Stochastic Differential Equations In Science And Engineering

m S T O H A D I F F E R E N T I A L EQUATIONS IN SCIENCE AND Douglas E N G I N E E R I N G Henderson • Peter Pl...

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m S

T

O

H

A

D I F F E R E N T I A L EQUATIONS IN SCIENCE AND Douglas

E N G I N E E R I N G Henderson

•

Peter

Plaschko

S T O C H A S T I C D I F F E R E N T I A L E Q U A T I O N S IN S C I E N C E AND

E N G I N E E R I N G

Douglas Henderson Brigham Young University, USA

P pc it cp if

rPIi3Qf*ihk"A IOOUIIIVV/

Uriiversidad Autonoma Metropolitans, Mexico

| | p World Scientific NEW JERSEY

• LONDON

• S I N G A P O R E • BEIJING • S H A N G H A I • H O N G KONG • T A I P E I » C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

STOCHASTIC DIFFERENTIAL EQUATIONS IN SCIENCE AND ENGINEERING (With CD-ROM) Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-296-6

Printed in Singapore by World Scientific Printers (S) Pte Ltd

To Rose-Marie Henderson A good friend and spouse

PREFACE

This book arose from a friendship formed when we were both faculty members of the Department of Physics, Universidad Autonoma Metropolitana, Iztapalapa Campus, in Mexico City. Plaschko was teaching an intermediate to advanced course in mathematical physics. He had written, with Klaus Brod, a book entitled, "Hoehere Mathematische Methoden fuer Ingenieure und Physiker", that Henderson admired and suggested that be translated into English and be updated and perhaps expanded somewhat. However, we both prefer new projects and this suggested instead that a book on Stochastic Differential Equations be written and this project was born. This is an important emerging field. From its inception with Newton, physical science was dominated by the idea of determinism. Everything was thought to be determined by a set of second order differential equations, Newton's equations, from which everything could be determined, at least in principle, if the initial conditions were known. To be sure, an actual analytic solution would not be possible for a complex system since the number of dynamical equations would be enormous; even so, determinism prevailed. This idea took hold even to the point that some philosophers began to speculate that humans had no free will; our lives were determined entirely by some set of initial conditions. In this view, even before the authors started to write, the contents of this book were determined by a set of initial conditions in the distant past. Dogmatic Marxism endorsed such ideas, although perhaps not so extremely. Deterministic Newtonian mechanics yielded brilliant successes. Most astronomical events could be predicted with great accuracy.

Vll

viii

Stochastic Differential

Equations in Science and

Engineering

Even in case of a few difficulties, such as the orbit of Mercury, Newtonian mechanics could be replaced satisfactorily by equally deterministric general relativity. A little more than a century ago, the case for determinism was challenged. The seemingly random motion of the Brownian motion of suspended particles was observed as was the sudden transition of the flow of a fluid past an object or obstacle from lamanar flow to chaotic turbulence. Recent studies have shown that some seemingly chaotic motion is not necessarily inconsistent with determinism (we can call this quasi-chaos). Even so, such problems are best studied using probablistic notions. Quantum theory has shown that the motion of particles at the atomic level is fundamentally nondeterministic. Heisenberg showed that there were limits to the precision with which physical properties could be determined. One can only assign a probablity for the value of a physical quantity. The consequence of this idea can be manifest even on a macroscopic scale. The third law of thermodynamics is an example. Stochastic differential equations, the subject of this monograph, is an interesting extension of the deterministic differential equations that can be applied to Brownian motion as well as other problems. It arose from the work of Einstein and Smoluchowski among others. Recent years have seen rapid advances due to the development of the calculii of Ito and Stratonovich. We were both trained as mathematicians and scientists and our goal is to present the ideas of stochastic differential equations in a short monograph in a manner that is useful for scientists and engineers, rather than mathematicians and without overpowering mathematical rigor. We presume that the reader has some, but not extensive, knowledge of probability theory. Chapter 1 provides a reminder and introduction to and definition of some fundamental ideas and quantities, including the ideas of Ito and Stratonovich. Stochastic differential equations and the Fokker-Planck equation are presented in Chapters 2 and 3. More advanced applications follow in Chapter 4. The book concludes with a presentation of some numerical routines for the solution of ordinary stochastic differential equations. Each chapter contains a set of exercises whose purpose is to aid the reader in understanding the material. A CD-ROM that provides

Preface

ix

MATHEMATICA and FORTRAN programs to assist the reader with the exercises, numerical routines and generating figures accompanies the text. Douglas Henderson Peter Plaschko Provo Utah, USA Mexico City DF, Mexico June, 2006

CONTENTS

Preface

vii

Introduction

xv

Glossary

xxi

1.

Stochastic Variables and Stochastic Processes

1

1.1. Probability Theory 1.2. Averages 1.3. Stochastic Processes, the Kolmogorov Criterion and Martingales 1.4. The Gaussian Distribution and Limit Theorems 1.4.1. The central limit theorem 1.4.2. The law of the iterated logarithm 1.5. Transformation of Stochastic Variables 1.6. The Markov Property 1.6.1. Stationary Markov processes 1.7. The Brownian Motion 1.8. Stochastic Integrals 1.9. The Ito Formula 1.9. The Ito Formula Appendix Exercises

9 14 16 17 17 19 20 21 28 38 38 45 49

2.

55

Stochastic Differential Equations

2.1. One-Dimensional Equations 2.1.1. Growth of populations 2.1.2. Stratonovich equations

1 4

56 56 58

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Engineering

2.1.3. The problem of Ornstein-Uhlenbeck and the Maxwell distribution 2.1.4. The reduction method 2.1.5. Verification of solutions 2.2. White and Colored Noise, Spectra 2.3. The Stochastic Pendulum 2.3.1. Stochastic excitation 2.3.2. Stochastic damping (/? = 7 = 0; a ^ 0) 2.4. The General Linear SDE 2.5. A Class of Nonlinear SDE 2.6. Existence and Uniqueness of Solutions Exercises

59 63 65 67 70 72 73 76 79 84 87

3.

91

The Fokker-Planck Equation

3.1. The Master Equation 3.2. The Derivation of the Fokker-Planck Equation 3.3. The Relation Between the Fokker-Planck Equation and Ordinary SDE's 3.4. Solutions to the Fokker-Planck Equation 3.5. Lyapunov Exponents and Stability 3.6. Stochastic Bifurcations 3.6.1. First order SDE's 3.6.2. Higher order SDE's Appendix A. Small Noise Intensities and the Influence of Randomness Limit Cycles Appendix B.l The method of Lyapunov functions Appendix B.2 The method of linearization Exercises 4.

Advanced Topics

4.1. Stochastic Partial Differential Equations 4.2. Stochastic Boundary and Initial Conditions 4.2.1. A deterministic one-dimensional wave equation 4.2.2. Stochastic initial conditions 4.3. Stochastic Eigenvalue Equations 4.3.1. Introduction 4.3.2. Mathematical methods

91 95 98 104 107 110 110 112 117 124 128 130 135 135 141 141 144 147 147 148

Contents

4.3.3. Examples of exactly soluble problems 4.3.4. Probability laws and moments of the eigenvalues 4.4. Stochastic Economics 4.4.1. Introduction 4.4.2. The Black-Scholes market Exercises 5.

Numerical Solutions of Ordinary Stochastic Differential Equations

xiii

152 156 160 160 162 164

167

5.1. Random Numbers Generators and Applications 5.1.1. Testing of random numbers 5.2. The Convergence of Stochastic Sequences 5.3. The Monte Carlo Integration 5.4. The Brownian Motion and Simple Algorithms for SDE's 5.5. The Ito-Taylor Expansion of the Solution of a ID SDE 5.6. Modified ID Milstein Schemes 5.7. The Ito-Taylor Expansion for N-dimensional SDE's 5.8. Higher Order Approximations 5.9. Strong and Weak Approximations and the Order of the Approximation Exercises

167 168 173 175 179 181 187 189 193 196 201

References

205

Fortran Programs

211

Index

213

INTRODUCTION

The theory of deterministic chaos has enjoyed during the last three decades a rapidly increasing audience of mathematicians, physicists, engineers, biologists, economists, etc. However, this type of "chaos" can be understood only as quasi-chaos in which all states of a system can be predicted and reproduced by experiments. Meanwhile, many experiments in natural sciences have brought about hard evidence of stochastic effects. The best known example is perhaps the Brownian motion where pollen submerged in a fluid experience collisions with the molecules of the fluid and thus exhibit random motions. Other familiar examples come from fluid or plasma dynamic turbulence, optics, motions of ions in crystals, filtering theory, the problem of optimal pricing in economics, etc. The study of stochasticity was initiated in the early years of the 1900's. Einstein [1], Smoluchowsky [2] and Langevin [3] wrote pioneering investigations. This work was later resumed and extended by Ornstein and Uhlenbeck [4]. But investigation of stochastic effects in natural science became more popular only in the last three decades. Meanwhile studies are undertaken to calculate or at least approximate the effect of stochastic forces on otherwise deterministic oscillators, to investigate the stability or the transition to stochastic chaos of the latter oscillator. To motivate the following considerations of stochastic differential equations (SDE) we introduce a few examples from natural sciences. (a) Pendulum with Stochastic Excitations We study the linearized pendulum motion x(t) subjected to a stochastic effect, called white noise x + x = (3£t, XV

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Engineering

where ft is an intensity constant, t is the time and £j stands for the white noise, with a single frequency and constant spectrum. For (3 = 0 we obtain the homogeneous deterministic (non-stochastic) traditional pendulum motion. We can expect that the stochastic effect disturbs this motion and destroys the periodicity of the motion in the phase space (x,x). The latter has closed solutions called limit cycles. It is an interesting task to investigate whether the solutions disintegrate into scattered points (stochastic chaos). We will cover this problem later in Section 2.3 and find that the average motion (in a sense to be defined in Section 1.2 of Chapter 1) of the pendulum is determined by the deterministic limit (/3 = 0) of the stochastic pendulum equation. (b) Stochastic Growth of Populations N(i) is the number of the members of a population at the time t, a is the constant of the deterministic growth and (5 is again a constant characterizing the intensity of the white noise. Thus we study the growth problem in terms of the linear scenario

The deterministic limit (/? = 0) of this equation describes the growth of a population living on an unrestricted area with unrestricted food supply. Its solution (the number of such a population) grows exponentially. The stochastic effects, or the white noise describes a stochastic varying food supply that influences the growth of the population. We will consider this problem in the Section 2.1.1 and find again that the average of the population is given by the deterministic limit. (c) Diffraction of Optical Waves The transfer function T(u>); UJ = (u\, U2) of a two-dimensional optical device is defined by oo

/

/-oo

dx / -OO CO

/

dy\F(x,y)\2,

dx -00

dyF{x,y)F*{x -wuy-

J —OO /*CO

J—00

u; 2 )/N;

Introduction

xvn

where F is a complex wave amplitude and F* = cc(F) is its complex conjugate. The parameter N denotes the normalization of |F(x,y)| 2 and the variables x and y stand for the coordinates of the image plane. In a simplified treatment, we assume that the wave form is given by F = |F|exp(—ikA);

|F|,fc = const,

where k and A stand for the wave number and the phase of the waves, respectively. We suppose that the wave emerging from the optical instrument (e.g. a lens) exhibits a phase with two different deviations from a spherical structure A = A c + A r with a controlled or deterministic phase Ac(x,y) and a random phase A r (x,y) that arises from polishing the optical device or from atmospheric influences. Thus, we obtain •1

/"OO

POO

T(u>) = —

dx

•••*• J—oo

dyexp{ifc[A(x-o;i,y-u;2) -

A(x,y)}},

J—oo

where K is used to include the normalization. In simple applications we can model the random phase using white noise with a Gaussian probability density. To evaluate the average of the transfer function (T(ui)) we need to calculate the quantity (exp{ik[AT(x - Ui,y - u2) - A r (x,y)]}). We will study the Gaussian probability density and complete the task to determine the average written in the last line in Section 1.3 of Chapter 1. An introduction to random effects in optics can be found in O'Neill [5]. (d) Filtering Problems Suppose that we have performed experiments of a stochastic problem such as the one in (a) in an interval t € [0, u] and we obtain as result say A(v), v = [0, u]. To improve the knowledge about the solution we repeat the experiments for t € [u,T] and we obtain A(t),t = [u,T]. Yet due to inevitable experimental errors we do not obtain A(i) but a result that includes an error A(i) + 'noise'. The question is now how can we filter the noise away? A filter is thus, an instrument to

xviii



Engineering

clean a result and remove the noise that arises during the observation. A typical problem is where a signal with unknown frequency is transmitted (e.g. by an electronic device) and it suffers during the transmission the addition of a noise. If the transmitted signal is stochastic itself (as in the case of music) we need to develop a non-deterministic model for the signal with the aid of a stochastic differential equation. To study basic the ideas of filtering problems the reader in referred to the book of Stremler [6]. (e) Fluidmechanical Turbulence This is the perhaps most challenging and most intricate application of statistical science. We consider here the continuum dynamics of a flow field influenced by stochastic effects. The latter arise from initial conditions (e.g. at the nozzle of a jet flow, or at the entry region of a channel flow) and/or from background noise (e.g. acoustic waves). In the simplest case, the incompressible two-dimensional flows, there are three characteristic variables (two velocity components and the pressure). These variables are governed by the Navier-Stokes equations (NSEs). The latter are a set of three nonlinear partial differential equations that included a parameter, the Reynolds number R. The inverse of R is the coefficient of the highest derivatives of the NSEs. Since turbulence occurs at intermediate to high values of the R, this phenomenon is the rule and not the exception in Fluid Dynamics and it occurs in parameter regions where the NSEs are singular. Nonlinear SDEs — such as the NSEs — lead additionally to the problem of the closure, where the equation governing the statistical moment of nth order contains moments of the (n + l)th order. Hopf [7] was the first to try to find a theoretical approach to solve the problem for the idealized case of isotropic homogenous turbulence, a flow configuration that can be approximately realized in grid flows. Hopf assumed that the turbulence is Gaussian, an assumption that facilitates the calculation of higher statistical moments of the distribution (see Section 1.3 in Chapter 1). However, later measurements showed that the assumption of a Gaussian distribution was rather unrealistic. Kraichnan [8] studied the problem again in

Introduction

xix

the 60's and 70's with the direct triad interaction theory in the idealized configuration of homogeneous isotropic turbulence. However, this rather involved analysis could only be applied to calculate the spectrum of very small eddies where the viscosity dominates the flow. Somewhat more progress has been achieved by the investigation of Rudenko and Chirin [9]. The latter predicted with aid of stochastic initial conditions with random phases a broad banded spectra of a nonlinear model equation. During the last two decades there was the intensive work done to investigate the Burgers equation and this research is summarized in part by Wojczinsky [10]. The Burgers equation is supposed to be a reasonable one-dimensional model of the NSEs. We will give a short account on the work done in [9] in Chapter 4.

GLOSSARY

AC

almost certainly

BC

boundary condition

dBj — dWj — £td£

differential of the Brownian motion (or equivalently Wiener process)

cc(a) = a*

complex conjugate of a

D

dimension or dimensional

DF

distribution function

DOF

degrees of freedom

Sij

Kronecker delta function

S(x)

Dirac delta function

EX

exercise at the end of a chapter

FPE

Fokker-Planck equation

r(x)

gamma function

GD

Gaussian distribution

GPD

Gaussian probability distribution

HPP

homogeneous Poisson process

H n (x)

Hermite polynomial of order n

IC

initial condition

IID

identically independently distributed

xxii



Engineering

IFF

if and only if

IMSL

international mathematical science library

C

Laplace transform

M

master, as in master equation

MCM

Monte Carlo method

NSE

Navier-Stokes equation

NIGD

normal inverted GD

N(jU, a)

normal distribution with \i as mean and a as variance

o

Stratonovich theory

ODE

ordinary differential equation

PD

probability distribution

PDE

partial differential equation

PDF

probability distribution function

PSDE

partial SDE

r

Reynolds number

RE

random experiment

RN

random number

RV

random variable

Re(a)

real part of a complex number

R, C

sets of real and complex numbers, respectively

S

Prandt number

SF

stochastic function

SI

stochastic integral

SDE

stochastic differential equation

SLNN

strong law of large numbers

Glossary

TPT

transition probability per unit time

WP

Wiener process

WS

Wiener sheet

WKB

Wentzel, Kramers, Brillouin

WRT

with respect to

W(t)

Wiener white (single frequency) noise

(a)

average of a stochastic variable a

a2 = (a2) — (a) (a)

variance

{x\y),{x,u\y,v)

conditional averages

s At

minimum of s and t

V

for all values of

€

element of

f f(x)dx

short hand for J^

X

end of an example

•

end of definition

$

end of theorem

f(x)dx

CHAPTER 1 STOCHASTIC VARIABLES AND STOCHASTIC PROCESSES 1.1. Probability Theory An experiment (or a trial of some process) is performed whose outcome (results) is uncertain: it depends on chance. A collection of all possible elementary (or individual) outcomes is called the sample space (or phase space, or range) and is denoted by f2. If the experiment is tossing a pair of distinguishable dice, then 0, = {(i,j) | 1 < i,j < 6}. For the case of an experiment with a fluctuating pressure 0, is the set of all real functions fi = (0, oo). An observable event A is a subset of f2; this is written in the form A c f2. In the dice example we could choose an even, for example, as A = {{i,j) \ i + J' = 4}. For the case of fluctuating pressures we could use the subset A = (po > 0,oo). Not every subset of £1 is observable (or interesting). An example of a non-observable event appears when a pair of dice are tossed and only their spots are counted, fi = {(i,j),2 < i + j < 12}. Then elementary outcomes like (1, 2), (2, 1) or (3, 1), (2, 2), (1, 3) are not distinguished. Let r be the set of observable events for one single experiment. Then F must include the certain event of CI, and the impossible event of 0 (the empty set). For every A C T, A c the complement of A, satisfies Ac C T and for every B C F the union and intersection of events, A U B and A D B, must pertain also to F. F is called an algebra of events. In many cases there are countable unions and intersections in F. Then it is sufficient to assume that oo

(J An e r,

if An e r.

1

2



Engineering

An algebra with this property is called a sigma algebra. In measure theory, the elements of T are called measurable sets and the pair of (F, Q,) is called a measurable space. A finite measure Pr(A) defined on F with 0 < Pr(A) < 1,

Pr(0) = 0,

Pr(fi) = 1,

is called the probability and the triple (I\ f2, Pr) is referred to as the probability space. The set function Pr assigns to every event A the real number Pr(A). The rules for this set function are along with the formula above Pr(A c ) = l - P r ( A ) ; Pr(A)• [0,1] and it is generally derived with Lebesque integrations that are defined on Borel sets. We introduced this formal concept because it can be used as the most general way to introduce axiomatically the probability theory (see e.g. Chung, [1.1]). We will not follow this procedure but we will introduce heuristically stochastic variables and their probabilities. Definition 1.1. (Stochastic variables) A random (or stochastic) variable ~X.(u),u £ Q is a real valued function defined on the sample space Q. In the following we omit the parameter u) whenever no confusion is possible. • Definition 1.2. (Probability of an event) The probability of an event equals the number of elementary outcomes divided by the total number of all elementary outcomes, provided that all cases are equally likely. • Example For the case of a discrete sample space with a finite number of elementary outcome we have, fi = {wi,... ,u>n} and an event is given by A = {LO\, ... ,u>k}, I < k < n. The probability of the event A is then Pr(A) = k/n. *

Stochastic

Variables and Stochastic

Processes

3

Definition 1.3. (Probability distribution function and probability density) In the continuous case, the probability distribution function (PDF) Fx(a;) of a vectorial stochastic variable X = ( X i , . . . , X n ) is defined by the monotonically increasing real function Fx(xi,...,xn)

= Pr(Xi < xi,...,Xn

< xn),

(1.1)

where we used the convention that the variable itself is written in upper case letters, whereas the actual values that this variable assumes are denoted by lower case letters. The probability density px(î, • • • ,xn) (PD) of the random variable is then defined by Fx(xi,...,xn)

=

•••

px(ui,...,-un)dn1---dun

(1.2)

and this leads to dnFx dxi...dXn

=!*(*!,...,*„).

