Random Integral Equations with Applications to Life Sciences and Engineering (Mathematics in Science and Engineering 108)

Random Integral Eqzlations with Applications to Lzfe Sciences and Engineering

To A . T. Bharucha-Reid

. . . yqpaatio 'mi~

Z ~ W K ~ ~ E V. O Socrates S . .

This is Volume 108 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southerri California The complete listing of books in this series is available from the Publisher upon request.

R.ANDOM INTEGRAL EQUATIONS W I T H APPLICATIONS T O LIFE SCIENCES A N D ENGINEERING Chris P . Tsokos DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH FLORIDA TAMPA. FLORIDA

W. J. Padgett DEPARTMENT OF MATHEMATICS A N D COMPUTER SCIENCE UNIVERSITY OF SOUTH CAROLINA COLUMBIA, SOUTH CAROLINA

A C A D E M I C P R E S S New York and London 1974 A Subsidiary of Harcourt Brace Jovunovich, Publishers

COPYRIGHT 0 1974, BY ACADEMIC PRESS,INC.

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ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published b y ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London N W 1

Library of Congress Cataloging in Publication Data Tsokos, Chris P Random integral equations with applications to life sciences and engineering. (Mathematics in science and engineering, v. 108) Bibliography: p . 1. Stochastic integral equations. 1. Padgett, W. J . , joint author. 11. Title. 111. Series. QA274.27.T76 5w . 4 5 73-2079 ISBN 0-12-702150-7

AMS (MOS) 1970 Subject Classifications: 60H20,45(399, 93E15

PRINTED IN THE UNITED STATES OF AMERICA

Contents PREFACE

ix

General Introduction

i

Chapter I. Preliminaries and Formulation of the Stochastic Equations 1.0 Introduction 1.1 Basic Definitions and Theorems from Functional Analysis 1.2 Probabilistic Definitions 1.3 The Stochastic Integral Equations and Stochastic Differential Systems Appendix l.A

6 7 12 18

22

Chapter 11. Some Random Integral Equations of the Volterra Type with Applications 2.0 Introduction 2.1 The Random Integral Equation 2.1.1 Existence and Uniqueness of a Random Solution 2.1.2 Some Special Cases 2.1.3 Asymptotic Stability of the Random Solution 2.2 Some Applications of the Equation 2.2. I Generalization of Poincare-Lyapunov Stability Theorem 2.2.2 A Problem in Telephone Traffic Theory 2.2.3 A Stochastic Integral Equation in Hereditary Mechanics

29 30 30 33 38 39 40 42 46

ui

CONTENTS

2.3 The Random Integral Equation 2.4 Applications of the Integral Equation 2.4. I A Stochastic Integral Equation in Turbulence Theory 2.4.2 Stochastic Models for Chemotherapy

49 55 55 57

Chapter 111. Approximate Solution of the Random Volterra Integral Equation and an Application to Population Growth Modeling 3.0 Introduction 3. I The Method of Successive Approximations 3.1. I Convergence of the Successive Approximations

3.1.2 Rate of Convergence and Error of Approximation 3.1.3 Combined Error of Approximation and Numerical Integration 3.2 A New Stochastic Formulation of a Population Growth Problem 3.2.1 The Deterministic Model 3.2.2 The Stochastic Model 3.2.3 Existence and Uniqueness of a Random Solution 3.3 Method of Stochastic Approximation 3.3.1 A Stochastic Approximation Procedure 3.3.2 Solution of Eq. (3.0.1) by Stochastic Approximation 3.3.3 Numerical Solution for a Hypothetical Population

Chapter IV. 4.0 4.1 4.2 4.3 4.4 4.5

65 66 68 71 74 78 79 81 84 87 87 89 94

A Stochastic Integral Equation of the Fredholm Type and Some Applications

Introduction Existence and Uniqueness of a Random Solution Some Special Cases Stochastic Asymptotic Stability of the Random Solution An Application in Stochastic Control Systems A Random Perturbed Fredholm Integral Equation

97

98

110 113 115 120

Chapter V. Random Discrete Fredholm and Volterra Systems 5.0 Introduction

Existence and Uniqueness of a Random Solution of System (5.0.1) 5.2 Special Cases of Theorem 5.1.2 5.3 Stochastic Stability of the Random Solution 5.4 An Approximation to System (5.0.I ) 5.5 Application to Stochastic Control Systems 5.5.1 A Discrete Stochastic System 5.5.2 Another Discrete Stochastic System 5.1

132 133 136 139 141 148 148

152

CONTENTS

rii

Chapter VI. Nonlinear Perturbed Random Integral Equations and Application to Biological Systems 6.0 Introduction 6.1 The Random Integral Equation 6.1.1 Existence and Uniqueness of a Random Solution 6.1.2 Some Special Cases 6.2 Applications to Biological Systems 6.2. I A Random Integral Equation in a Metabolizing System 6.2.2 A Stochastic Physiological Model 6.2.3 A Stochastic Model for Communicable Disease

Chapter VII. 7.0 7.1 7.2 7.3

i56 157 157

159 165 165 170 176

On a Nonlinear Random Integral Equation with Application to Stochastic Chemical Kinetics

Introduction Mathematical Preliminaries An Existence and Uniqueness Theorem A Stochastic Chemical Kinetics Model 7.3.1 The Concept of Chemical Kinetics 7.3.2 Stochastic Interpretation of the Rate of Reaction 7.3.3 Rate Functions of General Reaction Systems 7.3.4 A Stochastic Integral Equation Arising in Chemical Kinetics

180 181 194 197 198 20 1 20 I 204

Chapter VIII. Stochastic Integral Equations of the Ito Type 8.0 8. I 8.2 8.3

Introduction Preliminary Remarks On an Ito Stochastic Integral Equation On It+Doob-Typc Stochastic Integral Equations 8.3.1 An Existence Theorem

207 208 212 214 215

Chapter IX. Stochastic Nonlinear Differential Systems 9.0 Introduction 9. I Reduction of the Stochastic Differential Systems 9.1. I Stochastic System (9.0.1j(9.0.2) 9.1.2 The Random Differential System (9.0.3)-(9.0.4) 9.1.3 The Stochastic System (9.0.5)-(9.0.6) 9.1.4 The Random Differential System (9.0.7j(9.0.8) 9.2 Stochastic Absolute Stability of the Differential Systems Appendix 9.A 9.A. 1 Stochastic Differential System (9.0.1)-(9.0.2) 9.A.2 Stochastic Differential System (9.0.3j(9.0.4) 9.A.3 The Reduced Stochastic Integral Form of Systems (9.0.1H9.0.2) and (9.0.3)

217 219 219 220 22 I 222 225 239 239 240 240

viii

Chapter X.

CONTENTS

Stochastic Integrodifferential Systems 24 I 243 241 250

10.0 Introduction The Stochastic Integrodifferential Equation 10.1.1 Asymptotic Behavior of the Random Solution 10.1.2 Application to a Stochastic Differential System 10.2 Reduction of the Stochastic Nonlinear Integrodifferential Systems with Time Lag 10.2.1 The Integrodifferential System (10.0.2j(l0.0.3) 10.2.2 The Random Integrodifferential System (10.0.4j(10.0.5) 10.3 Stochastic Absolute Stability of the Systems

25 I 25 1 253 255

Bibliography

260

INDEX

215

10. I

Preface

Random or stochastic integral equations arise in virtually every field of scientific endeavor. Recently, attempts have been made by many scientists and mathematicians to develop and unify the theory of stochastic or random equations using the concepts and methods of probability theory and functional analysis. We have two main objectives in this book. First. we wish to give a complete presentation of various aspects of some of the most general forms of nonlinear stochastic integral equations of the Volterra and Fredholm types which have been studied, including the problems of existence, uniqueness, stochastic stability, and approximation of random solutions of the equations. In addition, we investigate stochastic integral equations of the It+Doob type. The second objective is to apply the theory developed to some very important problems in the life sciences and engineering. With respect to the applications, stochastic models for various phenomena in the biological, engineering, and physical sciences are obtained. For example, applications of the theory to the following areas are given: telephone traffic theory, hereditary mechanics, turbulence theory, chemotherapy, population growth, stochastic control systems, metabolizing systems, physiological models, communicable diseases, chemical kinetics, and stochastic integrodifferential systems. The book will be of value to mathematicians, probabilists, statisticians, and engineers who are working in the theoretical and applied aspects of random integral equations. The book can be used for a beginning graduate iA

x

PREFACE

course on random integral equations with emphasis being placed on probabilistic modeling of various problems in life sciences and engineering. It should be pointed out that this book differs in purpose considerably from the book of A. T. Bharucha-Reid [7]. He is concerned primarily with the overall theory of random integral equations, whereas we emphasize the stochastic modeling aspects and applications of certain types of random or stochastic integral equations. Thus, the two books are complementary. This book was written with the direct and indirect help of many people. We are grateful to Dr. J . Susan Milton for her helpful and stimulating discussions during the preparation of the manuscript. We would also like to acknowledge Dr. A. N. V. Rao for his assistance and valuable comments and Ms. Debbi Beach for her excellent typing of the manuscript. In addition we would like to express our appreciation to Professor Richard Bellman for his encouragement and interest in the subject matter. Finally, we would like to express our sincere thanks to our families for their understanding and patience during the writing of the book.

General Introdzlction7

Due to the nondeterministic nature of phenomena in the general areas of the biological, engineering, oceanographic, and physical sciences, the mathematical descriptions of such phenomena frequently result in random or stochastic equations. It is the aim of this book to present theoretical results concerning certain classes of stochastic or random equations and then to apply those results to problems that arise in the general areas just mentioned. In order to understand better the importance of developing such a theory and its application, it is of interest to consider first the various ways in which these equations may arise. Usually the mathematical models or equations used to describe physical phenomena contain certain parameters or coefficients which have specific physical interpretations but whose values are unknown. As examples, we have the diffusion coefficient in the theory of diffusion, the volume-scattering coefficient in underwater acoustics, the coefficient of viscosity in fluid mechanics, the propagation coefficient in the theory of wave propagation, and the modulus of elasticity in the theory ofelasticity, among others. The mathematical equations are solved using as the value of the parameter or coefficient the mean value of a set of observations experimentally obtained. However, if the

t Adapted from Pddgett and Tsokos [12]

with permission of Taylor and Francis, Ltd I

2

GENERAL INTRODUCTION

experiment is performed repeatedly, then the mean values found will vary, and if the variation is large, the mean value actually used may be quite unsatisfactory. Thus in practice the physical constant is not really a constant, but a random variable whose behavior is governed by some probability distribution. It is thus advantageous to view these equations as being random rather than deterministic, to search for a random solution, and to study its statistical properties. There are many other ways in which random or stochastic equations arise. Stochastic differential equations appear quite naturally in the study of diffusion processes and Brownian motion (Gikhmann and Skorokhod [I]). The classical Ito stochastic integral equation (Ito [l]) may be found in many texts, for example, Doob [l]. Integral equations with random kernels arise in random eigenvalue problems (Bharucha-Reid [7]). Stochastic integral equations describe wave propagation in random media (Bharucha-Reid [7]) and the total number of conversations held at a given time in telephone traffic theory (Fortet [l] and Padgett and Tsokos [4]). In the theory of statistical turbulence, stochastic integral equations arise in describing the motion of a point in a continuous fluid in turbulent motion (Bharucha-Reid [7], Lumley [l], Padgett and Tsokos [3]). Integral equations were used in a deterministic sense by Bellman, Jacquez, and Kalaba [l-31 in the development of mathematical models of chemotherapy. However, due to the nondeterministic nature of diffusion processes from the blood plasma into the body tissue, the stochastic versions of these equations are more realistic and should be used (Padgett and Tsokos [l, 2, lo]). Stochastic integral equations also arise in problems in chemical kinetics and metabolizing systems (Milton and Tsokos [l, 41). Random equations are also frequently encountered in a natural way in systems theory (for example, Morozan [l-51 and Tsokos [1-51). These examples point out the importance of random equations in diverse areas. However, in many instances the scientist tends to use a deterministic model to represent a process under investigation with the philosophy that there is a deterministic but unknown function x ( t ) which describes the phenomenon he observes. He then attempts by experimental methods to determine as accurately as possible the form of this function. A standard procedure is to obtain, at several specified values o f t , observations on the value of x ( t ) and then to use as the “true” value of x ( t ) some estimate based on these observations, the usual estimate being the mean. In this way a single trajectory can be constructed which is then taken as the true form of x ( t ) and is subsequently used in working with the model. This general technique characterizes the deterministic approach to a physical situation. However, if this procedure were repeated many times, even under the most carefully controlled conditions, the trajectories so obtained will differ, and in most cases

GENERAL INTRODUCTION

3

this variation could be quite signijicant. If this is indeed the case, then there is evidence that there is more than mere measurement error entering into the picture, and that, in fact, the function which governs the process is not a fixed unknown entity, but a random one. Thus it is more realistic in this situation to construct a stochastic model for the system rather than a determiiiistic model. This entails the basic assumption that at each point t , x ( t ) is not a fixed unknown value which should be estimated, but rather a random variable which we denote by x(t ;w), where w E R,the supporting set of a complete An important point to be made is that probability measure space (R, d,9). if a stochastic model is assumed when a deterministic model could be justified, nothing is lost ;but if a deterministic model is assumed when in fact the process is random, then the results obtained could be quite unsatisfactory. Recently attempts have been made by many scientists and mathematicians to develop and unify the theory of stochastic or random equations : Adomian [ l a ] , Ahmed [l], Anderson [l, 21, Bharucha-Reid [l-71, Hans [l], Tsokos [4], Padgett andTsokos [ l , 3,5-9, 1 1 , 12, 151, Rao [l], Milton et al. [l]. It was Antonin Spacek from Prague, Czechoslovakia, who began this work, utilizing the concepts and methods of probability theory and functional analysis. In fact, Bharucha-Reid [5] refers to probabilistic functional analysis as being concerned with the applications and extensions of the methods of functional analysis to the study of the various concepts, processes, and structures which arise in the theory of probability and its applications. Random or stochastic equations have been categorized into four main classes as follows : (1) (2) (3) (4)

Random or stochastic algebraic equations. Random differential equations. Random difference equations. Random or stochastic integral equations.

In this book we will be concerned with certain classes of random or stochastic integral equations of the Volterra and Fredholm types and a class of random integrodifferential equations of the Volterra type. For example, in Chapters I1 and I11 we will study various aspects of the stochastic integral equation of the Volterra type in the form x ( t ; w )= h ( t ; o )+

c

k(t,T;W)f(Z,X(T;W))dr

(0.1)

for t 2 0, where the integral is interpreted as a mean-square integral. We will consider the existence, uniqueness, asymptotic behavior, and approximation of random solutions of each type of stochastic integral equation studied in this book. In addition, the second aim of the book is to present numerous

4

GENERAL INTRODUCTION

applications of the theory of such equations to the problems in chemotherapy, chemical kinetics, physiological systems, population growth, telephone engineering, turbulence, and systems theory as previously mentioned. Furthermore, these equations are more general than any random Volterra or Fredholm equations of these forms that have been studied to date. The generality consists primarily in the choice of the stochastic kernel and the nonlinearity of the equations. This book includes the recent work of the authors, Padgett and Tsokos [l-151, and generalizes the work of Hans [l], Bharucha-Reid [l, 3-51, and Anderson [l, 21. In the area of systems theory we will apply the general theory which is presented concerning random integral equations to certain problems in random differential systems and random integrodifferential systems (Tsokos [l-3,5], Tsokos and Hamdan [l]). The nonlinear stochastic differential and integrodifferential systems will be reduced in a unified way to stochastic nonlinear integral equations. Then in each case the existence and uniqueness of a random solution of the stochastic system will be investigate.d.In addition, we will consider the concept of stochastic absolute stability of the systems. This type of stability has been studied in the deterministic case by many scientists, but to the knowledge of the authors it has not been utilized for random systems. The concept of absolute stability arose in the context of differential control systems and the general theory of stability of motion. The primary mathematical technique which was universally used to study absolute stability was Lyapunov’s direct method. However, in the late 1950’swhen Lyapunov’s method appeared to be exhausted V. M. Popov developed a new approach, obtaining very elegant and powerful results. Popov’s method is known as the frequency response method. In Chapter IX we successfully utilize the frequency response method with a random parameter to investigate the stochastic absolute stability of several stochastic differential systems. These results generalize the recent results of Morozan [ l ] in that he chose a specific form of the stochastic kernel, namely an exponential form. In Chapter I we present preliminary notations, definitions, lemmas, and theorems which are essential to the aims of this book. Further, we define and formulate the types of stochastic integral equations and the stochastic differential systems which will be investigated in later chapters. Finally, in Appendix l.A of Chapter I the proofs of some of the fixed-point theorems that are utilized throughout the book are given. As already mentioned, in Chapters I1 and 111 we will investigate the existence, uniqueness, asymptotic properties, and approximation of random solutions of certain stochastic integral equations of the Volterra type. In addition, several applications and examples of such equations will be presented in the areas of chemotherapy, telephone traffic theory, turbulence theory, hereditary mechanics, and

GENERAL INTRODUCTION

5

stochastic systems theory. Chapter IV will be devoted to stochastic integral equations of the Fredholm type. Certain random or stochastic discrete Volterra and Fredholm equations will be considered in Chapter V along with their approximate solutions and applications. Chapter VI will deal with perturbed versions of the random equations studied in Chapters I1 and IV and their application to biological systems. In Chapter VII an application of a nonlinear random integral equation to a problem in stochastic chemical kinetics is presented. A connection between Ito’s equation and certain classes of stochastic integral equations studied in earlier chapters is given in Chapter VIII; that is, Ito’s equation is studied by applying some aspects of the “theory of admissibility” (Corduneanu [l]). Several stochastic nonlinear differential systems and their stochastic absolute stability are studied in Chapter IX. Chapter X is devoted to the investigation of a class of random or stochastic integrodifferential equations and its application to nonlinear stochastic systems.

CHAPTER I

P reZiminaries and F ormzclution of the Stochastic Equations

1 .O

Introduction

In an attempt to make this book essentially self-contained, one purpose of the present chapter is to give some of the basic definitions and theory of functional analysis which will be used throughout the text. Therefore Section 1.1 will consist of the statements of several definitions concerning linear topological spaces and some important theorems which are needed in later discussions. In Appendix 1.A we will give the proofs of some of the classical fixed-point theorems of functional analysis for the interested reader, but otherwise most of the theorems will be stated without proof for the sake of brevity. The second purpose of this chapter is to present the probabilistic definitions, basic assumptions, and notations that are essential to the development of the theory in later chapters. These will be given in Sections 1.2 and 1.3. Section 1.2 will be devoted to certain probabilistic definitions and notations, while the specific types of stochastic or random integral equations which will 6

1.1

BASIC DEFINITIONS AND THEOREMS

7

be investigated are given in Section 1.3. Some definitions and notations concerning the stochastic differential systems to be studied also will be presented in Section 1.3. 1.1

Basic Definitions and Theorems from Functional Analysis

The following basic definitions and theorems are stated for the convenience of the reader. Definition 1.1.1 A real-valued measurable function f(x) defined on a closed interval [a, b] is said to be a square-summable function if lf(x)I2 dx
0 there exists a nonnegative integer N such that

Ilf” - f l l < E for all n > N . Definition 1.1.4 A nonempty set H is called a metric space if to an arbitrary pair x, y of elements in H there corresponds a real number p(x, y ) possessing the following properties :

(i) p(x, y ) 2 0, where p(x, y ) = 0 if and only if x = y . (ii) P(X, Y ) = P(Y, XI. (iii) p(x, z ) < p ( x , y ) + p ( y , z ) for any x, y , z E H (triangle inequality). The number p(x, y ) is the distance between the elements x and y . Definition 1.1.5 A sequence {xn} of elements in a metric space is said to be convergent in itself or a Cauchy sequence if for every E > 0 there exists an

8

I

PRELIMINARIES AND FORMULATION OF THE EQUATIONS

N such that for n > N and m > N we have P(Xm x),

< E.

Definition 1.1.6 A metric space H is said to be complete if every sequence of its elements which is convergent in itself has a limit in H . Definition 1.1.7 A set H of elements x, y , z, . . . is said to be a linear space if: (i) To every pair of elements x and y of H there corresponds a third element of H , z = x y, called the sum of x and y. (ii) T o every element x E H and every scalar a there corresponds an element, ax E H , which is called the product of a and x. (iii) The operations introduced possess the following properties for every x, y, z E H and scalars a and b :

+

+ y = y + x, i.e., addition is commutative. + y ) + z = x + ( y + z), i.e., addition is associative. x + y = x + z implies y = z.

x (x

(4) (5) (6) (7)

l x = x. a(bx) = (ab)x. ( a b)x = ax a(x y ) = ax

+ +

+ bx.

+ ay.

Definition 1.1.8 A linear space H is said to be normed if to each x E H there corresponds a real number IIx 11, called the norm of this element, possessing the following properties for every y E H and every scalar a : (i) llxll B 0, where llxll = 0 if and only if x = 0. (ii) llaxII = la( . Ilxll, and in particular 11 -XI/ = Ilxll. (iii) llx + Yll d IIXII + IIYII.

Definition 1.1.9 A complete normed linear space is called a Banach space. A FrPclzet space is a complete linear metric space. Definition 1.1.10 Let H be a given set, and let B be a set of subsets of H having the following properties : (i) H E E (ii) 4 E .9? (iii) The union of any nonempty family of sets from 9 belongs to 5 (iv) The intersection of any two sets of B belongs to E The ordered pair ( H , 9) is then called a topological space and 9 is the topology of the space. The sets belonging to B are called open sets.

Definition 1.1.11 If the set H in Definition 1.1.10 is a linear space and the two basic operations (addition and multiplication by a constant) are continuous, then H is a linear topological space or vector space.

1.1

BASIC DEFINITIONS AND THEOREMS

9

Definition 1.1.12 Let H and H , be metric spaces and let T be a rule which associates some point y E H , with every point x E H. Such a rule is called an operator which is defined on the space H and maps H into the space H,. If y E H, is the point which the operator T assigns to the point x E H, we write y = T ( x )and call y the value of the operator Tat the point x . Definition 1.1.13 Let the operator T map the metric space H into itself. If there exists a real number q, 0 < q < 1, such that for arbitrary points x and x’ of the space H we have

P ( T ( 4 ,T(x” d 4 . P ( X , x’), then we call T a (strict) contraction operator. Definition 1.1.14

Let f , g E L,( - 00,co). The function

1W

h(x) =

m

1a3

f ( x - Y)AY)dY =

m

f’(y)g(x- Y ) d Y

is defined almost everywhere on the real axis and is called the convolution of the functions f and g. Definition 1.1.15 Let ( H , 3)be a topological space. We shall say that a c 9 is a base for this topology if for every 0 E 9 collection of subsets 9‘ there exists a subset 0’ E 9such that 0’ c 0. Definition 1.1.16 A linear topological space is said to be locally conuex if it possesses a base for its topology consisting of convex sets. We now state several theorems which will be needed in later chapters. The proofs of the classical fixed-point theorems are presented in Appendix l.A for the convenience of the interested reader. Theorem l.l.l (Minkowski’s inequality) (Natanson [l]) If f ( x )E L , and g(x)E L,, then

Theorem 1.12 ( S . Banach’s $xed-point principle) (Natanson [2]) If T is a contraction operator on a complete metric space H, then there exists a unique point x* E H for which T ( x * ) = x*.

Theorem 1.13 (Closed-graph theorem) (Goldberg [ 11) A closed linear operator mapping a Banach space into a Banach space is continuous.

10

I

PRELIMINARIES AND FORMULATION OF THE EQUATIONS

Theorem 2.2.4 (Halanay [l]) If f ( x ) ,g(x)E L,( - co,co), then the convolution h(x) is defined for almost every x , h(x)E L,( - co,co), and we have

J-

m

1m

m

Ih(x)l dx G

J-

m

If(x)l dx

Ig(x)l dx.

