Henderson d., plaskho p. stochastic differential equations in science and engineering
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mS T O H AD I F F E R E N T I A LE Q U A T I O N S I N S C I E N C EA N D E N G I N E E R I N GD o u g l a s H e n d e r s o n • P e t e r P l a s c h k o
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S T O C H A S T I CD I F F E R E N T I A LE Q U A T I O N S IN S C I E N C EA N D E N G I N E E R I N G
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Douglas HendersonBrigham Young University, USAPp t p f PIi3Qf*ihk"Ac i c i r I O O U I I I V V /Uriiversidad Autonoma Metropolitans, Mexico| | p World ScientificNEW JERSEY • LONDON • SINGAPORE • BEIJING • S H A N G H A I • HONG KONG • TAIPEI » C H E N N A I
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To Rose-Marie HendersonA good friend and spouse
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PREFACEThis book arose from a friendship formed when we were both fac-ulty members of the Department of Physics, Universidad AutonomaMetropolitana, Iztapalapa Campus, in Mexico City. Plaschko wasteaching an intermediate to advanced course in mathematicalphysics. He had written, with Klaus Brod, a book entitled, "HoehereMathematische Methoden fuer Ingenieure und Physiker", thatHenderson admired and suggested that be translated into Englishand be updated and perhaps expanded somewhat.However, we both prefer new projects and this suggested insteadthat a book on Stochastic Differential Equations be written and thisproject was born. This is an important emerging field. From its incep-tion with Newton, physical science was dominated by the idea ofdeterminism. Everything was thought to be determined by a set ofsecond order differential equations, Newtons equations, from whicheverything could be determined, at least in principle, if the initialconditions were known. To be sure, an actual analytic solution wouldnot be possible for a complex system since the number of dynamicalequations would be enormous; even so, determinism prevailed. Thisidea took hold even to the point that some philosophers began tospeculate that humans had no free will; our lives were determinedentirely by some set of initial conditions. In this view, even beforethe authors started to write, the contents of this book were deter-mined by a set of initial conditions in the distant past. DogmaticMarxism endorsed such ideas, although perhaps not so extremely.Deterministic Newtonian mechanics yielded brilliant successes.Most astronomical events could be predicted with great accuracy.V l l
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viii Stochastic Differential Equations in Science and EngineeringEven in case of a few difficulties, such as the orbit of Mercury, New-tonian mechanics could be replaced satisfactorily by equally deter-ministric general relativity. A little more than a century ago, thecase for determinism was challenged. The seemingly random motionof the Brownian motion of suspended particles was observed as wasthe sudden transition of the flow of a fluid past an object or obstaclefrom lamanar flow to chaotic turbulence. Recent studies have shownthat some seemingly chaotic motion is not necessarily inconsistentwith determinism (we can call this quasi-chaos). Even so, such prob-lems are best studied using probablistic notions. Quantum theoryhas shown that the motion of particles at the atomic level is funda-mentally nondeterministic. Heisenberg showed that there were limitsto 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 macroscopicscale. The third law of thermodynamics is an example.Stochastic differential equations, the subject of this monograph,is an interesting extension of the deterministic differential equationsthat 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 thecalculii of Ito and Stratonovich.We were both trained as mathematicians and scientists and ourgoal is to present the ideas of stochastic differential equations ina short monograph in a manner that is useful for scientists andengineers, rather than mathematicians and without overpoweringmathematical rigor. We presume that the reader has some, but notextensive, knowledge of probability theory. Chapter 1 provides areminder and introduction to and definition of some fundamentalideas and quantities, including the ideas of Ito and Stratonovich.Stochastic differential equations and the Fokker-Planck equation arepresented in Chapters 2 and 3. More advanced applications follow inChapter 4. The book concludes with a presentation of some numeri-cal routines for the solution of ordinary stochastic differential equa-tions. Each chapter contains a set of exercises whose purpose is to aidthe reader in understanding the material. A CD-ROM that provides
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Preface ixMATHEMATICA and FORTRAN programs to assist the reader withthe exercises, numerical routines and generating figures accompaniesthe text.Douglas HendersonPeter PlaschkoProvo Utah, USAMexico City DF, MexicoJune, 2006
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CONTENTSPreface viiIntroduction xvGlossary xxi1. Stochastic Variables and Stochastic Processes 11.1. Probability Theory 11.2. Averages 41.3. Stochastic Processes, the Kolmogorov Criterionand Martingales 91.4. The Gaussian Distribution and Limit Theorems 141.4.1. The central limit theorem 161.4.2. The law of the iterated logarithm 171.5. Transformation of Stochastic Variables 171.6. The Markov Property 191.6.1. Stationary Markov processes 201.7. The Brownian Motion 211.8. Stochastic Integrals 281.9. The Ito Formula 381.9. The Ito Formula 38Appendix 45Exercises 492. Stochastic Differential Equations 552.1. One-Dimensional Equations 562.1.1. Growth of populations 562.1.2. Stratonovich equations 58
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xii Stochastic Differential Equations in Science and Engineering2.1.3. The problem of Ornstein-Uhlenbeck andthe Maxwell distribution 592.1.4. The reduction method 632.1.5. Verification of solutions 652.2. White and Colored Noise, Spectra 672.3. The Stochastic Pendulum 702.3.1. Stochastic excitation 722.3.2. Stochastic damping (/? = 7 = 0; a ^ 0) 732.4. The General Linear SDE 762.5. A Class of Nonlinear SDE 792.6. Existence and Uniqueness of Solutions 84Exercises 873. The Fokker-Planck Equation 913.1. The Master Equation 913.2. The Derivation of the Fokker-Planck Equation 953.3. The Relation Between the Fokker-Planck Equation andOrdinary SDEs 983.4. Solutions to the Fokker-Planck Equation 1043.5. Lyapunov Exponents and Stability 1073.6. Stochastic Bifurcations 1103.6.1. First order SDEs 1103.6.2. Higher order SDEs 112Appendix A. Small Noise Intensities and the Influenceof Randomness Limit Cycles 117Appendix B.l The method of Lyapunov functions 124Appendix B.2 The method of linearization 128Exercises 1304. Advanced Topics 1354.1. Stochastic Partial Differential Equations 1354.2. Stochastic Boundary and Initial Conditions 1414.2.1. A deterministic one-dimensional waveequation 1414.2.2. Stochastic initial conditions 1444.3. Stochastic Eigenvalue Equations 1474.3.1. Introduction 1474.3.2. Mathematical methods 148
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Contents xiii4.3.3. Examples of exactly soluble problems 1524.3.4. Probability laws and moments of the eigenvalues 1564.4. Stochastic Economics 1604.4.1. Introduction 1604.4.2. The Black-Scholes market 162Exercises 1645. Numerical Solutions of OrdinaryStochastic Differential Equations 1675.1. Random Numbers Generators and Applications 1675.1.1. Testing of random numbers 1685.2. The Convergence of Stochastic Sequences 1735.3. The Monte Carlo Integration 1755.4. The Brownian Motion and Simple Algorithms for SDEs 1795.5. The Ito-Taylor Expansion of the Solution of a ID SDE 1815.6. Modified ID Milstein Schemes 1875.7. The Ito-Taylor Expansion for N-dimensional SDEs 1895.8. Higher Order Approximations 1935.9. Strong and Weak Approximations and the Orderof the Approximation 196Exercises 201References 205Fortran Programs 211Index 213
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INTRODUCTIONThe theory of deterministic chaos has enjoyed during the last threedecades 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 systemcan be predicted and reproduced by experiments.Meanwhile, many experiments in natural sciences have broughtabout hard evidence of stochastic effects. The best known exampleis perhaps the Brownian motion where pollen submerged in a fluidexperience collisions with the molecules of the fluid and thus exhibitrandom motions. Other familiar examples come from fluid or plasmadynamic turbulence, optics, motions of ions in crystals, filtering the-ory, the problem of optimal pricing in economics, etc. The study ofstochasticity was initiated in the early years of the 1900s. Einstein[1], Smoluchowsky [2] and Langevin [3] wrote pioneering investiga-tions. This work was later resumed and extended by Ornstein andUhlenbeck [4]. But investigation of stochastic effects in natural sci-ence became more popular only in the last three decades. Meanwhilestudies are undertaken to calculate or at least approximate the effectof stochastic forces on otherwise deterministic oscillators, to investi-gate the stability or the transition to stochastic chaos of the latteroscillator.To motivate the following considerations of stochastic differentialequations (SDE) we introduce a few examples from natural sciences.(a) Pendulum with Stochastic ExcitationsWe study the linearized pendulum motion x(t) subjected to astochastic effect, called white noisex + x = (3£t,XV
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xvi Stochastic Differential Equations in Science and Engineeringwhere ft is an intensity constant, t is the time and £j stands forthe white noise, with a single frequency and constant spectrum. For(3 = 0 we obtain the homogeneous deterministic (non-stochastic) tra-ditional pendulum motion. We can expect that the stochastic effectdisturbs this motion and destroys the periodicity of the motion inthe phase space (x,x). The latter has closed solutions called limitcycles. It is an interesting task to investigate whether the solutionsdisintegrate into scattered points (stochastic chaos). We will coverthis 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 pendu-lum is determined by the deterministic limit (/3 = 0) of the stochasticpendulum equation.(b) Stochastic Growth of PopulationsN(i) is the number of the members of a population at the time t, ais the constant of the deterministic growth and (5 is again a constantcharacterizing the intensity of the white noise. Thus we study thegrowth problem in terms of the linear scenarioThe deterministic limit (/? = 0) of this equation describes the growthof a population living on an unrestricted area with unrestrictedfood supply. Its solution (the number of such a population) growsexponentially. The stochastic effects, or the white noise describes astochastic varying food supply that influences the growth of the pop-ulation. We will consider this problem in the Section 2.1.1 and findagain that the average of the population is given by the deterministiclimit.(c) Diffraction of Optical WavesThe transfer function T(u>); UJ = (u, U2) of a two-dimensional opticaldevice is defined by/oo /-oodx / dyF{x,y)F*{x -wuy- u;2)/N;-OO J —OO/CO /*COdx dyF(x,y)2,-00 J—00
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Introduction xvnwhere F is a complex wave amplitude and F* = cc(F) is its complexconjugate. The parameter N denotes the normalization of |F(x,y)|2and the variables x and y stand for the coordinates of the imageplane. In a simplified treatment, we assume that the wave form isgiven byF = |F|exp(—ikA); |F|,fc = const,where k and A stand for the wave number and the phase of thewaves, respectively. We suppose that the wave emerging from theoptical instrument (e.g. a lens) exhibits a phase with two differentdeviations from a spherical structure A = Ac + Ar with a controlledor deterministic phase Ac(x,y) and a random phase Ar(x,y) thatarises from polishing the optical device or from atmospheric influ-ences. Thus, we obtain•1 POO /"OOT(u>) = — dx dyexp{ifc[A(x-o;i,y-u;2) - A(x,y)}},•••*• J—oo J—oowhere K is used to include the normalization. In simple applicationswe can model the random phase using white noise with a Gaussianprobability density. To evaluate the average of the transfer function(T(ui)) we need to calculate the quantity(exp{ik[AT(x - Ui,y - u2) - Ar(x,y)]}).We will study the Gaussian probability density and complete thetask to determine the average written in the last line in Section 1.3of Chapter 1. An introduction to random effects in optics can befound in ONeill [5].(d) Filtering ProblemsSuppose that we have performed experiments of a stochastic problemsuch as the one in (a) in an interval t € [0, u] and we obtain as resultsay A(v), v = [0, u]. To improve the knowledge about the solution werepeat 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) buta result that includes an error A(i) + noise. The question is nowhow can we filter the noise away? A filter is thus, an instrument to
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xviii Stochastic Differential Equations in Science and Engineeringclean a result and remove the noise that arises during the observa-tion. A typical problem is where a signal with unknown frequencyis transmitted (e.g. by an electronic device) and it suffers duringthe transmission the addition of a noise. If the transmitted signalis stochastic itself (as in the case of music) we need to develop anon-deterministic model for the signal with the aid of a stochasticdifferential equation. To study basic the ideas of filtering problemsthe reader in referred to the book of Stremler [6].(e) Fluidmechanical TurbulenceThis is the perhaps most challenging and most intricate applicationof statistical science. We consider here the continuum dynamics of aflow field influenced by stochastic effects. The latter arise from initialconditions (e.g. at the nozzle of a jet flow, or at the entry region of achannel flow) and/or from background noise (e.g. acoustic waves). Inthe simplest case, the incompressible two-dimensional flows, there arethree characteristic variables (two velocity components and the pres-sure). These variables are governed by the Navier-Stokes equations(NSEs). The latter are a set of three nonlinear partial differentialequations that included a parameter, the Reynolds number R. Theinverse of R is the coefficient of the highest derivatives of the NSEs.Since turbulence occurs at intermediate to high values of the R, thisphenomenon is the rule and not the exception in Fluid Dynamics andit occurs in parameter regions where the NSEs are singular. Nonlin-ear SDEs — such as the NSEs — lead additionally to the problemof the closure, where the equation governing the statistical momentof nth order contains moments of the (n + l)th order.Hopf [7] was the first to try to find a theoretical approach tosolve the problem for the idealized case of isotropic homogenous tur-bulence, a flow configuration that can be approximately realized ingrid flows. Hopf assumed that the turbulence is Gaussian, an assump-tion that facilitates the calculation of higher statistical moments ofthe distribution (see Section 1.3 in Chapter 1). However, later mea-surements showed that the assumption of a Gaussian distributionwas rather unrealistic. Kraichnan [8] studied the problem again in
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Introduction xixthe 60s and 70s with the direct triad interaction theory in the ide-alized configuration of homogeneous isotropic turbulence. However,this rather involved analysis could only be applied to calculate thespectrum of very small eddies where the viscosity dominates the flow.Somewhat more progress has been achieved by the investigation ofRudenko and Chirin [9]. The latter predicted with aid of stochas-tic initial conditions with random phases a broad banded spectraof a nonlinear model equation. During the last two decades therewas the intensive work done to investigate the Burgers equation andthis research is summarized in part by Wojczinsky [10]. The Burgersequation is supposed to be a reasonable one-dimensional model ofthe NSEs. We will give a short account on the work done in [9] inChapter 4.
