The Probability Space of Brownian Motion

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1、Chapter2TheProbabilitySpaceofBrownianMotion2.1IntroductionAccordingtoEinstein’sdescription,theBrownianmotioncanbedefinedbythefollowingtwoproperties:first,ithascontinuoustrajectories(samplepaths)andsecond,theincrementsofthepathsindisjointtimeintervalsareindependentzeromeanGaussianrandom

2、variableswithvarianceproportionaltothedurationofthetimeinterval(itisassumed,fordefiniteness,thatthepossibletrajectoriesofaBrow-nianparticlestartattheorigin).Thesepropertieshavefar-reachingimplicationsabouttheanalyticpropertiesoftheBrowniantrajectories.Itcanbeshown,forex-ample(seeTheor

3、em2.4.1),thatthesetrajectoriesarenotdifferentiableatanypointwithprobability1[198].Thatis,thevelocityprocessoftheBrownianmotioncan-notbedefinedasareal-valuedfunction,althoughitcanbedefinedasadistribution(generalizedfunction)[152].Langevin’sconstructiondoesnotresolvethisdiffi-culty,becaus

4、eitgivesrisetoavelocityprocessthatisnotdifferentiablesothattheaccelerationprocess,Ξ(t)ineq.(1.24),cannotbedefined.OnemightguessthatinordertoovercomethisdifficultyinLangevin’sequationalldifferentialequationscouldbeconvertedintointegralequationssothattheequa-tionscontainonlywelldefinedvel

5、ocities.Thisapproach,however,failseveninthesimplestdifferentialequationsthatcontaintheprocessΞ(t)(whichinonedimen-Rt+∆tsionisdenotedΞ(t)).Forexample,ifweassumethat∆w(t)≡Ξ(s)ds∼tN(0,∆t)andconstructthesolutionoftheinitialvalueproblemx˙=xΞ(t),x(0)=x0>0(2.1)bytheEulermethodx∆t(t+∆t)−x∆t(

6、t)=x∆t(t)∆w(t),x∆t(0)=x0>0,(2.2)Z.Schuss,TheoryandApplicationsofStochasticProcesses:AnAnalyticalApproach,25AppliedMathematicalSciences170,DOI10.1007/978-1-4419-1605-1_2,©SpringerScience+BusinessMedia,LLC2010262.TheProbabilitySpaceofBrownianMotionthelimitx(t)=lim∆t→0x∆t(t)isnotthefunc

7、tionZtx(t)=x0expΞ(s)ds.0ItisshownbelowthatthesolutionisZt1x(t)=x0expΞ(s)ds−t.20ItisevidentfromthisexamplethatdifferentialequationsthatinvolvetheBrownianmotiondonotobeytherulesofthedifferentialandintegralcalculus.Asimilarphenomenonmanifestsitselfinothernumericalschemes.Con

8、sider,forexa