Theorem 2 . 2 5 (Halanay [l]) The Fourier transform of the convolution h(x) is the product of the Fourier transforms of the functions f ( x ) and A x ) . Theorem 2.2.6 ( P a r s e d equality) (Bochner [l]) Let f ( t )E L l ( - co,co) n L2(- co,co) m

f(U) =

Then

J-

e-'"(t)

Lemma 2.2.7

m

for A real.

dt,

m

lf(iA)I2 d A = 27t

and

J-

m

If(t)12dt.

(Barbalat [I]) If

(i) f ( t )is a continuous function, and its derivatives f ' ( t )are bounded for t 2 0; (ii) G(x)is a continuous function, G(x) > 0 for any x # 0, G(0) = 0 ; and (iii)

then

som

G [ f ( t ) dt ]
0 and 0 < u < 1 such that

{In

{ E l ~ , , ( o ) l=~ } ~ lxn(o)I2dP(w))'

< /3un,

n

=

1,2,3,. . . .

An important application of the theory which has been obtained for stochastic or random integral equations such as those given earlier is in the area of nonlinear stochastic systems. Throughout this book, particularly in Chapters IX and X, we will apply the theory obtained to stochastic differential systems or stochastic integrodifferential systems in various forms. As representative examples of such systems, we mention the following three : i ( t ;O ) =

A ( o ) x ( t ;O )

+ b(w)+(a(t;

0))

with ;w ) = i ( t ; o )=

(44, x(t ; 4) ; A(o)x(t;w)

+ b(~)4(a(t;o))

with

+

( ~ (t T;w),x(T;w))~T;

o ( t ; o )= f ' ( t ; ~ )

and i ( t ; o )=

+

A(o)x(t;w)

with

+

o ( t ; w )= j ( t ; o )

sd

J:

b(t - z ; o ) 4 ( a ( r ; o ) ) d r

(c(t - z;o),x(z;o))dz;

where (x, y ) denotes the scalar product in Euclidean space, . = d/dt, and (i) A(w) is an n x n matrix whose elements are measurable functions; (ii) x(t; o)is an n x 1 vector whose elements are random variables; (iii) b ( o ) and c ( o ) are n x 1 vectors whose elements are measurable functions;

22

I

PRELIMINARIES AND FORMULATION OF THE EQUATIONS

(iv) o ( t ;o)is a scalar random variable for each t E R , ; and (v) f ( t ; w ) is a scalar random variable for t E R , . Schematic diagrams illustrating some of the important stochastic differential systems that are presented in Chapter IX will be given in Appendix 9.A. Such diagrams are useful in the physical interpretation of stochastic differential systems. Finally, we state two definitions concerning systems such as those just given.

Definition 1.3.6 A matrix A ( @ )whose elements are measurable functions is said to be stochastically stable if .!3'{o:ReIl/k(o)
0. That is, the characteristic roots Gk(o), k negative real parts 9-a.e.

1,

= 1,2,. . . ,n,

have

Definition 1.3.7 A stochastic differential system is said to be stochastically absolutely stable if there exists a random solution x ( t ;o)of the system such that 9 { o :lim x ( t ; o)= O } = 1. 1-m

Appendix 1 .A

In order for this work to be more self-contained and complete, we will give in this appendix proofs of the fixed-point theorems of Banach and Schauder and the lemma of Barbalat which were introduced in Section 1.1. Several further definitions, lemmas, and theorems which are needed in the proofs but which were not given earlier will also be presented. We will begin with a proof of the fixed-point theorem of Banach. Theorem Z.A.2 ( S . Banach's fixed point theorem) If a contraction operator U is defined on a complete metric space E, then there exists a unique point x* in this space for which U(x*) = x*. PROOF

follows :

Let xo be any point in the space E . Construct a sequence as

XI =

U(x,),

x2 = U ( x , ) ,

. ..,

x, =

u(X"-J,

., ..

(This is called a sequence of successive approximations.) This sequence exists because U is defined on E into itself. Let p be the metric defined on E. Since

APPENDIX 1.A

U is a contraction operator on E, we have p(u(xn), u(xn- 1 ) ) = P(xn+ 1 ,x,) Q qp(x,,

- I),

X,

0 Q q < 1.

However, P(xn,xn-1) 6 q ~ ( x n l-, ~ n - 2 )

and hence 2

p(xn+lrxn) Q 4 ~ ( x n - l r x n - 2 ) .

Applying this argument repeatedly, we obtain (l.A.1)

P ( x ~ 1+r xn) Q qnP(xl XO). 9

Using the triangle inequality, we have for m > n

+ ... + ~ ( x m - 1 , ~ ~ ) .

P ( x ~ , x6~ P(xnrXn+l) ) + ~(xn+l,xn+z)

Therefore, using ( 1 .A.l), we obtain

+ qn+' + . . . + qm- ) = q"p(x,,x,)(l + q + . . . + qm-"-1)

p(xn,xm)Q p ( x l , x0)(qn

l

6 [qn/(l - ~ ) I P ( xXI O, )

+ +

since the sum of the series 1 q . . . is 1 / ( 1 - q) for 0 6 q < 1. This shows that {x,} is a Cauchy sequence. Since E is complete, then lim x,

n+ w

= x*

exists and x* E E. Since U is a contraction mapping from E into itself, we have P(&+ I , u ( x * ) ) = p( u ( x n ) ,U ( X * ) )6 qp(xn,x * )

which implies that as n + a, from the fact that p(x,, x * ) + 0 as n unique in E, we see that

+

co.Since the limit of {x,) is

x* = lim x n + , = U(x*). n- w

Therefore a fixed point of U exists in E. To show that x* is unique, suppose that x is another fixed point of U . Then P(X,

x * ) = d u b ) , U ( x * ) ) Q qp(x,x * )

< p(x, x * )

since 0 d q < 1, a contradiction. Thus x* is the unique fixed point of U in E, completing the proof.

24

I

PRELIMINARIES AND FORMULATION OF THE EQUATIONS

In order to prove the fixed-point theorem of Schauder, we will need a theorem of Brouwer which we now discuss.

Definition l.A.l A retraction is a continuous mapping from a space onto a subspace so that the subspace remains fixed. Theorem I.A.2 (Brouwer's jxed-point theorem) Any continuous map f :D" + D" has a fixed point x , f ( x ) = x , where D" = { x E En:1x1 d 1).

Theorem 1.A.2 is an immediate consequence of the following theorem. Theorem I.A.3 boundary.

There exists no retraction of a closed space onto its

PROOF OF THEOREM 1 .A.2 FOR n < 2 Suppose f : D" + D" has no fixed point. That is, for all x E D", f ( x ) # x . For x E D 2 we can draw a straight line through f ( x ) and x which intersects the circumference S' of the unit disk at some point p*. Then the transformation x + p* would be a continuous transformation of the whole disk D 2 onto its circumference S' and would leave each point on the circumference fixed, which contradicts the fact that such a transformation does not exist. Thereforef: D 2 + D 2has a fixed point. If n = 1, then D' = [-1, I ] and So = { -1, I}. Applying the same argument, we have f :D' + D' has a fixed point.

The proof for n > 2 is essentially the same. By assuming that f :D" + D" has no fixed point, it is shown that S"- is a retraction of D", which, by Theorem 1.A.3, is a contradiction.

'

Definition 1.A.2 Let B be a subset of a metric space and t > 0. A finite set of points { q l , q,, . . . ,4.} is called an &-netfor B if for every point x E B there exists a 4i,so that p ( x , qi,) < E . Definition 1.A.3 Let M be a compact set in a normed linear space and let

R be its closure. Let v l , . . . , v p be an &-netof R,and for x E R define

(1.A.2) where

Using this definition, we may obtain the following theorem, which is used in the proof of Schauder's fixed-point theorem. Theorem I.A.4 Let T be a compact transformation with domain W, a bounded subspace of H , a normed linear space, and let T ( W )c M . Let

25

APPENDIX 1.A

F, be defined on

R as given by Eq. (1.A.2). If x E M , then IITb) - F&T(X)ll< 6 .

PRooi:

By definition

by definition of mi(x),completing the proof. Theorem I.A.5 (Schauder's $xed-point theorem) Let M be a convex, bounded, closed set in a Banach space and let T be a compact transformation such that T ( M ) c M . Then T has a fixed point in M . That is, there exists an xo E M so that T(xo)= x o . PROOF Since T ( M ) c M , we have T ( M ) c R. The fact that M is closed implies that M = R,and hence T ( M ) c M . Let {E,} be a monotone decreasing sequence such that E, = 0. Let T, = F,,T be defined on M as described in Theorem 1.A.4. For x E M we have

T,(x) = F,,(T(X)).

But T(x)= y E M ; therefore Fen( T(x)) = F J y ) . Suppose { v l , . . . ,vp,} is an &,-netof T ( M ) .The function

which means that FJY)

=

m1(Y)V 1

mlb)

+ . . . + m,,(y)

Set mXy),"ml(y)

+ . . . + %(Y) + . . . + m,,(y)'

+ . . . + mp,(y)l

mP,(Y)vPn

=

Mi.

Therefore, since M is convex, F J y ) E M , which implies that T,(M) c M . Let H , be the finite-dimensional subspace of H which is spanned by { v l , . . . , v,,}. Let M , = M n H,. Now M is closed and H , is closed since it is a

26

I

PRELIMINARIES A N D FORMULATION OF THE EQUATIONS

finite-dimensional space. This implies that M , is closed since the intersection of two closed sets is closed. Also M is convex, and H , is convex since for XI E H,, i= 1

and for xz E H,,

=

2 y i v i € H,,

i= 1

where 0 < 4 < 1. Hence M , is convex (the intersection of two convex sets is convex). Therefore M , is a closed convex subset of H,. The transformation is defined on M and M , c M , which implies that T,, is defined on M , . Also, T,,(M,) c M , , for if x E M , , then x = ctivi, and

/

Pn

T,(x) = F,,v(X))

=

cpz

PF?

P"

i= 1

i= 1

1 mi[~(x)lvi 1 m i [ ~ ( x )=l 1 pivi

i= 1

E H,.

Also, T,,(x) E M , since if T,,(M) c M and M , c M , we have that x EM, c M and T,,(x) E M . Thus T,,(~)E M nH , = M , ,

which means that T,(M,) c M , . Since FE,is continuous and T is continuous, we have that T,, is continuous, and by using Brouwer's fixed-point theorem (Theorem 1.A.2),there is a point x, E M , such that T,,(X,) = x,.

-

The set ( T ( x , ) )is contained in the closed compact set T ( M ) . REMARK

The set T ( M )is compact since the &,-netgives a finite covering.

Now T ( M ) c M implies {T(x,)} c M . Thus T(x,) has a limit point xo and xo E M since M is closed. Either the sequence { T(x,)} converges to xo or there is a subsequence of { T(x,)} which converges to x o . For simplicity of notation, assume that { T(x,)) converges to xo. Then

(1 T(x,) - xo11 < E

for n > n ( ~ ) .

( 1 .A.3)

27

APPENDIX 1.A

T. we have

Also from the definition of

II T,(x,)

(1.A.4)

< En.

- T(x,)II

Then from (1.A.3) and (1.A.4) we have lIT#(x,) - xoll I

IIT.(X,)

- T(x,)ll

+ IIT(x,)

Since T,(x,) = x,, we obtain IIX,

- XgII

< En

+

- xoll

< c,

+ c.

E.

Let E‘ be given. T is continuous, and we have that there exists a d(d) > 0 such that

II V x , ) -

< E‘

T(X0)ll

whenever IIx, - xoll < d(~’).To make I/x, - xoII < d(~’),choose n large enough so that E , E < d(~’). We can d o this since E, = 0. Hence we have shown that

+

IIT(x,)

< -5

- W0)ll

whenever n is large enough, which means that T(x,) + T ( x o )as n Since the limit of the sequence { T(x,)} is unique, then we must have

+

co.

m,)= xo,

completing the proof. Finally, we present the proof of Barbalat’s lemma which was stated in Section 1. Lemma l.A.6 (Barbalat) If (i) f ( t ) is a continuous function, and its derivatives f ’ ( t ) are bounded fort 2 0; (ii) G(x) is a continuous function, G(x) > 0 for x # 0, and G(0) = 0;

(iii)

JOm

G [ f ( t ) ]dt
0 for k = 1,2,. . . , and some E > 0, such that If(tk)l

2

E

> 0.

28

I

PRELIMINARIES A N D FORMULATION OF THE EQUATIONS

We can further assume that for all k (1 .A.5)

tk+l - tk 2 m > 0

that is, the elements in sequence (1.A.5) are distinct, t , < t, < . . . < tk. If this condition does not hold, then we can choose a new sequence which would satisfy (l.A.5). Since f ‘ ( t ) is bounded for t 2 0, then using the mean value theorem, we can write I f ( t ) - f(tk)l

< blt

for all k.

- tkl

It is given that C ( x ) > 0 for x # 0, so we have

f {J

tk

JOm

G[f’(t)Idt 2

k=l

+( m i , )

G[l’(t)l d r } .

rr-(m/Z)

For the length of the interval m we can write t, -

tm d t d tk + i m

for all k .

(1.A.6)

On this interval (1.A.6) we can construct the following inequality:

r

=

min G ( x ) > 0 ;

&/ 0.

(2.2.9)

Inequality (2.2.9)can be majorized as follows : II(Tx)(t;o)II < Kllx(t; 4IIc,(a - PI- e-Of

(2.2.10)

because 0 < j3 < a. Dividing inequality (2.2.10) by e - P f , we have II(Tx)(t;o)llc, d K(a - P)-'Ilx(t; o)llc,.

Hence for x(t ;o)E C,(R+ , L,(R, d,9)) we have TC,(R+, L,(R, d,9)) c C,(R+ , L,(R, d, 9)), and the pair of Banach spaces (C,, C,) is admissible. The rest of the proof is due to Theorem 2.1.3.

2.2.2 A Problem in Telephone Traffic Theory In this subsection we shall examine a stochastic integral equation arising in the study of telephone traffic. We shall describe the problem in detail and then apply Corollary 2.1.5 to show existence of a unique random solution. Consider a telephone exchange, and suppose that calls arrive at the exchange at time instants t , , t,, . . . ,t,, . . . ,where 0 < t , < t , < . . . < t, < 00. These arrival times must be considered as random instants, so we denote the distribution function by A ( t ) on the time axis. For a call arriving at time t let the random variable H ( t ;o)denote the holding time, that is, the length of time that a "conversation" is held for a call arriving at the exchange at time t. The H ( t , ; o ) , H ( t , ; o ) , . .. are considered as being mutually independent for different times t , ,t, ,. . . and as being independent of the state of the exchange, where the state of the exchange is the number of busy channels.

2.2

SOME APPLICATIONS OF THE EQUATION

43

The number rn of trunks or channels of the exchange is assumed to be finite and large, so that we approximate a continuous process. It is also assumed that any channel not being used may be utilized by an incoming call and that the holding time for a channel begins at the time instant that the call arrives at the exchange. A conversation (or connection) is realized if a channel is not busy at the time a call arrives. If all channels are busy at the time t that a call arrives, then either the call is lost or a queueing problem develops. Only the first case will be considered here. Various problems have been studied in this situation. For example, the probability Pk(t)that at time t, k of the rn channels are busy has been examined in detail (Fortet [I]). We are concerned with the total number of “conversations” held (the number of busy channels) at time t, which for each t is a random variable and may be described by a stochastic integral equation. Let x ( t ; o)be the total number of conversations held at time t . That is, x(t; w ) is a random variable for each t E R , , and x(0; o)= 0. Let J ( t ; o) be a random function with value one if a call arising at time t > 0 is not lost and value zero if the call is lost. Let K ( t , z ; o )=

i

1

0

- z~[O,H(t;w)], if t - z $ [0, H ( 7 ; w ) ] ,

if

t

such that K ( t , z ; o)is equal to one if a conversation from a call arising at time z is still being held at time t 2 z and is equal to zero otherwise. Thus we may write

~ ( t0) ; =

sd

J ( z ; w)K(t,t ; W ) dA(z).

(2.2.1 1 )

Equation (2.2.11) is interpreted as the total number of telephone calls arising at times z, 0 d T d t , that were not lost such that the conversation is still being held at time t. Suppose that V(k) is any function such that V(k) =

i 1

if k = 0 , 1 , ..., r n - I ,

0

otherwise.

Clearly, x(t ;o)< rn for all t E R , and w E Q. Hence we may write V [ x ( t ;o)]=

i

1

if x ( t ; o ) = O , l , . . . , r n - 1 ,

0

otherwise,

which means that V [ x ( t ; o ) ]has value one if a call arising at time t is not lost and value zero otherwise. Then Eq. (2.2.11) may be written as the

44

I1

SOME VOLTERRA TYPE EQUATIONS WITH APPLICATIONS

nonlinear stochastic integral equation ~ ( to) ; =

J:

K ( t , 7 ;w ) V [ x ( z w ; ) ] dA(z),

sd

K ( t , T ; w ) V [ x ( z o; ) ] a ( z )dz.

(2.2.12)

which Bharucha-Reid [7] refers to as the Fortet integral equation. Suppose that the distribution A ( t ) of arrival times has a density function a(t).Then Eq. (2.2.12)reduces to a stochastic integral equation of the Volterra type x ( t ; o)=

If we let

f ( z , x(z ; 0)) = V [ x ( z; o)]a(z) =

a(r)

i o

if x(z; o)= 0, 1,. . . , m - 1, otherwise,

then we obtain a stochastic Volterra integral equation of the form of (2.0.1), with the stochastic free term identically zero, ~ ( tW;) =

JO

K ( t , z; o ) f ( z , x(z; w ) )dz.

(2.2.13)

K ( t , z ; o ) is the stochastic kernel .defined for 0 d 7 d t < co and taking only the value zero or one. Before showing that (2.2.13) possesses a unique random solution, we observe that the above description applies to many systems. If we replace the words “telephone exchange” with “serving mechanism,” and the words “channel,” “call,” and “conversation” with the words “server,” “customer,” and “service,” respectively, then we are dealing with a general system in which “customers” are being “served” by rn < co “servers.” If we assume that a customer does not wait when he finds all m servers busy so that no queue develops, then the random solution of the stochastic integral equation (2.2.13) gives the total number of “services” being performed at time t. Also, the functions in (2.2.13) may be any functions which behave as given in Chapter I and describe the physical situation. For example, the stochastic kernel may be of the form

K(t, r ; o)= I,,,,(t

- z) e-(‘-‘),

where ZX(,)( . ) is the indicator function of a random set X(w), which means that solutions at earlier times z ,< t have a decaying effect on the system. We now show that the stochastic integral equation (2.2.13) satisfies the conditions of Corollary 2.1.5. We first show that f ( t , x(t ; o)) E L,(R,

4 9) and

K ( t , z ;w )E L,(R, 4 9).

2.2

45

SOME APPLICATIONS OF THE EQUATION

Let t 2 0 be fixed. Since a(t) is a density function, it may be assumed to be bounded for all t except on a set of measure zero. Hence for some M > 0 and all t, 0 < a(t) < M 0 there is a 6 > 0 such that when I(t

[+h Therefore Also,

X ( t ; w ) is

J‘,

a(z)dz

+ h) - tl < 6

< E.

continuous on R + for each ~ Ix(t; w)12 d 9 ( w )

s m2
, 0, of the linear hereditary phenomenon. The linearized version of this problem leads to the integral equation

(2.2.14)

2.2

47

SOME APPLICATIONS OF THE EQUATION

Figure 2.2.1

Is

From Distefano [ I]

where f ( t , T ) is the “memory function” for the hereditary phenomenon and g(t) = c4t) +

sd

f’(u)dT)dT,

assuming initial straightness of the bars, that is, s ( t ) = 0 at t = 0. When the axial load has a small, randomly fluctuating component, that is, q ( t ;w), w E R, is a random variable for each fixed t > 0, then Eq. (2.2.14) reduces to the two stochastic Volterra integral equations u ( t ; w ) = g ( t ;w )

+

u ( t ; w )= G ( t ; w )

s:

4 ( ~o)f’(t, ; t)u(z;w ) dt,

+ &t;w)

where it is assumed that 0 d q(t ;w ) d

sd

f’(t,t)u(T;w)dz,

p < 1 P-a.e. for all t E R ,

. Also,

are random variables for each t E R , . Then the quantity of interest the sum of the two random solutions u(t ;w ) + u(t ;w). Let

s ( t ; w ) is

and

Then the random integral equations reduce to two stochastic integral equations each of the form (2.0.1) with f ( t , x ( t ; 0))= x ( t ; w), that is, linear

48

I1

SOME VOLTERRA TYPE EQUATIONS WITH APPLICATIONS

stochastic integral equations,

(2.2.15)

Suppose the "memory function" is of the form

Since 0 d q(t ;w ) < 9-a.e. for all t E R + , $(t ;o)is bounded 9-a.e. on R + , and we may assume a ( t ;o)and s(t ;w ) are bounded 9-a.e. and continuous on R , by the nature of the physical situation. Thus there is some M > 0 such that & t ; o)= q(t;o)/[l - q ( t ; o ) ]d M

9-a.e.

for all ~ E R ,

Hence

and

Also, f ' ( t ,0) = 0, since f ( t , u ( t ; 0)) = u ( t ; w) and f ( t , u ( t ; 0)) = v ( t ; w) by comparison of (2.0.1) and (2.2.15), and G ( t ; o ) and g ( t ; w )are continuous and bounded 9-a.e. on R + since $(t ;w )and a(t ;w )are assumed to have these properties. If in Corollary 2.1.6 we take the function g(t) = 1 for all t E R+,then sup

tsR +

{

l"lg(.)dz}

= 1 < co.

Also,

so that il = 1. We may take A2 = M Q > 0 and u = Q > 0, and then all conditions of the corollary are satisfied. Therefore there exists a unique random solution of each of the equations (2.2.15), provided that Ilg(t;o)ll and IIG(t;w)ll are small enough.

2.3 2.3

THE RANDOM INTEGRAL EQUATION

49

The Random Integral Equation

+

x ( t ; o)= h ( t ; o)

K(u, X ( U ; w ) ; o)du

This equation occurs in many important situations, as has already been stated. To complete the objectives of this part of the book, we shall investigate the existence and uniqueness of a random solution of the stochastic integral equation (2.0.2).Some results of Bharucha-Reid, Mukherjea, and Tserpes [ 11 will be given which use methods of probabilistic functional analysis from Chapter I. Also a theorem of the authors will be presented which utilizes some of the concepts of “admissibility theory” as was discussed in Chapter I and employed in Section 2.1. We shall first assume that the functions x(t; o), h(t;o), and K(u, x ; o)in Eq. (2.0.2)are real-valued, x and h are product-measurable on R + x R, and K(u, x ; o)is product-measurable on R , x R for each x. Theorem 2.3.2 Equation (2.0.2) with t E R + has a random solution if the following conditions are satisfied :

(i) The kernel K ( u , x ; o ) is a continuous function of u and x, U E R , and x E R for each o E R, and K(u, x ;o)is a random variable for each u and x. (ii) supx ( K ( u ,x ; o)l d g(w) E L,(R, d,P), 1 d p < 00. (iii) Let h(t ; o)bea continuous map from R , into L,(R, d,P),1 d p < 00, and Ih(t;o)l

+

1;

sup lK(u,X ; a)\ du X

1, and we have

j:

< ~ ( t+) mc= l J(c/v*)m m

tt(t)t

(s

Km(t, Y)G(Y) d y

1

m

1 (-

C / V * ) ~ ( Q 'u*/m!), ~~;~+

m=l

where t , = min(t, t,), 0 d t < t. The series on the right converges by the ratio test. Hence there exists a solution of (2.4.4) for 0 < t < t. Suppose that t < t < M , z < M < 00. Then Eq. (2.4.4) is u,(t; O ) =

G(t)

+

c

k(s,u,(s; 0); O )ds.