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GLOSSARYAC almost certainlyBC boundary conditiondBj — dWj — £td£ differential of the Brownian motion(or equivalently Wiener process)cc(a) = a* complex conjugate of aD dimension or dimensionalDF distribution functionDOF degrees of freedomSij Kronecker delta functionS(x) Dirac delta functionEX exercise at the end of a chapterFPE Fokker-Planck equationr(x) gamma functionGD Gaussian distributionGPD Gaussian probability distributionHPP homogeneous Poisson processHn(x) Hermite polynomial of order nIC initial conditionIID identically independently distributed
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xxii Stochastic Differential Equations in Science and EngineeringIFF if and only ifIMSL international mathematical science libraryC Laplace transformM master, as in master equationMCM Monte Carlo methodNSE Navier-Stokes equationNIGD normal inverted GDN(jU, a) normal distribution with i as mean and a as varianceo Stratonovich theoryODE ordinary differential equationPD probability distributionPDE partial differential equationPDF probability distribution functionPSDE partial SDEr Reynolds numberRE random experimentRN random numberRV random variableRe(a) real part of a complex numberR, C sets of real and complex numbers, respectivelyS Prandt numberSF stochastic functionSI stochastic integralSDE stochastic differential equationSLNN strong law of large numbers
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GlossaryTPT transition probability per unit timeWP Wiener processWS Wiener sheetWKB Wentzel, Kramers, BrillouinWRT with respect toW(t) Wiener white (single frequency) noise(a) average of a stochastic variable aa2= (a2) — (a) (a) variance{xy),{x,uy,v) conditional averagess At minimum of s and tV for all values of€ element off f(x)dx short hand for J^ f(x)dxX end of an example• end of definition$ end of theorem
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CHAPTER 1STOCHASTIC VARIABLES ANDSTOCHASTIC PROCESSES1.1. Probability TheoryAn experiment (or a trial of some process) is performed whoseoutcome (results) is uncertain: it depends on chance. A collec-tion of all possible elementary (or individual) outcomes is calledthe sample space (or phase space, or range) and is denotedby f2. If the experiment is tossing a pair of distinguishable dice,then 0, = {(i,j) | 1 < i,j < 6}. For the case of an exper-iment with a fluctuating pressure 0, is the set of all real func-tions fi = (0, oo). An observable event A is a subset of f2; thisis written in the form A c f2. In the dice example we couldchoose an even, for example, as A = {{i,j) i + J = 4}. For thecase of fluctuating pressures we could use the subset A = (po >0,oo).Not every subset of £1 is observable (or interesting). An exampleof a non-observable event appears when a pair of dice are tossed andonly their spots are counted, fi = {(i,j),2 < i + j < 12}. Thenelementary outcomes like (1, 2), (2, 1) or (3, 1), (2, 2), (1, 3) are notdistinguished.Let r be the set of observable events for one single experiment.Then F must include the certain event of CI, and the impossibleevent of 0 (the empty set). For every A C T, Acthe complement ofA, satisfies AcC T and for every B C F the union and intersectionof events, A U B and A D B, must pertain also to F. F is calledan algebra of events. In many cases there are countable unions andintersections in F. Then it is sufficient to assume thatoo(J An e r, if An e r.1
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2 Stochastic Differential Equations in Science and EngineeringAn algebra with this property is called a sigma algebra. In measuretheory, 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 with0 < Pr(A) < 1, Pr(0) = 0, Pr(fi) = 1,is called the probability and the triple (I f2, Pr) is referred to as theprobability space. The set function Pr assigns to every event Athe real number Pr(A). The rules for this set function are along withthe formula abovePr(Ac) = l - P r ( A ) ;Pr(A)<Pr(B); Pr(BA) = Pr(B) - Pr(A) for A C B € T.The probability measure Pr(r) on Q, is thus a function Pr(P) —>•[0,1] and it is generally derived with Lebesque integrations that aredefined on Borel sets.We introduced this formal concept because it can be used as themost general way to introduce axiomatically the probability theory(see e.g. Chung, [1.1]). We will not follow this procedure but we willintroduce heuristically stochastic variables and their probabilities.Definition 1.1. (Stochastic variables)A random (or stochastic) variable ~X.(u),u £ Q is a real valuedfunction defined on the sample space Q. In the following we omit theparameter u) whenever no confusion is possible. •Definition 1.2. (Probability of an event)The probability of an event equals the number of elementary out-comes divided by the total number of all elementary outcomes, pro-vided that all cases are equally likely. •ExampleFor the case of a discrete sample space with a finite number of ele-mentary outcome we have, fi = {wi,... ,u>n} and an event is givenby A = {LO, ... ,u>k}, I < k < n. The probability of the event A isthen Pr(A) = k/n. *
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Stochastic Variables and Stochastic Processes 3Definition 1.3. (Probability distribution function and probabilitydensity)In the continuous case, the probability distribution function(PDF) Fx(a;) of a vectorial stochastic variable X = (Xi,...,Xn )is defined by the monotonically increasing real functionFx(xi,...,xn) = Pr(Xi < xi,...,Xn < xn), (1.1)where we used the convention that the variable itself is writtenin upper case letters, whereas the actual values that this variableassumes are denoted by lower case letters.The probability density px(^i, • • • ,xn) (PD) of the randomvariable is then defined byFx(xi,...,xn) = ••• px (ui,...,-un )dn1 ---dun (1.2)and this leads todnFxdxi...dXn =!*(*!,...,*„). (1-3)Note that we can express (1.1) and (1.2) alternatively if we putPr(xn < Xi < X12,..., xnl < Xn < xn2)fX12 fXn2•••px(xi,...,xn)dxi •••dxn. (1.1a)rxi2 rxn-.JXn Jx„,The conditions to be imposed on the PD are given by the positivenessand the normalization conditionPxOci, ,xn)>0] / ••• / px(xi,...,xn)dxi •••dxn = 1. (1.4)In the latter equation we used the convention that integrals withoutexplicitly given limits refer to integrals extending from the lowerboundary — oo to the upper boundary oo. •In a continuous phase space the PD may contain Dirac deltafunctionsp(x) = Y^l(k)s(x- k) + P(x); q(k) = Pr(x = k), (1.5)
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4 Stochastic Differential Equations in Science and Engineeringwhere q(k) represents the probability that the variable x of the dis-crete set equals the integer value k. We also dropped the index X inthe latter formula. We can interpret it to correspond to a PD of a setof discrete states of probabilities q(fc) that are embedded in a con-tinuous phase space S. The normalization condition (1.4) yields now^2<ik+ p(x)dx = 1.1. J SExamples (discrete Bernoulli and Poisson distributions)First we consider the Bernoulli distribution(i) qf)= Pr(a; = fe) = 6(A:,n,p)=r™)pf c(l-p)l-f c; A; = 0,1,...and then we introduce the Poisson distribution(ii)7rfc(A0 = Pr(x = A; )=( A t ) f c eg) (-A t ); * = 0,1,....In the appendix of this chapter we will give more details about thePoisson distribution. We derive there the Poisson distribution as limitof Bernoulli distributionTTk(Xt) — lim b(k,n,p = Xt/n). *n—>ooIn the following we will consider in almost all cases only contin-uous sets.1.2. AveragesThe sample space and the PD define together completely a stochas-tic variable. To introduce observable quantities we consider now aver-ages. The expectation value (or the average, or the mean value)of a function G(xi,...,xn ) of the stochastic variables x,...,xn isdenned by(G(xi,...,xn)) = ••• G(zi,...,£n )px(xi,...,xn )dxi--dxn .(1.6)In the case of a discrete variable we must replace to integral in(1.6) by a summation. We obtain then with the use of (1.5) for p(x)<G(xi,..., xn)) = Y^ Yl G(fci> • • • M # i r • •, kn)- (1-7)
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Stochastic Variables and Stochastic Processes 5There are two rules for the application of the averages:(i) a and b are two deterministic constants and G(x,...xn)and H(xi,...,xn ) are two functions of the random variablesx,..., xn. Then we have(aG(xi,...,xn) + bK(xi,...,xn))= a(G(xi,..., xn)) + 6(H(xi,..., xn)), (1.8a)and(ii){(G{x1,...,xn))) = (G(x1,...,xn)). (1.8b)Now we consider two scalar random variables x and y, their jointPD is p(x,y). If we do not have more information (observed values)of y, we introduce the two marginal PDs px(x) and py(y) of thesingle variables x and yPx(ar) = / p{x,y)dy; pY(y) = / p(x,y)dx, (1.9a)where we integrate over the phase spaces S^ (Sy) of the variablesx(y). The normalization condition (1.4) yields/ px(x)dx = / pY(y)dy = 1. (1.9b)Definition 1.4. (Independence of variables)We consider n random variables x,..., xn, x to be independent ofthe other variables X2, - - -, xn if(xiX2 • • • xn) = (xi)(x2---xn). (1.10a)We see easily that a sufficient condition to satisfy (1.10a) isp(xu...,xn) = pi(xi)pn_i(x2,...,a;n), (1.10b)where p^(...), k < n denotes the marginal probability distribution ofthe corresponding variables. •
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6 Stochastic Differential Equations in Science and EngineeringThe moments of a PD of a scalar variable x are given by<*"> = /•><**•<* " e N-where n denotes the order of the moment. The first order moment(x) is the average of x and we introduce the variance a2bya2= ((x - {x))2} = (x2) - (re)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 thecharacteristic functionG(k,..., kn) = (ex.p(ikrxr)}p(xi,..., xn) ex.Y>(ikrxr)dx • • • dxn, (1-12)where we applied a summation convention krxr = ^?=i kjxj- Thisfunction has the properties G(0,..., 0)1; | G(ki,..., kn) < 1.ExampleThe Gaussian (or normal) PD of a scalar variable x is given byp(x) = (2vr)"1/2exp(-a;2/2); -co < x < oo. (1.13a)Hence we obtain (see also EX 1.1)<*2n> = | ? 7 ; "2= i; (^2n+1) = o. (l.isb)Li litA stochastic variable characterized by N(m, s) is a normal dis-tributed variable with the average m and the variance s. The vari-able x distributed with the PD (1.13a) is thus called a normaldistributed variable with N(0, 1).
