 2t − σσ11 −σ ⎪ +σ2 e 2πt ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ 2σ1 · √ 1 e− (y−x) 2t , σ1 +σ2 2πt

⎪ 2σ2 − (y−x) ⎪ 2t √1 , ⎪ σ1 +σ2 · 2πt e ⎪ ⎪ ⎪ 2 (y−x) ⎪ ⎪ 1 − ⎩ √1 e− 2t − σσ21 −σ +σ2 e 2πt

(y+x)2 2t

,

x ≥ 0, y > 0, x ≥ 0, y < 0,

2

x ≤ 0, y > 0, (y+x)2 2t

,

(A.26)

x ≤ 0, y < 0.

Let l (ζ (0)) = x, a < x < b and  τx [a, b] be the time . of the - of the first exit process  ξ from the interval (a, b), i.e.,  τx [a, b] = inf s :  ξ (s) ∈ / [a, b] . Since $ %  τx [a, b] = τl −1 (x) l −1 (a), l −1 (b) a.s., where τx [a, b] is the first exit time of the process ζ from the interval (a, b), then, having respective formulas for ζ (see [17, Chapter 3, § 15, Theorem 4]), we get . l −1 (b) − l −1 (x) , P  ξ ( τx [a, b]) = a = −1 l (b) − l −1 (a) . l −1 (x) − l −1 (a) P  ξ ( τx [a, b]) = b = −1 . l (b) − l −1 (a)

(A.27)

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Selected Facts and Auxiliary Results

In particular, . P  ξ ( τ0 [−ε, ε]) = −ε =

c1 c1 ε = , c1 ε + c2 ε c1 + c2 . c2 c2 ε = P  ξ ( τ0 [−ε, ε]) = ε = , c1 ε + c2 ε c1 + c2

(A.28)

and for x > ε > 0 . P  ξ ( τx [x − ε, x + ε]) = x + ε =

1 σ1 (x + ε) − σ1 x = , σ1 (x + ε) − σ1 (x − ε) 2

. P  ξ ( τx [x − ε, x + ε]) = x − ε =

1 σ1 x − σ1 (x − ε) = . σ1 (x + ε) − σ1 (x − ε) 2

Similar relations hold for x < 0 as well. Summarizing, in the case c1 = c2 , we have the trespassing at the point x = 0 of the exit symmetry of the process  ξ out of the level ε and out of the level −ε, while such symmetry is specific for the Wiener process. In particular, in the case c(x) = σ (x), and σ1 = σ2 , the diffusion process  ξ was called a skew Brownian motion by Itô and McKean [24, Section 4.2, Problem 1]. Therefore, it is quite natural to introduce the following object. Definition A.17 The process  ξ (t) = l(ζ (t)), where ζ(t) is the solution of x du Eq. (A.19), and l(x) = c(u) , c(x) = c1 for x ≥ 0 and c(x) = c2 for x < 0, 0

ci > 0, i = 1, 2, is called a process of skew Brownian motion type. Let us emphasize that the explicit form (A.26) of the transition density ρξ (t, x, y) plays a crucial role in the introducing of the notion of the generalized diffusion process provided by M.I. Portenko in [72]. It is a Markov process whose local Kolmogorov’s characteristics exist in the generalized sense. Definition A.18 A homogeneous Markov process with transition density ρ(t, x, y) is called a generalized diffusion process, if for any ε > 0 and any continuous finite function ϕ(x) the following relations hold:  lim t ↓0

 lim t ↓0

R

R

⎛ ⎜1 ϕ(x) ⎝ t ⎛

⎜1 ϕ(x) ⎝ t



⎞ ⎟ ρ(t, x, y) dy ⎠ dx = 0,

|y−x|>ε

 |y−x| 0, c1 = kc2 , σ1 = c1 b1 , σ2 = c2 b2 and k = b1 b2 (b1 +b2 )−2cb2 b1 b2 (b1 +b2 )+2cb1 . Indeed, it is easy to see that for the process  ξ of the form (A.23) with given constants c1 , c2 , σ1 , σ2 the transition density has the form (A.25). In particular, this generalized process is a process of skew Brownian motion type.

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Selected Facts and Auxiliary Results

A.4 Functions Regularly Varying at Infinity Recall the definition of class Ψ , see also Definition 4.1. Definition A.19 Let Ψ denote the class of functions ψ (r) > 0, r ≥ 0, that are nondecreasing and regularly varying at infinity of order α ≥ 0. It means that ψ (rT ) → rα, ψ (T ) as T → +∞, for all r > 0. Lemma A.16 Let the function ψ ∈ Ψ . Then for arbitrary N > 0 there exist constants 0 < CN < +∞ and 0 < TN < +∞ such that uniformly on T ≥ TN the following inequality holds:  √  ψ r T √  ≤ CN . sup 0≤r≤N ψ T

(A.32)

Proof It is clear that √ √ ψ(r T ) ψ(N T ) sup ≤ . √ √ ψ( T ) 0≤r≤N ψ( T ) √

Since for a regularly varying function ψ we have the convergence ψ(N√ T ) → N α , ψ( T ) as T → +∞, then for ε = 1 there exists a constant TN < +∞ such that for all T ≥ TN the following inequality holds true: √ ψ(N T ) √ ≤ N α + 1. ψ( T ) Hence the statement of the lemma is proved for CN = N α + 1.

 

According to Karamata’s theorem, the function ψ(r) ∈ Ψ for r > 0 admits the representation (for the proof see Appendix 1 in the book [22]) ψ(r) = r α c(r) exp

⎧ r ⎨ ⎩

a

⎫ ε(t) ⎬ dt , ⎭ t

(A.33)

where α > 0, c(r) → c0 = 0 for r → +∞ and ε(t) → 0 for t → +∞. Using the properties of this representation we prove the following statement.

A Selected Facts and Auxiliary Results

235

Lemma A.17 Let the function ψ ∈ Ψ . Then ! ! ! ψ (rT ) ! α! ! − r ! ψ (T ) ! → 0, 0 0 there exists Tδ1 > 0 such that for min(rT ·T , T ) > Tδ1 the inequality −δ1 < ε(t) < δ1 holds. Then we obtain !r T ! !r T ! ! T ! ! T ! ! ! ! ε(t) ! ! |ε(t)| !! ! ≤ dt dt ! ! ! ! ≤ δ1 | ln(rT T ) − ln T | = δ1 | ln rT | ≤ δ1 Cδ,N , t t ! ! ! ! T

T

and immediately, rT T

ε(t) dt → 0, t

T

as T → +∞. Taking into account the equality (A.34), we conclude that ψ(rT T ) − rTα → 0, ψ(T ) as T → +∞. The statement of the lemma now follows since, as it was mentioned above, the   set of numbers rT ∈ [δ, N] is arbitrary.

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