For o E SZ the concentrations uL(t; o)and u(l, t ;o)can be considered as continuous functions of time, and hence k(t, u L ( t ;w ) ;o)is continuous in t and uL. For each t and uL, k ( t , u , ; o ) is a constant. Hence k ( t , u , ; o ) E L ~ ( S Z , ~ 1, ~d )p, < GO, trivially, and Condition (i) of Theorem 2.3.2 is satisfied. By the nature of the physical situation, the maximum value of u,(t; o)and u(l, t - t ; o)is u*, and the minimum value is zero for all t E [t,MI. Thus suplk(t, u,; o)l = sup{(c/V*)lu,(t; UL

0)-

u(l, t - t ;o)l}= (c/V*)u*.

UL

Define g(0) = cu*/v*,

oER,

and then g ( o )E L,(Q, d,9),since

Therefore Condition (ii) of Theorem 2.3.2 is satisfied.

62

I1

SOME VOLTERRA TYPE EQUATIONS WITH APPLICATIONS

The stochastic free term h(t ;w ) is the deterministic function G(t),which is bounded, IG(t)l d /oT(f)(c/V*)u,(s)dsd t,cu*/V* < co

for all t E [0, MI. Hence h(t ;w ) is in L,(R, d,9). Obviously, h(t ;w ) is continuous in t since G(t) is an integral. Let t E [T, MI c [0,MI. Then Ih(r; w)l Take

+ J: suplk(s, u,; w)l ds d t,(cu*/V*) + t(cu*/V*) d (cu*/V*)(t,+ M ) .

+ M)]+ 1
0 there exists an N > 0 such that whenever n > N,

{In 4 luL,,(t;

-

<E

U L ( ~ ;w ) I P d W 4

~ .

lip

Therefore, letting E = (c/V*)tI > 0, we have for n > N

=

is,

(c/V*)

lUL,"(r; 0) -

1

u,(t; U ) ( P d Y ( 0 )

< ( C / V * ) E 1 = E.

lip

That is, k( t , uL,,(t ;w );w ) converges to k( t ; u,( t ;w ); w ) in L,(R, d,9'). Hence Condition (iv) of Theorem 2.3.2 is satisfied. Therefore for T d t d M < co, t, < M , there exists a random solution of Eq. (2.4.4). Thus we have shown that the semistochastic integral equation (2.4.4) possesses a semirandom solution for all t E [0, M I , where 0 < M < co, T < M , and t, M.

-=

2.4.2.b

TWO-ORGAN BIOLOGICAL

SYSTEMS

In this section we will extend the results ofthe previous section to biological systems which consist of two organs.

2.4

APPLICATIONS OF THE INTEGRAL EQUATION

63

Consider a closed system with a simplified heart, two organs or capillary beds, and recirculation of the blood with a constant rate of flow assumed, where the heart is considered as a mixing chamber of constant volume. With respect to the flow of blood, we assume that there is no mixing in the vessels as before. This system is not as simplified as it may appear at a casual glance, since we may consider one organ as a certain organ of the body, such as the lungs, and the other organ as being a collection of the remaining capillary beds or organs of the body. See Fig. 2.4.2 for a schematic description of the system. As in Section 2.4.2.a, it is assumed that an injection of drug, or a chemical agent, is given to the system directly at the entrance of the heart, producing a known concentration of drug in the blood plasma entering the heart; that is, drug particles are dissolved in the blood plasma. Also, as the blood passes through each of the organs, the particles of drug are assumed to enter the extracellular space only by the process of diffusion through the capillary walls. Due to the nondeterministic nature of a diffusion process, a random amount of drug is removed from the blood plasma in the organs, and hence the concentration of drug in the blood is reduced by a random amount. Therefore it is impossible to know exactly the point-by-point concentration of the drug in the blood plasma and in the extracellular space of the organs at any given time after the blood containing the drug passes through the organs. In addition to the notation already introduced in the previous section, the following will be used : uj(s,t ; o)is the random concentration of drug in organj at point s in the capillary at time t, for j = 1,2. c j is the constant-volume flow rate of blood in organ j , j = 1,2, and c = c 1 + c2 is the total constant-volume flow rate of blood in the system.

Again, suppose that the injection of drug produces an initial concentration

u,(t), 0 d t d t,, entering the heart, and let z denote the time lag due to the

64

11

SOME VOLTERRA TYPE EQUATIONS WITH APPLICATIONS

circulation of the blood as described in Section 2.4.2.a. Thus the drug enters each organ after a time z, and due to the diffusion process in the organs, the after time T is a random concentration of drug entering the heart, u , ( t ; o), variable for each t 2 T and is given by

u&;

4=

r

t < 0,

u,(t),

0 d t,

u,(t)

T

+ { [ c , u , ( l ,t - T ; w ) + c,u,(l, t - T ; o ) ] / c }

d t d M,

where uj(l, t ; o)is the concentration of drug in plasma leaving organ j (at the I end) at time t , j = 1,2, and ul(t) = 0 for t > t,. Also, the concentration of drug in plasma leaving the heart at time t 2 0 is given by the semirandom solution u,(t ; w ) of the semistochastic integral equation (2.4.3). We showed in the previous section that this semistochastic integral equation has a semirandom solution for each t, 0 d t d M , where M < cc is a constant. Then the concentration of drug in plasma entering organ j at time t is given by

where j = 1,2. These models for chemotherapy are realistic in that they retain such properties as recirculation of the blood, mixing in the heart, the presence of more than one organ, and randomness in the diffusion of the drug into the organ tissues, even though several simplifying assumptions were made in order to deal with the mathematics involved. Such assumptions seem necessary in obtaining mathematical descriptions of biological systems since they are in general very complex systems.

C H A P T E R 111

Approximate Solution of the Random Yolterra Integral Equation and an Application to Population Growth Modeling

3.0

Introduction

In this chapter we shall present some methods of approximating the unique random solution x ( t ; o ) of a stochastic integral equation of the Volterra type of the form ~ (; Wt ) = h(t ; W )

+

rl

J O

k(t, z ;w)f’(~,X(Z ;w ) )d ~ ,

(3.0.1)

where x ( t ; w), h ( t ; o),k(t, z ; o), and f’(t,x) behave as described in Chapters I and 11. We shall consider the problem of obtaining an approximation to a realization of x ( t ; o)by the method of successive approximations at each t E R , and also by applying some of the theory of stochastic approximation. 65

66

111

SOLUTION OF THE RANDOM VOLTERRA INTEGRAL EQUATION

A new stochastic formulation of a population growth model will also be given along with a numerical example. In the method of successive approximations we shall investigate the convergence of a generated sequence of random variables to the unique random solution x ( t ; w ) at each t E R+ . Also, the rate of convergence, the maximum error of approximation, and the combined error of approximation with the error of numerically evaluating the integral are considered. A general theorem of Burkholder [l] in the theory of stochastic approximation is also applied to Eq. (3.0.1) resulting in conditions under which a sequence of approximations converges with probability one to the unique random solution x ( t ; w ) at each t E R , . 3.1

The Method of Successive Approximations

Let C C ( R + , L 2 ( Q d , 9 ) )be the space described in Chapter I, and let B, D c C c ( R + , L 2 ( Q d , 9 ) )be Banach spaces with the norm in D defined such that

Let

s = { x ( t ; w ) : x ( t ; a ) E D , llx(t;w)II~d p } ,

as in Theorem 2.1.2, with x ( t ; w), h ( t ;w), f(t, x(t; w)), and k(t, T ; w), for 0 d z d t < 00, behaving as described previously. As in the existence proof of Theorem 2.1.2, let U be the contraction mapping from S into S defined by (Ux)(tw ; ) = h ( t ;w ) +

k ( t ,T

; ~ ) ~ ( T , x (w T ); )d z ,

which has the unique fixed point x ( t ;w). It is assumed here that the distribution of the random variable h(t ; w ) is known at each t E R + ,or that a value of h(t ;w )can be observed at each t E R + . Define the sequence of successive approximations {x,(t ;w ) } by x,(t; w ) = h ( t ;w),

x , , l ( t ; w ) = ( U x , ) ( t ;w),

n 3 0.

(3.1.1)

The sequence defined recursively here is contained in the set S, which is a result of the following lemma. Lemma 3.2.1

h ( t ;w ) E S and hence x , ( t ; a)E S, n = 0, 1,2,. . . . Also, x,(t;w)3 x(t;w)ES.

3.1 PROOF

THE METHOD OF SUCCESSIVE APPROXIMATIONS

67

From the condition of Theorem 2.1.2

Ilh(t;

w)llD


0, we have P { m :(X(w)l2 a } d EIX(o)lr/ar,

if EIX(w)l' exists. Theorem 3.1.2

(Lotve [l], p. 173) If for some r > 0

c EIXn(4 m

n= 1

then

X(W)l' < a,

-

X n ( 0 )".1.X ( 0 ) .

PROOF

By the Markov inequality, for every E > 0 P { 0 :I X n ( 0 )- X(W)l B

E}

d EIX,(o)

-

X(O)I'/E*

for every n 2 1, r > 0. Hence m

1 P {0 :IXn(0)

-

n= 1

m

X(0)l B E } d

1 [EIX,(u) - X(OJ)I~/E']
0, by hypothesis. But

and since

c P{o:(X,(o) X(0)I 2 m

-

n= 1

E}


0, the sum on the right in (3.1.2) must tend to zero as n Hence, B U [0:1xk(cC))-x(@)I 2 E] - 0 as n - , 00, {kkn

t Sections 3.1.1-3.1.3

Francis, Ltd.

I

+

co.

adapted from Padgett and Tsokos [S] with permission of Taylor and

3.1

69

THE METHOD OF SUCCESSIVE APPROXIMATIONS

and X,(w) -,a.s. X(w),by definition. We shall now show that the sequence of successive approximations (3.1.1) converges a s . to the unique random solution of (3.0.1). Theorem 3.1.3 For each t E R + ,xn(t;o)+a.s. x(t ;w ) under the conditions of Theorem 2.1.2. PROOF

By definition, for t E R , ,

Repeating the argument n - 1 times, we have

m

< 1(AK)”2p = 2p/(l n=O

-

AK)
0 such that for k > N

so that for k > N we have

+

co,

3.1

THE METHOD OF SUCCESSIVE APPROXIMATIONS

71

Therefore, by Theorem 3.1.2, for each t E R + x,( t ; o)"2x(t ;w).

Thus the sequence of successive approximations converges to the unique random solution x ( t ; o ) with probability one for each re R + . Therefore the sequence (x,(t ; o)}converges to x(t ; o)in probability and in distribution for each t E R + . As a by-product of this theorem, we also obtain that x,,(t ;o) converges to x(t;o) in mean-square (quadratic mean) for each t E R , , since m

1 Elx,(t;w)

- x(t;o)12


0 and c2 > 0 we may choose r so small (n large) and m so large that I(f:m+"(o) - X,(O)IID < E l 1- E* = E.

As an example, suppose that we use a quadrature such as the composite trapezoidal rule (Kopal [l], pp. 397-410). The error of approximating the integral at each W E Ris then of the order of r3. Hence, using r as small as possible gives a good approximation for sufficiently large m. In fact, if the interval [0, t ] is divided into n subintervals such that r = t/n, then the error of integration by the composite trapezoidal rule of a function g(t ;w ) is of the form P ( W ) = &nr3g"(5 ;o), where 5 is some point in [0, t] and the double prime indicates second derivative with respect to t . Hence here we must assume that k(t, T ;o)f'(z, x(z; 0)) has a second derivative with respect to t,and as r ---t 0 (n -P a), ll6(")(o)ll -+ 0.

3.2 A New Stochastic Formulation of a Population Growth Problem

In mathematical models for biological processes it is usually the case that a complex biological system is replaced by a simpler, idealized, hypothetical one (Bartlett [l, 21, Moran [l], Chiang [l], Bharucha-Reid [S]). Many simplifying assumptions on the actual biological system may still result in a very complicated mathematical or statistical model. In obtaining a mathematical model of a biological system, either the random or stochastic changes in the system may be ignored and a deterministic model reflecting "averages" of the random phenomena used, or the random variations may be accounted for, resultingin a stochastic model. The latter case is more realistic than the former and should be used even though in most situations stochastic models are much more difficult to work with mathematically. In formulating a mathematical model for the growth of a biological population consisting of a single species, such complications as age structure and random changes must be dealt with. In much of the classical theory of population growth in which the age structure is considered the changes produced in the population growth rate by the phenomena of birth, aging,

3.2

A NEW STOCHASTIC FORMULATION

79

and death are assumed to be deterministic (Feller [l], Kendall [I], Bartlett [I, 21, Moran [I]). In this deterministic theory the expected birth rate satisfies an integral equation which Feller [I] calls the renewal equation. This integral equation involves the expected reproduction rate of female individuals of a given age and the expected rate of reproduction of females at a given time by members of the parent population. In this section we shall present a formulation ofa population growth model which results in a stochastic integral equation that is similar in form to the deterministic integral equation just mentioned. However, the solution to the stochastic integral equation is a stochastic process giving the birth rate instead of the expected birth rate in the population. We will also show that the expected value of the birth rate process satisfies the deterministic integral equation previously mentioned. The stochastic integral equation obtained is of the Volterra type given by Eq. (3.0.1). The theory of random or stochastic integral equations of this type given in Chapter I1 and Section 3.1 permits one to show existence and uniqueness and to obtain an approximation to a realization of a random solution x(t ;a), of the process x ( t ; u) (also see Anderson [I], Tsokos [4],Padgett and Tsokos [5,6, 113, Milton and Tsokos [2,3], Hardiman and Tsokos [2], A. N. V. Rao and Tsokos [2]). This theory allows one to obtain the existence and uniqueness of a random solution x ( t ; o)without specifying the exact distributions of the stochastic processes at each time t . That is, the stochastic processes which constitute Eq. (3.0.1) may be very general processes. In Section 3.2.1 we shall discuss the deterministic model, and we give the stochastic formulation of the population growth problem in Section 3.2.2. We shall show that it completely specifies the state of the population at each time t and indicate the connection between the present formulation and the deterministic formulations mentioned earlier. In Section 3.2.3 we will show that the stochastic integral equation obtained in Section 3.2.2 has a unique random solution under certain conditions. Finally, we will give an example to indicate the fruitfulness of the stochastic integral equation formulation.

3.2.1 The Deterministic Model Consider the effect of age structure on the growth of a population consisting of a single species, in the absence of external influences such as emigration or immigration, dependence on the density of the population, and disease. In the classical theory the size of the population is treated as a continuous variable and it is supposed that the modifications produced in the population due to the phenomena of birth, aging, and death are deterministic (Kendall [I]).

80

111

SOLUTION OF THE RANDOM VOLTERRA INTEGRAL EQUATION

The variables such as the birth rate and number of individuals in the population are usually interpreted only for the female or reproducing portion of the biological population with the possibility of unequal sex ratio being ignored. Let A(t) dt be the expected number of females born during time (t,t + d t ) to females aged t as in Feller [I], Bartlett [l, 21, and Kendall [l]. Also, b(t)d t is the expected number of female births occurring in the time interval ( t , t + dt) (the expected birth rate), and n(s, t ) ds is the expected number of females in the population at time t in the age group (s, s + ds).Then the total number of females in the population, that is, the size of the female population at time t, is the quantity

where l ( t ) is the probability that an individual born at time zero is alive at time t > 0 and is given by (3.2.2) where p(s) is the death rate of individuals aged s, 0 (Kendall [ 11) n(s, t ) = l(s)b(t - s)

< s < t . Then we have that (3.2.3)

and that (3.2.4) where (3.2.5) The function g(t) is completely specified if the age structure and population size are known at the time observation of the population begins, that is, at epoch t = 0. The integral equation (3.2.4) was studied in detail by Feller [ 11. He showed that under certain conditions Eq. (3.2.4)possessed a unique solution, and also the asymptotic behavior of the solution was investigated. He suggested the method of successive approximations in order to approximate the expected birth rate b(t)at each time t > 0, after presenting a treatment of the equation by Lotka [l]. The paper by Feller [l] contains an extensive list of the research papers concerning Eq. (3.2.4) up to 1941.

3.2

A NEW STOCHASTIC FORMULATION

81

3.2.2 The Stochastic Model It should be evident that the deterministic theory ofpopulation growth does not provide an adequate and realistic description of the processes involved, because it does not take into account the random fluctuations which occur as the process develops. Using the theory of stochastic integral equations which was given in Chapter 11, we will formulate stochastic versions of Eqs. (3.2343.2.5). It will not be required to specify the exact behavior of the processes because of the very general theory involved. However, we will show that the stochastic formulation is related to the “expected value” (deterministic) formulation under the usual kind of assumptions that are made in the stochastic population models. Let m ( t ; o ) be a stochastic process which enumerates the number of offspring (female) produced by individuals (females) aged t at epoch t > 0. Then m(t ;u) dt is the number of offspring produced by individuals aged t in the interval ( t , t + dt). It is clear that m should be treated as a stochastic process, since at any epoch t the number of offspring produced by individuals aged t is random. We will consider m to be a continuous function o f t for almost all u,which approximates the number of offspring at each t 2 0. That is, for almost all o we assume that the stochastic process m(t;u) has continuous sample functions. Hence for each t 2 0 the number of female births at that epoch in the population is a random variable b(t;u).That is to say, b ( t ;o)is a stochastic process, and the number of individuals aged s at epoch t in the population is given by

n(s, t ; O)= I(s)b(t - S ; W )

(3.2.6)

and the process b(t ;u) must satisfy the stochastic integral equation of the type (3.0.1) given by b ( t ; w )= g ( t ; o )

where

+

J:

I(s)m(s;o)b(t- s ; u ) d s ,

g ( t ; w )= I(t)Jomrn(t+ u ; u ) n ( u , O ; u ) d u .

(3.2.7)

(3.2.8)

The process g ( t ; w ) is completely specified if the distribution of the population size at time zero and the behavior of the process m ( t ; o )are known. Thus the size of the population at time t 2 0 is a random variable and is given by n(t ;W ) =

s:

n(s, t i O)ds

the stochastic version of Eq. (3.2.1),

=

l(s)b(t - s ;W ) ds,

(3.2.9)

(12

I11

SOLUTION OF THE RANDOM VOLTERRA INTEGRAL EQUATION

Before investigating the stochastic integral equation with respect to the conditions which guarantee a unique random solution we will consider its relation to the classical stochastic theory of population growth. We assume that the process N ( t ; w ) is a discrete valued stochastic process giving the number of individuals in the population at time t. Assume the following :

(i) The subpopulations which are generated by two coexisting individuals develop independently of each other. (ii) An individual (female) age x existing at time t has a chance A(x)dt

+ o(dt)

of producing a single offspring (female) during any time interval of length dt, where A(x) is the same function for all individuals in the population. Let d N ( x , t ; o)denote the number of individuals in the small age interval ( x ,x d x ) at time t, and

+

E [ d N ( x ,t i o)]= ~ ( xt ),dx

+ o(dx).

This is also the variance of d N ( x , t ;w ) to the first order. [That is, d N ( x , t ;w ) is of Poisson character.] It is assumed that dN(x,t ;w ) = 0

with probability

1 - a(x, t ) d x

1

with probability

a(x, t ) dx

2

with probability

o(dx).

=

+ o(dx)

+ o(dx)

Also, let d M ( x ; o)denote the number of offspring produced by individuals aged x in the time interval ( x , x + dx). It is assumed that

+ o(dx)

d M ( x ;w ) = 0

with probability

1 - A(x) dx

= 1

with probability

A(x)dx + o(dx)

22

with probability

o(dx),

where E [ d M ( x ;o)]= A ( x ) dx

+ o(dx).

IfA(x) = A, a constant, then d M ( x ;w ) has the same probability distribution at each x . If the process dB(t ;w ) denotes the number of new births (integervalued) in the population in the time interval ( t ,t + dt), then d N ( x , t ;0)= l(x)dB(t - x ;w). Assume that the death rate p(s) is the same for all ages s.

3.2

83

A NEW STOCHASTIC FORMULATION

Then from Eq. (3.2.2) we have l(t) = e P p ' . Hence dN(x, t ; w ) = e - p xdB(t - x ; 0).

(3.2.10)

The joint probability distribution of dN(x, t ; w ) and dM(x; w ) is given to the first order by

I dM(x:oj

i

1 1

Thus

T$a,

dN(x,I :

o 1 - a ( x , r j dx 0 1 - a ( x , I ) dx

0)

1

Total

[ l - I ( x )d x ] a ( x ,1 ) dx

1 - I ( x )dx . a ( x , I ) dx R(x)dx . a ( x , t )dx

L(xj dx . a ( x , t ) dx a ( x , t ) dx

1

E[dM(x ;w ) dN(x, t ; w ) ] = A(x) dx . a(x, t ) dx

and hence cov[dM(x ; w),dN(x, t ; w ) ] = 0 ; that is, dM(x ; w ) and dN(x, t ; o)are uncorrelated. Then cov[dM(x ;w)f(x),dB(t - x ;w)] = cov[dM(x ;w ) e - p X ,dN(x, t ;w ) epx]

=o

from Eq. (3.2.10) and the given covariance. For the continuous case, therefore, we make the similar assumption that the processes m ( x ;w ) and n(x, t ;w ) are uncorrelated. Hence if we take the expectation of the stochastic integral equation (3.2.7), we obtain

E [ b ( t ; o ) ]= E[g(t;w)]+ or

b(t) = g ( t ) = g(t)

+

1;

Jo E[l(s)m(s;w)b(t

- s;w)]ds

f(s)E[m(s;w)] . E[b(t - s; w ) ] ds

+ J: f(s)i(s)b(t- s) ds,

since E[g(t;w)]= f(r)J"

E[m(t

som

= l(t)

0

A(t

+ u;w)n(u,O;w)]du

+ u)n(u,0) du = &).

(3.2.11 )

84

111

SOLUTION OF THE RANDOM VOLTERRA INTEGRAL EQUATION

Equation (3.2.11)is the deterministic integral equation (3.2.4)given by Feller [I], Bartlett [l, 21, Moran [I], and Kendall [l]. We make the assumption on the correlation of m and n in order to obtain (3.2.11). This assumption is not necessary in the general theory of stochastic integral equations of the form (3.0.1) as given in Chapter 11.

3.2.3 Existence and Uniqueness of a Random Solution As Feller [l] did for the deterministic integral equation (3.2.4),we will now obtain conditions under which the equation (3.2.7)possesses a unique random the population birth rate, bounded for t 2 0. solution b(t ;o), By changing the variable of integration we may rewrite the stochastic integral equation (3.2.7) as

b(t;o)= g ( t ;w )

+

J:

(3.2.12)

I(t - s)m(t - s; w)b(s;o ) d s .

Then the stochastic kernel in Eq. (3.2.12) is, for 0 d

T

dt
0 such that JOm

n(u,O;o)du< N o

for almost ail w e n .

Then g ( t ; o)= JOm I(t)m(t

+ u ; o)n(u,0 ;o)du

< e-”‘A*N0 c co

for almost all w

by the assumption that m(t ;o)is bounded by A* and I([) = e-”’. Therefore /!g(t; o)I2d.9’(o) 6

s,

le-”“A*NoI2W w )

= (e-”‘A*No)’
0, if Ix - 81 > E, then ( x - 8)M(x) > 0. (ii) suPx(lM(x)I/(l + Ixl)} < (iii) supx V ( x ) < co.

3.3

METHOD OF STOCHASTIC APPROXIMATION

89

(iv) If 0 < 6, < 6, < co,then inf

61s1x-e1$62

I,"=

IM(x)l > 0.