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Stochastic Variables and Stochastic Processes 7A Taylor expansion of the characteristic function G(k) of (1.13a)yields with (1.12)G(*) = E ^ V > . (L14a)n=0 U-We define the cumulants nm by mlnG(fc) = E^f-Km- (1.14b)A comparison of equal powers of k givesKi = (x); K2 = (x2) - (x)2= a2;K3 = {X3)-3(X2)(X)+2{X)3;....(1.14c)*Definition 1.5. (Conditional probability)We assume that A, B C T are two random events of the set ofobservable events V. The conditional probability of A given B(or knowing B, or under the hypothesis of B) is defined byPr(A | B) = Pr(A n B)/Pr(B); Pr(B) > 0.Thus only events that occur simultaneously in A and B contributeto the conditional probability.Now we consider n random variables x,... ,xn with the jointPD pn (xi,..., xn). We select a subset of variables x,..., xs. and wedefine a conditional PD of the latter variables, knowing the remainingsubset xs+i,... ,xn, in the formPs|n—sxli • • • ixs I Xs--, . . . , Xn)= pn(xi, . . . , Xn)/pn-s(xs+i, . . . , Xn). (1.15)Equation (1.15) is called Bayess rule and we use the marginal PDpn-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 Bayess rule (1.15) in the formP n l ^ l j • • • j Xn) = pn—syXs-^i, . . . , 3^nJPs|n—s v^l> • • • > xs Xs--lj • • • , XnJ.(1.15)
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8 Stochastic Differential Equations in Science and EngineeringWe can also rearrange (1.15) and we obtainP n ^ l i • • • > -En) =Ps(.-El> • • • ) •KsjPn—ss%s+1: • • • j %n Xi, . . . ,XS).(1.15")•Definition 1.6. (Conditional averages)The conditional average of the random variable x, knowingx2, • • •, xn, is defined by(Xi | X2, . • • , Xn) = / ZlPi|n _i(xi 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)DExampleWe consider a scalar stochastic variable x with its PD p(a;). An eventA is given by a; £ [a, 6]. Hence we havep(x | A) = 0 Vz^ [a, b],andp(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 JaFor 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) = / xexp(—Ax)ds / / exp(—Xx)dx = a + 1/A.JO / ./a JL
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Stochastic Variables and Stochastic Processes 91.3. Stochastic Processes, the Kolmogorov Criterionand MartingalesIn many applications (e.g. in irregular phenomena like blood flow,capital investment, or motions of molecules, etc.) one encountersa family of random variables that depend on continuous or dis-crete 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 andX(t7ui) £ Rn, as a stochastic process (random process or stochas-tic (random) function). If I is a discrete set it is more convenientto call X(t,u>) a time series and to use the phrase process only forcontinuous 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 arandom variable and for every fixed value of LO £ Q (hence for everyobservation) X(t, LO) is a real valued function. Any observation of thisprocess is called a sample function (realization, trajectory, path ororbit) of the process.We consider now a finite variate PD of a process and we definethe time dependent probability density functions (PDF) in analogyto (1.1) in the formFx (x,t) = Pr(X(t)<x);Fx.yfo t; y, s) = Pr(X(t) < x, Y(s) < y); ^1A^Fxu...,xn(xiiti---;xn,tn) = Pr(Xx(t) < xi,Xn(t) < xn),where we omit the dependence of the process X(t) on the chancevariable LO, whenever no confusion is possible. The system of PDFssatisfies two classes of conditions:(i) SymmetryIf {ki,..., kn} is a permutation of 1,..., n then we obtainFxlv..,xn (zfc! ,tkl;...;xkn,tkJ = FXl,...,x„ {x, h;...; xn, tn).(1.19a)(ii) CompatibilityFx1?...,x„ (xi,ti;...; xr, tr; oo, tr+i; ...;oo,tn)= FXl,...,xr(a;i, <i; •••xr, tr). (1.19b)
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10 Stochastic Differential Equations in Science and EngineeringThe rules to calculate averages are still given by (1.6) where thecorresponding PD is derived by (1.3) and where the PDFs of (1.19)are usedQnp(xi, ii; ...;xn, tn) = -—— —7r^FXll...,x„ (xi,h;...; xn, tn).dxi(ti) • • • dxn(tn)One would expect that a stochastic process at a high rate ofirregularity (expressed e.g. by high values of intensity constants, seeChapter 2) would exhibit sample functions (SF) with a high degreeof irregularity like jumps ore singularities. However, Kolmogorovscriterion gives a condition for continuous SF:Theorem 1.1. (Kolmogorovs criterion)A bivariate distribution is necessary to give information about thepossibility of continuous SF. If and only if (IFF)(|Xi(ti)-X2 (t2 )r> < c | i i - i 2 | 1 + b; a,6,c>0; tx,t2 G [t0,T],(1.20)then the stochastic process X(t) posses almost certainly (AC, thissymbol is discussed in Chapter 5) continuous SF. However, the lat-ter are nowhere differentiable and exhibit jumps, and higher orderderivatives singularities. &We will use later the Kolmogorovs criterion to investigate SF ofBrownian motions and of stochastic integrals.Definition 1.7. (Stationary process)A process x(t) is stationary if its PD is independent of a time shift rp(xi,h +T;...;xn,tn + T) = p(zi, tx;... ;xn,tn). (1.21a)Equation (1.21a) implies that all moments are also independent ofthe time shift(x(h + T)x(t2 + T) • • • x(tk + T))= (x(t1)x(t2)---x(tk)); forfc = l , 2 . . . . (1.21b)A consequence of (1.25a) is given by(x(t)) = (x), independent of t:(1.21c)(x(t)x(t + r)) = (x(0)x(r))=5 (r).•
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Stochastic Variables and Stochastic Processes 11The correlation matrix is defined byCik = (zi{h)zk(t2)); Zi(ti) = Xi(ti) - (xi(ti)). (1.22)Thus, we havecik = {xi(h)xk(t2)) - {Xi(ti))(xk(t2)). (1.23)The diagonal elements of this matrix are called autocorrelationfunctions (we do not employ a summation convention)Cii = {Zi{tl)Zi{t2)).The nondiagonal elements are referred to as cross-correlationfunctions. The correlation coefficient (the nondimensionalcorrelation) is defined byr. = (xj{ti)xk{t2)) - (xi(ti))(xk(t2)) ,x 2 4 )y/{xUh)) ~ (Xt(h))^(xl(t2)) - (Xk(t2)/For stationary processes we haveCik(h,t2) = (zi(0)zk(t2 - h)) = cik(t2 - h);(1.25)Cki(h,t2) = (zkit^Zifo)) = (zk(ti -t2)zi(0)) = Cik(ti -t2).A stochastic function with C{k = 0 is called an uncorrelatedfunction and we obtain(xl{h)xk(t2)) = (Xiihfiixkfa)). (1.26)Note that the condition of noncorrelation (1.26) is weaker than thecondition of statistical independence.ExampleWe consider the process X(i) = Ui cos t + l^sini. Ui,2 are inde-pendent stochastic variables independent of the time. The momentsof the latter are given by (Uk) = 0, (U|) — a = const; k — 1,2,(U1U2) = 0. Hence we obtain (X) — 0;cxx(s,t) = acos(t — s). JI»Remark (Statistical mechanics and stochastic differential equations)In Chapter 2 we will see that stochastic differential equations or"stochastic mechanics" can be used to investigate a single mechani-cal system in the presence of stochastic influences (white or colored
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12 Stochastic Differential Equations in Science and Engineeringnoise). We use concepts that are similar to those developed in statis-tical mechanics such as probability distribution functions, moments,Markov properties, ergodicity, etc. We solve the stochastic differen-tial equation (analytically, but in most cases numerically) and onesolution represents a realization of the system. Repeating the solu-tion process we obtain another realization and in this way we areable to calculate the moments of the system. An alternative way tocalculate the moments would be to solve the Fokker-Planck equation(see: Chapter 3) and then use the corresponding solution to deter-mine the moments. To establish the Fokker-Planck equation we willuse again the coefficients of the stochastic differential equation.Statistical Mechanics works with the use of ensemble averages.Rather than defining a single quantity (e.g. a particle) with a PDp(x), one introduces a fictitious set of an arbitrary large number ofM quantities (e.g. particles or thermodynamic systems) and these Mnon-interacting quantities define the ensemble. In case of interact-ing particles, the ensemble is made up by M different realizationsof the N particles. In general, these quantities have different charac-teristic values (temperature, or energy, or values of N) x, in a com-mon range. The number of quantities having a characteristic valuebetween x and x + dx defines the PD. Therefore, the PD is replacedby density function for a large number of samples. One observes alarge number of quantities and averages the results. Since, by defini-tion, the quantities do not interact one obtains in this way a physicalrealization of the ensemble. The averages calculated with this den-sity function are referred to as ensemble averages and a system whereensemble averages equal time averages is called an ergodic system.In stochastic mechanics we say that a process with the property thatthe averages defined in accordance with (1.6) equal the time averages,represents an ergodic process.An other stochastic process that posses SF of some regularity iscalled a martingale. This name is related to "fair games" and wegive a discussion of this expression in a moment.In everyday language, we can state that the best prediction ofa martingale process X(t) conditional on the path of all Brownian
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Stochastic Variables and Stochastic Processes 13motions up to s < t is given by previous value X(s). To make thisidea precise we formulate the following theorem:Theorem 1.2. (Adapted process)We consider a probability space (r, Q, Pr) with an increasing family(of sigma algebras of T) of events Ts £ Tt, 0 < s < t (see Section 1.1).A process X.(s,u);u) € Q,s £ [0, oo) is called Ts-adapted if it is Im-measurable. An rs-adapted process can be expanded into a (thelimit) of a sequence of Brownian motions Bu(u>) with u < s (but notu> s). ^ExampleFor n = 2, 3,... ; 0 < A < t we see that the processes(i) G1(t,co) = Bt/n(co), G2(t,uj) = Bt_x(u;),(ii) G3(t,Lj) = Bnt(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 thecondition<Xt |Ta) = 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 obtaina super (sub) martingale. We note that martingales have no otherdiscontinuities than at worst finite jumps (see Arnold [1.2]). ^Note that (1.27) defines a stochastic process. Its expectation((Xj | Ts)) = (Xs);s < t is a deterministic function.An interesting property of a martingale is expressed byPr(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
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14 Stochastic Differential Equations in Science and Engineeringwill be derived in EX 1.2. We apply later the concept of martingalesto Wiener processes and to stochastic integrals.Finally we give an explanation of the phrase "martingale". Agambler is involved in a fair game and he has at the start the capitalX(s). Then he should posses in the mean at the instant t > s theoriginal capital X(s). This is expressed in terms of the conditionalmean value (Xt | Xs) = Xs . Etymologically, this term comes fromFrench and means a system of betting which seeks the amount to bewagered after each win or loss.1.4. The Gaussian Distribution and Limit TheoremsIn relation (1.13) we have already introduced a special case of theGaussian (normal distributed) PD (GD) for a scalar variable. A gen-eralization of (1.13) is given by theN(m,o-2) PDp(x) = (2TTCT2)-1/2exp[-(x - m)2/(2a2)]; V i e [-oo, oo] (1.29)where m is the average and <72= (a;2) — m2is the variance. The mul-tivariate form of the Gaussian PD for the set of variables xi,...,xnhas the formp(xi,...,xn) = N e x p f --AikXiXk -bkxkj , (1.30a)where we use a summation convention. The normalization constantN is given byN = (27r)-"/2[Det(A)]1/2eX pf-^A-16i 6f e ) . (1.30b)We define the characteristic function of (1.30) has the formG(ki,...,kn) = expAn expansion of (1.31) WRT powers of k yields the moments(Xi) = -A^bfc, (1.32a)and the covariance is given byCik = {{xi - {xi)){xk - (xk)) = A^1. (1.32b)-l•uv (1.31)