(1-3)

Note that we can express (1.1) and (1.2) alternatively if we put Pr(xn < Xi < X12,..., xnl < Xn < xn2) fX12 rxi2

JXn

fXn2 rxn-.

•••px(xi,...,xn)dxi

•••dxn.

(1.1a)

Jx„,\

The conditions to be imposed on the PD are given by the positiveness and the normalization condition PxOci,

,xn)>0]

/ ••• / px(xi,...,xn)dxi

•••dxn = 1. (1.4)

In the latter equation we used the convention that integrals without explicitly given limits refer to integrals extending from the lower boundary — oo to the upper boundary oo. • In a continuous phase space the PD may contain Dirac delta functions p(x) = Y^l(k)s(x

- k) + P(x);

q(k) = Pr(x = k),

(1.5)

4

Stochastic Differential Equations in Science and Engineering

where q(k) represents the probability that the variable x of the discrete set equals the integer value k. We also dropped the index X in the latter formula. We can interpret it to correspond to a PD of a set of discrete states of probabilities q(fc) that are embedded in a continuous phase space S. The normalization condition (1.4) yields now

^2 0.

(1.11)

The random variable x — (x) is called the standard deviation. The average of the of the Fourier transform of a PD is called the characteristic function G(k\,...,

kn) = (ex.p(ikrxr)} p(xi,...,

xn) ex.Y>(ikrxr)dx\ • • • dxn,

(1-12)

where we applied a summation convention krxr = ^ ? = i kjxj- This function has the properties G ( 0 , . . . , 0)1; | G(ki,..., kn) \< 1. Example The Gaussian (or normal) PD of a scalar variable x is given by p(x) = (2vr)" 1/2 exp(-a; 2 /2);

- c o < x < oo.

(1.13a)

(^2n+1) = o.

(l.isb)

Hence we obtain (see also EX 1.1)

= | ? 7 ; Li

" 2 = i;

lit

A stochastic variable characterized by N(m, s) is a normal distributed variable with the average m and the variance s. The variable x distributed with the PD (1.13a) is thus called a normal distributed variable with N(0, 1).

Stochastic


Processes

7

A Taylor expansion of the characteristic function G(k) of (1.13a) yields with (1.12)

( L14a )

G(*) = E ^ V > . n=0

U

-

We define the cumulants nm by \m

lnG(fc) = E ^ f - K m -

(1.14b)

A comparison of equal powers of k gives Ki = (x);

= (x2) - (x)2 = a2;

K2

(1.14c)

{X3)-3(X2)(X)+2{X)3;....

K3 =

* Definition 1.5. (Conditional probability) We assume that A, B C T are two random events of the set of observable events V. The conditional probability of A given B (or knowing B, or under the hypothesis of B) is defined by Pr(A | B) = Pr(A n B)/Pr(B);

Pr(B) > 0.

Thus only events that occur simultaneously in A and B contribute to the conditional probability. Now we consider n random variables x\,... ,xn with the joint PD p n ( x i , . . . , xn). We select a subset of variables x\,..., xs. and we define a conditional PD of the latter variables, knowing the remaining subset xs+i,... ,xn, in the form Ps|n—s\ x li • • • i xs = pn(xi,

I Xs-\-\, . . . , Xn)

. . . , Xn)/pn-s(xs+i,

. . . , Xn).

(1.15)

Equation (1.15) is called Bayes's rule and we use the marginal PD pn-s(xs+i,...,xn)

=

pn{xi,...,xn)dxi---dxs,

(1.16)

where the integration is over the phase space of the variables x± • • • xs. Sometimes is useful to write to Bayes's rule (1.15) in the form P n l ^ l j • • • j Xn)

= pn—syXs-î,

. . . , 3^nJPs|n—s v^l> • • • > xs

\ Xs-\-lj

• • • , XnJ.

(1.15')

8


We can also rearrange (1.15') and we obtain =

P n ^ l i • • • > -En)

Ps(.-El> • • • ) •KsjPn—s\s\%s+1:

• • • j %n \ Xi, . . . ,XS).

(1.15")

• Definition 1.6. (Conditional averages) The conditional average of the random variable x\, knowing x2, • • •, xn, is defined by (Xi | X2, . • • , Xn) =

=

/ Z l P i | n _ i ( x i I X2, • • • , Xn)dXi

/ XxPnfa

\X2,...,

X n ) d x i / p n _ i ( x 2 , • • • , Xn).

(1.17) Note that (1.17) is a random variable. The rules for this average are in analogy to (1.8) (axi + bx2 | y) = a{xx \ y) + b(x2 \ y),

((x \ y)) = (x \ y). (1.18) D

Example We consider a scalar stochastic variable x with its PD p(a;). An event A is given by a; £ [a, 6]. Hence we have p(x | A) = 0 V z ^ [a, b], and p(x | A) = p(x) / / p(s)ds;

xe[a,b\.

The conditional PD is thus given by (x | A) = / xp(x)dx / / p(s)ds. Ja I Ja For an exponentially distributed variable x in [0, oo] we have p(x) = Aexp(—Arc). Thus we obtain for a > 0 the result / /-oo

/•oo

{x \ x > a) = / JO

xexp(—Ax)ds / / /

./a

exp(—Xx)dx = a + 1/A. JL

Stochastic


Processes

9

1.3. Stochastic Processes, the Kolmogorov Criterion and Martingales In many applications (e.g. in irregular phenomena like blood flow, capital investment, or motions of molecules, etc.) one encounters a family of random variables that depend on continuous or discrete parameters like the time or positions. We refer to {X(t,co),t £ l,u £ ft}, where I is set of (continuous or discrete) parameters and X(t7ui) £ Rn, as a stochastic process (random process or stochastic (random) function). If I is a discrete set it is more convenient to call X(t,u>) a time series and to use the phrase process only for continuous sets. If the parameter is the time t then we use I = [to, T], where to is an initial instant. For a fixed value of t £ I, X(£, a>) is a random variable and for every fixed value of LO £ Q (hence for every observation) X(t, LO) is a real valued function. Any observation of this process is called a sample function (realization, trajectory, path or orbit) of the process. We consider now a finite variate PD of a process and we define the time dependent probability density functions (PDF) in analogy to (1.1) in the form Fx(x,t) = Pr(X(t)<x); Fx.yfo t; y, s) = Pr(X(t) < x, Y(s) < y); Fxu...,xn(xiiti\---;xn,tn)

= Pr(Xx(t) < xi,Xn(t)

^1A^

< xn),

where we omit the dependence of the process X(t) on the chance variable LO, whenever no confusion is possible. The system of PDF's satisfies two classes of conditions: (i) Symmetry If {ki,..., kn} is a permutation of 1 , . . . , n then we obtain Fx lv ..,x n (zfc! ,tkl;...;xkn,tkJ

= FXl,...,x„ {x\, h;...;

xn, tn). (1.19a)

(ii) Compatibility Fx 1? ...,x„ (xi,ti;...;

xr, tr; oo, tr+i;

= F X l ,...,x r (a;i, ) with u < s (but not u> s). ^ Example For n = 2, 3 , . . . ; 0 < A < t we see that the processes (i) G1(t,co) = Bt/n(co), (ii) G3(t,Lj) = Bnt(u),

G2(t,uj) = Bt_x(u;), G4(t,w) = Bt+X(u>),

are Tj-adapted, respectively, not adapted.

*

Theorem 1.3. (martingale process) A process X(t) is called a martingale IFF it is adapted and the condition <Xt |T a ) = XS V 0 < s < t < o o ,

(1.27)

is almost certainly (AC) satisfied. If we replace the equality sign in (1.27) by < (>) we obtain a super (sub) martingale. We note that martingales have no other discontinuities than at worst finite jumps (see Arnold [1.2]). ^ Note that (1.27) defines a stochastic process. Its expectation ((Xj | Ts)) = (X s );s < t is a deterministic function. An interesting property of a martingale is expressed by Pr(sup | X(t) |> c) < (| X(6) \p)/cp;

c > 0;

p > 1,

(1.28)

where sup is the supremum of the embraced process in the interval [a, b]. (1.28) is a particular version of the Chebyshev inequality, that

14



Engineering

will be derived in EX 1.2. We apply later the concept of martingales to Wiener processes and to stochastic integrals. Finally we give an explanation of the phrase "martingale". A gambler is involved in a fair game and he has at the start the capital X(s). Then he should posses in the mean at the instant t > s the original capital X(s). This is expressed in terms of the conditional mean value (X t | X s ) = X s . Etymologically, this term comes from French and means a system of betting which seeks the amount to be wagered after each win or loss. 1.4. T h e G a u s s i a n D i s t r i b u t i o n a n d Limit T h e o r e m s In relation (1.13) we have already introduced a special case of the Gaussian (normal distributed) P D (GD) for a scalar variable. A generalization of (1.13) is given by t h e N ( m , o - 2 ) P D p(x) = (2TTCT 2 )- 1 / 2 e x p [ - ( x - m)2/(2a2)];

V i e [-oo, oo]

(1.29)

where m is the average and oo to a N(0, a2) variable with a PD given by (1.13a). To prove this we use the independence of the variables Xk and we perform the calculation of the characteristic function of the variable U with the aid of (1.12)

u=

Gu(fc) = / dxip(xi) • • • / dx n p(x n ) • • • exp [ik(xi -\

h

xn)/y/n\

= [Gx(A;/v^)]n kl2^2 a + 0(n ~2n~

-3/2.

exp(—k a 12)

for n —> oo. (1.38)

We introduced in the second line of (1.38) the characteristic function of one of the individual random functions according to (1.14a); (1.38) is the characteristic function of a GD that corresponds indeed to

Stochastic


Processes

17

N(0, a2). Note that this result is independent of the particular form of the individual PD's p(x). It is only required that p(a;) has finite moments. The central limit theorem explains why the Gaussian PD plays a prominent role in probability and stochastics. 1.4.2. The law of the iterated

logarithm

We give here only this theorem and refer the reader for its derivation to the book Chow and Teichler [1.4]. yn is the partial sum of n IID variables yn = xx -\

\-xn;

(xn) = /3,

{(xn - (if) = a2.

(1.39)

The theorem of the iterated logarithm states that there exists AC an asymptotic limit -a < lim / n ~ r a / ? < a. (1.40) rwoo v /2nln[ln(n)] Equation (1.40) is particular valuable in case of estimates of stochastic functions and we will use it later to investigate Brownian motions. We will give a numerical verification of (1.40) in program F18. 1.5. Transformation of Stochastic Variables We consider transformations of an n-dimensional set of stochastic variables x\,... ,xn with the PD pxi-x n (x\, • • •, xn). First we introduce the PD of a linear combination of random variables n

Z =

^2otkxk,

(1.41a)

fe=l

where the a^ are deterministic constants. The PD of the stochastic variable z is then defined by Pz(z) = / dxi ••• / dxn8 I z - YâkXk

1 PXi-x„(zi,--- ,xn).

(1.41b) Now we investigate transformations of the stochastic variables xi,..., xn. The new variables are defined by uk = uk(x1,...,xn),

k = l,...,n.

(1.42)

18



Engineering

The inversion of this transformation and the Jacobian are Xk = gk(ui,...,un),

J = d(x1,...,xi)/d(u1,...,u1).

(1.43)

We infer from an expansion of the probability measure (1.1a) that dpx!-x n = Pr(zi < Xi < xi + d x i , . . . , xn < X n < xn + dxn) = p X l ...x n (î, • • •, xn)dxi • • • dxn for dxk —> 0, k — 1,... ,n.

(1.44a)

Equation (1.44a) represents the elementary probability measure that the variables are located in the hyper plane n

Y[[xk,xk + dxk}. k=l

The principle of invariant elementary probability measure states that this measure is invariant under transformations of the coordinate system. Thus, we obtain the transformation d p ^ . . . ^ =dp X i ... X n .

(1.44b)

This yields the transformation rule for the PD's PUi-u„(Mi(a;i, • • -,xn),

• • •, un(xi,..

.,xn))

= | det(J) | px!-x„(a;i,---,a;n)-

(1-45)

Example (The Box-Miller method) As an application we introduce the transformations method of B o x Miller to generate a GD. There are two stochastic variables given in an elementary cube ,

,

(I P ( X 1 X 2 ) = ' U

V0<x1 = Yl(xk(tk)xk(sk)};

t = (h,...,tn);

s = (si,...,sn).

fc=i

The evaluation of the last line yields with (1.62) n

(M(tn)M^) = HtkAsk.

(1.70)

fc=i

The relations (1.69) and (1.70) show now that process (1.68) is an n-WP. In analogy to deterministic variables we can now construct with stochastic variables curves, surfaces and hyper surfaces. Thus, a curve in 2-dimensional WS and surfaces on 3-dimensional WS are given by c

« = Mt2f(ty

s

tut2 = M S 2 ) g ( t l > t 2 )-

We give here only two interesting examples. Example 1 Here we put (2)

Kt = M ^ ;

a = exp(i),

b = exp(—£);

—oo < x < oo.

This defines a stochastic hyperbola with zero mean and with the autocorrelation (K t K s ) =

(x1{et)x1{es))(x2{e-t)x2{e-s))

= (e* A e s )(e-* A e~s) = exp(-|t - s\).

(1.71)

The property (1.71) shows this process is not only a WS but also a stationary Ornstein-Uhlenbeck process (see Section 1.5.1). 4b Example 2 Here we define the process K t = exp[-(l + c)t]M^l;

a = exp(2£),

b = exp(2ct);

c > 0. (1.72)

Stochastic


27

Processes

Again we see, that stochastic variable defined (1.72) has zero mean and the calculation of its autocorrelation yields s)}(xl(e2t)x1(e2s)){x2(e2ct)x2(e2cs))

(K t K s ) = exp[-(l + c)(t +

= exp[-(l + c)(t + s)}(e2t A e2s){e2ct A e2cs) = exp[-(l + c ) | t - s | ] .

(1.73)

The latter equation means that the process (1.72) is again an Ornstein-Uhlenbeck process. Note also that because of c > 0 there is no possibility to use (1.73) to reproduce the result of the previous example. X Just as in the case of one parameter, there exist for WS's also scaling and translation. Thus, the stochastic variables H

- —M{2) a2u h2v • 'v ~ ab ' ''

T _ L>u,v -

M ( 2 ) M u+a,v+b

_M(2) M u+a,b

(1.74) _

M ( M

2) a,v+b

~

M(2) M

a,6>

are also WS's. The proof of (1.74) is left for EX 1.8. We give in Figures 1(a) and 1(b) two graphs of the Brownian motion. At the end of this section we wish to mention that the WP is a subclass of a Levy process L(t). The latter complies with the first two conditions of the Definition 1.8. However, it does not possess normal distributed increments. A particular feature of normal distributed process x is the vanishing of the skewness (x3) / (x2)3'2. However, many statistical phenomena (like hydrodynamic turbulence, the market values of stocks, etc.) show remarkable values of the skewness. This means that a GD (with only two parameter) is not flexible enough to describe such phenomena and it must be replace by a PD that contains a sufficient number of parameters. An appropriate choice is the normal inverted Gaussian distribution (NIGD) (see Section 4.4). The NIGD distribution does not satisfy the Kolmogorov criterion. This means that the sample functions of the Levy process L(i) is equipped with SF that jump up and down at arbitrary instances t. To get more information about the Levy process we refer the reader to the work of Ikeda k, Watanabe [1.6] and of Rydberg


28


Engineering

[1.7]. In Section 4.4 we will give a short description of the application of the NIGD in economics theories. 1.8. Stochastic Integrals We need stochastic integrals (SI) when we attempt to solve a stochastic differential equation (SDE). Hence we introduce a simple first order ordinary SDE ^

= a(X(t),t) + b(X(t),t)Zt;

X,a,b,t€R.

(1.75)

We use in (1.75) the deterministic functions a and b. The symbol £t indicates the only stochastic term in this equation. We assume = 0;

(tes) = 6(t-s).

(1.76)

The spectrum of the autocorrelation in (1.76) is constant (see Section 2.2) and in view of this £t is referred as white noise and any term proportional to £( is called a noisy term. These assumptions are based on a great variety of physical phenomena that are met in many experimental situations. Now we replace (1.75) by a discretization and we put Atk = tk+i — tk>0; AXfc = X fc+1 - X f c ;

Xfc = X(ifc); A; = 0,1,

The substitution into (1.75) yields AXfc = a(Xfc, tk) Atk + b{Xk,tk) ABfc = B f c + 1 - B f c ;

ABk;

A; = 1,2,...

where we used

A precise derivation of (1.77) is given in Section 2.2. Thus we can write (1.75) in terms of n-l

X n = X0 + Y^ [<xs,ts)Ats

+ b(Xa, ts)ABs] •

X0 = X(t 0 ).

s=Q

(1.78)

Stochastic


Processes

29

What happens in the limit Atk —> 0? If there is a "reasonable" limit of the last term in (1.78) we obtain as solution of the SDE (1.75) X(t) = X(0)+ / a(X(s),s)ds Jo

+ " / 6(X(s),s)dB 8 ". Jo

(1.79)

The first integral in (1.79) is a conventional integral of Riemann's type and we put the stochastic (noisy) integral into inverted commas. The irregularity of the noise does not allow to calculate the stochastic integral in terms of a Riemann integral. This is caused by the fact that the paths of the WP are nowhere differentiable. Thus we find that a SI depends crucially on the decomposition of the integration interval. We assumed in (1.75) to (1.79) that b(X,t) is a deterministic function. We generalize the problem of the calculation of a SI and we consider a stochastic function 1 = / i(w,s)dBs. Jo

(1.80)

We recall that Riemann integrals of the type (g(s) is a differentiable function)

'0

i(s)dg(s) = [ Jo

f(s)g'(s)ds,

are discretized in the following manner pT

/ JU

n—1

f(s)dg(s) = lim Vf(s fc )[g(sfc+i)-g(sfc)]. k=0

Thus, it is plausible to introduce a discretization of (1.80) that takes the form I = 53f(a f c ,a;)(B f c + 1 -B f c ).