Enw=

(v) a, = co and u," < 00. (vi) M ( x ) and V(x) are Bore1 measurable.

Then P{w:limn+wx,(w)

=

0) = 1.

The conditions in Theorem 3.3.1 seem to be more general than those given by Blum [l], although the proofs are similar. In the next section this theorem will be applied to the stochastic integral equation (3.0.1). 3.3.2 Solution of Eq. (3.0.1) by Stochastic Approximation We have that for each t and z satisfying 0 d z d t < 00 the variances existforh(t;w),k(t,r;w),andx(t;w),sinceforeacht~ R , , h ( t ; w ) a n d x ( t ; o ) are in L,(R, d,9)and for each t and z, 0 < z < t < co, k(t, z ; o)is in L,(R, d,9). Again we assume that the distribution function of h ( t ; o)is known for each t E R + or that a value of h(t ;w ) can be observed for each t . It is also assumed that x(t ; a),h(t ;w), and k ( t , z ; w ) are mutually independent, real-valued random variables at each t and z, 0 d z d t < co. As in Section 3.1, we write (Ux)(t;w)= h ( t ; w )+

sd

k(t,z;w)f'(r,~(r;o))dr

for t E R , . By Theorem 2.1.2 there exists a unique random solution x(t; o)of (3.0.1), (Ux)(t;0)= x(t; 0). Let

( Y x ) ( t ; w )= x ( t ; o ) - ( U x ) ( t ; o )

for x a continuous, real-valued function from R , into L,(R, d,P )contained in the set S of Theorem 2.1.2. For fixed r E R , define M[x(t)] = E[( Y x ) ( t ;o)lx(t; o)= x(t), =

x(z; w ) = O(r)

for z < t ]

E[x(t; w ) - (Ux)(t;w ) l x ( t ;o)= x(t), X(T;o)= H(r) for T < t ] ,

where O(z) is a realization of the unique random solution of Eq. (3.0.1). That is, we have already obtained values O(z) of the unique random solution for z < t . Hence we now wish to find the value O ( t ) such that

90

111

SOLUTION OF THE RANDOM VOLTERRA INTEGRAL EQUATION

That is, a value of the unique random solution is now to be obtained at tER,. Suppose that at the fixed t E R + we choose x,(t; o)= h ( t ;o), x,+ , ( t ; w ) = x,(t; o)- a,y,(t; o), n 2 1, (3.3.2) where y,(t ;w ) is the random variable (Yx,)(t; o).We apply Theorem 3.3.1 to obtain conditions for which the sequence {xn(t;o)}defined by (3.3.2) converges to e(t)with probability one. We have M[x(t)] = E[x(t;o) - (Vx)(t;w)lx(t;o) = x(t),

X(T;W) =

O(z) for T < t ]

= E[x(t; O ) - h(t ; O )

-Jo k(t,t;w)f(z,x(.r;o))dzlx(t;o)x(t),x(z;o) =

=

B(z)ifr < t ]

(3.3.3)

which exist by assumption. We will now show that Condition (i) of Theorem 3.3.1 holds at fixed t E R + . Let x(t) < e(t).Then from (3.3.3) we have M[x(t)l = x(t) - /dt)-

< e(t) - P h ( t )

-

=o

f f 0

Pk(t,

z)f(T, e(T)) dl

Pk([,

t ) f ( t , e ( T ) ) dr

0

since e(t)is a value of the unique random solution of (3.0.1) at t E R + . Likewise, if x ( t ) > O(t), then M[x(t)l = x(t) - P k t ) -

> e(t) - P h ( [ ) =

-

f f 0

P k ( [ , T)f(T,

e(T))

dz

P k ( t , T ) f ( T , e ( T ) ) dT

0

0.

Hence, Condition (i) of Burkholder's theorem holds.

3.3

91

METHOD OF STOCHASTIC APPROXIMATION

We now must show that Condition (ii) of the theorem is satisfied ; that is, SUP

X(f)

{IM[x(t)lli[l + Ix(t)ll) < a.

By definition, we have that M[x(t)l = x(t) - Ph(l) -

f

pk(t,t)

0

x E[f(t, x (t; o))lx(t;w ) = x(t), x ( t ; o)= 0(t) if

But by assumption the means of k(t, t ;a),h(t ; a),and and hence

t

< t] dt.

f ( t ,x(t ; a))exist,

IM[x(t)lI 1 + IxWl -

Ix(t )

- ph(t)

1 -

+ Ix(t)l Jb t)E[f’(t,X(T ;w))lx(f;0)= x(t), k ( t r

1 + Ix(t)l

if

t

< t] dt(

; 0)= e(7) if

t

< t]I dt

t

< t]

X ( t ; 0)= O(t)

+ Iph(t)l


0 and a positive continuous function

g(t) finite on R , such that JOm

lllko(t, T ; o)lIIg(ddr G 2,

tE R,

.

(ii) e(t, x) is continuous in t E R , and x E R such that le(t,O)l G

w(t)

and

14,x) - e(t,Y)I G 5gWlx - YI

for IIxllc, llyllc < p and y 2 0 and 5 2 0 constants. (iii) h(t ; o)E C , the set of continuous bounded functions from R , into L,(Q d ,9). Then there exists a unique random solution x(t; o)E C of Eq. (4.0.1) such that

provided that Ilh(t;o)llc,5, and y are small enough. PROOF We must show that under the given conditions the pair (C,, C) is admissible with respect to the integral operator

k,(t,~;w)x(r;o)dr,

Let x(t; w ) E C,. Then we have

C E R,.

4.2

SOME SPECIAL CASES

111

where Illko(t,t ; w)lll = IIko(t, t ; w)llLm(n,d.9) is a function only of ( t , t)E A 1 . Using the definition of the norm in C , , we have II(Yx)(t; w)IlLz(R,d,9)

Q Ilx(t; 4

C P Z '

by Condition (i) of the theorem. Thus Il(Yx)(t; is bounded, and Y is a continuous function of x from the proof of Theorem 4.1.6. Hence ( 9 x ) ( t ; w ) E C and (Cg,C) is admissible with respect to 9. From Condition (ii),

14,x(t; 0)) - e(t, Y O ; w))l Q tg(t)lx(t; 4 - Y O ; d l implies that

and hence

SUP 1Ile(t, x(t; 4) - e(t, I30

Q

W))IIL2(ntd.9)/g(t))

5 SUP Ilx(t; 0)- Y ( t ;4 I l L 2 ( Q , d * 9 ) 130

or Ile(t,x(t;w)) - e(t,y(t;w))llc,

Q t I l x ( t ; ~ )- Y ( ~ ; ~ I I c

for llxllc, llyllc d p. Likewise, le(t,O)I Q yg(t) implies that Ile(t,O)IIc,~d y. Therefore Corollary 4.1.7 applies with B = C , and D = C , provided that Ilh(t; w)llc, 5, and y are small enough in the sense that m

2

< 1,

Ilh(t;4llc

+ Kzy

Q P(1 - tK2).

Then there exists a unique random solution of Eq. (4.0.1) such that Ilx(t; d l l c Q PI

completing the proof. For g(t) = 1 for all ~ E R in, Theorem 4.2.1 we obtain the following corollary.

112

IV

A FREDHOLM TYPE EQUATION AND SOME APPLICATIONS

Corollary 4.2.2 Consider the random integral equation (4.0.1)under the following conditions :

(i) J$ lllko(t,r ;o)lll dz < Z , t E R + , where Z is some constant greater than zero. (ii) e(t, x) is a continuous function from R, x R into R such that 0)l

0, (i) Jllko(t,t ;o j J J and t , t E R + . (ii) e(t,x ) is defined on R , x R into R, continuous, and

on R , , y 2 0 a constant, and


0, t E R + .

Then there exists a unique random solution x(t ;o)of Eq. (4.0.1) satisfying

provided that H , 5, and y are sufficiently small. PROOF We must show that the pair (C,, C,) is admissible with respect to the integral operator

ko(t,t ;w)x(t; W ) d t

114

IV

A FREDHOLM TYPE EQUATION AND SOME APPLICATIONS

with g(t) = exp( - crt),a > 0, t E R + ,and the given conditions. For x ( t ;o)E C, we have

II(=w(t;4 1 1 L 2 ( R , . d , B ) d

s:

Illko(t9

W)lll . Ilx(t; W)IIL*(R,d,B) d t

x exp( - XT) dr

But from Condition (i) of the theorem, Joa Illko(t, 7 ; W)lll eXp( - a t ) d T d N

I0=

exp( - at 4-

/jt - a t ) d t

=

N(exp( - a t ) ) Joa exp[(/j - a ) t ] d t

=

“/(a -

p)] exp( -at).

(4.3.2)

Thus, combining inequalities (4.3.1) and (4.3.2), we obtain for every t E R , Il(Yx)(t ; 4 1 1L*(R..d.S) d IIx(t; W)ll “ / ( a -

811 exp( - at).

Therefore, by definition of C , with g(t) = exp( -at), ( Y x ) ( t ;o)E C,, and ( C , , C,) is admissible with respect to 9. Hence Condition (i) of Corollary 4.1.7 is satisfied with B = D = C,. Since we have (as in the proof of Theorem 4.2.1) that Condition (ii) implies lle(t, x ( t ; a))- e(t, y ( t ;w))IIC,

< tllx(t; 4 - I@;

w)lIC,

and IIe(t,O)\lCg d y, and that Condition (iii) implies that h ( t ; o)E C , = D for g ( t ) = exp(-crt) by definition of C,, all of the conditions of Corollary 4.1.7

are satisfied. Therefore, by Corollary 4.1.7, there exists a unique random solution x(t; O)E C , of Eq. (4.0.1) such that ~ ~ x ( r ; o ) l /dc gp, provided that H, 5, and y are small enough in the sense that 5K2 < 1 and

IlW; W)lIc, + K211e(t,0)llcgd H + Kzy d

P(1 - 5K2).

4.4

AN APPLICATION IN STOCHASTIC CONTROL SYSTEMS

I15

From (4.3.2) the norm of 9, by definition, is K 2 = N/(a - 8). Hence we must have

H + [ y N / ( a - fi)] Also, Ilx(t; w)(Ic,< p means that 5N < a - fi

and

II x ( t ; w)ll L2(R,.d.b) G p exp( - at)?

< p{1 - " / ( a

- fill)

ER+>

completing the proof. We therefore have that the unique random solution of (4.0.1) is stochastically asymptotically exponentially stable. In fact, with regard to the asymptotic behavior of the random variable x ( t ; w), we have

+O

as t - + m ,

that is, limt+mE { l x ( t ; w ) I 2 ) = 0, the second absolute moment of x ( t ; w ) approaches zero as t + co.Hence from Jensen's inequality we have that the expected value of the absolute solution approaches zero as t -+ a, lim,+mE { ( x ( t ;w)l} = 0. 4.4

An Application in Stochastic Control Systems

Corduneanu [3], Desoer and Tomasian [l], and Petrovanu [l] considered the stability properties of a nonrandom linear system described by the triple (E, F ; T ) , where E is the space of inputs to the system, F is the space of outputs from the system, and T is a linear operator from E to F given by (Tx)(t)=

jom k(t,

s)x(s) ds,

t 2 0.

Here we have altered the function k(t, s) in the paper of Corduneanu [3] to be zero whenever s 0. In this section we shall study a nonlinear stochastic feedback control system for which the random output is given in terms of the random input by the nonlinear operator T defined by

-=

for w E a. In a feedback control system the output, or a fraction of the output, is returned as input to the system. For a stochastic system this fraction may be a random function o f t in general. We shall consider the fraction of the

116

IV

A FREDHOLM TYPE EQUATION AND SOME APPLICATIONS

random output to be returned as q ( t ;w ) ( T x ) ( tw), ; where 0 9 q ( t ;w ) < 1 for all t 3 0 and w E R. See Fig. 4.4.1 for a schematic description. The following differential system with random parameters describes the stochastic feedback control system in Fig. 4.4.1 : with

+

A(t;to) = A ( t ; w ) x ( t ; o ) + ( t ; w ) 0) =

v(r; 4(Tx)(t;w),

(4.4.2) (4.4.3)

where the dot denotes the derivative with respect to t, T is the linear operator given by Eq. (4.4,1), x ( t ; w ) is an n x 1 vector whose elements are random variables, A ( t ; w ) and k(t, s; w ) are n x n matrices whose elements are measurable functions, q ( t ; w ) is a scalar random variable for each t 2 0, and e ( t , x ) is an n x 1 vector-valued function for each t and x. For n = 2 we have complex-valued random functions. By taking as the spaces E and F the space C,, we shall study the existence of a random solution x(t ;w ) and its stochastic stability properties by applying methods similar to those employed in the previous section and Theorem 4.1.6. Here we consider n = 1. The random differential system (4.4244.4.3)may be reduced to a stochastic integral equation of the mixed Volterra-Fredholm type in the form of Eq. (4.0.2). Integrating both sides of Eq. (4.4.2)and substituting the expression for &r; w ) given by (4.4.3), we obtain

J: =lor

x ( t ; w ) - x(0;o) =

+

A(z;o)x(z;w)dz

1:

+(z;w)dz

A ( z ; w)x(z ; w ) dz

+ J:

JOm

q ( z ; w)k(z,s; w)e(s,x(s; w ) )ds dz.

In the second integral on the right-hand side of Eq. (4.4.4) the integral JOm

k(z, s; o)e(s, x(s; w))ds

Figure 4.4.1.

(4.4.4)

4.4

AN APPLICATION IN STOCHASTIC CONTROL SYSTEMS

117

exists and is finite for each z and o ;otherwise, the output of the system is infinite. Also, if for each t >, 0 and s >, 0, yoq(z ;w)k(r,s ;w ) dz exists and is finite, that is, ifJb k(z, s; w ) dz exists and is finite, since 0 < q ( t ;w ) Q 1, then we may interchange the order of integration by Fubini’s theorem (Hewitt and Stromberg [l]) to obtain

1

[ / : q ( z ; w ) k ( z . s ; w ) d r e(s,x(s;w))ds. Then Eq. (4.4.4) may be written as x(t;w)=

1:

+

A(z;w)x(r;w)dz

where x(0 ;w ) = 0 and k * ( t , z ; w )=

J:

J0**

(

t , t ; w ) e ( z , x ( z ; w ) ) d r , (4.4.5)

t,zER+.

q(u;o)k(u,z;w)du,

The following theorem gives conditions under which a unique random solution of Eq. (4.4.5) exists and has the property of stochastic asymptotic exponential stability. Theorem 4.4.1 Suppose that the random equation (4.4.5) satisfies the following conditions :

(i) lllA(z;o)lll < N , exp( --at + 6 t ) for N , > 0 a constant, -a > 6 > 0, and 0 Q z < t < co. (ii) Illk*(t, z; w)lll Q N , exp( --at Bz) for N 2 > 0 a constant, -a > B > 0, and t , z E R + . (iii) e(t, x(t ;w)) is such that e(t,0) E C , , is continuous in t uniformly in x, and satisfies

+

le(t,

for Ilx(t; w)llc,, lly(t; w)llc,

x) - 4 4 Y)l 6 51x - Yl

< p, and t a constant.

Then there exists a unique random solution of Eq. (4.4.5) satisfying jElx(t; 0 ) 1 ~ } +

Q p exp(--at),

t 3 0,

provided that 5 and le(t, O)( are sufficiently small. PROOF We must show that the pair of spaces (C,, C,) is admissible with respect to the integral operators

A ( z ; O)X(Z;

W ) ~ T

118

IV

A FREDHOLM TYPE EQUATION AND SOME APPLICATIONS

and ( Y x ) ( t ; o)=

som

k*(t, r ;o)x(r ;o)dr,

t

2 0,

with g ( t ) = exp( - at) and Conditions (i) and (ii). For x(t ;w ) in C , we have I I ( = m ( t ;411L2(fi.d,b)

d

im

Illk*(t, t ; o)lll[Ilx(z;

o)llL,,,,d,b)/exp (-0rz)l exp( -015) dz

= Ilx(t; ~)IIC,[NZI(~ - P)1 exp(-Ut) < m,

t 2 0,

since a > ,!IThus, . by definition of C,, where g ( t ) = exp( - a t ) , ( Y x ) ( t ;w ) E C, for all x(t; o)E C,, and (C,, C,) is admissible with respect to 9. Likewise, II(.Xx)(t; 4 L 2 ( R , d * 9 )

=

Ilx(t;tu)IIC.NIexp(-axt)(I- exp[-(a - 6 ) t ] ) / ( a- 6)

6 llx(t; ( l j ) ) ) r p [ N , /( aS)] exp(-at) < a,

t 3 0,

from Condition (i) and the definition of the norm in C , . Hence (.Xx)(r;w ) E C, whenever x ( t ; o)E C,, and the pair ( C , , C,) is admissible with respect to X . Since the functionf(t, x) in Eq. (4.0.2) is the identity function in x in Eq. (4.43, the constant A in Theorem 4.1.6 is equal to one. From Condition (iii) we have, as before, that lIe(t, x ( t ; 0)) -

e(t,A t ; o))/Ic, 6 511x(t; 0)- ~ ( o)Ilc,. t ;

4.4

AN APPLICATION IN STOCHASTIC CONTROL SYSTEMS

119

Since the stochastic free term is identically zero, all of the conditions of Theorem 4.1.6 are satisfied for B = D = C,, and it follows that there exists a unique random solution of Eq. (4.4.5) in the set

s = { x ( t ; o ) : x ( t ; o ) E C g ,Ilx(t;o)llc, < p } for some p > 0, provided that random solution satisfies

< and le(t,O)I are small enough. Hence the

l l x ( ~ ; ~ ) l l L 2 ( ~ ,= d ,{EIx(t;W)l2)f 9)

< p exp ( - a t ) ,

by the definition of the space C,. The constants enough in the sense that KI

+ 5K2 < 1

and

t 2 0,

< and le(t,0)l must be small

K2Ile(t,O)Il~,d ~ ( 1 K i - 0 and a positive continuous function g(t) finite on R , such that

130

IV

A FREDHOLM TYPE EQUATION AND SOME APPLICATIONS

(ii) e ( t , x ) is continuous in t uniformly in x from R , x R into R such that le(t,0)(< yg(t) and I4L .u) - 4 t , Y)l G tg(t)lx - Y1 for y 2 0 and 5 2 0 constants. (iii) h(r, x) is continuous in t uniformly in x from R , x R into R such that (h(t,0)l < a and Ih(4-u) - h(t, y)1 < 4.x - L'I

for some I 3 0 and a 2 0.

Then there exists a unique random solution x ( t ; o ) C ~ of Eq. (4.5.1) such that IIx(t; o)llC< p provided that 5, A, y, and IIh(t,s(t;w))jlC are sufficiently small. PROOF We must show that under the given conditions the pair (Cg,C) is admissible with respect to the integral operator

(W.u)(t ;W) =

JOZ

k,(t, T ;(U)X(T ;O ) dr,

tE

R,

.

Let x(f ;to) E C,. Then we have

where Illko(t, T ;o)lll is a function only of ( r , T) E A , . Using the definition of the norm in C, we have

lo=

lllko(f,T ;o ) l l l g ( T ) d t ll(wx)(t;411,~2,Q..d,,l.) < I1.W; ~411~~

< llx(t; o4IIC,Z

by Condition ( i ) of the theorem. Thus W is a bounded operator and ( W x ) ( t CO) ; E C. Hence (C,, C) is admissible with respect to W . It can be shown that Conditions (ii) are sufficient for e ( t , x ( r ; c u ) ) to be in C, for x ( t ; w ) E S = I x ( t ; O ) E C : I I x ( t ; O ) l l c < p ] , and that Conditions (iii) are sufficient for h ( t , x ( t ; w ) )to be in C for x(t; W ) E S. Therefore Corollary 4.5.5 applies with B = C, and D = C provided that llh(t, x(t ;w)))Ic,(, y, and i. are small enough in the sense that K 2 5 + i. < 1 and Ilhk 4 t ; .Nlc

+ K,s < P(1 - 5K2X

where K 2 is the norm of the operator W .

For the special case g(t) = 1 for all t E R , in Theorem 4.5.6 we obtain the following corollary.

4.5

A RANDOM PERTIJRBED FREDHOLM INTEGRAL EQUATION

131

Corollary 4.5.7 Consider the random integral equation (4.5.1) under the following conditions :

(i) Illko(t,7 ; w)lll d7 d 2, r E R , , where 2 is some nonnegative constant. (ii) e(t,x) is continuous in t uniformly in x from R , x R into R such that le(t, 0)l 6 1' and Ir(t,x) - e(t,y)l d 51x - YI for some constants 1' 2 0 and 5 2 0. (iii) Same as Condition (iii) of Theorem 4.5.6. Then there exists a unique random solution x(t ; (0) E C of Eq. (4.5.1) such are sufficiently that Ilx(t; w)llc d p provided that t, A, y, and l~h(r,x(t;co))llc small. Corollary 4.5.8 Assume that the random integral equation (4.5.1)satisfies the following conditions :

(i) lllko(t, T ; w)lll d A for all t , T E R , and g(t)dr < m. (ii) Same as Condition (ii) of Theorem 4.5.6. (iii) Same as Condition (iii) of Theorem 4.5.6. Then there exists a unique random solution x(t ;w ) E C of (4.5.1) such that Ilx(t; w)llc d p provided that 5, A, y, and IIh(t, x ( t ; w))/Icare sufficiently small. We need only to show that the pair of Banach spaces (C,, C) is admissible with respect to the integral operator PROOF

(Wx)(t; w ) =

JOrn A,([,

T

; w)x(7 ; 0)d7,

tER

,,

along with Condition (i) of the corollary. For x(t; w ) E C, we have

From hypothesis (i) of the corollary we obtain


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

we obtain the random discrete Volterra system n

+ 1 ~ , , ~ ( w ) f , ( x ~ ( t on) )=, 1, 2 , . . . .

x,(w) = h n ( o )

(5.0.2)

j= 1

The discrete version of the random Volterra integral equation that was presented in Section 3.1.3 is analogous to the system (5.0.2) whenever the numerical integration error term d(")(w) is ignored. Some of the results presented here are stochastic versions of some results of Petrovanu [l]. 5.1 Existence and Uniqueness of a Random Solution of System (5.0.1 )

Let the spaces X , X , , X , , and X,, be as defined in Chapter I, Section 1.2. Let B* and D* be Banach spaces contained in X with the norm in B* denoted by IIxllB* = IIXn(w)llB*

and the norm in D* denoted likewise. That is, x,(w) E X is a function from N , the positive integers, into the space L,(R, d ,9) and X , , X , , and X,, are Banach spaces contained in X . Hence the random functions in X are discrete parameter second-order stochastic processes. Let T be a linear operator from X into itself. With respect to T and the Banach spaces B* and D*,we now state and prove a lemma analogous to Lemma 2.1.1.

Lemma 5.1.1 If T is a continuous operator from X into itself, B* and

D* are stronger than X , and the pair (B*,D*) is admissible with respect to T , then T is a continuous operator from B* to D*.

Suppose xi E B* such that x i +B* x, that is, xin(w)+'* x,(o) as co. Assume that T x i n ( o+D* ) y,(w) as i + co. But Txi,(w)+' Tx,(w), since T is continuous from X into itself, and xin(w)+B'x,(w) implies that xin(w)+' x,(w). However, Txin(w) +D* y,(w) implies that Txin(w) +' yn(to) as i + cc. Hence Tx,(w) = yn(w)because the limit in X is unique. Therefore PROOF

i

+

134

v

RANDOM DISCRETE FREDHOLM AND VOLTERRA SYSTEMS

T is closed and by the closed graph theorem it follows that T is continuous from B* into D*, completing the proof.