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Stochastic Variables and Stochastic Processes 15This indicates that the GD is completely given, if the mean value andthe covariance matrix are evaluated. The n variables are uncorrelatedand thus are independent if A - 1and hence A itself are diagonal.The higher moments of n-variate GD with zero mean are partic-ularly easy to calculate. To show this, we recall that for zero meanwe have bk = 0 and we obtain the characteristic function with theuse of (1.31) and (1.32) in form ofG = exp XUXV)KUKV1 -p ~ XUXV)ZUZV -- XuXvj XpXq]ZuZvZpZq -pA/y ( ( I f f . /u,v,p,q,r = 1,2,.... (1.33)A comparison of equal powers of z in (1.33) and in a Taylorexpansion of the exponent in (1.31) shows that all odd momentsvanishXaXbXcf — X aXl)X CX dX ej = • • • U.We also obtain with restriction to n — 2 (bivariate GD)( 4 ) = 3 ( 4 ) 2; (xlxp} = 3(xl)(x1x2), i,p= 1,2;(1.34){xlx22) = {x){xl) + 2{xlx2)2.In the case of a trivariate PD we face additional terms of the type(xkXpXr) — 2(xkXp){xpXr) + {xkXr){Xp). The higher order variate andhigher order moments can be calculated in analogy to the results(1.34).We give also the explicit formula of the bivariate Gaussian (seealso EX 1.3)withP(x,y)1N2expx- (x),e2 ( 1 - r 2) [av = y- (y),2T-£?7 rfVab b2Tr^ab(l - r2); a2x a;(1.35a)(1.35b)and where r = vi is defined as the cross correlation coefficient (1.24).For ax = ay = 1 and (x) = (y) = 0 in (1.35) we can expand the latter
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16 Stochastic Differential Equations in Science and Engineeringformula and we obtainp(x,y) = (27r)~1exp[-(x2+ y2)/2} £ -Hfc(x)Hfc(y), (1.36)A;!"fc=0where Hfc(x) is the fc-th order Hermite polynomial (see Abramowitzand Stegun [1.3]). Equation (1.36) is the basis of the "Hermitian-chaos" expansion in the theory of stochastic partial differentialequations.In EX 1.3 we show that conditional probabilities of the GD(1.35a) are Gaussian themselves.Now we consider two limit theorems. The first of them is relatedto GD and we introduce the second one for later use.1.4.1. The central limit theoremWe consider the random variableu = n{xk) = 0, (1.37)where x^ are identically independent distributed (IID) (but not nec-essarily normal) variables with zero mean and variance a2= {x2,).We find easily (U) = 0 and (U2) = a2.The central limit theorem says that U tends in the limitn —> oo to a N(0, a2) variable with a PD given by (1.13a). To provethis we use the independence of the variables Xk and we perform thecalculation of the characteristic function of the variable U with theaid of (1.12)Gu(fc) = / dxip(xi) • • • / dxnp(xn) • • • exp [ik(xi - h xn)/y/n= [Gx(A;/v^)]n2^2kla~2n~+ 0(n -3/2.exp(—k a 12) for n —> oo.(1.38)We introduced in the second line of (1.38) the characteristic functionof one of the individual random functions according to (1.14a); (1.38)is the characteristic function of a GD that corresponds indeed to
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Stochastic Variables and Stochastic Processes 17N(0, a2). Note that this result is independent of the particular formof the individual PDs p(x). It is only required that p(a;) has finitemoments. The central limit theorem explains why the Gaussian PDplays a prominent role in probability and stochastics.1.4.2. The law of the iterated logarithmWe give here only this theorem and refer the reader for its derivationto the book Chow and Teichler [1.4]. yn is the partial sum of n IIDvariablesyn = xx - -xn; (xn) = /3, {(xn - (if) = a2. (1.39)The theorem of the iterated logarithm states that there exists AC anasymptotic limit-a < lim / n~ r a / ?< a. (1.40)rwoo v/2nln[ln(n)]Equation (1.40) is particular valuable in case of estimatesof stochastic functions and we will use it later to investigateBrownian motions. We will give a numerical verification of (1.40)in program F18.1.5. Transformation of Stochastic VariablesWe consider transformations of an n-dimensional set of stochasticvariables x,... ,xn with the PD pxi-xn (x, • • •, xn). First we intro-duce the PD of a linear combination of random variablesnZ =fe=lwhere the a^ are deterministic constants. The PD of the stochasticvariable z is then defined byPz(z) = / dxi ••• / dxn8 I z - Y^akXk 1 PXi-x„(zi,--- ,xn).(1.41b)Now we investigate transformations of the stochastic variablesxi,..., xn. The new variables are defined byuk = uk(x1,...,xn), k = l,...,n. (1.42)^2otkxk, (1.41a)
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18 Stochastic Differential Equations in Science and EngineeringThe inversion of this transformation and the Jacobian areXk = gk(ui,...,un), J = d(x1,...,xi)/d(u1,...,u1). (1.43)We infer from an expansion of the probability measure (1.1a) thatdpx!-xn = Pr(zi < Xi < xi + dxi,..., xn < Xn < xn + dxn)= pXl...xn(^i, • • •, xn)dxi • • • dxnfor dxk —> 0, k — 1,... ,n. (1.44a)Equation (1.44a) represents the elementary probability measure thatthe variables are located in the hyper planenY[[xk,xk + dxk}.k=lThe principle of invariant elementary probability measurestates that this measure is invariant under transformations of thecoordinate system. Thus, we obtain the transformationdp^...^ =dpXi...Xn. (1.44b)This yields the transformation rule for the PDsPUi-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 Box-Miller to generate a GD. There are two stochastic variables givenin an elementary cube, , (I V 0 < x 1 < l , 0 < x 2 < l > n ^P ( X 1 X 2 ) =U elsewhere J ( L 4 6)Note that the bivariate PD is already normalized. Now we introducethe new variablesyi = y/—2 In x cos(27TX2),(1.47)y2 = V -21nxisin(27TX2).The inversion of (1.47) isxi = exp[-(yj + yl)/2] x2 = —arc tan(y2/yi).
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Stochastic Variables and Stochastic Processes 19According to (1-45) we obtain the new bivariate PDp(y1,y2) = p ( x 1 ^ 2 ) | | ^ | = i - e x P [ - ( y ? + y22)/2], (1.48)and this the PD of two independent N(0, 1) variables.Until now we have only covered stochastic variables that are time-independent or stochastic processes for the case that all variablesbelong to the same instant. In the next section we discuss a propertythat is rather typical for stochastic processes. Jf»1.6. The Markov PropertyA process is called a Markov (or Markovian) process if the condi-tional PD at a given time tn depends only on the immediately priortime tn-. This means that for t < t2 < • • • < tnPln-l(yn,tn | 2/1, h . . . ; y n - l , * n - l ) = Pl|l(?/n,£n I 2/n-l>*7i-l)>(1.49)and the quantity Pii(yn,tn yn-i,tn-i) is referred to as transitionprobability distribution (TPD).A Markov process is thus completely defined if we know the twofunctionsPi(yi,*i) and p2(y2,t2 | yi,ti) forti<t2.Thus, we obtain for t < t2 (see (1.15") and note that we use asemicolon to separate coordinates that belong to different instants)V2{yiMV2,t2) =pi(yi,*i)Pi|i(y2,*21 yi,*i), (1.50.1)and for t < t2 < £3P3(yi,*i; 2/2, £2; 2/3,^3)= Pi(yi,*i)pi|i(y2,*21 yi,<i)pi|i(y3,*31 y2,t2). (1.50.2)We integrate equation (1.50.2) over the variable y2 and we obtainP2(yi,*i;y3,*3) =pi(yi,h) / Pi|i(y2,*21 2/1, £I)PI 11(2/3^312/2,t2)dy2.(1.51)
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20 Stochastic Differential Equations in Science and EngineeringNow we usePi|i(2/3,*31 yi,*i) = P2(yi,*i;y3,i3)M(yi,*i),and we obtain from (1.51) the Chapman-Kolmogorov equationPi|i(z/3,*31 yi,h) = / Pi|i(y2,*21 yi,*i)pi|i(y3,*31 V2,t2)dy2.(1.52)It is easy to verify that a particular solution of (1.52) is given byPi|l(2/2,*2 I 2/1, *i) = [27r{t2-t1)}~1/2exV{-{y2-y1)2/[2{t2-t1)}}.(1.53)We give in EX 1.4 hints how to verify (1.53).We can also integrate the identity (1.50.1) over y and we obtainViiViM) = I Pi(z/i,*i)Pi|i(l/2,*2 I J/i,*i)dj/i. (1-54)The latter relation is an integral equation for the function pi(?/2, t2).EX 1.5 gives hints to show that the solution to (1.54) is theGaussian PDP l (y, t) = (27rt)-V2 exp[-j/7(2i)]; lim P l (y, t) = 8(y). (1.55)t—>U-|-In Chapter 3 we use the Chapman-Kolmogorov equation (1.52)to derive the master equation that is in turn applied to deduce theFokker-Planck equation.1.6.1. Stationary Markov processesStationary Markovian processes are defined by a PD and transi-tion probabilities that depend only on the time differences. The mostimportant example is the Ornstein—Uhlenbeck-process that wewill treat in Section 2.1.3 and 3.4. There we will prove the formulasfor its PDpi(y) = (2TT)-1/2 exp(-y2/2), (1-56.1)
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Stochastic Variables and Stochastic Processes 21and the transition probabilityPi|i(w,«2 I l/i, ti) = [2TT(1 - u 2) ] - 1" ^ (?/2~ " ^1 yJ (1.56.2)u = exp(-r); Pi|i(y2,*i I yi,*i) = <%2 - y{]The Ornstein-Uhlenbeck-process is thus stationary, Gaussian andMarkovian. A theorem from Doob [1.5] states that this is apart fromtrivial process, where all variables are independent — the only pro-cess that satisfies all the three properties listed above. We continueto consider stationary Markov processes in Section 3.1.1.7. The Brownian MotionBrown discovered in year 1828 that pollen submerged in fluids showunder collisions with fluid molecules, a completely irregular move-ment. This process is labeled with y := Bt(ui), where the subscriptis the time. It is also called a Wiener (white noise) process andlabeled with the symbol Wj (WP) that is identical to the Brownianmotion: Wt = Bt. The WP is a Gaussian [it has the PD (1.55)] anda Markov process.Note also that the PD of the Wiener process (WP) — givenby (1.55) — satisfies a parabolic partial differential equation (calledFokker—Planck equation, see Section 3.2)dp ld2pft =2 ft?" (L57)We calculate the characteristic function G(u) and we obtainaccording to (1.12)G{u) = (exp(mWt)) = exp(-n2t/2), (1.58a)and we obtain the moments in accordance with (1.13b)< w ? f c ) =2Wf f e ;(w?fc+1) = °; fcGN°- (L58b)We use the Markovian properties now to prove the independenceof Brownian increments. The latter are definedyi,y2-yi,---,yn-yn-i with yk := wtk; h<---<tn. (1.59)
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22 Stochastic Differential Equations in Science and EngineeringWe calculate explicitly the joint distribution given by (1.50) andwe obtain with the use (1.53) and (1.55)P2(yuh;y2,t2) = [(27r)2ti(t2 -t1 )]-1/2exp{-y2/(2t1 )- ( y 2 - y i ) 2/ [ 2 ( t 2 - i i ) ] } , (1-60)andP3(yi,*i; 2/2,^2; 2/3, *3)= [(27r)3*i(f2 - h)(t3 - t 2 ) ] - 1/ 2 e x p { _ y 2 / ( 2 i i )- (2/2 - yi)2/[2(t2 - h)] - (2/3 - y2)2/[2(«3 " *2)]},(1.61)P 4 ( y i , t i ; y 2 , * 2 ; y 3 , * 3 ; y 4 , * 4 )= [27r(*4 - * 3 ) r 1 / 2P 3 ( y i , * i ; 2 / 2 , * 2 ; 2 / 3 , * 3 )xexp{-(y4 -y3 )2/[2(i4-i3)]}-We see that the joint PDs of the variables 2/1,2/2 ~~ 2/1,2/3 ~~ 2/2,2/4 ~~ 2/3are given in (1.60) and (1.61) in a factorized form and this impliesthe independence of these variables. To prove the independence ofthe remaining variables y^ — 2/3,..., yn — yn~ we would only have tocontinue the process of constructing joint PDs with the aid of (1.49).In EX 1.6 we prove the following property(2/1^1)2/2^2)) = min(ti, t2) =hA t2. (1.62)Equation (1.62) also demonstrates that the Brownian motion is nota stationary process, since the autocorrelation does not depend onthe time difference r = t2 — t but it depends on t2 A t.To apply Kolmogorovs criterion (1.20) we choose a = 2 and weobtain with (1.58b) and (1.62) ([2/1 (*i) - 2/2to)]2) = 1*2 - *i|- Thuswe can conclude with the choice b = c = 1 that the SF of the WPare ac continuous functions. The two graphs Figures 1(a) and 1(b)are added in this section to indicate the continuous SF.We apply also the law of iterated logarithm to the WP. To thisend we consider the independent increments y^ — y^-i where weifc = kAt with a finite time increment At. This yields for the partialsum in (1.39)n]P(2/fc - 2/fe-i) = 2/n = 2/nAt! OL = (yk) = 0; ((yk - yk-i)2) = At.fc=i
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Stochastic Variables and Stochastic Processes 23Fig. 1(a). The Brownian motion Bt versus the time axis. Included is a graph of thenumerically determined temporal evolution of the mean value and the variance.Bv(t)Fig. 1(b). The planar Brownian motion with x = Bj and y = Bj • B^, k = 1, 2 areindependent Brownian motions.We substitute the results of the last line into (1.40) and we obtain- / A ! < limW,nAtn^°° v/2nln(ln(n))< VAt.
