Basics of Stochastic Calculus随机微积分基础

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1、Chapter2BasicsofStochasticCalculusLet.˝;F;F;P/beafilteredprobabilityspace.Weremarkagainthat,unlikeinstandardliterature,wedonotassumeFDfFtg0tTsatisfytheusualhypothesis.ThiswillbecrucialforthefullynonlineartheoryinPartIII,andforfixedPthisisaverymildrelaxationduetoPr

2、oposition1.2.1.2.1BrownianMotion2.1.1DefinitionDefinition2.1.1WesayaprocessBWŒ0;T˝!Risa(standard)BrownianmotionifB0D0,a.s.Forany0Dt0<

3、tT,Bs;tandFsareindependent.WenotethatasinthepreviouschapterwerestrictBtoafinitehorizonŒ0;T.ButthedefinitioncanbeeasilyextendedtoŒ0;1/,byfirstextendingthefiltrationFtoŒ0;1/.Whennecessary,wemayinterpretBasaBrownianmotiononŒ0;1/withoutmentioningitexplicitly.Moreover,wh

4、enthereisaneedtoemphasizethedependenceontheprobabilitymeasurePand/orthefiltrationF,wecallBaP-Brownianmotionor.P;F/-Brownianmotion.SinceBhasindependentincrements,©SpringerScience+BusinessMediaLLC201721J.Zhang,BackwardStochasticDifferentialEquations,ProbabilityTheory

5、andStochasticModelling86,DOI10.1007/978-1-4939-7256-2_2222BasicsofStochasticCalculusclearly.Bt1;;Btn/haveGaussiandistribution,orsayBisaGaussianprocess.Moreover,fromthedefinitionwecaneasilycomputethefinitedistributionofB.ThenbytheKolmogorovsExtensionTheoremweknowt

6、hatBrownianmotiondoesexist.Thefollowingpropertiesareimmediateandlefttothereaders.Proposition2.1.2LetBbeastandardBrownianmotion.Foranyt0andanyt0c1constantc>0,theprocessesBtWDBt0;tCt0andBOtWDpcBctarealsostandardBrownianmotions.Proposition2.1.3ABrownianmotionisMarkov

7、,andanF-BrownianmotionisanF-martingale.Inthemultidimensionalcase,wecallBD.B1;;Bd/>ad-dimensionalBrownianmotionifB1;;BdareindependentBrownianmotions.InmostcaseswedonotemphasizethedimensionandthusstillcallitaBrownianmotion.Fromnowon,throughoutthischapter,Bisad

8、-dimensionalF-Brownianmotion.Allourresultsholdtrueinmultidimensionalsetting.However,whileweshallstatetheresultsinmultidimensionalcase,fornotionalsimplic