If T is a continuous operator from Banach space B* into D*, then it is bounded and, as before, there exists a constant K > 0 such that IITxn(w)llD*

d Kllxn(w)llB*.

We make the following assumptions concerning the functions in the random system (5.0.1): The functions x,(w) and h,(o) are functions of n E N with values in L,(R, d,9). For each value of x,(o), n = 1,2,. . . ,J,(xn(w))is has values in the space L,(R, d,9). a scalar, and for each n = 1,2, . . . ,fn(xn(w)) For each n and j in N , cn,j(w)is assumed to be in Lm(Q d,9) so that the 9). Also, for each value product of c,,~(o) and fj(xj(to)) will be in L,(R, d, of n I l l ~ n , j ( ~ ~ )=l l l9 - e SUP ~ ~Icn,j(w)I = IICn,j(Q)IIL,(R,d,B) U>

and I I I c ~ , ~ ( ~ ) I.I IIX~(CU)II,~~(~,~,~) I are assumed to be summable with respect to j e N. Consider the linear operator T defined by m

Tx,(w) =

C c,,~(w)x~(o),

= 1,2,. . .,

t~

j= I

for x j ( o ) in X . I t may be shown that the operator T is continuous from the space X into itself by using an argument similar to that of Lemma 4.1.1. The following theorem gives conditions under which there exists a unique random solution of the system (5.0.1).

Theorem 5.2.2 Consider the random discrete equation (5.0.1) subject to the following conditions : (i) B* and D* are Banach spaces stronger than X such that the pair (B*, D*) is admissible with respect to the linear operator

c cn*j(o)xj(w), n m

Tx,(o) =

= 1,2,. . . ,

j= 1

where C , , ~ ( Whas ) the properties given previously. (ii) x,(w) +fn(x,(o)) is an operator on with values in B* satisfying for x,(o), y,(w) E S and A 2 0 a constant. (iii) h,(w) E D*.

5.1

EXISTENCE AND UNIQUENESS OF A SOLUTION OF(5.0.1)

Then there exists a unique random solution x,(Q) equation (5.0.1), provided that

~I~,,(~)~~+ D*~

AK < 1,

E S of

135

the random discrete

~ ~ ~ ( dOp(1 ) ~- ~AK), B *

where K is the norm of T . PROOF

Define the operator U from S into D* by m

uxn(0) = h,(w)

+ C cn,j(w)h{xj(w)),

fi

= 192, .

j= I

As in Theorem 2.1.2, we show that U(S) c S and that U is a contraction operator on S. Then Banach's fixed-point theorem applies. Let x,(to), y n ( w )E S . Then

It uxn(m)IID* =

m>

IIhn(w)

+ j=2I cn.j(w).fi(xj(w))II~

d llMw)llD* + KllL(-~n(w))ll~* from the result following Lemma 5.1.1 that T is a bounded linear operator from S into D*. From Condition (ii) of the theorem

Hence

/I f n ( x n ( w ) )11 B* d /I h(-xn((o))- fn(O) 11 B* + 11 f"(O) /IB* d AIIXn(to)IID* + IIh(o)IIB*.

+ K ~ l f n ( o ) ~ ~ B+* AKIIXn(W)IID* d p(1 - 3,K) + 3.Kp = p

11 UXn(w)llD* d

llhn(U)llD*

from the last hypothesis of the theorem. Thus U ( S ) c S . Since the difference of elements of a Banach space is in the Banach space, VX,(W) - U ~ , ( W )D* E

and

II u x n ( 0 ) -

m

~YA~)IID =*

v

136

RANDOM DISCRETE FREDHOLM AND VOLTERRA SYSTEMS

by Condition (ii). Since EX < 1, we have that U is a contraction mapping on S. Therefore, applying the fixed-point theorem of Banach, there exists a unique random solution x,(co) E S of Eq. (5.0.1),completing the proof.

5.2

Special Cases of Theorem 5.1.2

In this section we shall present some special cases of Theorem 5.1.2 which will be very useful in practice. We take as the spaces B* and D* the spaces x,,XI or Xb". 5

Theorem 5.2.1 Consider the random discrete equation (5.0.1) subject to the following conditions : (i) There exist a constant Z > 0 and a positive sequenceg,, n such that

c lllc~,j(~)lllgjd z, m

=

1,2,. . . ,

n = 192,. . . .

j =1

(ii) jJx) is a function defined for n E N and scalar x such that If,(O)l and

d yg,

for ~ ~ y , ( w )d~p~and x l 2 and y constants. (iii) h,(w) E X , . Then there exists a unique random solution of (5.0.1),

provided that ~~hn(co)[~xl, y, and 1, are small enough. PROOF If we show that the pair of Banach spaces ( X g ,X , ) is admissible with respect to the linear operator

n

=

1,2, ...,

then the conclusion follows from Theorem 5.1.2 with B*

=

(5.2.1)

X , and D* = X I .

5.2

137

SPECIAL CASES OF THEOREM 5.1.2

Let x,(o) E X , . Taking the norm of both sides of Eq. (5.2.1), we have by the generalized Minkowski inequality (Beckenbach and Bellman [ 1, p. 221)

m

by the definition of the norm in X,. Since the last sum on the right is less than or equal to Z by Condition (i), we have


0 and a positive-valued continuous function g ( t ) on R , such that

J: Illk(f,

5 ; to)lllg(T)

dz d A.

(ii) f ( t , x) is continuous in f uniformly in x from R , x R into R ; there exists a constant A such that I.f(t. 0)l d Ag(t), t E R , , and If(& x) - f ' ( t ,y)l d lg(t)lx - yl for some 3. > 0. (iii) h(t,x) is continuous in t uniformly in x from R , x R into R such that Ih(t, 0)l d Q, t E R , , and Q > 0 ; Ih(t, x) - h(t, y)( d y[x - yl for some y > 0. Then there exists a unique random solution x(t; o)E C of the random integral equation (6.0.1) such that Ilx(t; o)llcd p, provided lIh(t,x(t; 4 ) I I c , 2, IIf(t, O)IIc8 are sufficiently small. REMARK

By sufficiently small, we mean that y IIh(t, x(t; 4 ) I I c

+ Kllf(t, O)llC,

+ AK

d p(1

11,

and

< 1 and

- IK).

PROOF The proof consists basically in showing three things. First, that under the conditions assumed in (i) the pair (Cg,C) is admissible with respect to the operator

(Tx)(t;o)=

f

k(t, T ; W)X(T; w)d~;

second, that Conditions (ii) are sufficient for Conditions (ii) of Theorem 6.1.1 to hold; and third, that Conditions (iii) are sufficient for Conditions (iii) of Theorem 6.1.1 to hold. Letx(t;o)ECg.Then ,

6.1

161

THE RANDOM INTEGRAL EQUATION

Thus II(Tx)(t;w)ll is bounded, which implies that ( T x ) ( t; w ) E C for x ( t ; o ) ( C g ,C ) is admissible with respect to 7: Now let t, + t in R , . We must show that f (t,, x(tn;w ) ) + f (t, x(t ;w ) ) in L2(R,4 9). That is, we must show that given E > 0, there exists an N , such that n > N , implies E C g . Thus

II f (tn,x(tn ; w)) - f (t, ~

(; t

<E

or equivalently that t II lim II f (tn ~ ( t ;n01) - f(t9 ~ (;w))

n-rm

where

11. 1 1

=

II.IIL,cn,d,9,

=

0

3

throughout this section. Consider

We only need to show that each term on the right side of this inequality can be made arbitrarily small. Consider Il f ( t n ,x(tn; w ) ) - f (tn,x(t ; to))ll. For each n

by the Lipschitz condition given in (ii). Squaring and integrating over $2, we obtain that

Thus

Ilf ( t n , x(tn; 01)

-

f (t,, x ( t ; w))II G Ag(tn)llx(tn;W ) - x(t ;w)II

Taking limits, we obtain that

where the limit on the right is due to the fact that g is continuous at t and x ( t ; s)E C. Hence there exists an N , such that n > N, implies that

162

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

Since f (t,x ) is continuous in t uniformly in x, there exists an N , such that n > N, implies that Squaring and integrating over 0 , we obtain that

for n > N,. Hence for n > N , Let N, = max(N, , N,); then for n > N, inequality (6.1.3) becomes Thus under Conditions (ii), t + f (t,x ( t ; w)) is continuous from R + into L,(R, d, 9).Now fix o E R. For each t E R+

Again squaring and integrating over R, we obtain that

If

I*

(t,x(t ;w))I2dP(w) ,< A2g2(t) Ix(t; w)I2 Ix(t ; w)l d.Y(w)

Using the Cauchy-Bunyakovskii-Schwarz

II f (t,x(t ;0))1l2< A2g2(t)llx(t;0)Il Since

x ( t ; w ) E S,

where

< II x(t ; o)ll < P. Thus

+ A2g2(t).

inequality, we obtain

+ 2AAg2(t)llx(t;w))ll + A2g2(t).

S = { x ( t ;w ) E C : Ilx(t; w)IIC< p),

Ilx(t; o)ll

II f ( t , x ( t ; 0))Il < A2g2(t)p2+ 23.Ag2(t)p+ A2g2(t) =

z2g2(t)

where

Therefore

Il f (4 x(t ;w ) )II

z2 = AZp2

+ 21.Ap + A2.

< zg(t).

This implies that f (t,x(t ; w ) )E C , for x(t ;w ) E S. Let x(t ; w), y(t ; w ) E S. Then

6.1

163

THE RANDOM INTEGRAL EQUATION

implying that

II f ( t , x(t ; 4)- f ( t , Y o ; w))II d

Ag(t)llx(t ; 0)- Y o ; 4 1 1

or that

II f ( a 4; 0)) - f ( 4 YO ; dlI/g(t) 6 w 4 t ; 0)- Y(t ;w)ll. Hence by definition of the norms in C, and C we have that

IIfk

x ( t ; 0)) - f ( t ,y ( t ; w))llc,

6 Allx(t; 0)- A t : 0 ) l l C .

Thus Conditions (ii) are sufficient for Conditions (ii) of Theorem 6.1.1 to hold. The proof that Conditions (iii) are sufficient for Conditions (iii) of Theorem 6.1.1 to hold is analogous to that just given and will be omitted. The remainder of the proof is identical to that of Theorem 6.1.1. When g(t) 3 1 we have the following corollary. Corollary 6.1.3 Suppose that the random integral equation (6.0.1) satisfies the following conditions :

(i) (ii) exists d Alx (iii)

j b Illk(t, z ; w)lll d.r d

A for t E R , , A > 0. f ( t ,x) is continuous in t uniformly in x from R + x R into R ; there a constant A such that I f ( t , 0)l 6 A, t E R , ; and I f ( r , x) - f ( t , y)l

for some A > 0. Same as Theorem 6.1.2, Condition (iii).

- yl

Then there exists a unique random solution x ( t ; w ) E C of the random integral equation (6.0.1) such that Ilx(t; o)llc6 p , provided Ilh(t, x ( t ; o))llc, I., y, and II f ( t ,O)llc are sufficiently small. Corollary 6.2.4 Assume that the random integral equation (6.0.1)satisfies the following conditions :

At) d t = M < (i) Illk(t, z ;w)lll d Al for ( t ,z)E A and (ii) Same as Theorem 6.1.2, Condition (ii). (iii) Same as Theorem 6.1.2, Condition (iii).

cx).

Then there exists a unique random solution x ( t ; w ) E C of the random integral equation(6.0.l)such that Ilx(t; w)llc d p provided that IIh(t, x ( t ; w))IIc, 2, y, and 11 f ( t ,O)llcg are sufficiently small. PROOF It is necessary only to show that (Cg,C) is admissible with respect to the integral operator

( T x ) ( t ; o )= Corollary 6.2.5

sd

k(t,z;w)~(z;w)dz.

Consider Eq. (6.0.1) under the following conditions :

164

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

d A2 e-'('-'), 0 d z d t < (i) lllk(t,~;w)\ll < co,where A2 and tl are positive constants.

00, and

S U P ~ ~ ~ +g{( TJ ):d+z }~

(ii) Same as Theorem 6.1.2, Condition (ii). (iii) Same as Theorem 6.1.2, Condition (iii).

Then there exists a unique random solution x ( t ; w ) E C of Eq. (6.0.1) such that Ilx(t; w)llc d p, provided IIh(t, x ( t ; w))IIc, I , y, and IIf(t, O)llc, are sufficiently small. PROOF Since Conditions (ii) and (iii) are identical to those of Theorem 6.1.2, it is sufficient to show that (Cg,C ) is admissible with respect to the integral operator

( T x ) ( t ;w ) =

Let x ( t ; w ) E C,. Then

Sd

k(t, Z ;W)X(T;

0)d T .

Using the definition of the norm in C,, we have

However,

implies that ~ ~ e - " " - ' ) g ( . r ) d=i N
o. F(t - z, M ( z ) ) = e-'('-') , Thus in this case Eq. (6.2.2) reduces to the following form : M ( t ;w ) = M ( 0 ;o)e-"

+

1:

R ( z ;w ) e-'('-') dz,

C

2 0.

(6.2.3)

In order to facilitate our theoretical presentation of Eq. (6.2.3), we make the following identifications : h(t, x ( t ; o))= H ( t ; w ) = M ( 0 ;o)e-",

x ( t ; w ) = M ( t ;w),

k ( t , z ; w )= R(z;w)e-'('-'),

f ( t , x ( t ; w ) )= 1.

Thus Eq. (6.2.3) can be written as (6.0.1),that is, x ( t ; 0 ) = h(t, ~ ( t0)) ;

+

6

k(t, T ;w ) f ( z ,X(T; w ) )dz,

t 2 0. (6.2.4)

With respect to the functions which constitute Eq. (6.2.3) we shall make the following assumptions: For each t 2 0 the random variable M ( t ;o)has finite variance and there exists a constant Q independent of z such that IR(z;w)l < Q for almost all w. These restrictions are necessary for our theoretical presentations, and their validity in the physical sense will be commented on later. T o show that the given stochastic integral equation possesses a unique random solution, we must show that the basic assumptions concerning the functions which constitute the formulation of Eq. (6.0.1) and the conditions of Corollary 2.1.4 are satisfied. We shall assume that { e-C('-r)R(z ; o):w E a} is an equicontinuous family of functions from A into R and show that this

168

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

is sufficient for our purpose. To show that the functions of the model meet the given conditions, we proceed as follows : For t E R , ,M ( t ;o)by assumption has finite variance. This implies that for each t, M ( t ;o)E L2(Q,d,9). Since f ( t , x ( t ; o ) )= 1 and (Q,d,9) is a probability space and hence a for each t E R , . For fixed t , finite measure space, f ( t , x ( t ;w))is in L,(Q, d,9) ec' is a constant and since M(0 ;o)has finite variance, we can conclude that M ( 0 ;u)e-" E L,(Q, d,9). Fix (f, T) E A. Recall that k(t, T ; o)= R ( T ;o) x By assumption IR(T;w)l d Q, 9-a.e., (k(t,T ; o)l = ~ R ( To) ; e-c('-r)l < Qe-c('-'), 9-a.e. This implies that ( t , z) + k(t, T ; o)is a map from A into L,(Q, d,9). Now let (t,, T), -, ( t ,T). Choose E > 0. By the equicontinuity condition, there exists an N , such that n > N , implies le-C('n-rn)R(zn ., - e - C ( ' - r ) ~ ( ~ w)l ; <E for each o E Q. Thus by definition the infinity norm, for n > N , , becomes Il(e-C('n-'n)R(z n ,. w ) - e-'('-')R(z; o)lll < E . However, this implies that for n > N , we have

IIIW,, T, ; w ) - k(t, 7 ;0)lll < 6, as was to be shown. To show that the hypotheses of Corollary 2.1.4 are satisfied, we proceed as follows : For Conditions (i) we have

s:

Illk(t, z;

o)llldz

=

llle-c('-r)R(z;o)llldz

= (Qe-c'/c)(ec'- 1 ) = ( Q / c ) ( l - e-")

< Q/c. I t is easy to see that Condition (ii) is satisfied. To see that M ( 0 ;o)e-" E C , let t, -, t in R + and choose E > 0. If IIM(0;o)ll # 0, choose N such that n > N implies le-crn - ec'l < ~/llM(o; 411. Then for n > N

IIM(0;o)ePcrn- M ( 0 ; w ) e-"II

= le-crn -

e -cfl IIM(0; w)ll

< "mw;~)IIlIIM(O;4 1 1 = E.

6.2

APPLICATIONS TO BIOLOGICAL SYSTEMS

169

If IIM(0;u)II = 0, then IIM(0;w ) eWcrn - M ( 0 ;w ) e-"II = 0 < E . Thus t -,M ( 0 ;w)e-" is continuous from R , into L2(R,d,9 ) . To see that the map is bounded, consider IIM(0;w ) e-"II = e-"IIM(O; w)ll d IIM(0;w)ll. Thus the conditions of Corollary 2.1.4 are satisfied, and we can conclude that there exists a unique random solution of Eq. (6.2.3) provided IIM(0;u) x e-"jlc, 2, and IIf(t, 0)11, are sufficiently small. REMARK

that

When we say that the quantities are sufficiently small we mean

x < 1,

IIM(O;w)e-"Il,

+ KIIf(t,O)Il, < p ( 1 - m

where K = IITII*, the norm of the operator T defined in Section 6.1. Note also that

IIM(0;w ) e-c'llc = sup e-"(lM(O; w)ll = IIM(0;w)ll 0 8t

and that 11 f ( t , O)llc = 1. Hence we are actually requiring that AK < 1,

IIM(O;CO)II+ K

< p(1 - LK).

Since in this case 2 can be any positive number, the first condition can easily be satisfied. Hence we will have a unique random solution M ( t ;w ) such that EIM(t;w)I2 d p for each t provided EIM(0;w)I2 is sufficiently small. In formulating the stochastic model, we have been forced to make certain assumptions on M ( t ; w ) and R(t ;w). Namely, we assume that { M ( t ;w ):t E R , } is a second-order stochastic process and that for each t, R ( t ;w ) is 9-essentially bounded and furthermore that the bound is uniform over R , . These restrictions make our particular approach to the problem possible and may or may not be satisfied by a particular given metabolizing system under study by a biologist or biochemist. The feasibility of these assumptions must be determined in each instance by the experimenter. Although these requirements appear on the surface to be quite strong, in practice they are in many cases quite easily satisfied due to the physical or chemical characteristics of the system under study. For example, if the amount of metabolite present at any time were limited due to space considerations, we would automatically satisfy the condition that { M ( t;w ) :t E R , } be a second-order stochastic process. As a simple illustration, visualize the "metabolizing" system as being a reservoir and the "metabolite" as being the amount of water present at time t . If, on the other hand, the amount of metabolite

I70

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

present at any given time were limited due to some chemical characteristic of the system, we could come to the same conclusion. That is, visualize the metabolizing system as being perhaps a lake or stream and the metabolite of interest the amount of dissolved KCI present per gallon. This amount will be limited by chemical considerations due to the fact that there is a maximum amount of the salt which can be dissolved in a given amount of water at a given temperature. In any situation similar to this the assumption of finite variance on M ( t ; u)for each t will be quite naturally satisfied. Similarly, in many systems the rate at which metabolite enters the system from outside sources at any time will be restricted due to physical limitations, especially in systems where metabolite is simply being transported, or to chemical limitations in systems where metabolite is being consumed or produced. Hence the assumption that there exists a Q such that IR(z;w)l < Q 9-a.e. is an assumption which can often be realistically met. The point to be made here is twofold. First, the proposed random model is a more realistic description of a general metabolizing process than is the deterministic formulation and should be used whenever possible. Second, in many cases the restrictions placed on the functions in the random model are not extremely difficult to meet in practice, but whether or not they are met is a question which must be considered carefully by the experimenter in each case. Note that there is a certain degree of flexibility in the random formulation in that for each t we require no knowledge of the particular form of the distribution functions for the random variables involved but only that they are elements of certain or L,(Q d,9). spaces, namely either L,(Q d,9)

6.2.2 A Stochastic Physiological Model Physiologists quite often are faced with the problem of constructing mathematical models which attempt to describe the complex processes that take place within the human body. Such models are very difficult to obtain and must necessarily be oversimplified due to the inability of scientists to understand fully all of the factors which can influence even the simplest process taking place in a living organism. Thus mathematical models in use are constantly subject to refinement as more insight is gained into the true nature of the process taking place. Along this line, Milton and Tsokos [4] proposed a stochastic formulation for a model used to study blood flow in a simplified circulatory system which has been studied from the deterministic point of view by Stephenson [l]. The stochastic approach yields a more realistic characterization of the system than the nonrandom formulation and should be used whenever possible. A simplified circulatory system is visualized as consisting of the heart, a capillary bed, and connecting vessels. Schematically it is pictured in Fig.

6.2

APPLICATIONS TO BIOLOGICAL SYSTEMS

171

6.2.1. The points X and Y represent an inflow and an outflow point, respectively. It is assumed that a given amount M of indicator substance I is suddenly injected in the system at point X , the inflow to the capillary bed, thus producing a fixed concentration of indicator C,(O) > 0 at time t = 0. The fraction of the amount M flowing out of the capillary bed at time t is considered to be a deterministic function of time and is denoted by p(t). This function is determined by taking instantaneous measurements on the concentration of indicator at point Y. The concentration at time t is also considered to be a deterministic function of time which is denoted by C,(t). The model allows for recirculation and hence the concentration of indicator at point X will also depend on time and will be denoted C,(t).Stephenson [l] makes the following assumptions concerning these functions based upon experimental results and working experience with such models: (a) C,(t) and C2(t)are twice differentiable; (b) there exist constants G > 0 and H > 0 such that IC;(t)l d G

and

IC;(t)l d H

for all t, where C;(t) = dC,(t)/dt and C;(t) = dC,(r)/dt. Under the assumption that the model provides a reasonable description of the physical situation at hand, the following deterministic integral equation of the Volterra type is obtained : p ( t ) = [C;(t)/C,(O)l - [l/Cl(0)l

s’ 0

C;(t - ~ ) p ( ~ ) d t , t 2 0. (6.2.5)

Let us first consider the function C,(t). The usual technique for determining the value of this function at a given time t , is to obtain experimentally several observations on C2(t]). These experimental values are then averaged and the mean value of these experimental observations is used as the “true” value for C,(t,). Due to the possibility of some diffusion in the capillary bed together with the inherent difficulties of accurately measuring instantaneous concentrations at a given point, as well as the natural variability of the circulatory system in general, if the same experiment were repeated, the mean value obtained would most likely differ from the first determination. Heart

Capillary bed

Figure 6.2.1.

172

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

If this variability is large, the actual mean value used could be quite unsatisfactory. Therefore it is more realistic to assume that the concentration of indicator at Yis for each t a random variable and that C,(t) is in fact a random function which we shall denote by C,(t; w). This tacitly implies that the derivative of C,(t ; w ) is also random and it will be denoted by C;(t ; w). The function C,(t)can be considered random because of the fact that it is obtained experimentally in a manner identical to that described for C,(t) and also due to the fact that we are allowing for recirculation. Thus C,(t) will be influenced after a certain point by the same factors which influenced C,(t), namely diffusion and natural variability. The function p( t ) is being expressed in terms of the random functions C ; ( t ;w ) and C ; ( t ;w), and thus it should also be considered as random and will be denoted by p ( t ; w). In view of these remarks we have the following random version of Eq. (6.2.5):

x p ( ~w; ) d ~ ,

t 3 0.

(6.2.6)

We shall make the following identifications in order to simplify our theoretical investigation : x ( t ;w ) = p ( t ; w ) ; h(t,x ( t ; w)) = C#;

t~),’c,(O);

k ( t , r ; w )= C;(t - 7

; ~ ) ;

and

f ( r , x ( t ; w ) )= -x(t;w)/C,(O) = -[l/C,(O)]p(t;~).