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24 Stochastic Differential Equations in Science and EngineeringThe assignment of t := nAt into the last line and the approximationln(i/Ai) —> ln(i) for t —> oo gives the desired result for the ACasymptotic behavior of the WP- 1 < lim l< 1. (1.63)~ ™^°° ^/2tln(In(t))We will verify (1.63) in Chapter 5 numerically.There are various equivalent definitions of a Wiener process. Weuse the following:Definition 1.8. (Wiener process)A WP has an initial value of Wo = 0 and its increments Wj — Ws,t > s satisfies three conditions. They are(i) independent and (ii) stationary (the PD dependence on t — s)and (iii) N[0, t — s] distributed.As a consequence of these three conditions WP exhibits continu-ous sample functions with probability 1. •There are also WPs that do not start at zero. There is also ageneralization of the WP with discontinuous SF. We will return tothis point at the end of Section 1.7.Now we show that a WP is a martingale<BS | Bu> = Bu; s> u. (1.64)We prove (1.64) with the application of the Markovian property(1.53). We use (1.17) write(Bs | Bu) = (y2,s | yi,u) = / t/2Pi|iO/2,s I yi,u)dy2= /n . I 2/2exp{-(y2 - yi)2/[2{s - u)]}dy2J2lT{S — U) J= yi = Bu.This concludes the proof of (1.64).A WP has also the following properties. The translated quantityWi and the scaled quantity Wf defined byt , a > 0 : W t = W t + a - W a and Wt = 2 ( ) (1.65)
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Stochastic Variables and Stochastic Processes 25are also a Brownian motion. To prove (1.65) we note first that theaverages of both variables are zero (Wt) = (Wt) = 0. Now we haveto show that both variables satisfy also the condition for the autocorrelation. We prove this only for the variable Wt and leave thesecond part for the EX 1.7. Thus, we put(WtWs) = (Ba24Ba2s)/a2= ^ V 2^ =tAs.So far, we considered exclusively scalar WPs. In the study of par-tial differential equations we need to introduce a set of n independentWPs. Thus, we generalize the WP to the case of an independentWPs that define a vector of a stochastic processesxi(h),...,xn(tn); tk>0. (1.66)The corresponding PD is thenp(xu...,xn) = pXl(xi)...pXn(xn)= (27T)""/2I K 1 / 2e x p [-4/(2**)] • (1-67)fc=lWe have assumed independent stochastic variables (like the orthog-onal basic vectors in the case of deterministic variables) and thisindependence is expressed by the factorized multivariate PD (1.67).We define an n-dimensional WP (or a Wiener sheet (WS)) bynMin)= n **(**): t = (t1,...,tn). (1.68)k=Now we find how we can generalize the Definition 1.8 to the case ofn stochastic processes. First, we prove easily that the variable (1.68)has a zero mean(M(tn)) = 0. (1.69)Thus, it remains to calculate the autocorrelation (1.62). We use theindependence of the set of variables Xk(tk),k = l , . . . , n and weobtain with the use of the bivariate PD (1.61) with y = Xk(tk);
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26 Stochastic Differential Equations in Science and EngineeringV2 = xk(sk) and factorize the result for the independent variables.Hence, we obtainn(Mjn)M(")> = Yl(xk(tk)xk(sk)}; t = (h,...,tn); s = (si,...,sn ).fc=iThe evaluation of the last line yields with (1.62)n(M(tn)M^) = HtkAsk. (1.70)fc=iThe relations (1.69) and (1.70) show now that process (1.68) is ann-WP.In analogy to deterministic variables we can now construct withstochastic variables curves, surfaces and hyper surfaces. Thus, a curvein 2-dimensional WS and surfaces on 3-dimensional WS are given byc« = Mt2f(ty stut2 = MS2)g(tl>t2)-We give here only two interesting examples.Example 1Here we put(2)Kt = M ^ ; a = exp(i), b = exp(—£); —oo < x < oo.This defines a stochastic hyperbola with zero mean and with theautocorrelation(KtKs) = (x1{et)x1{es))(x2{e-t)x2{e-s))= (e* A es)(e-* A e~s) = exp(-|t - s). (1.71)The property (1.71) shows this process is not only a WS but also astationary Ornstein-Uhlenbeck process (see Section 1.5.1). 4bExample 2Here we define the processKt = exp[-(l + c)t]M^l; a = exp(2£), b = exp(2ct); c > 0.(1.72)
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Stochastic Variables and Stochastic Processes 27Again we see, that stochastic variable defined (1.72) has zero meanand the calculation of its autocorrelation yields(KtKs) = exp[-(l + c)(t + s)}(xl(e2t)x1(e2s)){x2(e2ct)x2(e2cs))= exp[-(l + c)(t + s)}(e2tA e2s){e2ctA e2cs)= exp[-(l + c)|t-s|]. (1.73)The latter equation means that the process (1.72) is again anOrnstein-Uhlenbeck process. Note also that because of c > 0 thereis no possibility to use (1.73) to reproduce the result of the previousexample. XJust as in the case of one parameter, there exist for WSs alsoscaling and translation. Thus, the stochastic variablesH - —M{2)•v~ ab a2uh2vT _ M ( 2 ) _ M ( 2 ) _ M ( 2 ) M ( 2 )L>u,v - Mu+a,v+b Mu+a,b Ma,v+b ~ Ma,6>(1.74)are also WSs. The proof of (1.74) is left for EX 1.8.We give in Figures 1(a) and 1(b) two graphs of the Brownianmotion.At the end of this section we wish to mention that the WP is asubclass of a Levy process L(t). The latter complies with the firsttwo conditions of the Definition 1.8. However, it does not possessnormal distributed increments. A particular feature of normal dis-tributed process x is the vanishing of the skewness (x3) / (x2)32. How-ever, many statistical phenomena (like hydrodynamic turbulence, themarket values of stocks, etc.) show remarkable values of the skew-ness. This means that a GD (with only two parameter) is not flex-ible enough to describe such phenomena and it must be replace bya PD that contains a sufficient number of parameters. An appropri-ate choice is the normal inverted Gaussian distribution (NIGD) (seeSection 4.4). The NIGD distribution does not satisfy the Kolmogorovcriterion. This means that the sample functions of the Levy pro-cess L(i) is equipped with SF that jump up and down at arbitraryinstances t. To get more information about the Levy process we referthe reader to the work of Ikeda k, Watanabe [1.6] and of Rydberg
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28 Stochastic Differential Equations in Science and Engineering[1.7]. In Section 4.4 we will give a short description of the applicationof the NIGD in economics theories.1.8. Stochastic IntegralsWe need stochastic integrals (SI) when we attempt to solve a stochas-tic differential equation (SDE). Hence we introduce a simple firstorder 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 £tindicates the only stochastic term in this equation. We assume<6> = 0; (tes) = 6(t-s). (1.76)The spectrum of the autocorrelation in (1.76) is constant (seeSection 2.2) and in view of this £t is referred as white noise andany term proportional to £( is called a noisy term. These assump-tions are based on a great variety of physical phenomena that aremet in many experimental situations.Now we replace (1.75) by a discretization and we putAtk = tk+i — tk>0; Xfc = X(ifc);AXfc = Xfc+1 -Xf c ; A; = 0,1,The substitution into (1.75) yieldsAXfc = a(Xfc, tk) Atk + b{Xk,tk) ABk;ABfc = Bf c + 1 -Bf c ; A; = 1,2,...where we usedA precise derivation of (1.77) is given in Section 2.2. Thus we canwrite (1.75) in terms ofn - lXn = X0 + Y^ [<xs,ts)Ats + b(Xa, ts)ABs] • X0 = X(t0).s=Q(1.78)
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Stochastic Variables and Stochastic Processes 29What 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 + " / 6(X(s),s)dB8". (1.79)Jo JoThe first integral in (1.79) is a conventional integral of Riemannstype and we put the stochastic (noisy) integral into inverted commas.The irregularity of the noise does not allow to calculate the stochasticintegral in terms of a Riemann integral. This is caused by the factthat the paths of the WP are nowhere differentiable. Thus we findthat a SI depends crucially on the decomposition of the integrationinterval.We assumed in (1.75) to (1.79) that b(X,t) is a deterministicfunction. We generalize the problem of the calculation of a SI andwe consider a stochastic function1= / i(w,s)dBs. (1.80)JoWe recall that Riemann integrals of the type (g(s) is a differen-tiable function)i(s)dg(s) = [ f(s)g(s)ds,Jo0are discretized in the following mannerpT n—1/ f(s)dg(s) = lim Vf(sfc)[g(sfc+i)-g(sfc)].JUk=0Thus, it is plausible to introduce a discretization of (1.80) thattakes the formI = 53f(afc,a;)(Bfc+1-Bfc). (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 thediscretization interval and we say that this decomposition does not
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30 Stochastic Differential Equations in Science and Engineeringlook into the future. We call this type of integral an Ito integraland writeI i = / {{s,uj)dBs. (1.82)JoAn other possible choice is to use the midpoint of the intervaland with this we obtain the Stratonovich integralIs = / f(s,w)odBs = ^f(sfc,w)(Bfc+i -Bfc); sk = -(tk+1 + tk).J° k(1.83)Note that the symbol "o" between integrand and the stochastic dif-ferential is used to indicate Stratonovich integrals.There are, of course, an uncountable infinity of other decomposi-tions of the integration interval that yield to different definitions ofa SI. It is, however, convenient to take advantage only of the Ito andthe Stratonovich integral. We will discuss their properties and findout which type of integrals seems to be more appropriate for the usein 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 +/JI2; h = [ f*(s,w)dBs.Ja J a(1.84)Note that (1.84) remains also valid for Stratonovich integrals. Theproof of (1.84) is trivial.In the following we give non-trivial properties that apply, how-ever, exclusively to Ito integrals. Now we need a definition:Definition 1.9. (non-anticipative or adapted functions)The function f(t, Bs) is said to be non-anticipative (or adapted,see also Theorem 1.2) if it depends only on a stochastic variableof the past: Bs appears only for arguments s < t. Examples for anon-anticipative functions arei(s,co)= [Sg(u)dBu; f(Jos,u) = B,. •