(1.81)

In Equation (1.81) we used s^ as time-argument for the integrand f. This is the value of s that corresponds to the left endpoint of the discretization interval and we say that this decomposition does not

30



Engineering

look into the future. We call this type of integral an Ito integral and write I i = / {{s,uj)dBs. (1.82) Jo An other possible choice is to use the midpoint of the interval and with this we obtain the Stratonovich integral Is = / f(s,w)odB s = ^f(s f c ,w)(B f c + i -Bfc); J° k

sk = -(tk+1

+ tk). (1.83)

Note that the symbol "o" between integrand and the stochastic differential is used to indicate Stratonovich integrals. There are, of course, an uncountable infinity of other decompositions of the integration interval that yield to different definitions of a SI. It is, however, convenient to take advantage only of the Ito and the Stratonovich integral. We will discuss their properties and find out which type of integrals seems to be more appropriate for the use in the analysis of stochastic differential equations. Properties of the Ito integral (a) We have for deterministic constants a < b < c, a, /3 G R. f [ah{s,u>) + /?f2(s,w)]dBs = all +/JI 2 ; Ja

h = [ f*(s,w)dB s . Ja

(1.84) Note that (1.84) remains also valid for Stratonovich integrals. The proof of (1.84) is trivial. In the following we give non-trivial properties that apply, however, exclusively to Ito integrals. Now we need a definition: Definition 1.9. (non-anticipative or adapted functions) The function f(t, B s ) is said to be non-anticipative (or adapted, see also Theorem 1.2) if it depends only on a stochastic variable of the past: B s appears only for arguments s < t. Examples for a non-anticipative functions are i(s,co)=

[Sg(u)dBu;

Jo

f( s,u) = B,.

•

Stochastic


Processes

31

Now we list further properties of the Ito integrals that include non anticipative functions f(s,B s ) and g(s,B s ). MlEE

(b)

W

f s B

( ' s)dBs)=°-

(L85)

Proof. We use (1.81) and obtain Mi =

/^f(sfc,Bfc)(Bfc+1-Bfc)

But we know that Bk is independent of B^+i — Bk. The function f(sfc,Bfc) is thus also independent of B^+i — B^. Hence we obtain Mi = ^2(f(sk,Bk)){Bk+1

- Bfc) = 0.

k

This concludes the proof of (1.85). (c) Here we study the average of a product of integrals and we show that M2 = I J

f(s,Bs)dBsJ

g(u,Bu)dBu\

=J

(i(s,Bs)g(s,Bs))ds. (1.86)

Proof. M2 = ] T < f ( s m , B m ) ( B m + 1 - B m ) g ( s n , B n ) ( B n + 1 - B n ) ) . m,n

We have to distinguish three subclasses: (i) n > m, (ii) n < m and (hi) n = m. Taking into account the independence of the increments of WP's we see that only case (hi) contributes non-trivially to M2. This yields M2 = ^ ( f ( S n , B n ) g ( s n , B n ) ( B ra+l ~ B n ) ). n

But we know that f(s n , B n )g(s n , B n ) is again a function that is independent of (B n + i — B n ) 2 . We use (1.62) and obtain ((B n + i — B n ) ) = ( B n + 1 — 2 B n + i B n + Bn) = tn+\ — tn — Ai n ,

32



Engineering

and thus we get M2 = ^ ( f ( S n , B n ) g ( S n , B n ) ) ( ( B n+l

n oo

—

^2(i{sn,Bn)g{sn,Bn))Atn. n=l

The last relation tends for Atn —» 0 to (1.86). (d) A generalization of the property (c) is given by /

M3 = M

pa

rb

i(s,Bs)dBsl

\

g(u,Bu)dBu\

rahb

= I

(f( S ,B s )g( S ,B s ))ds. (1.87)

To prove (1.87) we must distinguish to subclasses (i) b = a + c > a and (ii) a = b + c> b;c> 0. We consider only case (i), the proof for case (ii) is done by analogy. We derive from (1.86) and (1.87). M 3 = M2 + / / f(s,B s )dB s / \ J0

=

M

2 + Yl Yl ^ n

g(u,Bu)dBt

Ja

B n )g(s m , B m )AB n AB m ).

m>n

But we see that i(sn, Bn) and A B n are independent of f(s m , B m ) and A B m . Hence, we obtain M 3 = M2 + £ ( f ( s „ , B n )AB n ) Y, (g(sm, B m )AB m ) = M 2 , n

m>n

where we use (1.85). This concludes the proof of (1.87) for case (i). Now we calculate an example I(t) = / B s dB s . (1.88a) Jo First of all we obtain with the use of (1.85) and (1.86) the moments of the stochastic variable (1.88a) = 0;

(I(t)I(t + r)) = [\B2s)ds = f a d s = Jo Jo 7 = iA(t + r).

2 7

A

QQ^ (1.88b)

n

Stochastic


Processes

33

We calculate the integral with an Ito decomposition 1 = 2 ^ B*;(Bfc+i

—

Bfc).

k

But we have AB2k = (B2k+l - B2k) = (Bk+1 - Bkf + 2B f c (B f c + 1 - B fc ) = (ABfc)2 + 2B f c (B f c + 1 -B f c ). Hence we obtain

I(t) = lj2[A(Bl)-(ABk)% k

We calculate now the two sums in (1.90) separately. Thus we obtain the first place

I1(t) = E A ( B fc) = ( B ?- B 8) + ( B i - B ? ) + - + ( B N- B N-l) k N ~*

B

i>

where we used Bo = 0. The second integral and its average are given by

I2(t) = Y, ( AB *) 2 = E (B^+i " 2Bk+iBk + Bl); k

(I 2 (i))=E

k

Aifc = t

k

The relation (I 2 (i)) = t gives not only the average but also the integral I 2 (i) itself. However, the direct calculation of l2(t) is impractical and we refer the reader to the book of 0ksendahl [1.8], where the corresponding algebra is performed. We use instead an indirect proof and show that the quantity z (the standard deviation of I 2 (i)) is a deterministic function with the value zero. Thus, we put z = I2(£) — t. The mean value is clearly (z) — 0 and we obtain (Z2) = (l2(t)-2tl2(t)+t2)

=

(l2(t))-t2.


34


Engineering

But we have

ai(*)> = EEk

(i.88c)

rn

The independence of the increments of the WP's yields ((AB fc ) 2 (AB m ) 2 ) = ((ABk)2)((ABm)2)

+

5km{ABi),

hence we obtain with the use of the results of EX 1.6

$(«)>= (£ 0,

k

and this indicates that (z2) = 0. This procedure can be pursued to higher orders and we obtain the result that all moments of z are zero and thus we obtain l2(£) = t. Thus, we obtain finally I(*) = /

B s dB s = ^ ( B 2 - £ ) .

(1.89)

There is a generalization of the previous results with respect to higher order moments. We consider here moments of a stochastic integral with a deterministic integrand Jfc(t) = I ffc(s)dBs; k€N. (1.90) Jo These integrals are a special case of the ones in (1.82) and we know from (1.85) that the mean value of (1.90) is zero. The covariance of (1.90) is given by (see (1.86)) (Jfc(t)Jm(t)) = / h(s)fm(s)ds. Jo But we can obtain formally the same result if we put (dB s dB„) = 5(s - u)dsdu.

(1.91)

A formal justification of (1.91) is given in Chapter 2 in connection with formula (2.41). Here we show that (1.91) leads to a result that

Stochastic


Processes

35

is identical to the consequences of (1.86) <Jfc(*)Jm( = / h(s) f f m H ( d B , d B u ) Jo Jo -

\ fk(s) / fm(u)5(s - u)dsdu = / ffc(s)fm(s)d S. Jo Jo Jo We also know that B t and hence dB4 are Gaussian and Markovian. This means that all odd moments of the integral (1.90) must vanish <Jfc(*)Jm(*)Jr(t)> = ••• = ().

(1.92a)

To calculate higher order moments we use the properties of the multivariate GD and we put for the 4th order moment of the differential t2 = s,

with (xk) = 0;

(x\) = r(tfc) = rk;

k = 1, 2,

{xix2) = r 2 ,

has according to (1.35a) the joint PD p 2 (xi,x 2 ) = [(27r)ir1(r1 - r 2 )]

' exp

xi _ (xi - x2)2

"2n

2(n-r 2 )

Yet, the latter line is identical with the bivariate PD of the Wiener process (1.60) if we replace in the latter equation the t\- by rk. Hence, we obtain from Kolmogorov's criterion ([xi(ri) — x 2 i(r 2 )] 2 ) =| 7~i — r 2 | and this guarantees the continuity of the SF of the Itointegral (1.93a). A further important feature of Ito integrals is their martingale property. We verify this now for the case of the integral (1.89). To achieve this, we generalize the martingale formula (1.64) for the case of arbitrary functions of the Brownian motions (%2,«) I f(yi,i)> = / %2,s)Pi|i(y2,s | yx,t)dy2 = f(yi,t); yk = Btk; Vs>t,

(1.94)

Stochastic


37

Processes

where p ^ is given by (1.53). To verify now the martingale property of the integral (1.89) we specify (1.94) to (I(j/ 2 ,s) | Ifo!,*)) = - i = / {vl - s) exp[-(y 2 - yif /f5]dy2. The application of the standard substitution (see EX 1.1) yields (I(y2, s) | I(2/i,*)> = ^=J{y\-s

+ 2Vlz^

1 = -{yi-s

+ /3^2) exp(-z 2 )d^

+ P/2)=I(yi,t).

(1.95)

This concludes the proof that the Ito integral (1.89) is a martingale. The general proof that all Ito integrals are martingales is given by 0ksendahl [1.8]. However, we will encounter the martingale property for a particular class of Ito integrals in the next section. To conclude this example we add here also the Stratonovich version of the integral (1.89). This yields (the subscript s indicates a Stratonovich integral) rt

ls{t)=

i

/ B s odB s = - V ( B f c + 1 + B f c )(B f c + 1 -B f c ) Jo

2

k

k

The result (1.96) is the "classical" the Ito integral gives a non classical cant differences between the Ito and the moments do not coincide since we &(*)> = \

and

value of the integral whereas result. Note also the signifiStratonovich integrals. Even infer from (1.96)

(U*)I*(«)> = ^[tu + 2(t A u)2}.

It is now easy to show that the Stratonovich integral I s is not a martingale. We obtain this result if we drop the term s in second line of (1.95) (ls(y2,s) | I8(yi,t))

= \{y2 + P/2)Ûyut).

X

Hence, we may summarize the properties of the Ito and Stratonovich integrals. The Stratonovich concept uses all the transformation rules of classical integration theory and thus leads in many

38



Engineering

applications to an easy way of performing the integration. Deviating from the Ito integral, the Stratonovich integral does, however, not posses the effective rules to calculated averages such as (1.85) to (1.87) and they do not have the martingale property. In the following we will consider both integration concepts and their application in solution of SDE. We have calculated so far only one stochastic integral and we continue in the next section with helpful rules perform the stochastic integration. 1.9. The Ito Formula We begin with the differential of a function $(Bf, t). Its Ito differential takes the form d$(B t , t) = Qtdt + $ B t d B t + ^ B t B t (dB t ) 2 .

(1.97.1)

Formula (1.97.1) contains the non classical term that is proportional to the second derivative WRT Bt. We must supplement (1.97.1) by a further non classical relation (dB t ) 2 = dt.

(1.97.2)

Thus, we infer from (1.97.1,2) the final form of this differential d$(B t , t) = Ut + ^*B t B t ) dt + ^BedBj.

(1.98)

Next we derive the Ito differential of the function Y = g(x, t) where x is the solution of the SDE dx = a(x,t)dt + b(x,t)dBt.

(1.99.1)

In analogy to (1.97.1) we include a non classical term and put dY = gtdt + g x dz +

-gxx(dx)2,

We substitute dx from (1.99.1) into the last line and apply the non classical formula (dx) 2 = {adt + bdBt)2 = b2dt;

{dt)2 = dtdBt = 0;

(dB t ) 2 = dt, (1.99.2)

Stochastic


Processes

39

and this yields dY=(6+0b

+

£&.)BtBt = 0) given by the exact formula dY = d[f(t)Bt] = f(*)dBt + i'(t)Btdt.

(1.104b)

The integration of this differential yields i(t)Bt=

f f'(s)B s ds+ [ f(s)dB s . (1.105a) Jo Jo Subtracting the last line for t = u from the same relation for t — v yields rv

i{v)Bv - i(u)Bu =

rv

f'(s)Bsds+ Ju

f(s)dBs.

(1.105b)

Ju

E x a m p l e 2. (Martingale property) We consider a particular class of Ito integrals I(t) = / f(u)dBu, Jo

(1.106)

Stochastic


Processes

41

and show that I(i) is a martingale. First we realize that the integral l(t) is a particular case of the class (1.93a) with g(u,Bu) = i(u). Hence we know that the variable (1.106) is normal distributed and posses the intrinsic time given by (1.93b). Its transition probability Pi|i is defined by (1.53) with tj = r(tj); yj = I(i,); j — 1, 2. This concludes the proof that the integral (1.106) obeys a martingale property like (1.27) or (1.64). * S2 Here we consider the product of two stochastic functions subjected to two SDE with constant coefficients dxk(t) = ctkdt + frfcdBi; ak,bk = const; k ~ 1,2,

(1.107)

with the solutions xk(t) = akt + bkBt;

xfc(0) = 0.

(1.108)

The task to evaluate d(xiX2) is outlined in EX 1.9 and we obtain with the aid of (1.89) d{x\X2) = X2dxi + x\dx2 + b^dt.

(1.109)

The term proportional to 6162 in (1.109) is non classical and it is a mere consequence of the non classical term in (1.89). The relation (1.109) was derived for constant coefficients in (1.107). One may derive (1.109) under the assumption of stepfunction for the functions a and b in (1.106) and with that one can approximate differentiable functions (see Schuss [1.9]). We consider now two examples Example 1 We take put x\ = B i; X2 = Bf. Thus, we obtain with an application of (1.101b) and (1.109) dBt3 = B t dB? + B^dBj + 2B t di = 3(B t dt + B?dB t ). The use of the induction rule yields the generalization dBtfc = jfeB^dB* + ^ " ^ B ^ d t

(1.110)

42



Engineering

Example 2 Here we consider polynomials of the Brownian motion P n (B t ) = c0 + cxBt + • • • + c n B?;

ck = const.

(1.111)

The application of (1.110) to (1.111) leads to dP n (B 4 ) = P;(B t )dB t + ip£(B t )di;

' = d/dBt.

(1.112)

The relation (1.112) is also valid for all functions that can be expanded in form of polynomials. & S3 Here we consider the product *{Bt,t) = ip(Bt)g(t),

(1.113)

where g is a deterministic function. The use of (1.109) yields d*(B t ,t) = g(i)<MB t ) +

Equation (1.116) applies, in the first place, only to the function (1.113). However, the use of the expansion CO

$(B t ,t) = J> fc (B 4 )g fc (i),

(1.117)

fc=i

shows that (1.116) is valid for arbitrary functions and this proves (1.98).

Stochastic


Processes

43

S4 In this last step we do not apply the separation (1.113) or (1.117) but we use a differentiable function of the variables (x,t), where x satisfies a SDE of the type (1.107) $(B t ,t) = g(x,t) = g(at + bBt,t);

x = adt + bdBt; 0

*t = agx + gt;

*B t = &gx5

$B,B, = b lgxx.

a, 6 = const. (1.118)

Thus we obtain with (1.116)

The relation (1.119) represents the Ito formula (1.99.3) (for constant coefficients a and b). As before, we can generalize the proof and (1.119) is valid for arbitrary coefficients a(x,t) and b(x,t). We generalize now the Ito formula for the case of a multivariate process. First we consider K functions of the type y f c (B i 1 ,...,B t M ,t); fc = 1,2,... ,K,

Vk

where B ^ , . . . , B ^ are M independent Brownian motions. We take advantage of the summation convention and obtain the generalization of (1.97.1) dy f c (Bj,... , B f , , )

* £ d t + ^LdBr l + I ^ d B r d B f ; dt exp(—a)

for n —> oo.

n

J

Now we put b(k + l , n , a / n ) b(k,n,a/n) and this yields

a n — kf k+1 n \

6(1, n, a / n ) —> a exp(—a);

a

a

\~ n)

fc

+ l'

a2 6(2, n, a / n ) —> — exp(—a);...

an b(k, n, a/n) —y —- exp(—a) = ^ ( a ) . Definition. (Homogeneous Poisson process (HPP)) A random point process N(t), t > 0 on the real axis is a H P P with a constant intensity A if it satisfies the three conditions (a) N(0) = 0. (b) The random increments N(£&) — N(£fc_i); k = 1,2,... are for any sequence of times 0 < to < t\ < • • • < tn < • • • mutually independent. (c) The random increments defined in condition (b) are Poisson distributed of the form

Pr([N(tr+0-Nfa)] =

(A

fc)=

^7(-H

*• Tr = t r + i — tr,

k = (J, 1 , . . . ; r = 1, 2 , . . . .

(A.2) ^

To analyze the sample paths we consider the increment AN(£) — N(£ + At) — N(£). Its probability has, for small values of At, the form ' l - A A i f o r k = 0~ AAt for A; = 1 0(At2) forfc>2 (A.3) Equation (A.3) means that for At —> 0 the probability that N(t + At) is most likely the one of N(£) (Pr([N(t + At) - N(t)] = 0) ss 1). Pr(AN(£) = fe) = ih^fL fe!

exp(-AAt)

Stochastic


Processes

47

However, the part of (A.3) with Pr([N(t + At) - N(t)] = 1) « XAt indicates that there is small chance for a jump with the height unity. The probability of jumps with higher heights k = 2, 3 , . . . corresponding to the third part of (A.3) is subdominantly small and such jumps do not appear. We calculate of the moments of the HPP in two alternative ways. (i) We use (1.5) with (A.2) to obtain oo

m

/

p(x)s dx = J2km

Fr x

( = k)

fc=0 oo

= exp(-a)^fcma*/A;!;

a = At,

(A.4)

k=0

or we apply (ii) the concept of the generating function defined by g(z)

= J2zk

Pr x

( = *)>

fc=

° fc=0

with

g'w = (x)

^

(A.5)

dz 5"(l) = ( o : 2 ) - ( x ) , . . . ; * This leads in the case of an H P P to oo

oo

g(z) = ^2 zkak exp(-o;)/fc! = e x p ( - q ) y^(zq) f c /A:! fc=o fe=o

= exp[a(z-l)].

(A.6)

In either case we obtain (N(t)) = At,

(N 2 (t)) = (At)2 + At.

(A.7)

We calculate now the PD of the sum x\ + x^ of two independent HPP's. By definition this yields Pr([a;i + x2] = k) = Prl ^ [ x i = j , x 2 = A; - j] J fc = X P r ( X l = J > 2 = A; - j) A;

j=o


48


ex

Engineering

qk-j

ex

E p[-(^)]f p[-^)] (fc _ j)! k

exp[-(01 + l92)]£^'Q)/fc! 3=0

= exp[-(81 + 92)](01+92)k/kl

(A.8)

If the two variables are IID (0 = 6± = 02) (A.8) reduces to Pr([xi + x2] = k) = exp(-20){20)k/k\.

(A.9)

Poisson HPP's play important roles in Markov process (see Bremaud [1.16]). In many applications these Markov chains are iterations driven by "white noise" modeled by HPP's. Such iterations arise in the study of the stability of continuous periodic phenomena, in the biology and economics, etc. We consider the form of iterations x{t + s) = F(x(s),Z(t

+ s));

s,teN0

(A.10)

where t, s are discrete variables and x(t) is a discrete random variable driven by the white noise Z(t + s). An important particular case is Z(£ + s) := N(t + s) with a PD Pr(N(£ + 8) = k) = exp(u)uk/kl;

u = 9(t + s).