Using the notational changes, Eq. (6.2.6) takes the familiar form given by Eq. (6.0.1). With respect to the functions appearing in Eq. (6.2.6)we make the assumption that there exist constants G > 0 and H > 0 such that

for each t 3 0. T o show that the model possesses a random solution, we must verify that the random functions which constitute Eq. (6.2.6) meet the required assumptions of Eq. (6.0.1). We begin by assuming that the families {C;(t - T ; W ) : C O E R ) and

{C;(t;o):wER)

are equicontinuous families from A into R . Fix t E R , . Then ( p ( t ;o)l ,< 1 P-a.e. for each t due to the fact that p ( t ;w ) represents the fraction of the

6.2

APPLICATIONS TO BIOLOGICAL SYSTEMS

original amount of indicator in the outflow at time

1 Ic;(t; 1

= [1/Cl(O)]2

< [1/C1(O)]2

w)l2

n

t.

173

Hence

dY((0)

H 2 d.Y(to) = [1/C1(0)]2H2
Cj = 0 and hence that C ; ( u ;co) E LJR, .dP).Thus ( t , T ) + k(t, T ; Q) is a 9). I t remains only to show that this map is conmap from A + L,(R, d, tinuous. To this end, let ( t n ,T,) + ( t ,5). By the equicontinuity condition, given E > 0, there exists an N such that n > N implies

Ic;(t,- T , ;

O j ) - c’l(? -T;

to)\ < E

for each w E R. Hence for n > N , 9 { w : I c ; ( t n- Tn;-w)-

implying that for n > N ,

c;(t

-

t ; w ) (>

I;}

=

0,

174

VI

NONLINEAR PERTURBED RANDOM INTEGRAL EQUATIONS

Therefore the basic assumptions of our theoretical development are met. We shall now show that for an appropriate choice of the function g the pair (C,, C,) is admissible with respect to T as required in Theorem 2.1.2, Condition (i) and furthermore that Conditions (ii) and (iii) are also satisfied by the functions of the stochastic model. This allows us to apply the theorem to obtain the existence of a unique random solution. Choose g(t) = ct eB' for fi > 0 and c1 3 1. Note that g is positive valued and is well defined. continuous on R + and hence the space C,(R+ ,L,(R, d,9)) 9)) Consider . Let x(t ;o)E C,(R+ ,L,(R, d,

d [Gllx(t; W)Ilc,/P14eB' d [Gllx(t; 4IIc,/PIct

- 1)

eBt = [Gllx(t; W)llc,/Plg(t).

Thus for x ( t ; o ) ~C g ( R + , L 2 ( Q , d , 9 )()T, x ) ( t ; w ) ~C g ( R + , L 2 ( R , d , . 9 )im), plying that the pair (C,, C,) is admissible with respect to 7: Now, let x(t ; w ) E S, where S is as defined in Theorem 2.1.2, Condition (ii) with C, = D. Then which implies that IIf(t,x(t;

f(t,

4) = [l/Cl(0)lx(t; 4,

4 = [1/C,(O)lllx(t; 0)ll d

[1/C,(O)IAa eBt = Zg(t),

where Z is a constant, Z = [I/C,(O)]A. Thus by definition of C , ( R + ,L,(Q d,P)), f ( t , x(t ; 0)) E C,(R+ ,L,(R, d, 9)). To show that the Lipschitz condition is satisfied, consider

6.2

APPLICATIONS TO BIOLOGICAL SYSTEMS

175

Hence we can choose A = l/Cl(0)and Condition (ii) is satisfied. Let x(t ;w ) E S. Then

IIhk x(t; 4 1 1 = I l [ 1 / ~ 1 ( 0 ) 1 ~o)ll #~ = ~

~ / ~ l ~ do l l~ l l l ~ x ~ ~

d [1/C 1(0)IH d [ H / C*(O)l ( d a )epr = Zg(t).

Since the equicontinuity condition implies the continuity of the map t + h(t, x ( t ; w)), we have that h(t, x(t; o)) E C , ( R + ,L,(Q d,8)) as was desired. Therefore we have shown that Theorem 2.1.2 is applicable with B = D = C, where g(t) = aep’. We can thus conclude that there exists a unique random solution x ( t ; o)of Eq. (6.2.6) such that Ilx(t; w)IIc, d p provided 1.K < 1 and ll~(t,x(t;4)llcg + KIIf(t,O)llc, d ~ ( -1 W . Note that IIf(t,O)llc, = 0 and that IIh(t, x(t; w))IIc, = SUP {[l/Cl(O)l Ilc;(t; o)ll/a e B f ) O 0. Although the model has been formulated for communicable diseases, a similar model can be developed in the case of the spread of a rumor throughout a community, when the "news value" of the rumor decreases with time so that a person would be less likely to communicate the rumor, say, two weeks after hearing it than he would only a few hours after hearing it.

CHAPTER VII

On a Nonlinear Random Integral Eqzlation with A p plication to Stochastic Chemical Kinetics

7.0

Introduction

The aim of this chapter is to study a random vector integral equation of the form given by ~ ( w t ;) = h(t, x ( t ; w ) )

+

k(z,X ( T ; 0);W ) dz,

t 2 0,

(7.0.1)

where w E 0, the supporting set of a probability measure space (Q, d,P), x(t ; w ) is the unknown rn-dimensional vector-valued random function defined on R , , the stochastic kernel k(t, x ( t ; w ) ;w ) is an rn-dimensional vector-valued function on R,, and for each t E R , and each rn-dimensional vector-valued random function x ( t ; w),h(t, x ( t ; w ) )is an rn-dimensional vector-valued random variable. More specifically, we are interested in the existence and uniqueness of a solution, a random vector-valued function I80

7.1

MATHEMATICAL PRELIMINARIES

181

and some special cases which are important for studying certain physical problems. In addition, we shall give a stochastic formulation of a classical chemical kinetics problem. The formulation of such a model results in a random integral equation of the form given by (7.0.1). In Section 7.1 we shall introduce some topological spaces and definitions and state and prove certain lemmas which are essential in our study. An existence theorem and some special cases are given in Section 7.2. In Section 7.3 we shall give a complete and precise formulation of a chemical kinetics problem which is quite realistic for describing the physical phenomenon. The results given in this chapter are due to Milton and Tsokos [ l , 51. x ( t ; w),

7.1 M a thematica I Preliminaries

In this section we shall give some definitions and concepts which are basic to our study. Definition 7.1.1 Two random vectors x(w) = {xl(w),x 2 ( o ) ,. . . ,x,(w)} and y(w) = {yl(w),y2(o),.. .,y,(w)} are said to be equal if and only if xi(w)= yi(w)9-a.e. for each i = 1,2,. . . , m.

Definition7.1.2 We shall denote by +(a,&,!?) the set of all random where, for each vectors of the form z(w) = (zl(o), z2(o), . . . , z,(o)} i = 1,2,. . . , m, z i ( o )is an element of L,(Q d ,9).

Lemma 7.1.2 The space $(a,d , 9 )is a normed linear space over the real numbers with the usual definition of componentwise addition and scalar multiplication, where the norm in $(Q, d ,9) is defined by ll~(w)Il~(o,d,p) = max

Illzilll.

PROOF The fact that $(a, d,9)is a linear space follows from the fact is a linear space. That I( . ll*(o,d,p) is a norm follows from that L,(Q, d,9) the fact that 111 . 111 is a norm.

Definition 7.1.3 C,(R+ ,$(a,d ,9)) will denote the set of all continuous functions from R , into $(a,d,9).

We remark that Definition 7.1.3 simply states that t + x ( t ; w ) = {xl(t;w), each t~ R , and each fixed t E R ,

x 2 ( t ; o), . . . ,x,(t; o)}is continuous and that for i = 1,2,. . . ,rn, x i ( t ;o)E L,(Q d, 9). Therefore for

182

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

Also an element of the space C , ( R + , $ ( Q , d , P ) ) is a random function. We shall be assuming that for each i the sample function xi(t ;o)is continuous in t for each o E Q. Thus, since we are working with a finite measure space, as in for each t and i, E l x i ( t ; o ) l < co. Defining the norm of +(Q,d,9) Lemma 7.1.1 will enable us to obtain a relatively simple norm defined in terms of the components of the random vector.

9)) is a linear space over the Lemma 7.1.2 The space C,(R + ,$(a,d, real numbers with the usual definitions of addition and scalar multiplication for continuous functions. Lemma 7.1.3 The collection

F = { Ilx(t; m)lln: Ilx(t; o ) l l n = OSUP Ilx(t; m)Il,(~,d,~)} Qrdn for n = 1,2,. . . , is a family of semi-norms defined on C,(R+, +(Q, d ,9)).

d,9). Thus, PROOF By definition, x ( t ; o)is continuous from R+ + $(a, given the compact set [0,n] c R , there exists some constant M , such that t E [O, nl implies Ilx(t ;411,(R,d.B) d M , . Hence { Ilx(t ;W ) I I ~ ( ~ ,t ~ E LO, , ~ nl} ) : is bounded above by a constant which implies that the supremum exists, and thus that Ilx(t ; m)ll, is uniquely determined and also nonnegative. Furthermore,

IIW;w)ll,

=

SUP {IIW;W J I ( R , d , B J = OSUP (14 I M ; 4 J l ( R . d . B ) ) dt N , implies that p(X"(t;

O),X ( t ; W ) )

E

> 0,

< E.

That is, we need to show that for m > N ,

Since

I:==, 112"

Enrn= , +

is finite, there exists a natural number M such that 112" < ~ / 2 Also . for each n and m we have IIxrn(t;0)- x ( t ; w)llJ[1

Thus we have

+ IIxrn(t;0)- x ( t ; W)1ln1 < 1.

184

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

On an interval [O,M] there exists a natural number Ne,M such that for m > N e , Mwe have Ilxrn(t;0) - x ( t ; w)II$(R,d,S) < 4/(1 - 4 )

for every t E [0, MI where q = ( ~ / 2 )l/ (cC;"1/2"). = Since IIxrn(t;0) - x ( t ;wHM = SUP {IIx"(t; w ) - x ( t ;w)lI$(n,d,s)), OQtQM

for m > Ne,Mwe have IIxrn(t;w) - x(t;w)llw d 4/(1 - 4).

Because the sequence (11 . 1), of semi-norms is increasing, for n = 1 , 2 , .. . , M and rn > NE,M,Ilx"(t;w) - x(t;w)JI,,d q/(1 - 4). Choose N , = max(M, N E . J . Then for m > N , we have

c -)(-)( c) ; M

I

&

= (" = I 2" 2

M

-1

+z &

= &.

Therefore

uniformly on closed intervals implies that x m ( t ;w ) -, x ( t ; w ) in the metric 9)). Now assume that xm(t;w ) + x ( t ;w ) in the topology on C,(R+ ,$(a,d, metric topology on C,(R+, $(a,d,9)), but (Ixm(t;w ) - x ( t ; w)II$(R,d,9) does not converge uniformly to zero on some interval [O,Q]. Without loss of generality assume that Q is a natural number. Thus by definition of uniform convergence there exists an E > 0 such that for each natural number 2 there exists a point t , E [0, Q ] and a k > 2 such that IIXk(t,

;0) - x ( t , ;4 IIJI(R.d.9) > E .

7.1

185

MATHEMATICAL PRELIMINARIES

Since x m ( t ;o)-,x(t ;o)in the metric topology, there exists an M such that, for m > M , we have

c -2"1 1 +

1

l l x m ( t ; 4 - x(t;o)ll,

a

Ilx"(t;o)

M and t E [0, Q] the following inequality: 1

IIx"(t ;w ) - x(t ;4 1 1@(n,d,P, 2Q 1 + Ilxrn(t;0)- 4 1 ; ~ ) I l , ( R , d , 9 )

1

d2Q 1

Ilx"(t;o) - x(t;o)llQ IIXrn(t;o)- x(t; WllQ

+

1 E M such that IIx'(t0;

implying that 1

4 - x(t0; w)ll,(n,d,P) > E,

Ilx'(t0; w ) - x(t0;

2Q 1 + Ilx'(t0;

W)ll~(R.d,P)

4 - x ( t 0 ; o)IIJl(n,d,P)

1

E

zQ 1 +

6'

This is a contradiction to inequality (7.1.1) and hence the proof is complete. The following lemma is analogous to Lemma 2.1.1. Lemma 7.1.8 Let T be a continuous linear operator from C,(R + , $(Q, d ,9)) C,(R + $(Q, d,9)). 3

+

If B and D are Banach spaces stronger than C,(R+, +(a,d ,9)) and if ( B , D ) is admissible with respect to T , then T is continuous from B to D. Thus, as stated in Chapters I1 and IV, T :B + D is bounded and there exists a constant K such that

II(Tx)(t;

o)llD

Kllx(t;

o)llB.

Thus we use the usual method to define the norm of T by

M = 1) TI1 * = sup{ II(Tx)(t;o)llD/llx(t;0)ll~ : X ( t ; 0) E B,

IlX(t ; o)ll~ #

o}.

186

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

Definition 7.1.4 The random vector-valued function x(t; o)on R+ is a random solution of Eq. (7.0.1) if for each t~ R+ ,x(t ;o)is a vector random variable and satisfies Eq. (7.0.1)P-a.e. The following lemma is essential for obtaining the main results of the chapter.

Lemma 7.1.9 The operator T , ( Tx)(t ; W ) =

1:

X(Z ;W ) d t ,

defined on C,(R+ ,$(a,d,9)) is a continuous linear operator from C,(R + $(Q, d,9))into C,(R + $(a,d, 9)). 9

9

P). It is PROOF For fixed t we shall show that (Tx)(t; o)E $(a,d, sufficient to show that for fixed t and each i, the function of w

1: 1 < 1;

( Txi)(t ; o)=

xi(z ;O)d t

is P-essentially bounded. Consider

l(Txi)(t; 011 =

1 1:

xi(z ; 0)dr

IXi(t

;011 dr


1. Thus for z E [0, t E ] , maxiIllxi(z;w)lI/ < M,. This, in turn, implies that for Z E [0, t E ] and each i, I l l ~ ~ ( ~ ; ~ ) l l < M , . Hence

+

+

+

+

+

+

(xi(z;o)l < I~~x,(T; w)lll Q M E ,

8-a.e.

(7.1.2)

for z E [0, t E ] . Since t , + t in R + , there exists an N , such that for n > N , , It, - tl < &/ME < E. Consider I(Tx,)(t,;o)- (Tx,)(t;w)l for n > N , where i is

arbitrary but fixed. By definition

I(Txi)(t,;w)- (TxJ(t;w)J = Then for t , > t we have ( ( T ~ i ) ( tW) , ; - (Txi)(t;o)(=

Since [t, t,] E [0, t

I["

Isd'

1

J:

x i ( ~ ; ~ )-d t x ~ ( T ; w ) .~ T

X ~ ( Z;O ) d T

I < I"Ix~(s

;w)I dz.

+ E ] , we have by inequality (7.1.2) that

l(Txi)(t,;w ) - ( T x , ) ( t o)l ;
0 and pick an interval [0, Q] c R , where Q is arbitrary but fixed. Since ~ " ( 5w; ) -+'x(t; o),there exists an Ne,Qsuch that n > NE,Qimplies that x"(t; o)-+T x ( t ; o)as n -+

IIx"(t; W ) - x ( t ; ~

) I I ~ ~ NE,Q. Thus

Illxl(t; 4 - x i V ; dill < dQ for t E [0, Q] and n > NE,a.This implies that Ixr(t; W ) - xi(t ;0)1

< E/Q

for n > N,,,, and t E [0, Q] .9-a.e., Consider I(Tx;)(t;o)- ( T x i ) ( t ;o)l for n > NE,Q and t E [0, Q]. We can write

I( Tx;)(t;O)- (Txi)(t;w)I

= =

I/:

[x;(z ;o)- xi(z ;4

3 dz

< J: Ix;(r ;o)- xi(z ;o)ldz


NEVQ and

tE

9-a.e.

[0, Q]

I(Tx;)(t; o)- ( T x ) ( t ;o)l < E,

9-a.e.

Thus for n > NE,aand t E [0,Q ] YP(o:I(Tx;)(t; 0)- ( T x ) ( t ;w)[ > E } = 0. This implies that for n > NE,Qand t E [0, Q] III(Tx;)(t ;4- (Txi)(t;w)lll < E.

7.1

189

MATHEMATICAL PRELIMINARIES

Since the argument is independent of the choice of i, it holds for each i. Therefore we can write that for n > NE,aand t E [0, Q] II(Tx")(t; 0)- ( T W ; W)II,(~,~.~)= maxlll(Txl)(t;4 I

This implies that II(Tx")(t;w ) - (Tx)(t;W ) I I , ( ~ , ~ , ~ ) converges to zero uniformly on [0, Q], which is equivalent to saying that (Tx")(t;w ) +' (Tx)(t;w ) and the proof is complete. Definition 7.1.5 Cb = C J R + ,$(Q, d,9))) will denote the set of all 9)such that for g a continuous functions x(t; w ) from R + into $(a,d, positive-valued continuous function on R + we have

Ilx(t; ~ ) I I , ( R , d * S ) G Zg(t), where Z is some positive constant which depends on x(t ; w). Definition 7.1.6 C' = C'(R+,$(a,d,9)) will denote the set of all continuous and bounded functions x(t; w ) from R + into 1.4,9).

$(a,

Lemma 7.1.10 The space C i ( R + ,$(Q, d ,9)) is a normed linear subspace of C,( R + , $(Q, d,9)) where the norm in Ci is given by Ilx(t; 0)llc; =;tpx(-yJ PROOF

w)ll,(n,d,s)lg(t)).

Let x ( t ; w), y ( t ; o)E Ci(R+,$(Q d ,9)). Then we can write

Ilx(t; 4

+ YO; 41,(Q,d,g) G Ilx(t; 411,(R,d,B) + Ilv(t; 41,(n.d,9) G ZlAt) + Z2g(O G (Zl + Z2Mt).

Ilax(t; d l l = I4 Ilx(t; w)ll G I4W)= W t ) . Thus the space Ci(R+ ,$(Q, d,9)) is a linear subspace of C,(R + ,$(Q d,9)). Also from the definition we know that (I . (Ici 3 0. The fact that (1 . (Ir, = 0 if and only if x(t; w ) = 0 follows from the fact that 11 . IIS(R,d,9) is a norm by Lemma 7.1.1. The fact that IlMt; 4 c ; = 14 llx(t; 4 c ; also follows directly from the fact that 11 . Il,(R,d,S) the triangle inequality holds, consider 11x0; 4

+ Y o ; w)llc:,

is a norm. To show that

+

= sup{Ilx(t; 0) Y ( t ; 4,(n,d,9)/g(t)f O 0, there exists an N , such that s, n > N , implies that IIZYo) - Zfl(4Il(b(f2,d,9) < E.

Hence, by definition, for s, n > N , we have that maxlllZXd I

-

ZXdlll < E.

This implies that for s, n > N , and each i IllZXo) - -T(w)lll < E.

Therefore for each i the sequence Z l ( o ) is Cauchy in L , ( R , d , S ) . Since L,(Q, d, 9) is a Banach space, Z l ( o )converges in L,(Q, d,9) to an element Zi(w).We shall now show that the sequence Zn(o)converges in $(Q, d,9) to the element Z(o), where Z(o)= (Z,(o),Z,(o), . . . ,Z,(o)). Since for each i, Zl(w) -+Z,(o)given E > 0, there exists an N i , , such that n > N,,, implies that IllzX~) - Zi(o)lll < E. Choose N ,

=

max,(N,,,}; then for n > N , maxlllZl(o) - Z,(w)lll < E. i

Hence, by definition, for n > N ,

llZ"(4 - Z(4ll(b(*,d,B)< E? implying that Z"(o)converges to Z(w),and the proof is complete. Lemma 7.1.12

is complete. The space CC(R+,$(a,d ,9))

PROOF Let x " ( t ; w ) be a Cauchy sequence in t o E R , and consider x"(to ; 0). We shall show that

C i ( R + , $ ( Q , d 7 9 ) )Fix . the sequence x"(to; o)is Cauchy in $(a,d ,9). Choose E > 0. Since x " ( t ; o) is Cauchy in C;(R+ , $(Q, d,P)),there exists an N,, such that s, n > N,, implies Ilx"(t; 0)- x v ; 0)llc; < e/g(to).

Thus by definition SUP{ Ilx"(t; 4 - x S ( t ;4(b(n,d,s,/g(t))< E/g(to), OSr

7.1

MATHEMATICAL PRELIMINARIES

191

implying that in particular for n, s > N , , That is, for n, s > N,,, Ilx"(to;w ) - xS(to;O ) I I ~ ( ~ , ~ < , ~ E) , implying that x"(to; w) is a Cauchy sequence in $(Q d,9). Since $(Q, d ,9) is a Banach space, there exists an x(to;w ) such that x"(to ;w ) --* x(to;w). Since the argument is independent of the choice of t o ,x"(t ; w ) + x(t ;w ) in $(a,d, 9) for ) an element of each r E R + . We claim that the function t + x ( t ; ~ is C i ( R + ,I,!@, d ,9)). Choose E > 0. Since x"(t ; w ) is Cauchy in C i ( R + ,$(Q, d, 9)), there exists an N , such that s, n > N , implies IIxS(t;0)- x y t ; o)llc; < 4 3 . Choose t o E R , arbitrary but fixed and s(ro, E ) such that s ( t o , E ) > N , and also such that

11

XS('O,E)

( t o ;0) -

x(to ;w)ll$(n,d,s~)< (&/3)g(to),

Now we can write Ilx"(t0 ;4 - x(t0 ;W)II$(n,d,b)/g(to) =

+

Ilx"(t0; w ) - XS('O'&)(to; 0) XS('OqrO;w ) - x(t0; w)(($(n,srs))/g(to)

d [Ilx"(to; 0)- XS('o;E)(tO; w)ll$(n,d,s)/g(to)l

+ [IIxSirO.E)(tO; 0) - x(to;w)IIILiR,d,p)/g(tO)ld 4~+ 3~ < E for all n > N , . Note that the integer N , is independent of the choice of and since to was arbitrary, we may conclude that for n > N ,

to

SUP{ Ilx"(t; 0) - 4 ; w)II$(n,d,8)/g(t)) < E. OQ1

To show that there exists a constant 2 such that Ilx(t; w)IIsiR,d,B) d Zg(t), consider the following. By the preceding argument we can choose an n such that sup{Ilx"(t; 4 - x(t; w)ll$(n,d,9)/g(Q} < 1' OQr

Now IIx(t; w)ll$(R,d,b)/g(t) = Ilx(t; w ) - x"(t; 0)

+ x"(t; 4ll$(R,d,B)/&)

d [Ilx(t; 0) - x"(t; w)II$(n,d,tP)/g(t)l

+ [Ilx"(t; w)ll~(R,d9)/g(~)l d 1 + [IIx"(t;w)II$(n,d,9)/g(t)l

d 1 + Ilx"(t; 0)llc;

=

z.

