Chapter 4_Brownian motion.pdf

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1、Chapter4BrownianMotion4.1StochasticProcessesinContinuousTimeFortherestofthiscourse,wewillconcentrateourattentiononstochasticprocesseswhosetimeindexiscontinuous.So(refertosection1.4.5)wewilleithertakeI=[0;T]whereT>0is¯xedorI=R+.The¯rstcaseisusefulformodellingpr

2、ocesseswithina¯xedtimehorizonwhilethesecondismoregeneral.Inthissectionwe'llworkinthesecondcase.Werecallthatastochasticprocessincontinuoustimeisafamilyofrandomvariables(X(t);t¸0)de¯nedonacommonprobabilityspace(•;F;P).A(continuous-time)¯ltrationofFisanincreasing

3、familyofsub-¾-algebras(Ft;t¸0)ofFsothatifs·t;FsµFt.TheprocessX=(X(t);t¸0)issaidtobeadaptedtothe¯ltration(Ft;t¸0)ifeachX(t)isFtmeasurable.Thenatural¯ltrationassociatedtoXisde¯nedbyFX=¾fX(s);0·s·tg,i.e.thesmallestsub-¾-algebratofFwithrespecttowhichallX(s)aremeas

4、urablefor0·s·t.Everystochasticprocessisadaptedtoitsownnatural¯ltration.Themartingaleconceptincontinuoustimeneedsalittlethought.Wecannotgeneralisethemartingalepropertydirectly(whyshouldweworkwithintegers?)butwecaneasilygeneralisetheequivalentproperty(3.2.1).Wet

5、hensaythatanintegrableadaptedprocessisamartingaleifE(X(t)jFs)=X(s);(4.1.1)forall0·s·t<1.Theconceptsofsub-andsuper-martingalearede¯nedsimilarly.ArandomvariableTde¯nedon(•;F;P)andtakingvaluesin[0;1]iscalledastoppingtimewithrespecttothe¯ltration(Ft;t¸0)iftheevent

6、(T·t)2Ft;35forallt¸0,e.g.ifXisanadaptedprocessandA=(a;b)isanintervalinR,thenthe¯rstentrancetimeTAoftheprocessXtothesetAisde¯nedbyTA=infft¸0;X(t)2Ag:Nowthatwe'vegeneralisedkeyconceptsfromdiscretetocontinuoustime,we'llproceedtostudyaveryimportantexample.4.2Brown

7、ianMotionIn1828,thebotanistRobertBrowndiscoveredthatpollengrainsimmersedinwatermoveinanerraticmanner.AtthetimetherewasnogoodphysicalexplanationforthisandthephenomenonwasnamedBrownianmotion"afteritsdiscoverer.Asthenineteenthcenturydeveloped,thekinetictheoryofm

8、atterbecameestablishedasaresultofwhichBrownianmotionwasunderstoodasarisingfromrandombombardmentofpollengrainsbywatermolecules.ConsideraparticleexecutingBrownianmotionandletB(t)deno