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Stochastic Variables and Stochastic Processes 31Now we list further properties of the Ito integrals that include nonanticipative functions f(s,Bs) and g(s,Bs).(b) M l E EW f(sBs)d Bs)=°- (L85)Proof.We use (1.81) and obtainMi = /^f(sf c ,Bf c )(Bf c + 1 -Bf c )But we know that Bk is independent of B^+i — Bk. The functionf(sfc,Bfc) is thus also independent of B^+i — B^. Hence we obtainMi = ^2(f(sk,Bk)){Bk+1 - Bfc) = 0.kThis concludes the proof of (1.85).(c) Here we study the average of a product of integrals and we showthatM2 = I J f(s,Bs)dBsJ g(u,Bu)dBu = J (i(s,Bs)g(s,Bs))ds.(1.86)Proof.M2 = ]T<f(sm ,Bm )(Bm + 1 -Bm )g(sn ,Bn )(Bn + 1 - B n ) ) .m,nWe have to distinguish three subclasses: (i) n > m, (ii) n < m and(hi) n = m.Taking into account the independence of the increments of WPswe see that only case (hi) contributes non-trivially to M2. This yieldsM2 = ^(f(S n ,Bn )g(sn ,Bn )(Bra+l ~ B n ) ).nBut we know that f(sn, Bn)g(sn, Bn) is again a function that is inde-pendent of (Bn+i — Bn)2. We use (1.62) and obtain((Bn+i — Bn) ) = (Bn + 1 — 2Bn + iBn + Bn) = tn+ — tn — Ain,
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32 Stochastic Differential Equations in Science and Engineeringand thus we getM2 = ^(f(S n ,Bn )g(S n ,Bn ))((Bn+l —noo^2(i{sn,Bn)g{sn,Bn))Atn.n = lThe last relation tends for Atn —» 0 to (1.86).(d) A generalization of the property (c) is given by/ pa rb rahbM3 = M i(s,Bs)dBsl g(u,Bu)dBu = I (f(S,Bs)g(S,Bs))ds.(1.87)To prove (1.87) we must distinguish to subclasses (i) b = a + c > aand (ii) a = b + c> b;c> 0. We consider only case (i), the proof forcase (ii) is done by analogy. We derive from (1.86) and (1.87).M3 = M2 + / / f(s,Bs)dBs / g(u,Bu)dBt J0 Ja= M2 + Yl Yl ^ Bn)g(sm, Bm )ABn ABm ).n m>nBut we see that i(sn, Bn) and ABn are independent of f(sm, Bm) andABm . Hence, we obtainM3 = M2 + £ ( f ( s „ , Bn)ABn) Y, (g(sm, Bm)ABm) = M2,n m>nwhere we use (1.85). This concludes the proof of (1.87) for case (i).Now we calculate an exampleI(t) = / BsdBs. (1.88a)JoFirst of all we obtain with the use of (1.85) and (1.86) the momentsof the stochastic variable (1.88a)<I(t)> = 0; (I(t)I(t + r)) = [B2s)ds = f a d s = 72A n QQ^Jo Jo (1.88b)7 = iA(t + r).
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Stochastic Variables and Stochastic Processes 33We calculate the integral with an Ito decomposition1 = 2 ^ B*;(Bfc+i —Bfc).kBut we haveAB2k = (B2k+l - B2k) = (Bk+1 - Bkf + 2Bfc(Bfc+1 - Bfc)= (ABfc)2+ 2Bfc(Bfc+1-Bfc).Hence we obtainI(t) = lj2[A(Bl)-(ABk)%kWe calculate now the two sums in (1.90) separately. Thus weobtain the first placeI1(t) = EA(Bfc) = (B?-B8) + ( Bi - B? ) + - + (BN-BN-l)kN ~* Bi >where we used Bo = 0. The second integral and its average aregiven byI2(t) = Y, (AB*)2= E (B^+i "2Bk+iBk + Bl);k k(I2(i))=EAifc = tkThe relation (I2(i)) =t gives not only the average but also theintegral I2(i) itself. However, the direct calculation of l2(t) is im-practical and we refer the reader to the book of 0ksendahl [1.8],where the corresponding algebra is performed. We use instead anindirect proof and show that the quantity z (the standard deviationof I2(i)) is a deterministic function with the value zero. Thus, we putz = I2(£) — t. The mean value is clearly (z) — 0 and we obtain(Z2) = (l2(t)-2tl2(t)+t2) = (l2(t))-t2.
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34 Stochastic Differential Equations in Science and EngineeringBut we haveai(*)> = EE<(AB*)2(AB™)2>- (i.88c)k rnThe independence of the increments of the WPs yields((ABfc)2(ABm)2) = ((ABk)2)((ABm)2) + 5km{ABi),hence we obtain with the use of the results of EX 1.6$(«)>= (£<(ABfc)2)) +]T<(Bfc+1+Bfc)4} = £2+ 5>+1-£fc)2.V k J k kHowever, we have^2(tk+1 - tkf = J^(At)2= tAt^0 for At -> 0,k kand this indicates that (z2) = 0.This procedure can be pursued to higher orders and we obtainthe result that all moments of z are zero and thus we obtain l2(£) = t.Thus, we obtain finallyI(*) = / BsdBs = ^ ( B 2- £ ) . (1.89)There is a generalization of the previous results with respect tohigher order moments. We consider here moments of a stochasticintegral with a deterministic integrandJfc(t) = I ffc(s)dBs; k€N. (1.90)JoThese integrals are a special case of the ones in (1.82) and we knowfrom (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.JoBut we can obtain formally the same result if we put(dBsdB„) = 5(s - u)dsdu. (1.91)A formal justification of (1.91) is given in Chapter 2 in connectionwith formula (2.41). Here we show that (1.91) leads to a result that
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Stochastic Variables and Stochastic Processes 35S.is identical to the consequences of (1.86)<Jfc(*)Jm(<)> = / h(s) f fm H(dB,dBu )Jo Jo- fk(s) / fm(u)5(s - u)dsdu = / ffc(s)fm(s)dJo Jo JoWe also know that Bt 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 ofthe multivariate GD and we put for the 4th order moment of thedifferential<dBpdB9dBudB„) = <dBpdB9>(dBudB„) + (dBpdBu)(dBgdB^)+ (dBpdB^)(dB(?dBu) = [S(p - q)5{u - v)+ S(p — u)S(q — v) + 5(p — v)8(q — u)]dpdqdudv.Note that the 4th order moment of the differential of WPs has aform similar to an isotropic 4th order tensor. Hence, we obtain<Jj(t)Jm(i)Jr(*)J*(<)> = / f » f m ( a ) d a f ir{(3%((3)d(3Jo Jo+ / ij{a)ir{a)da f fm(/3)f,(/3)d/3Jo Jo+ / f » f s ( a ) d a [ fm(/3)fr(/3)d/3.Jo JoThis leads in a special case to<j£(i)> = 3<J2(i)>2. (1.92b)Again, this procedure can be carried out also for higher ordermoments and we obtain<J2"+1(i)) = 0; <J2^)) = 1.3....(2/ ,-l)<J2(i))^ ^ N .(1.92c)Equation (1.92) signifies that the stochastic Ito-integral (1.90) withthe deterministic integrand ffc(s) is N[0, fQ f|(s)ds] distributed. How-ever, one can also show that the Ito-integral with the non-anticipative
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36 Stochastic Differential Equations in Science and EngineeringintegrandK(i) = / g(s,Bs)dB„ (1.93a)Jois, in analogy to the stochastic integral with the deterministicintegrand,N[0,r(t)]; r(t)= [ (gu,Bu))du, (1.93b)Jodistributed (see Arnold [1.2]). The variable r(t) is referred to asintrinsic time of the stochastic integral (1.93a). We use this vari-able to show with Kolmogorovs Theorem (1.20) that (1.93a) possescontinuous SF. The Ito integralrtkXk= / g(«,Bu)dBu, ti = t > t2 = s,Jowith(xk) = 0; (x) = r(tfc) = rk; k = 1, 2, {xix2) = r2,has according to (1.35a) the joint PDxi _ (xi - x2)2"2n 2(n-r2)p2(xi,x2) = [(27r)ir1(r1 - r2)] expYet, the latter line is identical with the bivariate PD of the Wienerprocess (1.60) if we replace in the latter equation the t- by rk. Hence,we obtain from Kolmogorovs criterion ([xi(ri) — x2i(r2)]2) =|7~i — r2 | and this guarantees the continuity of the SF of the Ito-integral (1.93a). A further important feature of Ito integrals is theirmartingale 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)
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Stochastic Variables and Stochastic Processes 37where p ^ is given by (1.53). To verify now the martingale propertyof the integral (1.89) we specify (1.94) to(I(j/2,s) | Ifo!,*)) = - i = / {vl - s) exp[-(y2 - yif /f5]dy2.The application of the standard substitution (see EX 1.1) yields(I(y2, s) | I(2/i,*)> = ^=J{y-s + 2Vlz^ + /3^2) exp(-z2)d^1= -{yi-s + 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 by0ksendahl [1.8]. However, we will encounter the martingale propertyfor a particular class of Ito integrals in the next section.To conclude this example we add here also the Stratonovich ver-sion of the integral (1.89). This yields (the subscript s indicates aStratonovich integral)rt ils{t)= / BsodBs = - V ( B f c + 1 + Bfc)(Bfc+1-Bfc)Jo 2kkThe result (1.96) is the "classical" value of the integral whereasthe Ito integral gives a non classical result. Note also the signifi-cant differences between the Ito and Stratonovich integrals. Eventhe moments do not coincide since we infer from (1.96)&(*)> = a n d(U*)I*(«)> = ^[tu + 2(t A u)2}.It is now easy to show that the Stratonovich integral Is is not amartingale. We obtain this result if we drop the term s in secondline of (1.95)(ls(y2,s) | I8(yi,t)) = {y2+ P/2)^Uyut). XHence, we may summarize the properties of the Ito andStratonovich integrals. The Stratonovich concept uses all the trans-formation rules of classical integration theory and thus leads in many
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38 Stochastic Differential Equations in Science and Engineeringapplications to an easy way of performing the integration. Deviat-ing 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 followingwe will consider both integration concepts and their application insolution of SDE.We have calculated so far only one stochastic integral and wecontinue in the next section with helpful rules perform the stochasticintegration.1.9. The Ito FormulaWe begin with the differential of a function $(Bf, t). Its Ito differen-tial takes the formd$(Bt, t) = Qtdt + $B t dBt + ^ B t B t (dBt)2. (1.97.1)Formula (1.97.1) contains the non classical term that is proportionalto the second derivative WRT Bt. We must supplement (1.97.1) bya further non classical relation(dBt)2= dt. (1.97.2)Thus, we infer from (1.97.1,2) the final form of this differentiald$(Bt, t) = Ut + ^*BtBt) dt + ^BedBj. (1.98)Next we derive the Ito differential of the function Y = g(x, t)where x is the solution of the SDEdx = a(x,t)dt + b(x,t)dBt. (1.99.1)In analogy to (1.97.1) we include a non classical term and putdY = gtdt + gxdz + -gxx(dx)2,We substitute dx from (1.99.1) into the last line and apply the nonclassical formula(dx)2= {adt + bdBt)2= b2dt; {dt)2= dtdBt = 0; (dBt)2= dt,(1.99.2)