The transition probability is the matrix governing the transition from state i to state k. Examples (i) Random walk This is an iteration of a discrete random variable x(t) x(t) = x(t-l)

+ N(t);

x{0) = xoeN.

N(t) is HPP with Pr([N(t) = A;]) = exp(-Xt){Xt)k/k\. obtain the transition probability Pjl

(A.ll) Hence, we

= Pr(x(t) = j , x(t - 1) = i) = Pr([i + N(j)] = 3) = Pr(N(j)=j-l).

Stochastic


Processes

49

(ii) Flip-Flop processes The iteration takes here the form x(i) = ( - l ) N ( t ) .

(A.12)

The transition matrix takes the form p _ u = Pr(x(t + s) = 1 | x(s) = - 1 ) = Pr(N(t) = 2k + 1) = a; p M = p r (x(t + s) = 1 | x(s) = 1 = Pr(N(t) = 2k) = 0, with a = ^Texp(-\t)(\t)2k+1/(2k

+ 1)! = exp(-At) sinh(Ai);

k=o oo

0 = êxp(-Ai)(Ai) 2 f c /(2fc)! = exp(-Ai) cosh(At).

X

fc=0

Another important application of HPP is given by a ID approach to turbulence elaborated by Kerstein [1.17] and [1.18]. This model is based on the turbulence advection by a random map. A triplet map is applied to a shear flow velocity profile. An individual event is represented by a mapping that results in a new velocity profile. As a statistical hypothesis the author assumes that the temporal rate of the event is governed by a Poisson process and the parameter of the map can be sampled from a given PD. Although this model was applied to ID turbulence, its results go beyond this limit and the model has a remarkable power of prediction experimental data. Exercises E X 1.1. Calculate the mean value Mn(s,t) = ((B t - B s ) n ), n G N. Hint: Use (1.60) and the standard substitution y^ = yi + Z\j2{t2 — t\), where z is a new variable. Show that this yields [2(£ 2 -*i)] n / 2

Mn

/ exp(—v2)dv / exp(—z 2 )z n dz.

/it

The gamma function is defined by (see Ryshik & Gradstein [1.15]) /

ex.p(-z2)zndv

r ( ( n + l)/2)Vn = 2fc; ^0Vn = 2ife + l;

r ( l / 2 ) =-S/TF,

fceN,

r ( n + l) = nT(n).


50


Engineering

Verify the result M 2n = ir-V2[2(t2 - ii)] n r((2n + l)/2). E X 1.2. We consider a ID random variable X with the mean fi and the variance a2. Show that the latter can be written in the form (f x (x) is the P D ^ = ( s ) ; e > 0 ) (

+ \J fi+e

) fx(x)(x -

For x < /x — e and X>/J, + £^(X a2>e2

1 - /

tfdx;

e.

J—oo /

ix(x)dx

— fi)2 > e2, this yields = e 2 P r ( | X - ^ | > e),

J ix—e

and this gives the Chebyshev inequality its final form P r { | X - / z | >e} 0;

£3 - t\ = v + u > 0,

introduce (1.53) into (1.52) and put T = (47r 2 m;)- 1/2 exp(-A); A = (2/3 - y2?/(2v) + (3/2 - 2/i) 2 /( 2u ) = a 22/| + ai2/2 + «o, with afc = ctk(yi,y3),k = 1,2,3. Use the standard substitution (see EX 1.1) to obtain / Tdt/2 = (47rra)" 1/2 exp[-F(y 3 ,y2)] / exp(-K)djy 2 ;

F=

4a 0 a 2 - af

-^2—

/

;

K= a2 y2+

l

ai \ 2

2^J'

and compare the result of the integration with the right hand side of (1.52). E X 1.5. Verify that the solution of (1.54) is given by (1.55). Prove also its initial condition. Hint: To verify the initial condition use the integral oo

/

exp[-y 2 /(2£)]H(y)dy,

-00

where H(y) is a continuous function. Use the standard substitution in its form y = \/2tz. To verify the solution (1.55) use the same substitution as in EX 1.4. E X 1.6. Calculate the average (yf (£1)^(*2)>; 2/fc = Btfc, k = 1,2; n, m G N with the use of the Markovian bivariate PD (1.60). Hint: Use standard substitution of the type given in EX 1.2. E X 1.7. Verify that the variable B t defined in (1.65) has the autocorrelation (BtB s ) = £ A s. To perform this task we calculate for a

52



Engineering

fixed value of a > 0 k{z)Az,

(E.l)

where A^ is a set of IID N(0,1) variables and ipi-; k = 1, 2 , . . . is a set of orthonormal functions in [0, 1] »i 1/Jk(z)lpm(z)dz

=

Skm.

Jo '0

Show t h a t ( E . l ) defines a W P . Hint: T h e autocorrelation is given by (BtB s ) = t A s. Show t h a t d_ (B t B a ) = — (t A s) = V Vfc(0 / ~dt

Mz)dz.

Multiply the last line by ipm(t) and integrate the resulting equation from zero to unity. E X 1.18. A bivariate PD of two variables x,y is given by p(x,y). (a) Calculate the PD of the "new" variable z and its average for (i) z = x ± y\ (ii) z = xy. Hint: Use (1.41b). (b) Find the PD iuy(u,v) for the "new" variables u = x + y; v = x - y. E X 1.19. The Ito representation of a given stochastic processes F(t,(j) has the form F(t,u) = (F(t,u)) + [ f(s,u)dBs, Jo where i(s,u) is an other stochastic process. Find i(s,u) for the particular cases (i) F(t,iu) = const; (ii) F(t,u) = BJ1; n = 1,2,3; (hi) F(t,u) = exp(B t ). E X 1.20. Calculate the probability of n identically independent HPP's [see (A.8)].

CHAPTER 2 STOCHASTIC DIFFERENTIAL EQUATIONS There are two classes of ordinary differential equations that contain stochastic influences: (i) Ordinary differential equations (ODE) with stochastic coefficient functions and/or random initial or boundary conditions that contain no stochastic differentials. We consider this type of ODE's in Chapter 4.3 where we will analyze eigenvalue problems. For these ODE's we can take advantage of all traditional methods of analysis. Here we give only the simple example of a linear 1st order ODE dx — = -par;

p = p(w),

ar(0) =

x0(u),

where the coefficient function p and the initial condition are xindependent random variables. The solution is x(t) = xoexp(—pt) and we obtain the moments of this solution in form of (xm) = {XQ1 exp(—pmt)}. Assuming that the initial condition and the parameter p are identically independent N(0, a) distributed, this yields (*2m> = ^ ^ e x p ( 2 a m 2 t 2 ) ;

( x 2 ^ 1 ) = 0.

*

(ii) We focus in this book — with a few exceptions in Chapter 4 — exclusively on initial value problems for ordinary SDE's of the type (1.123) that contain stochastic differentials of the Brownian motions. The initial values may also vary randomly xn(0) — xn(u). In this chapter we introduce the analytical tools to reach this goal. However, in many cases we would have to resort to numerical procedures and we perform this task in Chapter 5. The primary questions are: (i) How can we solve the equations or at least approximate the solutions and what are the properties of the latter? 55

56



Engineering

(ii) Can we derive criteria for the existence and uniqueness of the solutions? The theory is, however, only in a state of infancy and we will be happy if we will be able to answer these questions in case of the simplest problems. The majority of the knowledge pertains to linear ordinary SDE, nonlinear problems are covered only in examples. Partial stochastic differential equations (PSDE) will be covered in Chapter 4 of this book. 2.1. One-Dimensional Equations To introduce the ideas we begin with two simple problems. 2.1.1. Growth of

populations

We consider here the growth of an isolated population. N(i) is the number of members of the population at the instant t. The growth (or decay) rate is proportional to the number of members and this growth is, in absence of stochastic effects, exponential. We introduce additionally a stochastic term that is also proportional to N. We write the SDE first in the traditional way dN — = rN + uW(t)N;

r,u = const,

(2.1)

where W(t) stands for the white noise. It is, however, convenient to write Equation (2.1) in a form analogous to (1.99.1). Thus, we obtain dN = adt + bdBt;

a = rN,

b = uN;

dB t = W(t)dt.

(2.2)

Equation (2.2) is a first order homogeneous ordinary SDE for the desired solution N(B 4 ,i). We call the function a(N, t) (the coefficient of dt) the drift coefficient and the function fe(N, t) (the coefficient of dBi) the diffusion coefficient. SDE's with drift coefficients that are at most first order polynomials in N and diffusion coefficients that are independent of N are called linear equations. Equation (2.2) is hence a nonlinear SDE. We solve the problem with the use of the


Equations

57

Ito formula. Thus we introduce the function Y = g(N) = InN and apply (1.99.3) (see also (1.129)) dY = d(lnN) = (ag N + 6 2 g NN /2)di + bg^dBt = (r - u2/2)dt + udBt.

(2.3)

Equation (2.3) is now a SDE with constant coefficients. Thus we can directly integrate (2.3) and we obtain its solution in the form N = N 0 exp[(r-M 2 /2)£ + uB t ];

N 0 = N(i = 0).

(2.4)

There are two classes of initial conditions (ICs): (i) The initial condition (here the initial population No) is a deterministic quantity. (ii) The initial condition is stochastic variable. In this case we assume that No is independent of the Brownian motion. The relation (2.4) is only a formal solution and does not offer much information about the properties of the solutions. We obtain more insight from the lowest moments of the formal solution (2.4). Thus we calculate the mean and the variance of N and we obtain (N(t)> = (N0) exp[(r - u2/2)t] (exp(uB4)) = (N0) exp(rt),

(2.5)

where we used the characteristic function (1.58a). We see that the mean or average (2.5) represents the deterministic limit solution (u — 0) of the SDE. We calculate the variance with the use of (1.20) and we obtain Var(N) = exp(2ri) [(N§> exp(u2t) - (N 0 ) 2 ].

(2.6)

An important special case is given by the combination of the parameters r = u2/2. This leads to N(t)=N0exp(«Bt);

(N(t)> = (N0) exp(u 2 i/2);

Var(N) = (N§> exp(2u2t) - (N 0 ) 2 exp(u 2 f).

, ^ (2.7)

58



Engineering

We generalize now the SDE (2.2) and we introduce the effect of two independent white noise processes. Thus, we put (see also (1.129)) dN = rNdi + (udBJ + udB 2 ) N;

u, v = const.

(2.8)

We obtain now in lieu of (2.4) the solution N = N 0 exp{[(r - (u2 + v2)/2)\t + vBlt + VB2}.

(2.9)

Taking into account the independence of the two Brownian motions yields again the mean value of the exponential growth (2.5) and we obtain for the variance Var(N) = exp(2rt){(Nl)exp[(u2 2.1.2. Stratonovich

+ v2)t} - (N 0 ) 2 }.

(2.10)

equations

Now we compare the results of the Ito theory in Section 2.1.1 with the one derived by the concept of Stratonovich. We use here the classical total differential (1.100). To indicate that we are considering a Stratonovich SDE we rewrite (2.2) in the form dN = rNdt + «NodB t ,

(2.11)

where the symbol "o" is used again [see (1.83) or (1.96)]. To calculate the solution of (2.11) we use again the function g = ln(N) and we obtain d(ln(N)) = dN/N = rdi + u o d B t ,

(2.12)

We can directly integrate this (2.12) and we obtain N = N 0 exp(ri + uBt)

with (N) = (N0) exp[(r + u2/2)t}.

(2.13)

The calculation of the variance gives Var(N) = exp(2ri){(N^)exp(2u 2 t) - (N 0 ) 2 exp(u2t)}. Note that result of the Stratonovich concept is obtained with the aid of a conventional integration.


2.1.3. The problem the Maxwell

Equations

of Ornstein-Uhlenbeck distribution

59

and

We consider here the SDE dX = (m - X)dt + 6dB t ;

m,b = const.

(2.14)

Equation (2.13) is a first order inhomogenous ordinary SDE for the desired solution X(Bt,t). Its diffusion coefficient is independent of the variable X and this variable only appears linear in the drift coefficient. Therefore, we will classify (2.2) as a linear equation. We use the relation dX+(X-m)dt-e-*d[ei(X-m)],

(2.15)

to define an integrating factor. Thus, we obtain from (2.14) d[e*(X — m)] = 6e*dBt. Now we have a relation between just two differentials and we can integrate X(t) = m + e-*(X0 ~m) + b J exp(s - i)dB s .

(2.16)

Jo

Relation (2.16) is the formal solution of the SDE (2.14). To obtain more information we continue again with the calculation of the first moments. The integral on the right hand side of (2.17) is a non anticipative function. The mean value is hence given by (771 and Xo are supposed to be deterministic quantities) ( X ( i ) ) = m + e - i ( X 0 - m ) ; (X(0)) = X 0 , <X(oo)) = m, (2.17a) and it varies monotonically between Xo and m and it represents the deterministic limit of the solution (2.16). The calculation of the variance yields Var(X) = b2(l2);

I = / exp(s - t)dB a . Jo 2

We use (1.86) to calculate (I ) and thus we obtain ft 2

u2

Var(X) = b / exp[2(s - t)]ds = —(1 - e~2t). (2.17b) Jo 2 In Figures 2.1 and 2.2 we compare the theoretical predictions of average (2.17a) and the variance (2.17b) with numerical simulations.

60



Engineering

Fig. 2.1. Theoretical prediction (2.17a) and numerical simulation of the average of the Ornstein-Uhlenbeck process. The irregular curve belongs to a particular simulation of (2.14). Parameters: m = 3; b = 1; Xo = 0. Numerical parameters: step width h = 0.005, and a number Ensem = 200 of repetitions of individual realizations.

Fig. 2.2. Theoretical prediction (2.17b) and numerical simulation of the variance of the Ornstein-Uhlenbeck process. Parameters as in Figure 2.1.

The numerical computations of the moments are performed repeating individual realizations (Ensem times). We see from Figure 2.1 that the theoretical and numerical averages are almost coinciding. The values of the moments are calculated from averages of the random


Equations

61

numbers Xj, (x) = EnLm Z^=i e m x j• However, we find that in the case of the variance a difference arises between numerics and theory and this discrepancy grows with increasing values of the time. In the latter case it is advisable to use a higher value of the parameter Ensem, or to change the routine and use higher order techniques (see Sections 5.5 and 5.6). More details about the Ornstein-Uhlenbeck problem can be found in Risken [2.1]. Now we consider the Maxwell distribution. We investigate the three-dimensional SDE du = — udt + \/26dB+,

6 — const; /

u=(Ul,u2,u3y,

1 1

9 2

^

(2-

1 8 a

dB t =(dB t ,dB t ,dB?),

where u stands for a velocity field and B^, k = 1, 2, 3 are independent WP's. Equation (2.18a) degenerates into three individual SDE's of the Ornstein-Uhlenbeck type (2.14). The individual solutions are uk = ukoexp(-t)

+ V2bexp(-t)

exp(s)dB*;

ukQ = uk(0).

Jo

We obtain as mean value uk(t) = uk(0) exp(—t) and the standard deviation has the form Vfe(t) = uk(t) - «fc(0)exp(-£) = V2bexp(-t)

/ exp(s)dB^. Jo

(2.18b) We know that the integral in (2.18b) has a GD (see (1.90)). We calculate the moments of V/t and we obtain (see (2.17b)) (V2k(t)) = a2 = 6(1 - A2),

A = exp(-t)].

(2.18c)

Now we use (1.92) and this yields = 3(V 2 (t)) 2 ;

(Yt(t))

= 15 d^2 = — xidt + (7 — ax2 — (3xi)dBf

(2.60)

To solve (2.60) write it in its vectorial form dx = Axdt + bdBj,

We introduce an integrating factor and set dx - Axdt = exp(At)d[exp(-At)x],

(2.62)

where the exponential matrix is defined by a Taylor expansion 00

exp(At) = V k-Aktk

to -

= I + At + 2- A 2 t 2 + • • • ;

( 2 - 63 )

d[exp(At)] = Aexp(At)dt, where I is the identity matrix. Thus, we obtain the SDE d[exp(-At)x] = e x p ( - A t ) b d B i ;

(2.64)

where the term exp(—At) represents the integrating factor. An integration yields x = exp(At)xo + / exp[A(t — s)]b(s)dB s ; Jo

xo = x(t = 0). (2.65)

We assume that the initial condition xo is deterministic quantity and we obtain from (2.65) the mean value (x(t)} = exp(At)x 0 ;

x 0 = (x0,y0)

(2.66)

that is again the solution of the deterministic limit of the SDE (2.59). We obtain from (2.63) e x p ( A t ) = ( C°S\ V —suit

S i a t

\ cost /

(2.67)

72


Thus, we infer from (2.65) xi =

XQ

cos t + yo sin t

+ [ sin(i-s)[7-ax2(s)-#c1(s)]dB8, Jo X2 = —XQ sin t + yo cos t

(2.68.1)

+ / cos(t-s)[7-aa:2(s)-/foi(s)]dBs. Jo

(2.68.2)

Thus we can infer from (2.68) that the solution of the stochastic pendulum is given in general by a set of two coupled Volterra integral equations. 2.3.1. Stochastic

excitation

Here we assume (a = j3 = 0; 7 / 0) and equation (2.59) contains only additive noise and the problem is linear. The solution (2.68) is thus not an integral equation anymore, but it involves only a stochastic integral. We put for the mean value of the initial vector (x\) = xocosi + yosint;

(^2) = —XQ sin t + j/o cost,

(2.69)

and we calculate the autocorrelation of the x\ component ru(t,T)

= {z(t + r/2)z(t-T/2));

z(t) = xi(t) - (xi(t)>.

The application of the rule (1.87) to the last line yields (see (2.47)) r

ii(£>r)=72 /

sin(i + r / 2 — u) sin(t — r / 2 — u)du;

Jo a = t-

(2.70)

|T|/2.

The evaluation of this integral leads to ru{t, T) = \l2{2{t

- |r|/2) cos(r) + [sin(|r|) - sin(2t)]}.

(2.71)


2.3.2. Stochastic

damping

Equations

73

(/3 = 7 = 0; a / 0)

We assume here /3 = 7 = 0 ; a ^ 0 and the SDE (2.59) contains a multiplicative noise term. We obtain from (2.68) xi = xocost+

yosmt — a

sin(i — s)x 2 (s)dB s ,

(2.72.1)

cos(t — s)x2(s)dBs.

(2.72.2)

Jo

X2 =—XQsint + yocost — a Jo

We recall that the solution of the problem with additive stochastic excitation was governed only by a stochastic integral. We see that (2.72) represents an integral equation. This means that multiplicative noise terms influence a process in a much more drastic way than additive stochastic terms do. The mean value is again given by (2.69). However, the determination of the variance is an interesting problem. We use Zk = xk - (xk);

Vjk = (zj(t)zk(t));

j , k = l,2.

(2.73)

Thus we obtain from (2.72) Vn(t) = s

Jo

sin 2 (t - u)K(u)du;

K(«) = (xl(u));

Vi 2 (t) = V 2 i(t) =S- I sin[2(t - u)]K(u)du; 1 Jo

s = a2; (2.74)

V 22 (t) = s / cos 2 (i - u)K(u)du. Jo We must determine all the four components of the variance matrix and we begin with V 22 (t) = K(t) - (x2)2 = s [ cos2(t - u)K{u)du.

