07_Brownian_Motion

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1、BrownianMotionImportanceofMartingaleinFinanceApplicationsChangeofMeasureonBinomialTreeFinancialEngineeringMartingaleandStoppingTimeHaiLanDept.ofManagementSciencesShanghaiJiaoTongUniversity.October31,2012H.LanFinancialEngineeringBrownianMotionImportanceofM

2、artingaleinFinanceApplicationsChangeofMeasureonBinomialTreeOutlineBrownianMotionImportanceofMartingaleinFinanceApplicationsChangeofMeasureonBinomialTreeH.LanFinancialEngineeringBrownianMotionImportanceofMartingaleinFinanceApplicationsChangeofMeasureonBino

3、mialTreeDefinitionAreal-valuedstochasticprocess{B(t):t>0}iscalleda(linear)Brownianmotionwithstartinx∈Rifthefollowingholds:1B(0)=x2theprocesshasindependentincrements,i.e.foralltimes0≤t1≤t2≤···≤tntheincrementsB(tn)−B(tn−1),B(tn−1)−B(tn−2),...,B(t2)−B(t1)arei

4、ndependentrandomvariables.3forallt≥0andh>0,theincrementsB(t+h)−B(t)arenormallydistributedwithexpectation0andvarianceh.4almostsurely,thefunctiont→B(t)iscontinuous.H.LanFinancialEngineeringBrownianMotionImportanceofMartingaleinFinanceApplicationsChangeofMea

5、sureonBinomialTreey−100102030405060−30−20−10010H.LanFinancialEngineeringxBrownianMotionImportanceofMartingaleinFinanceApplicationsChangeofMeasureonBinomialTreeFactsonBrownianMotion1.Existence.2.IfastochasticprocessXhascontinuouspathsandstationary,independ

6、entincrements,thenXisaBrownianMotion.3.AGaussianProcessX(i.e.allfinitedimensionaldistributionsarenormaldistributions.)havingcontinuouspaths,mean0andcovariancefunctionCov(Xs,Xt)=s∧tisastandardBrownianMotion.H.LanFinancialEngineeringBrownianMotionImportanceo

7、fMartingaleinFinanceApplicationsChangeofMeasureonBinomialTreeScalingInvarianceTheoremSuppose{B(t):t>0}isastandardBrownianmotionandleta>0.Thentheprocess{X(t):t>0}definedbyX(t)=1B(a2t)isalsoastandardBrownianmotion.aProof:Continuityofthepaths,independenceands

8、tationarityoftheincrementsremainunchangedunderthescaling.ItremainstoobservethatX(t)−X(s)=1(B(a2t)−B(a2s))isnormallyadistributedwithexpectation0andvariance(1/a2)(a2t−a2s)=t−s.H.LanFinancialEngineeringBrownianMotionIm