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Stochastic Variables and Stochastic Processes 39and this yieldsd Y = ( 6 + 0 b + £& .)< K + t b d B , . (1.99.3)The latter equation is called the Ito formula for the total differen-tial of function Y = g{x,t) given the SDE (1.99.1). (1.99.3) containsthe non classical term b2gxx/2 and it differs thus from the classical(or Stratonovich) total differentialdYc = (gt + agx)dt + bgxdBt. (1.100)Note that both the Ito and the Stratonovich differentials coincide ifg{x,i) is a first order polynomial of the variable x.We postpone a sketch of the proof of (1.99) for a moment andgive an example of the application of this formula. We use (1.99.1)in the formdx = dBt, or x = Bt with a = 0, 6 = 1, (1.101a)and we consider the functionY = g(x) = x2/2; gt = 0; gx = x; gxx = 1. (1.101b)Thus we obtain from (1.99.3) and (1.101b)dY = d(x2/2) = dt/2 + BtdBt,and the integration of this total differential yieldsd(x2/2) = / d(B2s/2) = B2/2 = t/2+ [ BsdBsJo Joand the last line reproduces (1.89). XWe give now a sketch of the proof of the Ito formula (1.99) andwe follow in part considerations of Schuss [1.9]. It is instructive toperform this in detail and we do it in four consecutive steps labeledwith Si to S4.I
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40 Stochastic Differential Equations in Science and EngineeringSiWe begin with the consideration of the stochastic function x(t)given byrv rvx(v)-x(u) = a(x{s),s)ds + b(x(s),s)dBs, (1.102)Ju Juwhere a and b are two differentiate functions. Thus, we obtain thedifferential of x(t) if we put in (1.102) v = u + dt and let dt —• 0dx(u) = a(x(u),u)du + b(x(u),u)dBu. (1.103)Before we pass to the next step we consider two important examplesExample 1. (integration by parts)Here we consider a deterministic function f and a stochastic func-tion Y and we putY(Bt,t) = g(Bt)t) = f(i)Bt. (1.104a)The total differential is in both (Ito and Stratonovich) cases (see(1.98) with 3>BtBt = 0) given by the exact formuladY = d[f(t)Bt] = f(*)dBt + i(t)Btdt. (1.104b)The integration of this differential yieldsi(t)Bt= f f(s)Bsds+ [ f(s)dBs. (1.105a)Jo JoSubtracting the last line for t = u from the same relation for t — vyieldsrv rvi{v)Bv - i(u)Bu = f(s)Bsds+ f(s)dBs. (1.105b)Ju JuExample 2. (Martingale property)We consider a particular class of Ito integralsI(t) = / f(u)dBu, (1.106)Jo
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Stochastic Variables and Stochastic Processes 41and show that I(i) is a martingale. First we realize that the integrall(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 andposses the intrinsic time given by (1.93b). Its transition probabilityPi|i is defined by (1.53) with tj = r(tj); yj = I(i,); j — 1, 2. This con-cludes the proof that the integral (1.106) obeys a martingale propertylike (1.27) or (1.64). *S2Here we consider the product of two stochastic functions subjectedto two SDE with constant coefficientsdxk(t) = ctkdt + frfcdBi; ak,bk = const; k ~ 1,2, (1.107)with the solutionsxk(t) = akt + bkBt; xfc(0) = 0. (1.108)The task to evaluate d(xiX2) is outlined in EX 1.9 and we obtainwith the aid of (1.89)d{xX2) = X2dxi + xdx2 + b^dt. (1.109)The term proportional to 6162 in (1.109) is non classical and it is amere 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 step-function for the functions a and b in (1.106) and with that one canapproximate differentiable functions (see Schuss [1.9]).We consider now two examplesExample 1We take put x = Bi;X2 = Bf. Thus, we obtain with an applicationof (1.101b) and (1.109)dBt3= BtdB? + B^dBj + 2Btdi = 3(Btdt + B?dBt).The use of the induction rule yields the generalizationdBtfc= jfeB^dB* + ^ " ^ B ^ d t (1.110)
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42 Stochastic Differential Equations in Science and EngineeringExample 2Here we consider polynomials of the Brownian motionPn(Bt) = c0 + cxBt + • • • + cnB?; ck = const. (1.111)The application of (1.110) to (1.111) leads todPn(B4) = P;(Bt)dBt + ip£(Bt)di; = d/dBt. (1.112)The relation (1.112) is also valid for all functions that can beexpanded in form of polynomials. &S3Here we consider the product*{Bt,t) = ip(Bt)g(t), (1.113)where g is a deterministic function. The use of (1.109) yieldsd*(Bt,t) = g(i)<MBt) + <p(Bt)g!(t)dt= LdBt + ^"dtg +Vg(t)dt (1.114)= Ug + ^"g]dt + gipdBt.But we also haveThus, we obtain1 d2 , 1 ,,+25Bfj$ = g V +2 g^ ^L115)(d 1 d2 <9$d* = ( » +2 8 B ? j W f +8 B ; d B" <L116>Equation (1.116) applies, in the first place, only to the function(1.113). However, the use of the expansionCO$(Bt,t) = J>fc(B4)gfc(i), (1.117)fc=ishows that (1.116) is valid for arbitrary functions and this proves(1.98).
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Stochastic Variables and Stochastic Processes 43S4In 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), wherex satisfies a SDE of the type (1.107)$(Bt,t) = g(x,t) = g(at + bBt,t); x = adt + bdBt; a, 6 = const.0 (1.118)*t = agx + gt; *Bt = &gx5 $B,B, = b lgxx.Thus we obtain with (1.116)The relation (1.119) represents the Ito formula (1.99.3) (for constantcoefficients 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 multivariateprocess. First we consider K functions of the typeVk yfc(Bi1,...,BtM,t); fc = 1,2,... ,K,where B^,...,B^ are M independent Brownian motions. We takeadvantage of the summation convention and obtain the generaliza-tion of (1.97.1)dyfc(Bj,... ,Bf,,) = * £ d t + ^LdBr + I ^ d B r d B f ;dt <9B[ l2dWtWt(1.120)/c = l,...,K; r,s = l,...,M.We generalize (1.97.2) and putdB[dB? = Srsdt, (1.121)and we obtain (see (1.98))d s t ( B , , . . . , B ( - t ) = ( ^ + i ^ ) d . + ^ d B E . (1.122)Now we consider a set of n SDEsdXfe = afc(Xi,... ,Xn,t)dt + 6fer(Xi,... ,Xn,i)dB£;Jfe = l,2,...,n; r = l,2,...,R.[1.123)
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44 Stochastic Differential Equations in Science and EngineeringWe wish to calculate the differential of the functionZfc = Zfc(Xi,...,Xn,t); fc = l,...,K. (1.124)The differential readsdt M+dXm^m+2dXmdX1JdZfc = - ^ dt + T^dXm + - v * dXmdXM-dt + ^r— (amdt + 6mrdBj)mdt dX,1 d2Z+ o oV ov (a™di+ bmrdBrt) (audt + busdBst);m,n = 1,2,... ,n; r = 1,2,... ,R. (1.125)The n-dimensional generalization of the rule (1.99.2) is given byd(B[dBJ1) = <5rudt; (dt)2= dB[ dt = 0. (1.126)Thus, we obtain the differential of the vector valued function (1.124)dZfc = ( -XT + amT^rp - T.bmrKr ^ r ^ r I d tdt + amdxm+2bmrburdxmdxu+ h dZkaw+ t>mr flv ar3t.Now we conclude this section with two examples.Example 1A stochastic process is given by(1.127)Yi = B j + B ? + Bt3; Y2 = (B?)2-BjBWe obtain for the SDE in the form (1.120) corresponding to the lastlinedYi = dB! + dB2+ dB3;dY2 = dt + 2B2dBt2- (B^dBj + BjdB?). *
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Stochastic Variables and Stochastic Processes 45dY 5,+ «gI + ^ + 7 2) §Example 2Here we study a single stochastic process under the influence of twoindependent Brownian motionsdx = a(x, t)dt + (3{x, t)dB] + j(x, i)dBf2. (1.128)The differential of the function Y = g(x, t) has the formdt + g^dBJ+jdB2).We consider now the special caseg = In x; a = rx; (5 — ux; 7 = ax; r,u,a = const,and we obtaind(lnar) = [r - (u2+ a2)/2]dt + (udBJ + odB2). (1.129)We will use (1.129) in Section 2.1 of the next chapter. XWe introduced in this chapter some elements of the probabilitytheory and added the basic ideas about SDE. For readers who wishto get more deeply involved in the abstract theory of probabilityand in particular with the measure theory we suggest they considerthe following books: Chung & Aitsahia [1.10], Ross [1.11], Mallivan[1.12], Pitman [1.13] and Shiryaev [1.14].Appendix: Poisson ProcessesIn many applications appears there a random set of countable pointsdriven by some stochastic system. Typical examples are arrival timesof customers (at the desk of an office, at the gate of an airport, etc.),the birth process of an organism, the number of competing buildingprojects for a state budget. The randomness in such phenomena isconveniently described by Poisson distributed variables.First we verify that the Poisson distribution is the limit of theBernoulli distribution. We substitute for the argument p in theBernoulli distribution in Section 1.1 the value p = a/n and this
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46 Stochastic Differential Equations in Science and EngineeringyieldsKt,n,a/n) = i ; ) ( 2 ) ( l - 2( aV6(0, n, a/n) = 1 1 —> exp(—a) for n —> oo.V nJNow we putb(k + l,n,a/n) a n — kf a~ a(A.l)b(k,n,a/n) k + 1 n n) fc + land this yieldsa26(1, n, a/n) —> a exp(—a); 6(2, n, a/n) —> — exp(—a);...anb(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 HPP witha 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 forany sequence of times 0 < to < t < • • • < tn < • • • mutuallyindependent.(c) The random increments defined in condition (b) are Poisson dis-tributed of the formPr([N(tr+0-Nfa)] = fc)=(A^7(-H*• (A.2)Tr = t r + i — tr, k = (J, 1 , . . . ; r = 1, 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~Pr(AN(£) = fe) = ih^fL exp(-AAt)fe!AAt for A; = 10(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).