(2.75)

Jo

Equation (2.75) is a convolution integral equation that governs the function K(t). It is convenient to solve (2.75) with an application of the Laplace transform POO

H(p) = £K(£) = / Jo

exp(-pt)K(t)dt.

(2.76)

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Engineering

The Laplace transform given by (2.76) and the use of the convolution theorem S, f A(tJo

s)B(s)ds = £A(t)£B(t),

(2.77)

lead to H(p) = £ ( - x 0 sin t + y0 cos i) 2 + sH(p)£ cos 2 t = xl £ sin2 t - xoyo £ sin It + [y2, + «H(p)] £ cos 2 1. To reduce the algebra we consider only the special case XQ = 0, yQ ^ 0. This yields H(P)=[W8 + « H ( P ) ] ^ ± ^ ) and solving this for the function H(p) we obtain H(p)=(p!1+2)y°;

F ( p , S ) = p ( p 2 + 4 ) - S ( p 2 + 2).

(2.79)

To invert the Laplace transform we need to know the zeroes of the function F(p, s). In EX 2.5 we give a simple special case of the parameter s to invert (2.79). However, our focus is the determination of weak noisy damping. Thus, we consider the limit s —> 0+. In this case we can approximate the zeroes of F(a,p) and this leads to F(p,s) = [ ( p - s / 4 ) 2 + 4 ] ( p - s / 2 ) + 0(s 2 )

Vs = a 2 ^ 0 .

(2.80)

A decomposition of (2.79) into partial fractions yields with (2.80) H(p) = vi

1/2 p - s/2

(p - s/4)/2 + 5s/8 {p - s/4) 2 + 4

(2.81)

With (2.81) and (2.75) we obtain the inversion of the Laplace transform (2.76) V22(£) = K ( t ) - y 2 l c o s 2 i = - ^ e x p ( s t / 2 ) { l + exp(-st/4) x [cos(2i) + 5ssin(2t)/8]} - ygcos 2 ^

(2.82)

Note that the variance (2.82) has the order of unity for O(st) < 0(1).


Equations

75

We give in Figures 2.3 and 2.4 a comparison of the theoretically calculated average (2.69) and the variance (2.82) with numerical simulations. We note again that the theoretically predicted and the numerical averages almost coincide. However, we encounter a discrepancy between the theoretical and numerical values. The latter

y(t),

Fig. 2.3. Theoretical prediction (2.69) and numerical simulation of the average of Equation (2.72). Parameters: a = 0.3; XQ = 0; yo = 1. Numerical parameters: step width h = 0.005, and a number Ensem = 200 of repetitions of individual realizations.

Fig. 2.4. Theoretical prediction (2.82) and numerical simulation of the variance of Equation (2.72). Parameters as in Figure 2.3.

76


might be caused by a value of a that violates the condition s —> 0 in Equation (2.80). However, we will repeat the computation of the variance in Chapter 5 with the use of a different numerical routine. We assign the calculation of the two other components of the variance matrix to EX 2.6. We also shift the solution of the pendulum problem with a noisy frequency to EX 2.7. 2.4. The General Linear SDE We consider here the general linear inhomogeneus n-dimensional SDE dxj = [Ajk{t)xk + a,j(t)]dt + bjr(t)dBl; j,k = l,...,n;

r = l,...,m,

(2.83)

where the vector dj(t) represents a inhomogeneous term. Note that the diffusion coefficients are only time-dependent. (2.83) represents thus a problem with additive noise. Multiplicative noise problems are (for reasons discussed in Section 2.3) included in the categories of nonlinear SDEs. To introduce a strategy to solve (2.83) we consider for the moment the one-dimensional limit of (2.83). Thus we analyze the SDE dx = [A(t)x + a(t)]dt + b(t)dBt.

(2.84)

To find an integrating factor we use the integral l(t) = exp

o

A(s)ds

(2.85)

We find with (2.85) d[x(t)l(t)} = I(i)[a(t)dt + b(t)]dBt and after an integration we obtain x(t)=[x0+l

l{s)a(s)ds+

I(s)b(s)dBs\

/l(t);

x0 = x(t = 0). (2.86)

Thus, we determine the mean value m{t)

=IXQ+

f l{s)a(s)ds \/l(t),

(2.87)


Equations

77

and this is again the solution of the deterministic limit of the SDE (2.84). The covariance has the form c(t,u) = ([x(t) — m(t)][x{u) — m(u)]) =

TZATT^ / Kt)i\u) JO

l2(s)b2(s)ds;

a = uM.

(2.88)

i

To solve the n-dimensional SDE we start with the homogeneous deterministic limit of (2.83) ±j = Ajk(t)xk;

xm(0) = x0m.

(2.89)

We use the matrix of the fundamental solutions, 0,

where the negative sign of the drift coefficient is the ID remainder of the condition of negative imaginary parts of the matrix A. The stationary Fokker—Planck equation (3.43a) governing the PD p(x) reads d f°° A(xp)' + b2p"/2 = 0; ' = — ; / p(x)dx = 1. dz J-oo


Equations

79

A first integral leads to d —p = da;

2A tr

- T Q - ^ P + T;

7 = const.

We put 7 = 0 and this leads to p = (27ra 2 )~ 1 / 2 exp[-x2/(2a2)};

a2 = b2/(2A),

where we applied the normalization condition of a normal distribution with zero mean and the variance a2 and these values coincide with the ID limit of (2.95). 2.5. A Class of Nonlinear SDE We begin with a class of multiplicative one-dimensional problems dx = a(x, t)dt + h(t)xdBt.

(2.96)

Typical for this SDE is a nonlinear drift coefficient and a diffusion coefficient proportional to x: b(x, t) = h{t)x. We define an integrating factor with the function G{But)

= ex.pl- f h(s)dBs + ]- f h2(s)ds\.

(2.97a)

An integration by parts in the exponent gives with (1.105)

G(Bt,t) = expl-h(t)Bt+

f

L

ti(s)B " ^ s + l ^h2(s) ds\.

(2.97b)

The differential dG is performed with the Ito formula (1.98) yields G t = \h2G; G Bt = -hG, G B t B t = h2G. Thus we obtain dG = Gh2dt + b2dBt,

b2 = -hG

(2.98a)

and the application of (1.109) yields d(xG) = G(adt + hxdBt) + Gx(h2dt - hdBt) - Gxh2dt = Gadt. (2.98b) Now we introduce the new variable Y = xG.

(2.99)

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Engineering

The substitution into (2.98) yields dY — = Go(x = Y/G,t).

(2.100)

It is important to realize that (2.100) is a deterministic equation since the differential dBt is missing. Example 1. ax =

h axdBt; a = const. x We see that with a = 1/x, h = a this example belongs to the class (2.96). The function G is given by G = exp (a2t/2-aBt). Thus we obtain with from (2.100) dY /"* Y — = G2 ^> Y 2 = const + 2 / exp(a 2 s - 2aB s )ds. Jo

d*

Finally we obtain after the inversion of (2.99) x = e x p ( - a 2 t / 2 + aB^VD;

D = XQ + 2 / exp(a 2 s -

2aBs)ds.

Jo

Example 2. We consider a generalization of problem of population growth (2.2) da; = rx(l — x)dt + axdBt;

a,r = const.

Note this SDE coincides with the SDE (2.35a) for k = 1. The function G is defined as in the pervious example and (2.100) yields dY _ rY(G - Y) ~dt ~

CJ

'

This is a Bernoulli differential equation. We solve it with substitution Y = 1/z. Thus we obtain the linear problem z — —rz + r/G, where dots indicate time derivatives. An integration yields 1 zG

U(t) l/x0 + rl(t)'

with U(t) = e x p [ ( r - a 2 / 2 ) i + aB t ];

I(t) = / U(s)ds. Jo

*


Equations

81

The class (2.96) of one-dimensional SDE's is of importance to analyze bifurcation problems and we will use the technique given here in Chapter 3. It is, however, easy to generalize the ideas to n-dimensional SDEs. Thus we introduce the problem dXfc = ak(X,t)dt

+ hk{t)XkdBt;

k = l,...,n.

(2.101)

Note that we do not use the summation convention. We introduce the integrating factor (see (2.95)) G fc (B t) t) = e x p j - J hk{s)dBs + ~ f h2k{s)ds\ and we obtain the differential dGk(Bt,t) we have

(2.102)

= {h\dt — hkdBt)Gk.

d{Gkxk) = akGkdt.

Thus

(2.103)

Thus, we obtain the system of n deterministic differential equations — - = Gkak(X\ = Y i / G i , . . . ,X n = Y n /G n ,£);

Y^ = GkXk. (2.104)

Example: The Nonlinear Pendulum. We consider nonlinear pendulum oscillations in the presence of stochastic damping. The corresponding SDE is (compare the linear counterpart (2.59)) d2x . dx ^ + a 6 ^ + s m x = 0. Passing to a system of first order equations leads to dx\ = xzdt;

da>2 = —ctX2dBt — sinxidi.

(2.105)

First, we find that the fixed points (FPs) of the deterministic problem (a — 0) given by Pi = (0,0)(center); P2,3 = (±7r,0) (saddle points) are also FPs of (2.105). To find the type of the FPs of the SDE we linearize (2.105) and we obtain d 6

= 6di;

, A d^2 = —a^2dBt — cos(x{ )£id£;

f x{ = 0, ± TT.

(2-106)


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Engineering

In case of the deterministic center (x[ — 0) we rediscover the linear problem (2.59) with j3 = 7 = 0. Its variance V22 is already given by (2.82). To be complete we should also calculate the variances Vn and V12. An inspection of the Laplace transform of the first and second integral equations (2.74) shows 2 2

2

£ V n = a H(j9)£sin i;

£Vi 2 =

?-ll(p)£sm(2t).

These additional variances exhibit the same structure as (2.82). This means that for a2t < 0(1) the perturbations remain small and the FP (the deterministic center) is still an elliptic FP. It is also clear that the deterministic saddles become even more unstable under the influence of stochastic influences. It is convenient to study the possibility of the occurrence of heteroclinic orbits. To consider this problem we first observe that (2.105) belongs to the class (2.101) of SDEs. We obtain Gi = 1; G2 = G = exp(aB 4 + a 2 t / 2 ) , and we use the new variables Vi=xi;

y2 = Gx2.

(2.107)

This yields ill = y 2 /G;

y2 = - G s i n y i ,

(2.108)

These are the canonical equations of the one-degree-of-freedom nonautonomous system with the H a m i l t o n function P2 H = —-Gcosg;

q = yi, p = y2.

(2.109)

In the deterministic limit (a = 0) we obtain G = 1 and (2.109) reduces to the deterministic Hamilton function H = p2/2 — cosq, where deterministic paths are given by H = H(go,Po) with qo,po as initial coordinates (see e.g. Hale and Kocak [2.6]). We return now to the stochastic problem. There, it could be argued that we may return to the "old" coordinates p = Gx2;

q = xu

(2.110)


Equations

83

the substitution of which into (2.109) gives the transformed Hamiltonian (the Kamiltonian) K = G(x|/2-cosxi).

(2.111)

However, it is clear that (2.108) are not the canonical equations corresponding to the Kamiltonian (2.111). The reason for this inconsistency is that (2.110) is not a canonical transform. To see this we write the old coordinates in the form x\ = x$q,p) = q; %2 — %2(q,p) = p/G. A transform is canonical if its Poisson bracket has the value unity. Yet we obtain in our case dxi 3x 2

3xi 3x 2

1 /r, _LI

Hence, we return to the Hamiltonian (2.109) and continue with the investigation of its heteroclinic orbits. The latter are defined by H(q,p,t) = H(±7r,0, i) and this leads to p = ±2Gcos{q/2).

(2.112)

In the stochastic case we must evaluate in general the heteroclinic orbits numerically. However, it is interesting to note that we can also generalize the concept of parametrization of the heteroclinic path which in the case of the stochastic pendulum is given by p = ±2Gsechi;

q = 2arctan(sinht).

(2.113)

The variables in Equation (2.113) comply with (2.112). To verify that p and q tend to the saddle points we obtain asymptotically q —> 7rsign(t) for |i| —> oo and this gives the correct saddle coordinates. As to the momentum we obtain p —> ± 4 exp (aBt + a2t/2 - \t$

for |t| -> oo.

(2.114)

Using (1.58a) we obtain the mean of the momentum (p) - • ± 4 e x p ( a 2 i - |t|)

for \t\ -> oo.

(2.115)

To reach the saddle (ir, 0) in the mean we must comply with \a\ < 1. If we consider the variance of p we obtain (p2) ^ 1 6 e x p ( 3 a 2 t - 2 | t | ) .

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Engineering

-1 -2 "

Fig. 2.5. The motion of the oscillator (2.105) in the phase space. The non-noisy curve is the corresponding deterministic limit cycle. Parameters: a = 0.3, (xo,yo) = ( T / 3 , 1), h = 0.005.

The latter relation tells us that to reach the saddle (IT, 0), we need to ask for a2 < 2/3. Thus, we can conclude that the higher the level of the considered stochastic moment, the lower we have to choose the value of the intensity parameter a that allows the homoclinic orbit to reach for t —> oo, the right hand side saddle. Finally, we show in Figure 2.5 for one specific choice of initial conditions a particular numerical realization of the solution to (2.105) in the phase space. We juxtapose this stochastic solution to the corresponding deterministic solution. Thus, we see that the stochastic solution remains close to the deterministic limit cycle only in the very first phase of its evolution from the initial conditions. For later times the destructive influence of the multiplicative noise becomes dominant and any kind of regularity of the stochastic solution disappears, no matter how weak the intensity of the noise. $

2.6. Existence and Uniqueness of Solutions The subject considered in this section is still under intensive research and we present here only a few available results for one-dimensional


Equations

85

SDEs. First, we consider a deterministic ODE x = a(x);

x,a E R .

A sufficient condition for the existence and uniqueness of its solution is the Lipschitz c o n d i t i o n \a(x) — a(y)\ < K\x — y\; K > 0 where K is Lipschitz's constant. There is, however, a serious deficiency because the solutions may become unbound after a small elapse of time. A simple demonstration of this is given by an inspection of t h e solutions of the equation x — x2. Here, we have \x2 — y2\ < K\x — y\ and K is the maximum of \x + y\ in the (x, y)-plane under consideration. The solution of this O D E is x(t) = XQ/{1 — x^t) t h a t becomes unbounded after the elapse of the blow up time £& = \/XQ. To ensure the global existence of the solutions of an O D E (the existence for all times after the initial time) we need in addition to the Lipschitz's condition, the growth bound condition \a{x)\ < L ( l + |x|),

L>0;

Vt>t0

where L is a constant. Now, we consider as an example x — xk, and we obtain from the growth bound condition \x\k < L ( l + |a;|); we can satisfy this condition only for k < 1. We t u r n to a general class of one-dimensional SDE's given by (2.20). Now, the coefficients a and b must satisfy the Lipschitz and growth bound conditions t o guarantee existence, uniqueness and boundedness. This means t h a t the drift and diffusion coefficients must satisfy the stochastic Lipschitz condition \a(y,t)-a(z,t)\

+ \b(y,t)-b(z,t)\

0,

(y,z)EK2,

(2.116)

and the stochastic growth bound condition \a(y,t)-b(y,t)\

< L ( l + |y|),

L > 0;

Vt>t0,

xGR.

(2.117)

E x a m p l e 1. dy = ~ 2 e x p ( - 2 y ) d t + e x p ( - y ) d B t . This SDE does not satisfy the growth bound condition for y < 0. Thus we have to expect t h a t the solution will blow up. To verify this we use the reduction method. The SDE complies with

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Engineering

(2.34a) and we obtain the solution y = lnjBj + exp(yo)}- This solution blows up once the condition B t = —exp(yo) is met for the first time. £ Example 2. As a counter-example where the SDE satisfies the growth bound condition, we reconsider EX 2.1(iii) and we obtain 1 fy

du

1, .

. .„

,„,.,?/

„

and this yields 7T

y(t) = — — + 2arctan[êxp(rB()]; z = tan(yo/2 + 7r/4). £ These examples suggest that we should eliminate the drift coefficient. To achieve this we start from (2.20) (written for the variable x), we use the transformation y = g(x) and the application of Ito's formula (1.99.3) gives dy = [ag'(x) + b2{x)g"{x)/2]dt + b(x)g'(x)dBt.

(2.118)

Hence, we can eliminate the drift coefficient of the transformed SDE if we put g"(x)/g'(x) = —2a{x)/b2{x) and the integration of this equation leads to the transformation g(x) = J e x p j - 2 J [a(v)/b2(v)}dv\du,

(2.119)

where the parameter c is appropriately selected. Thus we obtain from (2.118) a transformed SDE dy = b(x) g'(x)dB i ; x = g~l(y) where we have replaced x by the inverse of the function g. Hence, we obtain ft xs = x(s). (2.120) V = Vo+ / g'(xs)b(xs)dBs; Jo L/0

Karatzas and Shreve [2.7] showed that although the original process x may explode in time, the process y will not explode. Example (Bessel process) We consider the SDE a - 1 dx=——di 2x

+ dBj;

a = 2,3,....

(2.121)


Equations

87

Thus (2.119) yields with c = 1 g(s) = ln(s)

for a = 2 and

g(x) = ( x 2 _ a - l ) / ( 2 - a )

for a > 3.

Hence, we obtain for a = 2: x = exp(y);g' = exp(—y). Thus we get the solution y = Vo+

exp(-y s )dB s ,

ys = y(s).

4

Jo Exercises E X 2.1. We consider the SDE (2.20). Find which of the following cases is amenable to the reduction technique of Section 2.1.4 (i) a = ry;b = u, (ii) a = r; b = ^ ; 7 = 1/2,1, 2, (hi) a = — ^-sin(2y); b = rcos(y); r,u = const. E X 2.2. Verify the solutions (2.4) and (2.17) with the use of the Ito formula (1.98) and (2.37). E X 2.3. Verify that the SDE dx = -/32x(l

- x2)dt + /3(1 -

x2)dBt,

has the solution x

aexp(z) + a —2

(*) = —

ô—;

a

l

.,

x

. .

z

^

B

= + °); = ^ *-

aexp(z) + 2 — a E X 2.4. Verify the solution of the stochastic pendulum (2.65). Use the Ito formula (1.122) and a generalization of (2.37). E X 2.5. Find a parameter s that allows a simple determination of the zeros of (2.79) and thus an inversion of (2.79). E X 2.6. Determine the components V n , V12 of the variance matrix for the pendulum with stochastic damping. Use (2.74) and (2.82). E X 2.7. Solve the problem of the pendulum with a stochastic frequency. Use the (2.68) with the parameters a = 7 = 0 ; / 3 ^ 0 t o calculate the variance V n .

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Engineering

We obtain from (2.68) xi — xocost + yosint — f3 / sin(i — s)xi(s)dB s , Jo

X2 = —xosint + yocost — f3 / cos(£ — s)xi(s)dB s . Jo

We introduce here variance Vn(i) in terms of Vn(t) = K(t) - 2/0 sin2 * = (? / sin 2 (i - u)K(u)dit;

K(t) = (x 2 ).

The determination of Vn is performed in analogy to the calculations in Section 2.3.2 and we obtain

H(p) = vl p - A / 2

p+ \ ( p + A/4) 2 +4_

2;

\ = (3,22.