192

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

Therefore

d mt). To show that the function t -+ x(t ;o)is continuous, we must show that t, -+ t in R , implies that x(t,;w) -+ x ( t ; m ) in $(Q, d , P ) as rn -+ co.Fix t o € R,. Let t, + t o . Choose E > 0. There exists an N 1 such that m > N , implies Ig(t,) - g(to)l < g(to).By the first argument we can choose an n large enough so that Ilx"(t ; 0)- x(t ; dll~(*,d,9)/g(t)} < a&/g(to). Ilx(t; W)IIS(R,d,S)

zy'

Since x"(t; w ) E CC(R+,$(Q, d,P)),there exists an N 2 such that rn > N 2 implies /Ix"(t,; 0)- x"(t0; t0)II < be. Let N = max(Nl, N 2 ) .Then for rn > N we have

However,

Hence

7.1

MATHEMATICAL PRELIMINARIES

193

This in turn implies that for m > N This is simply the definition of convergence in $(Q, d,9). Since the choice of to was arbitrary, the same argument will suffice for each t and we have that the function t + x(t ;w ) is continuous. We can thus conclude that the 9)) The . conclusion of function t + x ( t ; o)is an element of C J R + ,$(a,.d, the first part of the proof that there exists an N , such that n > N , implies that SUP{ IlxYt; 0)- x(t; w)ll,(n,d.9.rP)ig(t); < I:. o N , IIx"(t ; 4

-

x(t ; w)llC; < 8,

which implies that the Cauchy sequence x"(t; w ) + x(t; w ) in C;(R+,$(a,d, 9)). Therefore the space C;(R+,$(a,d, P))is complete, as was to be shown.

Lemma 7.1.13 The space C'(R+,$(a,d,9)) is a normed linear subspace of C,(R+ ,$(Q, d, 9)) where the norm in C' is given by Ilx(t; w)llc,

=

sup{IIx(t; o)ll,(*,d*9)}. OQt

The proof is similar to that of Lemma 7.1.10 with g(t) = 1.

Lemma 7.1.14 The space C'(R+,$(Q;d, 9)) is complete. The proof is analogous to that given for Lemma 7.1.12.

Lemma 7.1.15 The Banach spaces

$(a,,rQ, 9))and C'(R+ $(a,d,9)) are stronger than C,(R+ ,$(a,d, 9)). C;(R+

7

9

as n + co. Select PROOF Let x " ( t ; o) x ( t ; w ) in C'(R+, $(Q, .d 9)) Q > 0 and consider the interval [O,Q]. We must show that Ilx"(t;w) - x(t; w ) I I , ( ~ , ~ , ~ ) converges to zero uniformly on [0, Q]. Choose E > 0. Since x"(t; o) x(t; o) in C'(R+, $(Q, d, 9)), there exists an N such < E . Hence for that n > N implies that supOat(Ilx"(t;o)- x ( t ; o)JI,cn,d,9,} every t E [0, Q] and n > N , -+

-+

IlxYt; 0)- x(t; W)ll,(*,d,b)

< 69

implying that C'(R+ , $(Q, d, 9)) is stronger than C,(R + , $(Q, d,9)). Let x"(t ; o)+ x(t ; co) in C I ( R + ,$(Q, d, 9)). Pick Q > 0 and consider the interval

[0, Q]. Choose E > 0. Since g is continuous, g assumes a maximum at some

194

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

point t o E [0, Q ] . Since x " ( t ; (0) -+ x ( t ; w ) in C ; ( R + ,$(O, d,9)), there exists an N , such that n > N , implies SUP{ llx"(t ;4 - x ( t ;w)llJl(*.d*B)/g(t)) < E/g(to).

Odt

Hence for t E [0, Q] and n > N , we have IIXYt ;4

- x ( t ;w)ll,(n,d*s)/g/g(t)< E M t 0 ) . This in turn implies that for t E [0, Q ] and n > N , IIx"(t; 0) - x ( t ; 4ll,(*,d,9) < &g(t)/g(to).

Since g(t) < g(to),g(t)/g(ro) < 1 and we conclude that for t E [0, Q] and n > N,

Ilx"(t; 4 - X(t ;w)ll$(n,d.B)c E.

Thus the space C i ( R + ,$(Q d ,9)) is stronger than C,(R+ ,$(a,d ,9)). 7.2

An Existence and Uniqueness Theorem

With respect to the partial aim of this chapter we state and prove the following theorem, which gives sufficient conditions for the existence of a unique random solution of (7.0.1). Also, a special case will be given which is useful when studying applications.

Theorem 7.2.1 Suppose that the random integral equation (7.0.1)satisfies the following conditions: (i) B, D E C,(R + ,$(a,d,9)) are Banach spaces stronger than C , ( R + , +(Q, d,9)) and the pair (B, D)is admissible with respect to

(Tx)(t;o)=

1:

x(r ;o)dr.

(ii) k(t, x(t ;0); w ) is a mapping from the set

W = ( x ( t ; o ) : x ( t ; o ) E D , llx(t;o)ll,,

< PI

into the space B for some p >, 0, such that IIk(t, x ( t ; 4; 4 - k(t, Y O ; 4; 4IB G Allx(t; 4 - ~ ( td ;l l D

for x ( t ; w), y ( t ; w ) E W and 1 2 0. (iii) x ( t ; o)-+ h(t, x ( t ; a))is a mapping from W into D such that

11 h(t, x(t; w)) - h(t, Y ( l ;w))ll D for some y 2 0.

< 711x(t;

- Y ( t ; w)ll

D

7.2

AN EXISTENCE AND UNIQUENESS THEOREM

195

Then there exists a unique random solution of Eq. (7.0.1) provided that y +AM < 1 and

IIh(t,x ( t ;w))llD + MIIk(t, x ( t ;0 ) ;w)IIB G p, where M = 11 T(I*. PROOF

Define the operator U from W into D by

(Ux)(t; w ) = h(t,x(t ; w)) +

Sd

k(r,X ( T ; w ); w ) d t .

We need to show that U ( W )G W and that for some r E [0, 1 ) Let x(t ; w),y(t ;to)E W . Since (Ux)(t; w) and (Uy)(t; w) E D and D is a Banach space, (Ux)(t; w ) - ( Uy)(t;w) E D. Thus we can write

where the last inequality is due to the Lipschitz condition given in (iii) and the fact that T is continuous from B to D by Lemma 7.1.9, and therefore bounded. However,

+

by the Lipschitz condition given in (ii). Since y Ml, < 1, the first condition of the definition of a contraction map is satisfied.

196

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

We must now show that the inclusion property holds. Let x(t ;o)E W . We can thus write

6

+

Its:

~ ~ ~ ( ~ , X ( ~ ; ~ k) ()~~, ~~ ( D z;

W);o)dT

d IIh(t,x(t;w))llDf M \ l k ( t , x ( f ; a ) ; w ) l l B d p. Hence ( U x ) ( t; w ) E W , implying U( W ) E W . Applying Banach's fixed-point theorem, we conclude that there exists a unique point x ( t ; o)E W such that ( u x ) ( t ; o )= h ( t , x ( t ; o ) )4-

and the proof is complete.

sd

k ( ~ , x ( ~ ; w )= ; ~x () td; o~)

The following theorem is a useful special case of Theorem 7.2.1. Theorem 7.2.2 Assume that Eq. (7.0.1) satisfies the following conditions : (i) k(t, x ( t ; 0); w ) is a mapping from the set W = { ~ (;0) t :x ( t ;0)E C'(R+

7

+(a,.r$, g)),I l 4 t ;o)llc' d P }

into the space Cb(R+,+(Q, d,9')) for some p 2 0; I I k ( t , x ( t ; o ) ; w )- k(t,y(t;w);w)llc;,

d W 4 t ; m ) - y(t;w)llc.

for x ( t ; w), y ( t ; o)E W , A 2 0 a constant; g is also integrable on R + . (ii) x ( t ;w ) 4 h(t, x ( t ;w ) ) is a mapping from W into C' such that Ilh(t,x(t;4 - h(t,y(t;o))II,.

d ~ l l x ( t ; w-) y(t;w)llc,

for some y 2 0. Then there exists a unique random solution of Eq. (7.0.1) provided that y AM < 1, where M = 11 TI[*,and

+

IlW, x ( t ;W))IIC' + Mllkk x ( t ;0); w)llc;, d

P.

PROOF The proof consists in showing that under the assumption that g is integrable the pair (CL(R+,+(a, d,Y)), C'(R+,$(Q, d,9))) is admissible with respect to the operator T given by

( T x ) ( t ;w ) =

s:,

X(T;

o)dz.

7.3

rm

r m

=

197

A STOCHASTIC CHEMICAL KINETICS MODEL

p < co,

9-a.e.

By definition of the norm in Lm(Qd,9), we can conclude that III(7'xi)(t;o)lll d

p for each i.

This in turn implies that II('x)(t;

u)ll$(o,d,y) = max{lll(7'xi)(t;~ ) l l l }< B,

which is the condition needed for ( T x ) ( t ;w ) to be an element of C'(R+,$(a,d, 9)). Since the remaining conditions are identical to those of Theorem 7.2.1, the proof is complete.

7.3 A Stochastic Chemical Kinetics Model Gavalas [ 11 has formulated a deterministic model which characterizes a chemically reacting system. It is the aim of this section to formulate a stochastic version of this model and thus make it describe the physical situation more realistically (Milton and Tsokos [l]). The basic formulations of such a model involve a random or stochastic integral equation of the type discussed theoretically in the previous sections. Thus we shall illustrate the applicability of the theoretical results, mainly Theorems 7.2.1 and 7.2.2, to obtain conditions under which a chemically reacting system will possess a unique random solution. The stochastic approach to the study of chemical kinetics is relatively new and has been developing rapidly during the last ten years. One of the main reasons for studying the classical chemical kinetics problem from a statistical point of view is that the evolution in time of a chemically reacting system is indeed random rather than deterministic. Blanc-Lapierre and Fortet [l] give a general discussion of how randomness enters into many physical systems. A brief description of how randomness enters into a

198

VII

AN APPLICATION TO STOCHASTIC CHEMICAL KINETICS

chemically reacting system is given by McQuarrie [I]. An excellent bibliography of recent work in the area of stochastic chemical kinetics is also given in McQuarrie’s paper. Bartholomay [l] gives a strong argument relative to the desirability of viewing chemical kinetics from a statistical point of view. In Section 7.3.1 we shall give a brief description of the classical chemical kinetics problem and give a basic definition and introduce some notation of the subject area to set the stage for the stochastic formulation. The stochastic interpretation of the rate of reaction of a simple system is given in Section 7.3.2. In Section 7.3.3 we shall give some basic concepts of the rate functions of a general reacting system and describe the manner in which an integral equation arises in chemical kinetics. The stochastic formulation of the chemical kinetics is given in Section 7.3.4. 7.3.1

The Concept of Chemical Kinetics

Chemical kinetics is that branch of chemistry which deals with the rate and mechanism of chemical reactions and attempts to discover and explain those factors which influence the speed and manner by which a reaction proceeds. The reaction of a system under study which takes place in a single phase can be characterized at each point by the following variables: the velocity, the concentrations of all chemical species, and a thermodynamic variable such as the internal energy or temperature. A chemical system is called uniform if there are no space variations within the system. In our brief discussion of the subject area we shall assume that we are dealing with a homogeneous, uniform system at constant volume and constant temperature. We shall be concerned with the evolution of the system in time, and the variables used to study this evolution will be the concentrations of the species involved in the reactions. Such variables are called state variables. Thus chemical kinetics is concerned with the manner by which a reacting system gets from one state to another and with the time required to make the transition. In what follows we shall give certain notation and ideas which are fairly standard in chemical kinetics and stoichiometry. We shall be concerned with a mixture of N chemical species M 1,M, , . . . , MN. For example, the chemical reaction 2HzO

-+

2Hz

consists of three species M , = HzO, M, reaction is usually written as N

+ 0, =

H,, and M 3

C “Mi = 0,

i= 1

(7.3.1) =

0,. A chemical (7.3.2)

7.3

A STOCHASTIC CHEMICAL KINETICS MODEL

199

where vi is called the stoichiometric coeficient of species M iin the balanced equation for the reaction. That is, reaction (7.3.1)can be written as 2 H 2 0 - 2H2

- 0,= 0.

We shall consider a species M i to be a reactant if vi > 0 and a product if vi < 0, for i = 1,2,. . . ,N. A similar convention is given by Gavalas [l]. Note that from the stoichiometric equation we have 6ni/6n, = vJv,,

1

0

+ iAq)[l - h7(iA;o)(iAl - A(o))-'?(iA;o)]-' d'(iJ.;cu)(iJ.l


0 is some constant depending on t We shall make use of the integral operators TI and T2on C,(R+ , L,(R, d, P)), defined as follows : (10.1.1) and (T,x)(t;o)= where K ( t , z ; w )=

sd

s:

K(t,r; w ) x ( T ; o ) ~ T ,

(10.1.2)

k(s,~;o)ds.

(10.1.3)

244

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

These integral operators will be needed in obtaining existence and uniqueness of a random solution of Eq. (10.0.1). It is clear from the given conditions and Lemma 4.1.1 that the integral operators Tl and T, are continuous mappings from C , ( R + ,L,(R, d, 9)) into itself. If we integrate Eq. (10.0.1) from zero to t, we obtain

=

xo(w)

sd

+ J ; ~ ( T , X ( T ; W ) +) ~ T K ( t , ~ ; o ) f ( x ( z ; w ) ) d z ,(10.1.4)

where xo(w) = x(0; w ) and K(t, 7 ; w ) is given by Eq. (10.1.3). We now prove the following existence theorem (Padgett and Tsokos [15]). Theorem 20.2.1 Suppose the random equation (10.0.1) satisfies the following conditions :

9)) and (i) B and D are Banach spaces stronger than C , ( R + ,L,(R, d, the pair ( B , D) is admissible with respect to each of the integral operators (Tlx)(t;w)=f0x(7;w)d7 and (T,x)(t;w)=& K(t,z;w)x(t;w)dt, t 2 0 , where K(t, 7 ; w ) is given by (10.1.3). (ii) x(t;w)+ h ( t , x ( t ;0))is an operator on S = { x ( ~ ; ~ ) E D : I I x ( ~ ; o6 ) (pI D }

with values in B satisfying Ilh(t, x(t; 0)) - h(t, ~ ( tw))IlB ; 6

I1 Ilx(t; ~ 0 ) ~ ( tw)llD ;

for x(t; w ) , y ( t ; w ) E S and I l constant. (iii) x(t;w)+f(x(t;w)) is an operator on S with values in B satisfying f ( 0 ) = 0 and Ilf(x(t; 0))- f ( y ( t ; w))IIB 6 A211x(t;O ) - y([; w)llD for x(t; w ) , y(t ; w ) E S and I , constant. (iv) xo(w)E D. Then there exists a unique random solution of (10.1.4),x(t ;w ) E S , provided

where K1 and K , are the norms of Tl and T,, respectively.

10.1

THE STOCHASTIC INTEGRODIFFERENTIAL EQUATION

245

PROOF By Condition (1) TI and T, are continuous from B into D. Hence their norms K , and K , exist. Define the operator U from S into D by

=

x0(w) +fh(r,x(r;w))dr 0

+

s:

K ( t , ~ ; w ) f ( x ( ~ ; w ) ) d t(10.1.5) .

We must show that U ( S ) c S and that U is a contraction operator on S. Then we may apply Banach’s fixed-point theorem to obtain the existence of a unique random solution. Let x(t;w ) E S . Taking norms in Eq. (10.1.5),we get

By Condition (ii)

and by Condition (iii)

=P by the last condition of the theorem. Thus U ( S ) c S .

246

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

Let y(t ;w ) be another element of S . We have, since the difference of two elements of a Banach space is in the Banach space,

Il(ux)(t;

W,

=I(

- ( U y ) ( t ;w)llD

II

dz + Jo h(7, x(7 ;a))d7 + Jo K(t, z ;w ) f ( x ( z;0)) ff

f l

xo(w)

G K II

x(t ;4) -

W ,YO ;w ) )II

+ K 2 l l f ( x ( t ; 4 )- f ( Y ( t ; 4 ) l l e

< (AlK1 + &&)llx(t;w)

- y(t;w)IID

by Conditions (ii) and (iii). Since by hypothesis l , K l + L2K2 c 1, U is a contraction operator on S . Applying Banach’s fixed-point theorem, there exists a unique element of S so that (Ux)(t;w ) = x(t ;w), that is, there exists a unique random solution of the random equation (lO.O.l), completing the proof. Now, when the stochastic perturbing term h(t, x(t ;0)) is zero we obtain a stochastic version of the integrodifferential equation studied by Levin [l] as a corollary to Theorem 10.1.1.

s:

Corollary 10.1.2 Consider the stochastic integrodifferential equation k(t ;O ) =

k(t, z ;w ) ~ ( x ( 0)) T ; dz

(10.1.6)

under the following conditions :

(i) B and D are stronger than C,(R + , L,(R, d ,P))and (B,D)is admissible with respect to the operator ( T x ) ( t ;w ) = So K ( t , r ;w)x(r;w ) dr, t 2 0, where K(t, r ;w ) is given by Eq. (10.1.3) and behaves as described. (ii) x(t ;w ) + f ( x ( t ; 0))is an operator on S = ( x ( t ; o ) ~ D : I l ~ ( t ; w I~

10.1

24 7

THE STOCHASTIC INTEGRODIFFERENTIAL EQUATION

with values in B satisfyingf(0) = 0 and

< AI\x(t;

IIf(x(t ;0))- f ( y ( t ;w ) ) \ l B

O) -

Y ( t ; w)llD

for x ( t ; w), y ( t ; w ) E S and 1constant. (iii) xo(o) E D. Then there exists a unique random solution of Eq. (10.1.6)provided AK and IIx0(w)IID ,< p(1 - AK), where K is the norm of T.

-= 1

Since Eq. (10.1.6) is the equivalent of Eq. (10.1.4) with h ( t , x ) equal to zero, the proof follows from that of Theorem 10.1.1with T, the null operator. 10.1.1 Asymptotic Behavior of the Random Solution

Using the space C,(R + ,L,(O, d,P)),we now give some results concerning the asymptotic behavior of the random solution of Eq. (10.0.1). We shall first consider the unperturbed equation (10.1.6). Theorem 20.1.3 Suppose Eq. (10.1.6) satisfies the following conditions :

(i)

J: Illk(s,T ; w)lll ds < A,

for some constants A 1 > 0 and tl > 0,

O < T < t .

(ii) x ( t ;w)+f(x(t ;0)) satisfies, for some A2 > 0 and f(0) = 0, Ilf(x(t; w))ll < A, e-O', t 2 0, and IIf(x(t; w)) -f(y(t; 0))ll for Ilx(t; w)I( and Ily(t; w)ll (iii) xo(w) = 0, 9-a.e.

tl

>

> 0,

< Allx(t; 0 ) - y ( t ;4 1

< p e-O' at each t 2 0 and 1 constant.

Then there exists a unique random solution of (10.1.6) which is stochastically exponentially stable, llx(t; w)ll

0)1~]}+

= {E[lx(t;

< p e-p',

t 2 0,

where E [ . ] is the mathematical expectation, provided that A is small enough. PROOF It is sufficient to show that Condition (i) implies the admissibility of the pair of spaces (Cg,C,) with respect to the operator

(Tx)(t;w) = with

J: S:

K ( t , T ;w ) =

K ( t , ~ ; o ) x ( ~ ; w ) d T t, 2 0,

k(s, T ; w ) ds,

0d

T

(10.1.7)

< t,

and that Condition (ii) is equivalent to Condition (ii) of Corollary 10.1.2, g(t) = e-O*,p > 0. with B = D = C,(R+, L,(O, d,9)),

248

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

by Condition (i). But x ( t ; w ) E C,(R+ , L,(R, d ,9)) with At) = e-", t 3 0, fl > 0, and we get by definition of the norm in C , that

a>

= A 1IJx(t;w))1,--(a - 8)-l ( e P p t- e-")

< l M t ; w ) l l c , [ ~ l M~ flu since a > > 0. Hence for x ( t ; W ) E C , , (Tx)(t; w ) C ~ , ; that is, (C,, C,) is admissible with respect to T. From Condition (ii),J'(x(t ;w))E C,(R + ,L,(Q d,P)),and sup{llf(x(t; 230

0)) - f(y(t;

o)ll/e-P'}

< A sup{ IIx(t; w ) - y(t; o)II/e-B2} 250

implies that the Condition (ii) of Corollary 10.1.2 holds. Therefore by Corollary 10.1.2the conclusion follows. If h(t, x) is not identically equal to zero, then we can still obtain the result that there is a unique random solution of Eq. (10.0.1) which is bounded in mean square for all f E R + . Theorem 10.1.4 Assume that Eq. (10.0.1) satisfies the following conditions : (i) $, Illk(s, z; w)lll ds d A , for some constant A, > 0,O < T < t. (ii) x(t ; o)+ h(t, x(t ;0))satisfies, for some A2 > 0 and fl > 0,

and Ilh(t,x(t; o))(I< A, e - p t Ilh(t,x(t;w)) - h(t,y(t;w))Il d A, e-Prllx(t;4 - v(t;w)ll for ( I x ( t ;w)ll and IIy(t; w)ll < p , t 2 0, and A1 constant.

10.1

THE STOCHASTIC INTEGRODIFFERENTIAL EQUATION

249

(iii) x(t; w)+f'(x(t; w))satisfiesf'(O) = O,IIf'(x(t; w))II d A3 e-01,A3> 0, and II f ' ( x ( t ;w ) ) - f'(y(t; w))Il < ;I., e-PtllX(t; 0) - ~ ( tw)ll ; for Ilx(t; w)ll and Ily(t; w)ll d p, t 2 0, and 1, constant. (iv) xo(w) E C , . Then there exists a unique random solution of Eq. (10.0.1) satisfying Ilx(t;4ll = { E [ l x ( t ; ~ ) l 2 l } *d P,

tER+,

(bounded in mean square on R + ) , provided that ;I.,,A,, IIh(t, O)llcg are sufficiently small.

(Ixo(w)IIc,, and

PROOF It will suffice to show that the pair of spaces (C,(R+, L,(R, d ,9)), C , ( R + ,L,(R, d, 9'))) is admissible with respect to the integral operators defined by equations (10.1.1)+10.1.3) under Condition (i). Then . from Eq. (10.1.1) we have that Let x(t; w ) E C,(R+, L,(R, d,9))

= lIx(t; w)llcg(l//w

-e-9

< Ilx(t;w)IIcg(l/P) < a , by definition of the norm in C,(R + , L,(Q d,9)). Hence ( T , x ) ( t ;0)E C I ( R + ,

w-&9)) &I

and the pair (Cg,C , ) is admissible with respect to TI. Now from Eqs. (10.1.2) and (10.1.3) for x(t; w )E C,(R+, L,(R, d,9)) we obtain

< A,

1;

[[Ix(T;

~ ) ( I / e - ~e-P'dz 'l

d A, Ilx(t; w)IIcg

f e - p r dz 0

< A, Ilx(t; 411cg(1/P) < a from Condition (i). Thus (T,x)(t;w ) E C , ( R +,L,(Q, d,9)) and the pair

(C,, C , ) is admissible with respect to T,.

250

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

Therefore the conditions ofTheorem 10.1.1 hold with B = C , , g ( t ) = e - B 1 , = C,, and there exists a unique random solution of (lO.O.l), x(t ;w ) , bounded in mean square by p for all t E R + .

p > 0, and D

10.1.2 Application to a Stochastic Differential System Consider the following nonlinear differential system with random parameters : (10.1.8) n(t; 0) = A ( o ) x ( t ;0) b ( o ) 4 ( o ( t ;w ) ) ,

+

b ( t ;o)= cT(t; w)x(t ; o),

(10.1.9)

where A(w) is an n x n matrix of measurable functions, x(t ;o)and c ( t ; o) are n x 1 vectors of random variables for each t E R , , b ( o ) is an n x 1 vector of measurable functions, 4(o) is a scalar function, o(t ; o)is a scalar random variable for each t E R , , and T denotes the transpose of a matrix. Note that Eq. (10.1.9) can be written as o ( t ; o )= o(0;o)

+

J-:

cT(s;o)x(s;w)ds,

which is similar to a system studied in Chapter IX. The system (10.1.8H10.1.9) may be reduced to a stochastic integrodifferential equation of the form (10.0.1). We may write (10.1.8) as

J-:

x ( t ; w ) = eA(W)rxO(w) + eA(w)(t-r)b(w)+(o(r; w ) )dr.