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Stochastic Variables and Stochastic Processes 47However, the part of (A.3) with Pr([N(t + At) - N(t)] = 1) « XAtindicates that there is small chance for a jump with the height unity.The probability of jumps with higher heights k = 2, 3,... correspond-ing to the third part of (A.3) is subdominantly small and such jumpsdo not appear.We calculate of the moments of the HPP in two alternative ways.(i) We use (1.5) with (A.2) to obtain/oop(x)smdx = J2km Fr(x= k)fc=0oo= exp(-a)^fcma*/A;!; a = At, (A.4)k=0or we apply (ii) the concept of the generating function defined byg(z) = J2zk Pr(x= *)> withgw = (x)fc=05"(l) = ( o : 2) - ( x ) , . . . ; *fc=° ^ (A.5)dzThis leads in the case of an HPP tooo oog(z) = ^2 zkakexp(-o;)/fc! = exp(-q) y^(zq)fc/A:!fc=o fe=o= exp[a(z-l)]. (A.6)In either case we obtain(N(t)) = At, (N2(t)) = (At)2+ At. (A.7)We calculate now the PD of the sum x + x^ of two independentHPPs. By definition this yieldsPr([a;i + x2] = k) = Prl ^ [ x i = j , x 2 = A; - j] Jfc= X P r( X l= J>2 = A; - j)A;j=o
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48 Stochastic Differential Equations in Science and Engineeringqk-jEexp[-(^)]fexp[-^)](fc_j)!kexp[-(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 toPr([xi + x2] = k) = exp(-20){20)k/k. (A.9)Poisson HPPs play important roles in Markov process (seeBremaud [1.16]). In many applications these Markov chains are iter-ations driven by "white noise" modeled by HPPs. Such iterationsarise in the study of the stability of continuous periodic phenomena,in the biology and economics, etc. We consider the form of iterationsx{t + s) = F(x(s),Z(t + s)); s , t e N 0 (A.10)where t, s are discrete variables and x(t) is a discrete random variabledriven by the white noise Z(t + s). An important particular case isZ(£ + s) := N(t + s) with a PDPr(N(£ + 8) = k) = exp(u)uk/kl; u = 9(t + s).The transition probability is the matrix governing the transi-tion from state i to state k.Examples(i) Random walkThis is an iteration of a discrete random variable x(t)x(t) = x(t-l) + N(t); x{0) = xoeN. (A.ll)N(t) is HPP with Pr([N(t) = A;]) = exp(-Xt){Xt)k/k. Hence, weobtain the transition probabilityPjl = Pr(x(t) = j , x(t - 1) = i) = Pr([i + N(j)] = 3)= P r ( N ( j ) = j - l ) .
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Stochastic Variables and Stochastic Processes 49(ii) Flip-Flop processesThe iteration takes here the formx(i) = (-l)N ( t ). (A.12)The transition matrix takes the formp _ u = Pr(x(t + s) = 1 | x(s) = -1) = Pr(N(t) = 2k + 1) = a;p M = pr(x(t + s) = 1 | x(s) = 1 = Pr(N(t) = 2k) = 0,witha = ^Texp(-t)(t)2k+1/(2k + 1)! = exp(-At) sinh(Ai);k=ooo0 = ^exp(-Ai)(Ai)2fc/(2fc)! = exp(-Ai) cosh(At). Xfc=0Another important application of HPP is given by a ID approachto turbulence elaborated by Kerstein [1.17] and [1.18]. This modelis based on the turbulence advection by a random map. A tripletmap is applied to a shear flow velocity profile. An individual event isrepresented by a mapping that results in a new velocity profile. Asa statistical hypothesis the author assumes that the temporal rateof the event is governed by a Poisson process and the parameter ofthe map can be sampled from a given PD. Although this model wasapplied to ID turbulence, its results go beyond this limit and themodel has a remarkable power of prediction experimental data.ExercisesEX 1.1. Calculate the mean value Mn(s,t) = ((Bt - Bs)n), n G N.Hint: Use (1.60) and the standard substitution y^ = yi +Zj2{t2 — t), where z is a new variable. Show that this yields[2(£2-*i)]n/2Mn / exp(—v2)dv / exp(—z2)zndz./itThe gamma function is defined by (see Ryshik & Gradstein [1.15])r((n + l)/2)Vn = 2fc;^0Vn = 2ife + l; fceN,r(l/2) =-S/TF, r(n + l) = nT(n)./ ex.p(-z2)zndv
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50 Stochastic Differential Equations in Science and Engineeringa2>e2Verify the resultM2n = ir-V2[2(t2 - ii)]nr((2n + l)/2).EX 1.2. We consider a ID random variable X with the mean fi andthe variance a2. Show that the latter can be written in the form(fx(x) is the P D ^ = ( s ) ; e > 0 )<r2>( + ) fx(x)(x - tfdx; e.J fi+e J—oo /For x < /x — e and X>/J, + £^(X — fi)2> e2, this yields1 - / ix(x)dx = e 2P r ( | X - ^ | > e),J ix—eand this gives the Chebyshev inequality its final formPr{|X-/z| >e} <a2/e2.The inequality governing martingales (1-28) is obtained with con-siderations similar to the derivation of the Chebyshev inequality.EX 1.3.(a) Show that we can factorize the bivariate GD (1.35a) with zeromean and equal variance ((x) = (y) — 0; a2= a = b) in the formp(x, y) = J~1/2p(x)p((y - rx)/y/T); 7 = (1 - r2),where p(x) is the univariate GD (1.29).(b) Calculate the conditional distribution (see 1.17) of the bivariateGD (1.35a). Hint: (c is the covariance matrix)PiliO* I V) = V <W(27rD) exp[-cTO(x - ycxy/cyy)/(2D)};Verify that the latter line corresponds to a N[ycxy/cyy,cxx —ciy/cyy) distribution.EX 1.4. Prove that (1.53) is a solution of the Chapman-Kolmogorovequation (1.52)
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Stochastic Variables and Stochastic Processes 51Hint: The integrand in (1.52) is given byT = Pi|i(y2,t2 | ?/i,*i)Pi11 (2/3,^3 I 2/2, £2)-Use the substitutionu = t<i - t > 0, v = £3 - £2 > 0; £3 - t = v + u > 0,introduce (1.53) into (1.52) and putT = (47r2m;)-1/2exp(-A);A = (2/3 - y2?/(2v) + (3/2 - 2/i)2/(2u) = a22/| + ai2/2 + «o,with afc = ctk(yi,y3),k = 1,2,3. Use the standard substitution (seeEX 1.1) to obtain/ Tdt/2 = (47rra)"1/2exp[-F(y3,y2)] / exp(-K)djy2;4a0a2 - af / ai 2F =- ^ 2 — ; K = a 2ly 2 +2^Jand compare the result of the integration with the right hand sideof (1.52).EX 1.5. Verify that the solution of (1.54) is given by (1.55). Provealso its initial condition.Hint: To verify the initial condition use the integral/ooexp[-y2/(2£)]H(y)dy,-00where H(y) is a continuous function. Use the standard substitutionin its form y = /2tz.To verify the solution (1.55) use the same substitution as inEX 1.4.EX 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.EX 1.7. Verify that the variable Bt defined in (1.65) has the auto-correlation (BtBs) = £ A s. To perform this task we calculate for a
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52 Stochastic Differential Equations in Science and Engineeringfixed value of a > 0<BtBs) = (Bt + a Bs + a ) - (Bt+aBa) - (Bs+aBa) + {B2a)= sAf + a - a - a + a = sAt.EX 1.8. Prove that the scaled and translated WSs defined in (1.74)are WSs.Hint: To cover the scaled WSs, putHUl„ = ^M-alpv = ^x1(a2u)x2(b2v).Because of (xi(a)x2((3)) = 0 we have (H.u,v) = 0. Its autocorrelationis given by(HUi„HPi9) = —-^(x1(a2u)x1(a2p))(x2(b2v)x2{b2q))[ao)(a2u) A (a2p)(b2v) A (b2q) = (u A p)(v A q).(ab)2For the case of the translated quantity use the consideration ofEX 1.7.EX 1.9. Verify the differential (1.109) of two linear stochasticfunctions.Hint: According to (1.89) we have dBt2= 2BtdBt + dt...EX 1.10. Show that the "inverted" stochastic variablesZt = tB1/t; H8it = stM{2lA/t,are also a WP (Zt) and a WS (HS)t).EX 1.11. Use the bivariate PD (1.60) for a Markov process, to cal-culate the two-variable characteristic function of a Brownian motion.Verify the resultG(u,v) = (exp[i(uBi + uB2)])exp -(u2h + v2t2) + 2uv(ti A£2) Bfc = Btfcand compare its ID limit with (1.58a).EX 1.12. Calculate the probability P of a particle to stay in theinterior of the circle D = {(a;, y) G R2 | x2+ y2< R}.
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Stochastic Variables and Stochastic Processes 53Hint: Assume that components of the vector (x, y) are statisticallyindependent use the bivariate GD (1.35) with zero mean to calculateP[Bt€D] = J J p(x,y)dxdj/.EX 1.13. Consider the Brownian motion on the perimeter of ellipsesand hyperbolas(i) ellipsesx(t) = cos(Bi), y(t) = sin(Bt),(ii) hyperbolasx{t) = cosh(Bt), y(t) = sinh(Bt).Use the Ito formula to obtain the corresponding SDE and calcu-late (x(t)) and (y(t)).EX 1.14. Given the variablesZ1 = (Bj - B2)4+ (Bj)5; Z2 = (B,1- Bt2)3+ (B,1)6,where B^ and B2are independent WPs. Find the SDEs governingdZx and dZ2.EX 1.15. The random functionRW = [(Bt1)2+ --- + (BD2]1/2,is considered as the distance of an n-dimensional vector of indepen-dent WPs from the origin. Verify that its differential has the formn— 1dR(t) = J2BtdBt/R+T^Tdt-EX 1.16. Consider the stochastic functionx(t) = exp(aBt — a2t/2); a = const.(a) Show thatx(t) = x(t — s)x(s).Hint: Use (1.65).(b) Show that x(t) is a martingale.
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54 Stochastic Differential Equations in Science and EngineeringEX 1.17. The Wiener-Levy Theorem is given byoo »tBt = V A f c / rl>k{z)Az, (E.l)where A^ is a set of IID N(0,1) variables and ipi-; k = 1, 2,... is a setof orthonormal functions in [0, 1]»i1/Jk(z)lpm(z)dz = Skm.0Show that (E.l) defines a WP.Hint: The autocorrelation is given by (BtBs) = t A s. Show thatJod_~dt(BtBa) = — (t A s) = V Vfc(0 / Mz)dz.Multiply the last line by ipm(t) and integrate the resulting equationfrom zero to unity.EX 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.EX 1.19. The Ito representation of a given stochastic processesF(t,(j) has the formF(t,u) = (F(t,u)) + [ f(s,u)dBs,Jowhere i(s,u) is an other stochastic process. Find i(s,u) for the par-ticular cases(i) F(t,iu) = const; (ii) F(t,u) = BJ1; n = 1,2,3; (hi) F(t,u) =exp(Bt).EX 1.20. Calculate the probability of n identically independentHPPs [see (A.8)].
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CHAPTER 2STOCHASTIC DIFFERENTIAL EQUATIONSThere are two classes of ordinary differential equations that containstochastic influences:(i) Ordinary differential equations (ODE) with stochastic coefficientfunctions and/or random initial or boundary conditions that con-tain no stochastic differentials. We consider this type of ODEs inChapter 4.3 where we will analyze eigenvalue problems. For theseODEs we can take advantage of all traditional methods of analysis.Here we give only the simple example of a linear 1st order ODEdx— = -par; p = p(w), ar(0) = x0(u),where the coefficient function p and the initial condition are x-independent random variables. The solution is x(t) = xoexp(—pt)and we obtain the moments of this solution in form of (xm) ={XQ1exp(—pmt)}. Assuming that the initial condition and the param-eter p are identically independent N(0, a) distributed, this yields(*2m> = ^ ^ e x p ( 2 a m 2t 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 SDEs 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 thischapter we introduce the analytical tools to reach this goal. However,in many cases we would have to resort to numerical procedures andwe perform this task in Chapter 5.The primary questions are:(i) How can we solve the equations or at least approximate thesolutions and what are the properties of the latter?55
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