Finally, we obtain the variance in the form Vn(t) = |{exp(/3 2 i/2) - exp(-/3 2 i/4) x [cos(2i) + 3/32 sin(2i)/8] - 2 sin2 t}. Note that the variance has the same structure as (2.82) and it is of order unity if for 0(f32t) = 1. E X 2.8. Calculate the mean and the covariance function of the stochastic vectorial integral K

j(J) = / bjr(s)dBrs, Jo

j = l,...,n; r = l,...,m.

E X 2.9. Take advantage of the multivariate Ito formula (1.127) to prove the (2.91). E X 2.10. Show that the ODE y = y/x satisfies the growth bound conditions, but not the Lipschitz condition. Solve this ODE and discuss the singularities of its solution in connection with the mentioned conditions.


Equations

89

E X 2.11. Solve the SDE da; = ydt + adB], dy = ±xdt + bdBt;

a,b = const,

where B^; k = 1,2 are independent WP's. The case with the negative sign of the drift coefncient is a generalization of the linearized pendulum. E X 2.12. Solve the SDE and find the positions where the solution blows up dx = xkdt + bxdBt;

b,k = const; k > 1; x(0) > 0.

E X 2.13. Solve the generalized population growth model SDE K

da; = rxdt + x N J frfcdBt; r,b^ — const. fe=i

E X 2.14. Solve the damped oscillation SDE x + ax + UJ x = b£t(t);

a,u> = const.

Hint: Write the SDE in form of scalar equations and use the matrix

A4°2 \ -or

M; -a I

exp(Ai) = ÊL^!'J-[l v Cos{vt) + {lu + A) sin(vt)]; v 2 2 u = a/2; v = y/co - (a/2) , where I is the 2D unit matrix. See also the undamped case (2.59) with (2.61).

CHAPTER 3 THE FOKKER-PLANCK EQUATION In Section 3.1 we derive the master equation and we use this equation in Section 3.2, where we focus on the derivation of the FokkerPlanck equation that is a PDE governing the distribution function of the solution to a SDE. In the following sections we treat important applications of this equation in the field of bifurcations and limit cycles of SDE's. 3.1. The Master Equation We consider a Markovian, stationary process and our goal is to reduce the integral equation of Chapman-Kolmogorov (1.52) to a more useful equation called the master equation. We follow here in part the ideas of Van Kampen [3.1] and we simplify in the first place the notation. The transition probability depends only on the time difference and we can write Pi|i(y2,*2|yi,*i) = T(y 2 |yi;r);

T = t2-t1.

(3.1)

An example was already given for the Ornstein-Uhlenbeck process (1.56.2). There we got also the generally valid initial condition T(y 2 bi;0) = (z,t) —T(z\x;t) + OUAt)2). J ot The right hand side of (3.22) can be written in the form RHS=

dwT(w\x;t)J(w,t);J(w,t)

=

(3.24)

dzT(z\w;At)i/>(z,t). (3.25)

We approximate now also ip(z,t) by a Taylor series and this leads to J(w, t) = ip(w, t) / dzT(z\w; At) + At A(w,t)ip'(w,t) ijk(\,t)

=

+ -B(w,t)x/>"(w,t) + •••

dkiP(\,t)/d\k.

(3.26)

We use in (3.26) the abbreviations

A{w,t)At=

dzT(z\w;At)(z-w); (3.27)

B(w,t)At=

2

dzT(z\w;At)(z-w) .

Note that the normalization condition of T(z|u;; At) gives the first integral on the left hand side of (3.26) the value unity.

The Fokker-Planck

Equation

97

The substitution of (3.24) to (3.27) into (3.23) leads now to /

dzi/)(z, t)T(z\x; t) + At f dz^{z, t)—T(z\x; t) + 0((Ai) 2 ) dwip(w,t)T(w\x;t) +-B(w,

+ At

dw A(w,t)ip'(w,t)

t)ip"(w,t) T(w\x;t).

The first terms on both sides cancel and we obtain in the limit At —> 0 dz{^(z,t)-T(z\x;t) A(z,t)rl>'{z,t) +

-B(z,tW(z,t) T(z\x;t)}=Q.

(3.28)

where we replaced the dummy integration variable w by z. As a further manipulation we introduce partial integrations and we put A(z,t)iJ>'(z,t)T(z\x;t)

=

[A(z,t)ip(z,t)T(z\x;t)]> -TP(z,t)[A(z,t)T(z\x;t)]',

and another relation for the term proportional to B(z, t) (see EX 3.3). Thus, we obtain 3

dz->P(z,t) i ^_T{z\x;t)

+

-±[B(z,t)T(z\x;t)]"}

= 0.

[A(z,t)T(z\x;t)}'

The braces in the last line must disappear, since ip(z, t) is an arbitrary function. This yields 7\

- T ( z | x ; t ) = -[A(z,t)T(z\x;t)]'

1

+ -[B(z,t)T(z\x;t)]"

(3.29)

Equation (3.29) is the Fokker—Planck equation (FPE). Since the spatial derivatives correspond to the forward spatial variable z, (3.29) is sometimes also called the "forward ChapmanKolmogorov" equation. In EX 3.4 we derive from (3.21) also a "backward Chapman-Kolmogorov" equation with spatial derivatives with respect to the backward variable a;. It is also more convenient to use

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Engineering

the nomenclature (3.10) and we obtain in this way the FPE in its usual form

^ T

1

= -|;[ A (w.*)P(y.t)] +

\~2my,t)P(yM (3.30)

P(y,t) =

T(y\x;t).

To apply the FPE we introduce boundary conditions, an initial condition and the normalization [(a, j3) are the boundaries of the interval] 9

V tfc oy

= °> fc = 0,1;

A= a

and

A = /?,

P(y,0) =

(3.43a)

where a and b are the drift and diffusion coefficients of the SDE (3.36a). Let us focus now on the calculation of stationary solutions to (3.43a). There is only a stationary solution P s (y) if the coefficients are time-independent a = a(y); b = b(y). We use the boundary conditions for a — — oo; j3 = oo and we obtain a first integral in the form d(6 2 P s )/d,z — 2aPs. A further integration yields the stationary solution

Ps(yH

?Sy e x p

Wa

/o

d«

(3.43b)

V{u

The constant in (3.43b) serves to normalize the distribution. Example We consider the Ornstein-Uhlenbeck problem (2.14) and (3.43a) reduces to 3P dt

=

82P dy2

3[(m-s/)P] dy

h =

^

The stationary solution is governed by dP/dy — (m — x)P = 0 and we obtain P = C exp(my — y2/2);

C = const.

We get the normalization (3.38) and this yields C = exp(—m 2 /2)/ \/2~7r (see also (1.56.1) for m — 0). A

The Fokker-Planck

Equation

105

We conclude now this section with the derivation of nonstationary solutions of autonomous SDE's with a = a(y),

b = b(y).

(3.44)

We solve the Fokker-Planck equation with the use of a separation of variables. Thus we put (the function A(y) should not be confused with the coefficient A(z,t) in (3.27) through (3.30)) P(y,i) = exp(-At)A(2/).

(3.45)

The substitution of (3.44) and (3.45) into (3.43a) gives [b2A]" - 2[aA}' + 2AA = 0;

' = d/dy;

A(±oo) = 0.

(3.46)

The ODE (3.46) and the BC's are homogeneous, hence (3.46) constitutes an eigenvalue problem. We will satisfy the initial condition (3.31b) later when we replace (3.45) by an expansion of eigenfunctions and apply orthogonality relations. The rest of the algebra depends on the particular problem. We will consider, however, only problems that satisfy the conditions of the Sturm-Liouville problem (see e.g. Bender and Orszag [3.3]). The latter boundary value problem has the form

[p(yK(y)}' + m(y)Uy) = o; Uvi) = to) = o.

(3.47)

The eigenfunction is denoted by in(z) and /i is the eigenvalue. The orthogonality relation for the eigenfunctions is expressed by /

^{z)ln{z)lm{z)dz

= NmSnm,

(3.48)

where N m and Snm stand for the norm of the eigenfunctions and the Kronecker-Delta. To illustrate these ideas consider the following example. Example We consider an SDE with the coefficients a(y) = -f3y;

b(y) = VK;

P,K = const.

(3.49)

The coefficients in (3.49) correspond to an Ornstein-Uhlenbeck problem with a scaled Brownian movement, see EX 3.9.

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Engineering

The separated equation (3.46) reads KA" + 2(3(yA)' + 2AA = 0. We use a scale transformation and this yields d2 az

d(zA) az

A

_A

/2/3 v rv

n

( 3 5Q)

A* = ^ ; A(z = ±oo) = 0. We transform (3.50) into the Sturm-Liouville operator (3.47) and we obtain d d~z

P(z)—An(z)

+ (A* + l)p(z)An(z)

p(z) = q(z) = exp(z 2 /2),

= 0; (3.51)

/x = (A* + 1).

The eigenvalues and eigenfunction of (3.51) have the form A; = 0 , 1 , . . . ;

An(z) = £êxV(-z2/2).

(3.52)

Now we use an eigenfunction expansion and replace (3.45) by oo

P(y, t) = ^2 exp(-n/3i)B n A n (y v / 2/3/K);

B n = const.

(3.53)

n=0

We can satisfy the initial condition (3.31b) and we obtain oo

Y, BnAn(y^/2j/K)

= 5(y).

(3.54)

n=0

The evaluation of B n is performed with the aid of (3.48) and we obtain

Bm=f^J

Am(0)/Nm.

The explicit evaluation of the eigenfunctions and the calculation of the norm is assigned to EX 3.10.

The Fokker-Planck

Equation

107

3.5. Lyapunov Exponents and Stability To explain the meaning of the Lyapunov exponents, we consider a n-dimensional system of deterministic ODE's x = f(x,t);

x(t = 0) = XQ;

X,XQ,

f G Rn.

(3.55)

We suppose that we found a solution x(t); x(0) = XQ. NOW we study the stability of this solution. To perform this task we are interested in the dynamics of a near trajectory y(t). Thus we use the linearization x(t) = x(t) + ey(t);

0 < e < 1,

y G Rn

(3.56)

where s is a small formal parameter. The substitution of (3.56) into (3.55) leads after a Taylor expansion y = J(x,t)y;

3pq = dfp(xi,...

,xn, t)/dxq;

p,q =

l,...,n, (3.57)

where the matrix J is the Jacobian and (3.57) is the linearized system. We can replace the vector-ODE (3.57) by a matrix-ODE for the fundamental matrix $fcm(i) that takes the form $fcm(*) = hs(x, t)<S>sm(t); $sm{t = 0) = 8sm.

(3.58)

The solution to the linearized problem (3.57) is now given by yk{t) = <s>km(t)ym(0).

(3.59)

We define now the coefficient of expansion of the solution y(t) in the direction of y(0)

In the last line we use the symbol | • • • | to label the norm in R n . The Lyapunov exponent corresponding to the direction y(0) is then

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Engineering

defined by A

_

lim

NM»(Q).0l.

t—>oo

( m

t

The solution x(t) is therefore stable (in the sense of Lyapunov) if the condition Re(A) < 0 is met. Note also that in case of onedimensional ODE (n = 1 in (3.55)) we obtain A = limiln[y(i)];

y(t) G R.

(3.61)

t—>oo t

Equation (3.61) is sometimes called the Lyapunov coefficient of a function. The stability method of Lyapunov can be generalized to stochastic differential equations. We will see that it is plausible to calculate the Lyapunov exponents with the use of the stationary FPE equation of the corresponding SDE. A rigorous treatment to calculate the Lyapunov exponents is given by Arnold [3.4]. To illustrate this idea we consider a class of first order SDE's dx = f(x) +

\g(x)g\x) dt + g{x)dBt;

' = d/dx,

(3.62)

where f and g are arbitrary but differentiable functions. The stationary FPE (3.43) takes the form Ps(x) =

C X 2 -—-exp 2 [ f(s)g- (s)ds Jo g{x)

C = const.

(3.63)

Now we suppose that we know the solution z(t) to (3.62). We study its stability with the linearization x(t) = z(t) + ey(t). The substitution into (3.62) leads to y(t) = ( j f ' ( z ) + \[g(z)g'(z)}>} dt + g'(z)dB?j y.

(3.64)

The SDE (3.64) is analogous to the one for the population growth (2.1). Thus we use again the function ln(y) and we obtain d ln(y) = (f + i g g " ) dt + g'dB t .

(3.65)

The Fokker-Planck

Equation

109

We integrate (3.65) and this yields t /

ln[y(t)} =

I

\

?+tgg»ys+^

rt

g'dB,

(3.66) f' + ^gg" + g ^ ds, / Jo where £s is the Wiener white noise. Now we calculate the Lyapunov exponent of the one-dimensional SDE (3.62). Thus we use (3.61) and obtain A = lim t—»oo

gs +g &

Ul'(H " ' )4

(367)

Equation (3.67) is the temporal average of the function f + ^gg" + g'£s. For stationary processes we can replace this temporal average by the probabilistic average that is performed with the stationary solution of the FPE (3.63). Thus, we obtain A

f + ^ ' + l f t - W f + ^gg" |p s (*)(V + lgg")dz,

(3.68)

which used the fact that (g'£s) = 0. We substitute P s from (3.63). An integration by parts yields A = [Psf] oo —oo

fP's - 2gg" P ^ da;.

(3.69)

The first term of the right hand side of (3.69) vanishes. We give in EX 3.11 hints how we can rearrange with the use of the (3.63) the integral (3.69). This yields finally (P s > 0)

\ =

-2f((/g)2Psdx 1. (B.16)

The Lyapunov function takes now the form

L = y+

( T) 2 + T i1 ~ I) x2 + 4(1 ~ a(3S)sin2^/2)-

^B-17)

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Engineering

To achieve the condition L > 0 for (x, y) ^ (0,0) we require that 0 < B < 25;

a(38 < 1.

(B.18)

A recombination of condition (B.16) yields finally «

2

< ^ 2

with/3 = a 7 ,

7

>0;

B< - ^ .

(B.19)

With constants complying with (B.19) we can conclude now that the FP in the origin is stable. B.2. The method of

linearization

We begin with a linear nth order ODE with constant coefficients j/(")(x) + biy^-l\x) bj = const.;

+ ••• + bn.iy'{x)

+ bny(x) = 0;

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

(B.20)

where primes denote derivations with respect to the variable x. (B.20) has the FP (y(0),y'(0),...,^-1)(0))=0.

(B.21)

The Routh-Hurwitz criterion (see [1.2]) states that this FP is asymptotically (for t —> oo) stable if and only if the following determinants satisfy the relations (see also Ex 3.16) Ai = 6i > 0;

A 2 = bib2 > 0;

A3 =

h 1 0

b3 b2 6i

0 0 > 0. b3

(B.22)

We generalize now the concept of the Routh-Hurwitz criterion for the following class of second order SDE's y" + [ai + Ci(t)]y' + [aa + Q2{t)y = 0; aj = const.;

Q(t)dt = const. dB];

j = 1,2,

[a.26)

with the two white noise processes [see (2.41)] (Cj(t)Ck(s)) = 6(t - s)Qik.

(B.24)

This stochastic extension was derived by Khasminskiy [3.17]. It states that the FP at the origin is asymptotically stable in the mean square IFF the following conditions are met a1 > 0;

a2> 0;

2axa2 > Qna2 + Q22.

(B.25)

The Fokkei—Planck

Equation

129

E x a m p l e (Stochastic Lorentz equations) We consider a Lorentz attractor under the influence of a stochastic perturbation. The SDE has the form dx = s(y — x)dt, dy = (rx — y — xz)dt + ay dBt, dz = (xy — bz)dt;

b:s,r

(B.26)

> 0.

This system of SDE's is a model for the development of fluid dynamical disturbances and their passage to chaos in the process of heat transfer. The dimensionless parameters in (B.26) are given by s (the P r a n d t l number) r (proportional to the Rayleigh number) and b (a geometric factor). (B.26) has the deterministic F P ' s (0,0,0);

(±y/b{r-l),

±y/b(r-l),

r - 1).

However, in the stochastic case (a ^ 0) only the F P in the origin survives. We use now the linearization routine to study the stability of the F P in the origin. The linearization of (B.26) leads to du = s(v — u)dt, dv = (ru - v)dt + avdBt,

(B.27)

dw = —bwdt. We infer from (B.27) t h a t the component w is decoupled from the rest of the system and because of w(t) = w(0) exp(-bt) this disturbance is stable for b > 0. Hence we will investigate only the stability of the system (u,v) taking advantage of the first two equations of (B.27). However, this system is not a member of the class (B.23). To achieve a transformation we eliminate the variable v by use of v = u + (du/dt)/s. This yields u + [(1 + s) - aC{t)}u + [s(l - r) - as((t)]u

= 0.

(B.28)

A comparison with (B.23) and (B.24) leads then to ai = l+s; Q

_ / a2 "Va2S

a2 = s(l-r); a2s \ a2s2)-

(i =-a((t);

(2 =

-as((t); (B.29)

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Engineering

The Khasminskiy criterion applied to our example reads now l + s>0;

s(l-r)>0;

2(l + s)(l -r)

> a2(l -r + s).

(B.30)

Since we have by definition s > 0 the first part of (B.30) is automatically satisfied, while the second part gives r < 1. Finally, the third part of (B.30) yields for small intensity constants

r 0; At -»• 0; D = (Ax)2/(2At)

= 0(1)

to derive a diffusion PDE dpi dt

=r)9

2

Pi dx2 '

This is the FPE with A = 0; B = 2D. E X 3.2. Calculate the expression for d(y2)/dt to the derivation of (3.17), the relation

^(y 2 ) = jdyjdy'[(y

and use, in analogy

- y1)2 + 2y(y' - y)]W(y'\y)Z(y,t).

Verify that the evaluation of the last line leads to ^t(y2}

=

(a2(y))+2(ya1(y)).

The variance a satisfies a2' = (y2)' — 2(y)(y)'. Verify that this yields *2' =

(a2(y))+2((y-{y))al(y)).

E X 3.3. To calculate the transformation of the third term in (3.28) use the relation (B^T)" = (BT)'V + 2(BT)>' + ( B T ) < , to verify that f(BT)^"dz

= /"(BT)'Vdz.

E X 3.4. Derive the backward Chapman Kolmogorov equation. Use in (3.4) u = At and v = t, y% = z; y2 = w and y\ = x (x, w and z are backward, intermediate and forward spatial variables, respectively).

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Engineering

Hint: This yields T(z|x; t + At)=

/ dwT(w\x; At)T(z\w;

t).

Multiply this equation by an arbitrary function ip{x, t) and note that x < w. Apply the other considerations that lead from (3.21) to (3.29). The result ^T{z\x;t)

= -^[A(x,t)T(z\x;t)]

E X 3.5. Consider [see (3.32)] 9P

the

-

three-term

^^[B(x,t)T(z\x;t)]. Kramers-Moyal

1 32

3

equation

Id3

Multiply this formula by Ay and (Ay) 2 (see (3.33a)). Show that the third term in this Kramers-Moyal equation does not affect the first and second order terms in (3.34). Hint: V

d

3

= {yv)

- Sv ;

y

d

3

= (yv)

- 6{yv) + 6^ .