Substituting this expression for x ( t ; w ) in (10.1.9),we obtain

+

&(t ; w ) = cT(t;o)eA(uJ)rxo(w)

J-:

cT(t;w ) eA(")('-')b(w)4(o(r; 0))dr.

Assume that I((cT(t; w)lll < K for all t 2 0 and K , 2 0 a constant. Also let x o ( o )E C,, +(O) = 0, and b ( w )E L,(R, d,9).If we assume that the matrix A ( o ) is stochastically stuble, that is, there exists an ci > 0 such that S ( w : R e & ( o ) < -a,

k = 1,2,. . . , n ] = 1,

where +k((w), k = 1,2,. . . ,n, are the characteristic roots of the matrix, then it has been shown by Morozan [3] that


0. We also let &o(t; 0)) be in the space t 2 0, and C,(R + ,L,(Q at,9)) with At) = I&o,(t ; 0))- 4(02(t;o))l

< 1 e-atlol(t;4 - 0 2 ( t ;w)l.

10.2

Let

REDUCTION OF SYSTEMS WITH TIME LAG

25 1

h(t, a(t ;w ) ) = cT(t;O)e A ( w ) r ~ o ( ~ ) .

Then,

llh(t, a([; o))ll d

o)IIIK, e-a'llxo(o)ll d KIK,

lIICT(f;

e-"Z

where Z > 0 is a constant, since xo(w)E C , . Thus h(t, d t ;0)) E Cg(R + L,(Q, d , 9)) 9

by definition. Also, IIh(t, o,(t;

0)) -

h(t, o,(t; 0))ll

=

0

so that it satisfies a Lipschitz condition. b(o), and A(w) we Now, by the assumptions we have made on cT(t;o), have k(s, t ; o)= cT(s;o)eA('U)(S-7)b(o), satisfying

1

Ill&,

f;

o)lllds d l

IllcT(s; o)lllK2e-a(s-7)lllb(o)lll ds

d K K, e"'llIb(o)lll

f e-as ds 7

=

K1K2111b(w)lll(l/cc)[l - e-"('-')]

< K,K,IIIb(~)Ill(~/~). Therefore all conditions of Theorem 10.1.4 are satisfied and there exists a unique random solution of the system (10.1.8H10.1.9) which is bounded in mean square on R + . 10.2 Reduction of the Stochastic Nonlinear lntegrodifferential Systems w i t h Time Lag 10.2.1

The Integrodifferential System (10.0.2)-( 10.0.3)

The random system with time lag (10.0.2) can be written as a stochastic integral equation of the form x(t;w)

=

+

sp,

X ( t ; ~ ~ ) x 0 ( ~ ) X ( t - t - u;w)B(o)x(u;o)du

+ J; X ( t - u ; o)b(o)&a(u ;0)) du,

t > 0,

(10.2.1)

252

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

X

where xo(w) = x(0; w) and X ( t ; w) is the stochastic fundamental matrix solution of the homogeneous system

+

x'(t; 0)= A(w)x(t;0) B(w)x(t - z; w),

t

> 0,

with the initial condition X ( 0 ; w) = I , the identity matrix, and X ( t ; w) = 0 for t < 0. Let $(ti

0) =

+ s_". X ( t -

X ( t ; w)x~(o)

-

u ;~ ) B ( ~ ) ;xW)( udu.

Then the random equation (10.2.1) becomes

+

s:

~ ( t0) ; = f ( t ; 0)

+

~ ( W) t ; = $ ( t i W)

with

X ( t - u ; w)b(w)+(a(u ;0))du

s:

cT(t -

U ;O)X(U; W)

Substituting Eq. (10.2.2) into (10.0.3),we get

+

a(t;w) = f ( t ; w )

+

s:

h(t;w)

du.

i

CT(t - u ; o ) $ ( u ; w )

I

X ( u - s ; w)b(w)+(a(s; w)) ds du.

(10.2.3)

+ yocT(t - u ;w)$(u ;w) du. Then Eq. (10.2.3) becomes

Let h(t ;w) = f ( t ;w) o ( t ;w) =

si

sd

(10.2.2)

+

1: si

cT(t - u ; w)

sI

X ( u - s; w)b(w)+(a(s;

0))ds

du.

Using the property of the convolution integral, we have cT(t - u ; w) =

=

=

X(u

-

1; I:-' 1:s:-'

0))ds

du

X ( t - u - s ; w)b(w)+(o(s; 0))ds du

cT(u;w)

/: s:-'

s ; w)b(w)4(a(s;

cT'(u;o ) X ( t - u - s ; w)b(w)+(a(s; w)) ds du cT(u;w ) X ( t - u

Define

k(t ;W) =

s:

-

s; w)b(w)+(a(s; 0))du ds.

C~(U ;o ) X ( t -

u ; w)b(w)du.

(10.2.4)

10.2

REDUCTION OF SYSTEMS WITH TIME LAG

253

Then

which is the same as Eq. (10.2.4). Therefore the equation for the error signal in the presence of a random parameter o ( t ; o)can be written as ~ ( O t ;) = h(t ;O )

+

k(t - u ; L O ) ~ ( O ( U;w ) )du.

(10.2.5)

Thus we know that there exists a unique random solution to Eq. (10.2.5) under the conditions given in Chapter 11. 10.2.2 The Random Integrodifferential System (10.0.4)-( 10.0.5)

The random equation (10.0.4)can be written as X ( t ; 0)

x(t;w)Xo(w) +

+

1;

X(t

+ j:X(t

-

s_". x(t

- T - U ;w)B(w)X(u;0) du

u ;o)b(w)4(o(u ; 0)) du

- U;W)

q(t - s ; o ) 4 ( a ( s ; o ) ) d s d u ,

(10.2.6)

where x0(o) and X ( t ;o)behave as defined earlier. We shall write

+

$ ( t ; W ) = X ( t ; o)x,(o)

s_",

X(t - T

-

u ; o ) B ( o ) x ( u ;o)du.

Then the stochastic integral system (10.2.6) becomes

+

x ( t ; o )= $(t;o)

+ /;X(r

J:

X ( t - u;o)b(w)f)(o(u;w))du

- u;w)[:q(t - s;w)+(a(s;o))dsdu.

(10.2.7)

254

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

Applying the well-known result that the convolution product commutes, we can reduce part of Eq. (10.2.7) as follows: J : X ( t - u ; ~ ) S d ( q (-t s ; o ) ~ ( a ( s ; w ) ) d s d u =

J ; X ( U ; W ) J : - ' ~ ( I- u - s;w)4(D(s;W))dsdu

=

fJ;-' X ( U;~ ) q ( -t u - s ;w)&o(s; w ) )ds du X ( U; ~ ) h ( t u - s ;w ) ~ ( D ( w s ;) )du ds.

(10.2.8)

Let k,(t;o)=

i'd r

i':

X(t

-

s;w)q(s;w)ds.

Then Eq. (10.2.8) can be written as follows :

1;

X ( t - s - u ; ~ ) r l ( ~ ; o ) 4 ( a ( s ; ~ ) ) d=u d sk , ( t - s ; w ) & ~ ( s ; w ) ) ~ s .

Therefore Eq. (10.2.7) becomes x(t;w)= $(t;w)

+

sd

X(t - u ; w ) ~ ( w ) ~ ( D ( u ; w ) ) ~ u

+ /;k,(t - u;u)4(a(u;w))du or ~ ( tW; ) =

$ ( t ;W )

where

+

sd

K2(t -

U ;W ) + ( C ( U ; w ) )du,

(10.2.9)

+

k,(t ;W ) = X ( t ;w ) b ( ~ ) k , ( t ; w ) .

Substituting stochastic Eq. (10.2.9) into Eq. (10.03, we have a ( t ; o )= f ( t ; W )

+

+ J;cT(t

cT(t - u ; w ) $ ( u ; w ) ~ u - u;r.)J0'

k,(t - s ; w ) 4 ( a ( s ; w ) ) d s d u . (10.2.10)

10.3

255

STOCHASTIC ABSOLUTE STABILITY OF THE SYSTEMS

Define

Sd

+

h ( t ; o )= f ( t ; o )

CT(t

- u;o)~(u;o)du.

Then random equation (10.2.10) can be written as

+

a ( t ; o )= h ( t ; o )

J-: I:-"

cT(u;o ) k 2 ( t - u - s; o)4(a(s; w ) )ds du.

Let k(t ;W ) =

Then Eq. (10.2.11) becomes a ( t ; o )=

(10.2.11)

l

C'(U

+

h(r;w)

sd

; o ) k , ( t - u ;W ) du.

k(t -

U ; W ) ~ ( ~ ( U ; O ) ) ~ U ,

which is the same as the nonlinear stochastic integral equation (10.2.5) that we obtain by reducing the stochastic system with time lag (10.0.2H10.0.3). 10.3

Stochastic Absolute Stability of the Systems

The following theorems give the conditions under which the stochastic differential systems with lag time (10.0.2H10.0.3) and (10.0.4)+0.0.5) are stochastically absolutely stable. Theorem 10.3.1 Suppose that the stochastic system with time lag (10.0.2H10.0.3) satisfies the following conditions :

+

(i) The equation det{A(o) e-"B(o) - 21) = 0 has all its roots in the semiplane Re 1 < - a < 0. (ii) (a) The random vector function c(t ;o)is defined for all t 2 0, o E R, and is such that c ( t ; W ) E L I ( R + , L , ( S Z , , ~ , 9 ) ) nL2(R+,L,(SZ,&',9)); (b) f ( t ;o) is defined for t 2 0, W E Q, and is such that f ( t ; o ) , f ' ( t ; o ) E L I ( R + L,(Q, d,9)). (iii) There exists a q 3 0 such that 9

+ il.q)ET(iA; o)[i11- A ( o ) - B ( o )eiAr]-' b ( o ) },< 0, where ZT(iA; o)= I," cT(t;o)e-i2.'dt. Re((1

Then the stochastic system (10.0.2H10.0.3) is stochastically absolutely stable.

256

X

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

PROOF We shall prove the theorem by showing that the assumptions of Theorem 9.2.1 are satisfied. By definition,

h ( t ; o )= f ( t ; w )

+

J:

CT(t -

u;w)$(u;w)du

+ J~cT(u;w)$(t

=f(t;u)

-

u;w)du.

From Condition (ii) we know that the functions f ( t ;w),c ( t ;0) E Li(R + , L,(Q ,d, sq)).

The definition of $(t ;w), together with assumption (i) of the theorem, Thus the convolution product implies that $(t ; w )E L , ( R + , L,(Q d,9)). 9)). Hence of cT(t;o)and $ ( t ; w ) also belongs to L I ( R + ,L,(R, d,

4 t ; w )E L , ( R + ,L$4 d,9)). Differentiating h(t ; w ) with respect to t , we have

h ( t ; w )= f ( t ; w )+

J:

c'(u;w)$(t - u ; w ) d u .

(10.3.1)

Each term of Eq. (10.3.1) belongs to L I ( R +L,(R, , d, 9)).Thus, h ( t ;w ) d, 9)) and assumption (i) of Theorem 9.2.1 is satisfied. We shall consider the stochastic kernel

E L , ( R + ,L,(R,

k ( t ; 0)=

and L ( t ;w ) =

It is given that

J: J:

cT(s;w ) X ( t - s; 0)ds

cT(s; w ) X ( t - s; 0) ds.

~(w t ;) E L i ( R + L,(R d ,9)) n L,(R+ Lm(Qd, 9)) 7

9

and from assumption (i) we conclude that X ( t ;w ) belongs to L 1 ( R + L,(R, , d,9)). Using the fact that the convolution product of c T ( s ;w ) and X ( t ;w ) belongs to L I ( R + L,(R, , d,9))n L 2 ( R + ,L,(Q d,Y)), we have

k ( t ; 0)E L , ( R + ~ , ( Q d,9))L,(R + L,(Q d,9)). Applying a similar argument, it can be easily seen that 3

7

4 E L , ( R + , L,(Q d,9)) n L,(R + , L,(Q d, P)), and Condition (ii) of Theorem 9.2.1 is satisfied. Assumption (iii) of this theorem is the same as (iii) of Theorem 9.2.1. Knowing the fact that the

10.3

STOCHASTIC ABSOLUTE STABILlTY OF THE SYSTEMS

25 7

Fourier transform of the convolution product is equal to the product of the Fourier transforms, we have

&iA; w ) = Z T ( i l ; w ) [ i l l - A(w) - B(w)elLr]-'b(o). From Condition (iv) of the theorem, we obtain

Re((1 + ilq)?(iA; w)[iAl - A ( o ) - B(w)e ' L r ] - l b ( w ) 0 for

r~ # 0. (iv) There exists a q b 0 such that

Re((1

+ iAq)ZT(iA;o)[iAI- A ( o ) - B(o)e'"']-'(b(o)+ q ( i A ; o ) ) },< 0,

where ZT(iA;o)=

cT(t;w ) e-lL' dt and q(iA; o)= J; q(t ;o)e-"' dr.

Then the stochastic system of equations (10.0.4)+0.0.5) absolutely stable.

is stochastically

PROOF This theorem will also be proved by demonstrating that the conditions of Theorem 9.2.1 are satisfied. By definition

h ( t ; o )=

+

f ( t ; ~ )

cT(t -

u;W)$(U;W)du,

258

x

STOCHASTIC INTEGRODIFFERENTIAL SYSTEMS

where

From the hypothesis of the theorem, we have Also, because of the fact that from Condition (i), and x(t ; o) is a continuous function for t E [ - T, 01 and a.e. with respect to o. Differentiating h(t ; w ) with respect to t , we have

Using a similar argument, it is easy to see that h(t ; 4 E L ( R+ , La@, d, 9))n L2(R+ ,L,(R, d, 9)).

Thus assumption (i) of Theorem 9.2.1 holds. We have defined the stochastic kernel k(t ; w ) as follows :

with and

It is given that X(t ;w), c(t ;o ) , and q(t ; w ) belong to Also, because each of its terms belongs to L l ( R+ , La@, 4 9)) n L 2 ( R + ,L,(R,

d,P)),which implies that

k2(t;a)E L , ( R + La(Q d, 9)) n L z ( R + ,Lm(Q,d, 9)).

10.3

STOCHASTIC ABSOLUTE STABILITY OF THE SYSTEMS

k l ( t ; w )=

259

6

X ( t - s;w)q(s;w)ds.

Using a similar argument, it can be seen that k ( t ; ~ ) E L ~ ( LRm+( Q d ,9)) n L,(R+ 3

9

Lm(Q

d ,9)).

Thus Condition (ii) of Theorem 9.2.1 is satisfied. Assumption (iii) of Theorem 9.2.1 is the same as Condition (iii) of this theorem. It can be shown that the Fourier transform of the stochastic kernel k ( t ; w ) is given by

l(iA;w)= ET(iA;w)[iAl- A(w) - B(w)e'"]-'[b(w) + q(iA;w)].

From hypothesis (iv) of the theorem, we have Re((1

+ iAq)k(iA;w))< 0,

which satisfies Condition (iv) of Theorem 9.2.1. Therefore we conclude that the stochastic system with time lag is stochastically absolutely stable.

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Index Numbers in italics refer to the pages on which the complete references are listed.

A Admissibility theory, 30,49,54, 102 Adomian, G., 3,266 Ahmed, N. U., 3,266 Almost sure convergence, 68 Anderson, M. W., 3,4,79,266 Approximate solutions, 65-78, 87-96, 141-148 Arithmetic fixed-point problem, 74 Asymptotic stability, 20-21, 38-39

B

Banach, S., 9 Banach space(s), 8, 14, 193 admissible, 14, 3 1 of sequences, 17-18, 133 Barbalat, I., 10, 27,260 Barbalat lemma, 10,27-28 Bartholomay, A., 198,266 Bartlett, M. S., 78,80,267

Beckenbach, E. F., 137,260 Bellman, R., 2,50,57,59, 137,260,261 Bharucha-Reid, A. T., 2, 3,4, 14,44,49, 78,267 Biological system, 57-64, 165-179 one-organ, 57-62 two-organ, 62-64 Bochner, S., 10,261 Bounded variation, 18 Branson, H., 165, 166,261 Brownian motion, 2 Brownian motion process, 207-209,214 Burkholder, D. L., 66,87,88,267

C Cauchy sequence, 7 Chemical kinetics, 180, 198-200 rate functions, 201-204 rate of reaction, 201 stochastic integral equation in, 204-206 275

276

INDEX

Chemotherapy models, 57-64 Circulatory system, see Physiological models and Chemotherapy models Closed-graph theorem, 9,31, 134 Communicable disease model, 176-179 random integral equation in, 178 Contraction operator, 9, 22, 32 Contraction mapping, 66 Control systems discrete, 148-155 feedback, 115 stochastic, 115-1 19, 148-155 Convolution, 9,227,256 Corduneanu, C., 5,98,115,261

D Desoer, C. A., 115,261 Discretized equation, 132 Distefano, N., 46, 268 Doob, J. L., 208,209,211,268 Dunford, N., 11,262 Dynkin, Y. B., 208,268

E Error signal, 253 Essentially bounded function, 14, 18. Eulerian velocity field, 56 Existence and uniqueness theorems, 30-39,49-55,8L87,98-113,

120-131, 157-164,194-197,215,244 for random discrete equations, 133-141

F Feller, W., 79, 80, 268 Fixed-point theorem(s), 9,22-27 Banach’s, 9,22-23,54, 109 Brouwer’s, 24,26 Krasnosel’skii’s, 10, 102 Schauder’s, 10,25-26,106 Fortet integral equation, 44 Fortet, R., 2,43, 197,267,268 Fourier transforms, 10,225 FrCchet space, 8,99, 101 Fredholm random integral equation, 18, 97-1 3 1 discrete version, 132-155 Free random variable, 18

G Gavalas, G. R., 197, 201,204, 262 Gikhmann, I. I., 2,208,268

H Halanay, A., 10,262 Hamdan, M. A., 4,273 Hans, O., 3,4,268 Hardiman, S., 79,268 Hearon, J., 165, 166,262 Hereditary mechanics, 46-48 Hilbert space, 11, 16, 101, 102 Hildebrand, F. B., 149, 153,262

I Inner product, 1 1, 16 Ito, K., 2, 208, 269 Ito stochastic integral, 208-21 1

J Jacquez, J. A., 2,57,59,260,261 Jazwinski, A. H., 208,211,269

K Kalaba, R., 2,57,59,260,262 Kendall, D. G., 79, 80,269 Kernel, 18,33, 180,243 Kotkin, B., 59,261

L L,, 7 Landau, H. G., 165,176, 177,263 Levin, J. J., 243, 263 Linear hereditary phenomenon, 46 Linear space, 8 normed, 8 topological, 8 Lipschitz condition, 32,52, 103 Lobve, M., 68,269 Lumley, J. L., 2, 56, 270 Lur’e, A. I., 242, 264 Lyapunov, 4,40

277

INDEX

M Markov inequality, 68 McKean, H. P., 208,270 McQuarrie, D., 198,270 Mean-square convergence, 71 Mean-square integral, 3, 18 Metabolizing systems, 165-170 random integral equation in, 167 Method of successive approximations, 60, 66-78, 208 Metric space, 7 Mikhlin, S. G., 59,264 Miller, R. K., 98, 243, 264 Milton, J. S., 2,3,79, 120, 129, 156, 157, 165,170,176,181,197,200,270 Minkowski’s inequality, 9, 13, 50, 137 Morozan,T., 2,4,88,218,270 Mukhejea, A., 14,49,270,271

N Nohel, J. A., 98,243,263,265 Norm, 7,181 of operator, 157, 185 0

Operator, 9 bounded, 31,102,134 closed, 14, 31 completely continuous, 10, 102, 105, 106, 121,125 continuous, 14, 30-31, 99, 133, 185, 186,214 contraction, 9,22, 32 linear, 31,99, 115, 134, 185, 186 nonlinear, 115

P Padgett, W. J., 2, 3,4,57,79, 120,244, 271,272 Parseval equality, 10,227 Petrovanu, D., 18,98, 115,265 Physiological models, 170-175 random integral equation in, 172 PoincarC-Lyapunov stability theorem, 40-42

Popov, V. M., 4,218 Popov frequency response method, 4, 149, 152,225 Population growth problem, 78-87 numerical solution, 94-96 Postnikov, V. N., 242,264 Probability measure space, 12 complete, 12

R Rall, L. B., 72, 74,265 Random Arzela-Ascoli theorem, 15,51 Random differential system, 40, 116 Random equations, 3 algebraic, 3 difference, 3 differential, 3 discrete, 20-21, 132-155 Random integral equation(s), approximate solutions, 65-78,87-96 Fredholm, 18,97-131 ItO, 207-216 mixed Volterra-Fredholm, 19, 98, 101, 116-117, 120 perturbed, 19, 120-131, 156-164 vector, 180-197 Volterra, 18,29-64, 156-164 Random integrodifferential equation, 241, 243-251 Random Lipschitz condition, 52 Random solution, 20, 186 asymptotically stable in mean-square, 20,249 existence and uniqueness theorems, 30-39,49-55,84-87,98-113, 120-131,133-141,157-164, 194197,244-247 stochastically asymptotically exponentially stable, 20, 38-39, 113-115, 117,247 stochastically geometrically stable, 21, 139,141,149,152 Random vector, 181 Rao, A. N. V., 79,272 Rao, B. L. S. Prakasa, 3,272 Rapoport, A., 165,176,177,263 Renewal equation, 79 Retraction, 24

278

INDEX

S

Saaty, T. L., 208,265 Schwartz inequality, 162 Schwartz, J., 11,262 Semi-norm, 11-13, 182-184 Semirandom solution, 57,59 Semistochastic integral equation, 59 Skorokhod, A. V., 2,208,268 Spacek, A., 3 Square-summable function, 7 Stability, 20-21, 113-115, 139-141, 149,225,247,255-259 Stephenson, J., 170,175,265 Stochastic absolute stability, 4,225-240, 255-259 Stochastic approximation, 87-94 Stochastic chemical kinetics, 180 model, 197-206 Stochastic control, see Control systems Stochastic differential systems, 21-22, 116,250 absolute stability of, 225-239 nonlinear, 148,152,217-240 reduction of, 2 19-225 schematic representations of, 239-240 stochastically absolutely stable, 22 Stochastic discrete equations, 20-21 Stochastic free term, 18 Stochastic fundamental matrix solution, 252 Stochastic integral, see Ito, K. Stochastic integral equations, see also Random integral equations Ito, 212-213 Ito-Doob, 214-216 Stochastic integrodifferential systems, 21, 241-259 with time lag, 251-259 Stochastic kernel, 18,33, 180,243 Stochastic process, 14 continuous in mean-square, 14 second-order, 14, 133,157,215,243 Stochastically stable matrix, 22,41,229, 231,234,236 Stoichiometry, 198

Stratonovich, R. L., 208,272 Successive approximations, 22-23, 60, 145 almost sure convergence of, 68-71 error of approximations, 71-74, 147 rate of convergence of, 71-74, 145 sequence of, 22,66,75,77,94

T Telephone exchange, 44 Telephone traffic, 42-46 Telionis, D. P., 273 Tomasian, A. I., 115,261 Topological space, 8 linear, 8, 181 locally convex, 9, 12, 183 Tserpes, N., 14,49,270,272 Tsokos, C. P., 2, 3,4,40,57,79, 120, 129, 149, 152, 156, 157, 165, 170, 176, 181,197,200,218,244,272,273 Turbulence theory, 55-56

V Vector space, 8 Volterra random integral equation, 18, 29-64 discrete version, 133

W Wasan, M. T., 88,273 Wijsman, R., 165, 166,167,266 Wong, E., 208,273 Wong, J. S. W., 98,264

Y Yosida, K., 17,266

2 Zakai, M., 208,273