E X 3.6. Continue the calculation of (3.33b) for k = 4 and 5 and find formulas in analogy to (3.33c) to (3.33e). E X 3.7. Consider the average S= / J

a(y(s),s)dsf

b(y(w),w)dBl

= ^2(a(ym, s m ) ( W i - tm)b{yn, s n )(B n+ i - B„)). n,m

The integrand is a non anticipative function. Thus, we obtain from

the independence of the increments of Brownian motion that the average written above vanishes: (S) = 0. E X 3.8. Use the third line of (3.38) to calculate dT 3 /di; T 3 = ((Ay) 3 ). Take advantage of (2.14') and compare the results with T3 calculated from (2.19a').

The Fokker-Planck

Equation

133

E X 3.9. Compare the Ornstein-Uhlenbeck-problem (2.14) with SDE that arises from the coefficients (3.49). Use a scaled Brownian movement (see (1.65)) to perform this juxtaposition. E X 3.10. Consider (3.54) and verify Bm =

1 N,m

6l(2(3/K)1/2z}exp(z2/2)Am(z)dz -1/2

A m (0)/N, The symmetry of the eigenfunctions (3.50) leads to B2m+i = 0. Eventually we obtain p(x,t) = y^

exjp(-2nt)B2nMn

n=0

We observe that (3.38a) tends for t —• oo to its asymptotic limit. This is the stationary solution l i m ^ o o p ^ i ) = Boexp(—(5x2/K). Note also that we can determine the normalization constant Bo in the form 1 = / p(x, oo)dx =$> B 0 =

\ZKTT/(3.

E X 3.11. To prove (3.70) we put the first term in (3.69) to zero. We obtain with the use of the first integral of the PD that leads to (3.63) A

1 f(2f/g2-g'/g)--gg" -2|(f/g): Psdx+ /

P s dx

fg'/g

P,dx.

The first part of the last line gives Equation (3.70). In the second integral we apply an integration by parts for the second term where substitute (3.63). This shows that the second integral vanishes. E X 3.12. (a) Show that (3.62) is equivalent to da; g(x)odB t . (b) Solve with the use of (2.35) the transcritical case dx = (ax — x2)dt +

axodBt-

f(x)dt +

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Engineering

E X 3.13. The Hamilton function H is defined by (A.14). Using (A.10) show that H = const, along W = const. E X 3.14. Find possible FP's of the 1-D SDE's (3.72), (3.74) and (3.76) of Section 3.6.1. E X 3.15. Verify that all moments of Bt/i vanish for t —> oo. E X 3.16. The Routh-Hurwitz matrices An = (ay) are in (B.22). They are constructed in the following way. First we set the coefficient of ?/n) to bo = 1. Then we put together the first row as a sequential array of odd coefficients 6i,6 3 ,...,b2fc+i,0,..., Vl7r

rt

$ = a / dy Jo Jo

dsG(x,t,y,s)Wys

oo OO

„OT

„i

a ; V u f c ( x ) e x p ( - A f c t ) / dBj,Ufc(y) / dB s exp(A fc s). u-n JO JO fc=o

(4.16)

Note t h a t we set $o = 0 since (4.16) already satisfies the initial condition [ T + -AM 0 $/«o

(4.34)

is the solution of the PDE (4.26) and satisfies the initial condition. We replace now $ by v and use a Fourier series expansion to investigate the frequency components of its discrete spectrum. We write (4.34) in the form v(x, t) =

VQ

sin(u;r +

/3V/VQ);

(5 =

UJXMOX/C.

(4.35)

We obtain a Fourier series from oo

A

v/v0 = Y^

n(/?) sin(amr).

(4.36)

71=1

The coefficients of this series are given by

2 r An(/?) = - /

sin(iOT +(3v/vo)sm(ionT)d(u>r).

(4.37)

71" Jo

We substitute £ =

UIT+(3V/VO, V/VQ

= sin£ and we obtain from (4.37)

A„(/3) = - r s i n ( £ ) s i n [ n £ - n/3sin(0][l - /?cos(0]d£.

(4.38)

7T JO

To evaluate (4.38) we take advantage of the theorems of the harmonic functions and of the recurrence formulas of the Bessel function (see Abramowitz and Stegun [1.3]). We use the integral representation of the Bessel function

i r Jfc(n/3) = — / 7T Jo

cos[ka — n/3sin(a)]da,

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Engineering

where 3n(z) denotes the Bessel function of order n. Thus, we obtain from (4.38) Ki{P) = Jn-l(-2) - Jn+l(z) - -[Jn-2(z) =

~ Jn+2^)]

2

^±z = n(5. z The substitution of (4.39) into (4.36) yields -} = f > ^ )nJ1 /

v0

^

2

sin(no;r); v

"

E K{ nJ

^

(4.39)

= H ^ M .

(4.40)

nf3

n=l

The relation (4.40), t h a t is sometimes called t h e Bessel-Fubini formula, represents now the explicit solution t o t h e P D E (4.27) for small values of the Mach number Mo- The latter condition limits the validity of (4.40) to a space variable in the order of 0(x) = 1. 4 . 2 . 2 . Stochastic

initial

conditions

Now we concentrate on a stochastic wave problem where the stochasticity is introduced by the application of a stochastic initial condition t h a t includes slowly varying amplitudes and phases. We follow here the ideas of Rudenko & Soluyan [4.6], Akhmanov & Chirkin [4.7] and Rudenko & Chirkin [1.9]. First, and we apply t h e initial condition v(x = 0,i) = vo(£lt)sm[u>t

+ ip(ttt)},

Q = ecj,

0 < £ « 1. (4.41)

The relation (4.41) represents a randomly modulated wave with a stochastic phase ip. In the following we will assume t h a t the parameter e is sufficiently small such t h a t we can use the solution v(x,

T) — VQ(CIT) sin LOT

+

-ÛJV(X, T)X

(4.42)

+ ip(Qr)

where r is again the convection time defined in (4.27). (4.42) complies with the PDE (4.27) and with the initial condition (4.41). We rewrite (4.42) with the use of new variables V{z,9) = A(e)sm[e + zV(z,6) + G(x)) and use this to verify (4.94b). E X 4.11. Applying the rule of l'Hospital to verify (4.99). Hint: Show that the PD (4.96) gives rise to lim /" P e (0)H(0)d0 = H(O). E X 4.12. We consider the 4th order boundary value problem

yw - ^

y = o;

s/(°>A) = y"(°> A ) = y&A) = s/"(i>A) = °-

Verify that the eigenvalue equation has the form sm[y/9/(l + A)]. Use the PD (4.96) to find numerically the mean and the standard variance for Ai and A2.

CHAPTER 5 N U M E R I C A L SOLUTIONS OF O R D I N A R Y STOCHASTIC D I F F E R E N T I A L EQUATIONS In this chapter we will familiarize the reader with some numerical routines to solve ordinary SDE and we shall introduce these routines with the F O R T R A N programs t h a t are included on the attached CD. The latter programs are labeled with an " F " . So, for example, F 3 is the third F O R T R A N program.

5.1. R a n d o m N u m b e r G e n e r a t o r s a n d A p p l i c a t i o n s A prerequisite to solve SDE are r a n d o m n u m b e r s (RN's). There are various techniques to construct "random numbers". In fact, the numbers repeat after a period P. As a result, we should call these numbers pseudo-random numbers. To find an algorithm for a random number generator, we recall t h a t ever since the work of Feigenbaum [5.1] it has been known t h a t the iteration of numbers on the real axis (ID maps) can lead to irregular outcomes. It is convenient to apply a congruential iteration Nfc+i = (aN/- + u) mod v;

k = 0,1,...,

where a, u, i>,No are given positive integers and the symbol u m o d t ; (the modulo) denotes the remainder of u after dividing u by v. T h e choice of v is determined by the capacity of the computer and there are various ways to select a and u. The period P of this iteration was investigated by K n u t h [5.2] and it was found t h a t the condition t h a t u and v are relatively prime makes it important to increase the value of P . The IMSL (international mathematical scientific library) proposes for RN's with a uniform P D in the interval (0, 1) the iteration N f c + i = (16807 Nfc) mod (2 3 1 - 1); 167

k = 0,1,...,

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Engineering

where 0 < No < 1 is an initial number and the factor 2 31 — 1 is a prime number. The numbers in the above formula are not RN's in (0, 1) but the sequence R fe+ i = N f c + 1 /(2 3 1 - 1), serves to produce a sequence of RN's, Rfc+i, in (0, 1). Other techniques are discussed by Press et al. [5.3] and we recommend in particular the routines called rani, ran2 and ran3, that are based on congruent iterations. The question of a uniform distribution and the independence of the RN's is investigated in the next section. RN's that vary in arbitrary intervals are obtained with transformations (see Section 1.5, where also Gaussian distributed numbers are introduced). 5.1.1. Testing of random

numbers

We will investigate here the property of uniformity and the independence of stochastic variables. To perform this task we introduce a few elements of the testing of stochastic (or probabilistic) models Ho on random variables X. The basic idea is that we perform a random experiment (RE) of interest and group the outcomes into individual categories k with K as the total number of categories. One particular outcome of the RE is given by Xi and this particular outcome is called a test statistic. We categorize the predictions of an application of the model Ho in the same way. If, under a given model Ho, values of the test statistic appear that are likely (unlikely), we are inclined to accept (reject) the model HoAn important statistical test is x 2 -statistics. The quantity X is a X2-random variable of m degrees of freedom (DOF) with the PD px(x,m) = [2sr(s)]-V-1exp(-:r/2);

s = m/2;

x > 0.

(5.1a)

The mean and variance of (5.1a) are calculated in EX 5.1. It can be shown that (5.1a) is the PD of a set of m independent identical N(a, a) distributed RV's, X i , . . . ,X m , with m

X = J](X p -a) 2 /^ 2 . P=i

(5.1b)

Numerical Solutions of Ordinary Stochastic Differential

Equations

169

We assume now that under a given hypothesis Ho (e.g. the uniformity of RN's) one particular simulation falls with the probability Pfc in the category k. We perform N simulations and the expected number of events under Ho in the category k has the value Npfc. This contrasts with the outcome of a RE where Z& is the number of events in the category A; of a particular simulation. Now we calculated the sum of the K squares of the deviation of Z^ from Npfc and divided each term by Npfc. This test statistics yields the RN K

DK-i = ^(Zfc-NPfc)2/(NPfc).

(5.2)

k=l

It can be shown that for N > 1 the RN in (5.2), X = D K _i approximately has the PD (5.1a) with the DOF m = K — 1. To verify this value of the DOF we note that we can sum RV's in (5.2) but we must consider the compatibility relation Ylk=i ^fc = NA close agreement between the observed (Z&) and the predicted values (under Ho) Npfc yields (Z^ — Npfe)2 6) = l - /

p x (:r,K-l)d:r = a;

b = *cK-i-

(5.3)

Jo

In Equation (5.3) we must substitute the PD (5.1a) and use a given and small level c t « l . Conveniently one chooses a = O(10~ 2 ) and solves (5.3) numerically to obtain %2 K - 1 . As a goodness criterion for Ho we employ now the condition that D K - I must not exceed the critical value. Thus, we accept (reject) the hypothesis for D K - I < Xc,K-l(DK-l

>Xc,K-l)-

Example (Uniformity test) As an application of the x 2 -test statistics we investigate now the hypothesis Ho of the uniformity of RN's. We categorize by dividing the interval (0, 1) into K subintervals of equal length 1/K. Then we generate N RN's and count the numbers that fall into the fcth

170


subinterval. Now we assume that Zfc RN's in: ((k - 1)/K, Jfe/K);

k = 1 , . . . , K.

Under Ho, the probability of simulation RN's in each subinterval is pfc = 1/K. Thus, we infer from (5.2) K D

N

K-i = N ^ ( Z fc=l

f c _ N / K ) 2

(5 4)

-

-

We investigate in program F l the uniformity of RN's using 10, 30 and 50 subintervals and applying various RN generators. The results are given in Table 5.1. £ Now we discuss independence tests. Suppose that we have obtained a random sample of the size of n pairs of variables (xi, yj);j — 1,.. -, n. To investigate the independence of the two sets of variables X = (x\,..., xn), Y = (j/i,..., yn) and we calculate their covariance R = I>P-)(%>-)/D; P=i

n

n

p—X

p=l

I

n

(5.5)

p=X

If the variables X and Y are uncorrelated we would find R = 0 and the variables are independent. We focus now on a test of the independence of N RN's x\,..., xn and we group them into consecutive pairs (x\,X2),{x2,x%),..., (xn-i,xn). We regard the first member of such a pair as the first Table 5.1. The evaluation of the uniformity test with the r a n i , ran3, the IMSL routine and a congruent method (CM) for a = 0.05. DK-I

D

DK-I

DK-I

K

XC,K-I

(ranl)

(ran3)

(IMSL)

(CM)

10 30 50

16.92 42.56 66.34

9.240 37.100 44.200

9.200 23.900 33.900

9.300 47.780 43.100

9.300 47.780 43.340

K

-I

Numerical

Solutions

of Ordinary Stochastic

Differential

Equations

171

variable (say X) and the second member of the pair as another variable (say Y). The correlation coefficient Ri of these consecutive pairs is called the correlation coefficient with lag 1 n-l

Ri = E ( ^ - ^ ( 1 ) ) ) ( ^ + i - ^ ( 2 ) ) ) / D ; P

=i

n—l

2

D =5>

2

n—1

- p + 1 - oo

where the PD's are supposed to be continuous. (ii-2) The weak convergence in the narrow sense is expressed by lim / g(x)pn(x)dx = / g(x)p(x)dx, (5-17) nôoj J where we include in the integrals classes of test functions g(x) (e.g. polynomials). An important application of the concept of strong convergence is given in the following example: Strong law of large numbers (SLLN) We consider a sequence of RV xi,X2,--- with finite means fi^ ~ (xk), k = 1, 2 ... and we define a new RV 1

n

An = - V ] x k

n

1

with

M n = - V'/ifc.

fc=i

(5.18)

k=i

Then, provided that l i m S n = 0 with

n-ôo

Sn = ^ V a r ( y x f c ) n

\ *•—' \/c=l

(5.19)

/ /

the SLLN says that lim (|An - M n |) = 0.

(5.20)

n—too

Note that for IID RV we have /ife = /i = const. =>• M n = lim A n = /x.

(5.21)

n—>oo

Proof. First we note that (A„) = M,,,

Var(A n ) = S n .

Now we apply the Chebyshev inequality (see EX 1.2) and we obtain P r ( | A n - M n | > £ ) < ^ 4 ^ = %->0

forrwoo.


Equations

175

This proof can be made more rigorous (see Bremaud [1.16]). The convergence in (5.20) is of mean square type. * An interesting application of the SLLN is given in the following: Example (Estimation of the probability distribution function, PDF) It is required to find (or numerically approximate) the PDF defined by (1.1) of the RV X. To this end we generate a sequence of independent samples of the RV X i , X 2 , . . . and we define the indicator function

{

1, n 0, Thus we obtain with the use of the

V Xi. < x, ^ otherwise. SLLN (5.20)

l . This is where the MCM is superior to traditional numerical calculations. Consider the computation of a volume in a 10D space. If we employ a minimum of 100 points in a single direction we would need for the traditional computation 1020 points. By contrast, the MCM would need according to (5.28) 10 points for one indicator variable and a collection of say 107 points would be sufficient for an accurate computation. Thus we generalize the present example and we calculate the volume of an m-D unit sphere. The indicator variable is now given by m

Zfc — 2_]X2A < 1; Zfc = 0 otherwise. i=i Thus, the MCM gives the volume V m of the m-D unit sphere the value (2 m is the number of sectors that make up the m-D unit sphere: 4 quadrants in case of a circle, 8 quadrants in case of a 3D sphere,...) tym

n

- V z ^ V n ^—'

m

.

(5.30)

k=i

Note that the exact value V ^ is given by (see Giinther and Kusmin [5.8])

v

" = w^wy

(5 31)

-

The numerical computation is made using program F5. We summarize the results of the computation of a 10-D sphere in Table 5.2. The error of the computation is defined by err = 100|(VfS-Vi 0 )/VfS|%.

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Engineering

Table 5.2. The results of a MCM calculation of the volume of a 10-D unit sphere using ran3. n

Vio

err

60000 1100000 1150000

2.54123 2.54203 2.55029

0.35048 0.31884 0.00508

There are several improvements of the MCM for multiple integrals. In the case of the determination of the area between an irregular shaped curve i(x) and the x-axis we can write b

rb

i(x)dx =

dx p(x)i(x)/p(x),

(5.32a)

Ja

where p(x) is an arbitrary PD defined in x = [a, b]. If N RN's £&, k = 1 , . . . , n are generated from the PD p(x), then

One of the most widely used application is in statistical mechanics. The PD of a statistical mechanical state with energy E is given by the Gibbs distribution function lK

'

exp(-/?E) /exp(-/3E)dE'

where (3 is the inverse temperature. The computation proceeds by starting with a finite collection of molecules in a box. It can be made to imitate an infinite system by employing periodic BC's. Algorithms exist to generate molecular configurations that are consistent with p(E). The most popular one is the Metropolis algorithm. A presentation of this algorithm would take us too far afield. Details are given in Metropolis et al. [5.9]. Once a chain of configurations has been generated, the PD is determined and various thermodynamic functions can be obtained by suitable averages. The most simple example is (E) = V E f c p ( E f c ) .


Equations

179

5.4. The Brownian Motion and Simple Algorithms for SDE's We recall that the Brownian motion is a non-stationary Markovian N(0, t) distributed process. We remind also the reader of the BoxMuller method (1.46) to (1.48) of generating a N(0, 1) distributed variable. The latter method contains the computer time-consuming use of trigonometric functions [see (1.47)]. Hence, it is more convenient to use the polar Marsaglia method. We start from a variable X that is uniformly distributed in (0, 1) and use the transformation V = 2x — 1 to obtain a variable V that is uniform in (—1,1). Then we apply two functions of the latter type Vi and V2 with W = V? + V | < 1 ,

(5.33)

that is distributed again in (0, 1) and the angle 8 is such that 6 = arc tan(V2/Vi) that varies in (0,2ir). The area-ratio between the inscribed circle (5.33) and the surrounding square has the value 7r/4. For this reason we see that a point (Vi, V2) has the probability 7r/4 of falling into this circle. We consider only these points and disregard the others. We put now cos 9 = V i / \ / W ; sin^ = "V^/VW and we obtain as in (1.47) yi = Vi V / -21n(W)/W;

y2 = V 2 y / -21n(W)/W.

(5.34)

This yields in analogy to (1.48) to the PD p(yi,y 2 ) = (27r)- 1 / 2 exp[-(y? + y22)/2}.

(5.35)

We calculate with this generation of the Brownian (or Wiener) processes in program F6 the numerical data pertaining to Figures 1.1 and 1.2 of Chapter 1. We present in Figure 5.2 the graph of the solution to population growth problem (2.4) and calculate the corresponding numerical data in F7. We give now a simple difference method to approximate solutions of ID deterministic ODE o\x — = a(x, t); x(t = 0) = XQ (5.36) is the Euler method. The latter discretizes the time into finite steps to = 0 < h < ••• < tn < tn+r,

Ak = tk+1-tk.

(5.37)

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Engineering

Fig. 5.2. A graphical evaluation of the results of the population growth model (2.4), r = 1, u = 0.2. The figure reveals also one realization and the predicted mean (2.5) that coincides with the numerically calculated mean using 50 realizations.

This procedure transforms the ODE (5.36) into a difference equation 2-n+l — Xn -\- cl\Xn, tn)L\n]

Xn = X[tn).

(Ô.OOJ

The difference equation must be solved successively in the following way xi = x0 + &(x0, t0)A0 = x0 + a(x 0 , 0)ii; x2 = xi + &(x1,t1)A1 = x0 + &(x0,0)ii + a(xi,tj)(t2-ti);

a(xi, h) = a(s 0 +

a(x0,0)ti,ti).

Equation (5.39) represents a recursion. Given the initial value XQ we obtain with the application of (5.39) the value of Xk for every following time £&. Now we propose heuristically the Euler method for the ID autonomous SDE dx — a(x)dt + b(x)dB 4 ;

x(t — 0) — XQ.

(5.40)

In analogy and as a generalization to (5.39) we propose the stochastic Eulerian difference equation xn+i =xn + a(xn)An + b(xn)ABn;

AB n = Bn+l

•Br

B, = B t (5.41)


Equations

181

In the majority of the examples in this chapter we will use equidistant step widths, tn = to + n/S. = nA;

A n = A = const.

This yields (AB n ) = 0

and

((AB n ) 2 ) = A.

In the next section we shall derive (5.41) as the lowest order term of the Ito-Taylor expansion of the solutions to the SDE (5.40). 5.5. The Ito—Taylor Expansion of the Solution of a I D SDE We start here from the simplest case of an autonomous ID SDE (5.40). In the case of a non-autonomous N-dimensional SDE, that is covered in Section 5.7, we need only to generalize the ideas of the present derivations. We integrate (5.40) over one step width and we obtain /•*n+l

x(tn+i)

/"tn + 1

= x(tn) + / &(xs)ds+ Jtn Jt„ XQ = x(0); xs = x(s).

b(x s )dB s ; (5-42)

Now we apply Ito's formula (1.99.3) to the function f(x), where the variable x satisfies the ODE (5.40). This leads to df(s) = L°f(x)dt + L 1 f(x)dB i ; 1 L°f(x) = a(x)f'(x) + - b2(x)f"(x);

L : f(x) - b(x)f (x).

(5-43)

The integration of last line leads to f(x n + i) = f(x n ) + / " +1 [L°f(xs)ds + iMixJdB,].

(5.44)

Jtn

If we specify (5.44) for the case i(x) = x, this equation reduces again to (5.42). The next step consists in the application of (5.44) for f(xs): = &(xs) and i(xs): =b(xs) the substitution of the corresponding result

182



Engineering

into (5.42). Hence, we put g(x s ) = g(xn) + / [L°g(xu)du + L 1 g(x u )dB u ]; Jtn

g= a

or

g = b.

Proceeding along these lines we find xn+i =xn + a(x n )A n + b(x n )AB n + Ri; ftn + l

R1=

Jtn

fS

/ [L°&{xu)dsdu + L 1 a(a; u )dsdB u Jt„

(5.45)

+ L 0 b ( ^ ) d B s d u + L 1 b(x u )dB s dB n ]. Equation (5.45) is the simplest nontrivial Ito-Taylor expansion of the solutions of (5.40). Neglecting the (first order) remainder Ri we obtain the stochastic Euler formula (5.41). We repeat this procedure and this leads, with the application of the Ito formula the function i(xu) = L 1 b(x„) = b(xu)b'(xu), to x(tn+i)

= x(tn) + a(iEn)An + b(x n )AB n + b(xn)b'(xn)lijl

Ii.i = /

n+1

dB s f dB u = f " +1 dB s (B s - Btn)

+ R2; (5.46)

= i(AB2-An) where we have used the Ito integral (1.89). If we proceed to second order, the remainder R2 contains two second and one third order multiple Ito integrals that are based on the following integrals rtn+1

Ii,o= / Jtn ftn + l

Io,i= / Jtn

rs

ftn+i

/ dBsdu = / Jtn

(s-tn)dBs,

<Jtn

fS

(5.47)

ftn + i

/ dsdBu- / Jtn Jtn

(Bs-BtJds,

with Io,i + Ii,o = A„AB„,

(5.48)


Equations

183

and rtn+i

Ii,i,i = / Jtn

rs

ru

ptn+i

/ / dBsdB„dB, = / Jt„ Jtn Jtn

fs

ru

d B s / dB„ / d B , Jtn Jtn

= f "+1 dB s J &BU{BU - B t n ) = ^[(AB n ) 2 - 3A n ]AB n . (5.49) We will verify (5.48) and (5.49) in EX 5.3. The application of (5.46) with R2 = scheme xn+1

0 yields the M i l s t e i n

= xn + M(xn,An,ABn),

(5.50a)

with the Milstein operator M(xn,An,ABn) — a(xn)An + b(xn)ABn + ^b(xn)b'(xn)[(ABn)2

- An}.

(5.50b)

Note t h a t the Milstein scheme (5.50) is again a recursive method and can be considered as a generalization of the Eulerian scheme (5.41). The numerical d a t a revealed in Figures 2.1 and 2.2 for the Ornstein-Uhlenbeck problem (2.14) are computed with the Milstein routine (5.50) and the corresponding program is F8. The latter is a general solver for first order autonomous SDE's. We calculate in particular the solution of the generalized population problem (2.35a). A sample p a t h of numerical solution of this equation is plotted in Figure 5.3. With the inclusion of more stochastic integrals, we obtain more information about sample paths of a solution. Hence we include more terms of the expansion of &(x) and b(x) into the remainder R2 of (5.46). We postpone this cumbersome algebra to treat the general case of non-autonomous N-D SDE's. We can, however, infer from (5.45) t h a t one term resulting from the expansion of Loa(x u ) must be given by pv

ps

/ ds / duL°&(xu) Jw Jw

1

= - &{w)a!(w) + -b2(-u;)a"(u;) A 2 ^ tn+l, W = tn.

(5.51)

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Engineering

Fig. 5.3. Lower line: numerical average of the solution to (2.35a), middle (irregular) line: a SF of (2.35a). Upper line: graph of the exponential, exp(krt); r = 0.01, k = 100, u = 0.2, Ensem = 200, h = 0.001.

Proceeding along this lines we end up with the scheme x„+i = xn + M(x n , A n , AB n ) + H n (A n , AB n , AZ n ),

(5.52a)

with the operator R(xn,An,

AB n ,AZ n )

= a'bAZn + ~ LB! + ±b2A A2n + Lb' + \b2bA (ABnAn - AZn) + ^b(bb" + b /2 )[(AB n ) 2 - 3A n ]AB n ,

(5.52b)

b

where the drift and the diffusion coefficients depend on the argument xn. In Equation (5.52b) a new RV appears that is defined by the first of the integrals in (5.47) AZ n = I i , 0 = /

dBa(s-io),

Jw

where the integration boundaries are defined in (5.51).

(5.53)


Equations

185

We easily find the moments (AZ n ) = 0 and Var(AZ n ) = M

dBs(s - w)

\Jw

dBr(r - w) Jw

I

= [ ds(s - wf = \A3U, Jw

(5.54a)

"J

and the covariance (AZ n AB n ) = / fV dBs(s - w) fV dB u = [ ds(s-w)

= lA2n.

Jw

(5.54b)

^

Note also that Equation (1.93) states that the variable AZ n is N(0,A£/3) distributed. In order to perform a numerical simulation of the RV AZ n we formulate the following theorem: Theorem 5.2 (Numerical simulation of the RV AZ n ) First we note that we can generate the Wiener variable AB n in form of N(0, A n ) distributed variable. We obtain, from the polar Marsaglia (or the Box-Miller) method (5.34), two independent N(0,1) distributed RV's, yi and y2- Now we put

ABn = v/Aîl

AZn = ±A3J2 L + 4 p V

(5-55)

We obtain from (5.55) (AZ n ) = 0;

= ±A3n;

(AZ n AB n ) = ~A2n.

and this proves (5.54).

&

We verify (5.52) to (5.54) numerically in program F9A. There we use the traditional mean a of RV x by averaging over K realizations of this variable 1 K (x) = Jim — V xk,

(5.56a)

fc=l

where the RV's x^ are individual samples of x. We contrast this average in program F9B with the batch average. There we calculate

186


Equations

in Science and

Engineering

the mean of x by computing K batches, each of which consists of M samples of x and has a different seed of the random generator. This yields 1

^

K

M

(5.56b)

= KMEE^ fc=l m = l

(k)

where x^ represents the mth realization of the variable x in the kth batch. Note that (5.56a) does not coincide with (5.56b), even not for identical seeds of the individual batches. We propose to use F9B with K = M = 25 and compare the results with the ones of F9A with K = 625. The numerical data for Figure 5.3 were calculated with the Milstein scheme (5.50) as well as with the higher order routine (5.52) and the SF's as well as the averages that coincide up to the first three digits. We tried also to compare the growth of the solution for generalized population growth (2.35a) with the solutions of the usual growth problem (2.1). To accomplish this we use in (2.35a) rk = 1 and compare the corresponding numerical results with to those of (2.1) with to those of (2.1) with r = 1. We also note that (2.35a) can be written as dX = rkX(l - X/k) + uXdBt, and the deterministic limit of this SDE has stationary point at X = k. The latter property prevents the solutions from tending asymptotically to each other. We can only expect an approximate coincidence in the first phase of the growth. Example The application of the routine (5.52) to the problem of OrnsteinUhlenbeck (2.14) with a — m — x; b = const, yields H(x n , A n , AB n , AZ n ) = -bAZn - - ( m xn+l

=xn + (m-x)

xn)A2n.

( 1 \ I An - - A n J + b{ABn - AZ n ).

(5 57)

-

* The schemes discussed in this section belong to the class of explicit schemes, where the advanced solution xn+\ is given by


Equations

187

functions of the solution xn. Explicit routines have the disadvantage of including a great variety of functions and their derivatives. An additional disadvantage of these schemes is their instability. Hence, we replace these routines in the next section with modified routines. A particular modification is given by an implicit scheme that is known in deterministic problems to be less unstable. However, a general comment about the use of numerical routines to approximate the solutions of SDE'S must be given. To eliminate the possibility of spurious results, it is always advisable to confirm numerical results of a particular routine with the numerical data obtained from another method. We note also that we introduced at the beginning of this section the Ito-Taylor expansion that is based on the application of the Ito formula (1.99.3). In the case of Stratonovich SDE's there is also a Stratonovich-Taylor expansion that relies on the Stratonovich differential (1.100). The latter approach is not covered in this book and interested reader should consult the book of Kloeden et al. [2.5].

5.6. Modified I D Milstein Schemes We used the Milstein scheme in program F8 to solve the population growth problem (2.1) in the regime of the variable 0 < x < 10 and depicted the results in Figures 2.1 and 2.2. There we found that the numerically calculated moments agree very well with the theoretical ones. We did not also come across with any problems of a divergence of the numerical solution. However, the situation changes dramatically if we would like to extend the regime of this solution beyond x > 10. The Milstein routine is in this regime unstable and results in a function that diverges rapidly. The question of how to improve the stability of the Milstein scheme arises. Basically, there are two possibilities to modify given routine and we exemplify this for the Milstein routine. In the first case (i) we perturb the drift coefficient. In the second alternative (ii) we modify the coefficients that are based on the diffusion coefficient. We begin with the first alternative.

188



Engineering

(i) An implicit ID scheme Here, we replace the Milstein routine (5.50) by xn+i = xn + [aa(x n + i) + (1 -

a)&(xn)]An

+ b(x n )AB n + ^b(xn)b'(xn){(ABn)2

- An}.

(5.58)

Note that (5.58) is for a <E [0,1] a family of SDE's. We call (5.58) in the limit a = 1 (a = 0) a fully implicit (an explicit) SDE, and a is the degree of implicitness. Note also, that implicit effects refer exclusively to the drift coefficient, whereas the diffusion coefficient is always taken at xn. In an implicit scheme we must calculate the solution xn+\ as the numerical solution of the (generally nonlinear) Equation (5.58) with given xn. To solve this aspect of the problem one usually uses a Newton-iteration routine. However, convergence problems arise in this iteration, in particular if the stochastic influences are too strong. Thus, it is convenient to take advantage of this method, despite its popularity in deterministic problems, only for the case of SDE's with a linear drift coefficient, where no iterations are needed. In EX 5.4 we use (5.58) for the Ornstein-Uhlenbeck SDE (2.14). However, the employment of the corresponding program F l l shows that the scheme (5.58) again leads to unstable results in the regime x > 10. This contrasts with the result of the higher order explicit scheme (5.52) for the same problem and we must conclude that the implicit routine (5.58) is inconsistent with a ID autonomous Ito SDE. (ii) Modification of the diffusive terms Here we use for an SDE with a variable drift coefficient the scheme proposed by Kloeden (see [2.5]) xn+i = xn + a(x n )A n + b(x n )AB n + ^[(AB n ) 2 - A n ][b(S n ) - h{xn)]/y/E~n,

(5.59)

with the support value S n = xn + &(xn)An + b(xn)^/A^.

(5.60)

To motivate (5.69) we note that a Taylor expansion of b(S n ) for the usually small parameter An reproduces (5.50) to leading order.

Numerical Solutions of Ordinary Stochastic Differential Equations

189

X, <X>

20

30

t

Fig. 5.4. A sample function and the mean value of the SDE (5.61); m = 3, 0 = 1, A„ = 0.005, Ensem = 200.

We test successfully (5.59) and (5.60) in program F12 for the SDE with variable drift coefficients dx = (m — x)dt + /3sin(x)dB i ;

m, f3 = const.

(5.61)

A particular test criterion for applicability is the agreement with respect to the average, since we obtain from (5.61) [in analogy to (2.17a)] (x) = m— (m — XQ)exp(—t). In Figure 5.4 we reveal a graph of a particular SF and depict the averages where the exact and t h e numerical one are almost coinciding. Thus, we may conclude t h a t the routine (5.59) represents a stable modification of the Milstein scheme.

5.7. T h e Ito—Taylor E x p a n s i o n for N - d i m e n s i o n a l S D E ' s We consider now the general case of non-autonomous N-dimensional SDE's with R > 1 independent Wiener processes. We introduce here the nomenclature t h a t Sk(tt,t)

=gk(xi,...,XN,t);

tit =

{xi(t),...,xN(t)),

is the fcth component of the vector function g at the space-time position (&,i).

190



Engineering

We write now the N-dimensional SDE (1.123) in the form dxk(t) = afc(£fc, t)dt + bkr(£t, t)dBrt; Jfc = l , . . . , N ; r = l , . . . , R .

(5.62)

For the differential of the vector function ffc(£t,i) we find now, with (1.127), the expression dffe(&,t) = [U&,t)ik{Zt,t)]dt+

[L^ t ,*)f fc (&,t)]dB[

(5.63)

with T _

9

9

dt

dx

k 2

U

^ dx dx '

(5.64)

9 Li — b m r OXr,

where the functions afc,bmr,ffc are taken at the space-time position (Ct,t) and B | , . . . , B ^ represent a set of R independent Brownian motions. We can integrate (5.63) formally and the result is ffc(6, t) = ffc(Co,«) + /

{Lo(6, s)ffe(6, «)ds,

+ L;(6,s)f f c (6,s)dBS}, *o = a;

€o =

(5 65)

"

(xi(a),...,xu(a)).

We apply now (5.65) to the functions xk to obtain Lo^^ = afe(a),L^Xfc = bkr and we infer from (5.65) Xk(t) = xk(a) + I {a fe (£ s ,s)ds + b fcr (£ s ,s)dB£}.

(5.66)

./a

However, we could obtain Equation (5.66) in a more simple way as the result of a formal integration of (5.62). We apply (5.65) to calculate the functions a&(£,£) and bkr{t],t) and substitute the results into (5.66). Proceeding in this way


Equations

191

we obtain afc(6>s) = afc(£o,a) + /

{L0(£u,u)a,k(£,u,u)di lit

Ja + Llû,u)akû,u)dBru},

(5.67a)

and bto(£ s , s ) = bfcu;(£o, a) + /

{M&*, «)bfcu,(û, *t)du

./a

+ L^(^,B)bto(^,«)dB;}.

(5.67b)

The substitution of (5.67) into (5.66) yields + (Brt-Bra)bkr^0,a)

xk(t) = xk(a) + (t-a)a.k^0,a)

+ R^)

(5.68)

with R4 1} = f

f {{L0ak)dsdu + (L> f c )dsdB^ + (L 0 b fcr )dB^d«

Ja J a

+ (L^b fcr )dB^dB-}

(5.69)

where all functions and operators are taken at the space-time location (£u,u). Equation (5.68) with Kk = 0 represents now the non-autonomous N-dimensional Euler-scheme [see (5.45) for the ID autonomous case]. To obtain the N-dimensional Milstein routine we expand the 3rd order tensor L ^ b ^ in (5.69) into an Ito-Taylor series and truncate after the first term. This leads to TZ(£u,u)

= bmw(tiu,u)—-bkr(Zu,u)

1

=

M N

MN £

i=l

£ |X*,*(T) " Yj>fc(T)|.

(5.89)

j=l k=l

We take advantage of the latter average to obtain the estimate of the variance of the batch averages 1

M

It is convenient to apply the Student distribution with M-l DOF to achieve a 100(1 — a)% confidence interval for r\ in the form ( T J - A T / . T J + ATJ),

A77 = K 1 _ a , M - i ^ / M ,

(5.91)

where the parameter Ki- a i M-i is calculated from the Student distribution. As an example we list the following data: For a = 0.05 and M = 20 we obtain Ko.95,19 = 2.09 and the absolute error will lie in the interval (5.91) with the probability 1 — a = 0.95. The Student or (£-) distribution governs an N(0,1) distributed RV X and x 2 is a RV following the chi-squared distribution (5.1a) with s DOF-parameter. Then we consider the ratio X / y x V s . The latter variable has the PDF [see (1.1)] F ( X ^ x V ^ ) = PrflX, ^fW*\ < t) •

(5.92)

198



Engineering

We give here the following represention of this PDF for even values of the DOF (see Abramowith & Stegun [1.3]) F(X/^Ts)

= sin(0) | l + \ cos2(0) + ^ | cos4(0)

+- ^ e f K H 9 = arctan(i/y / s),

(593>

(s even),

with an analogous formula for odd values of s. Definition 5.2 (Strong convergence and order of convergence) Strong convergence is a pathwise convergence characterized by the error criterion (5.87). A discrete approximation scheme with a maximum step size 5 converges with the order 7 > 0 at a time T if there is a constant C independent of S such that ea = < | X ( T ) - Y ( T ) | ) < G F

(5.94) D

In the framework of weak convergence we are not interested in pathwise approximation of the stochastic process. We focus there on the calculation of the PD, or its lowest moments, or functionals of the PD. Hence we do not compare the difference between exact solution and approximation as in (5.87), yet we concentrate on the errors of the moments. For the lowest moment, the mean, we introduce the mean error £m

= |(X(T))-(Y(T))|.

(5.95)

Definition 5.3 (Order of weak convergence) We use the weak convergence criterion (5.95) we define the order 7 > 0 by ||

Stochastic Differential Equations In Science And Engineering

Stochastic Differential Equations in Science And Engineering

Stochastic Differential Equations in Science And Engineering