08_Brownian_Motion

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1、BrownianMotion(2)FinancialEngineeringMartingaleandStoppingTimeHaiLanDept.ofManagementSciencesShanghaiJiaoTongUniversity.November6,2012H.LanFinancialEngineeringBrownianMotion(2)OutlineBrownianMotion(2)H.LanFinancialEngineeringBrownianMotion(2)MarkovianPropertyB(t)isaSBM,thenB(t+s)B(s)

2、isalsoaSBMandindependentoffB(t):0tsg.Proof:B(t+s)B(s)hasthesamenitedimensionaldistributionsasB(t)B(0)=B(t)a.s.anditisalsoa.s.continuous.ThenB(t+s)B(s)isaSBM.TheindependencedirectlycomesfromtheindependentincrementpropertyofSBM.BrownianMotionrestart"atanyxedtimes0.Thisresultcan

3、beextendedtorandomstoppingtime,i.e.B(t+)B()isalsoaSBM.Butnotalwaystrueforanyrandomvariable.ForexampleM(t)=maxfB(s):0stgLet=infft:B(t)=M(1)g,thenX(t)=B(t+)B()isnotaSBM.H.LanFinancialEngineeringBrownianMotion(2)MarkovianPropertyB(t)isaSBM,thenB(t+s)B(s)isalsoaSBMandindependentof

4、fB(t):0tsg.Proof:B(t+s)B(s)hasthesamenitedimensionaldistributionsasB(t)B(0)=B(t)a.s.anditisalsoa.s.continuous.ThenB(t+s)B(s)isaSBM.TheindependencedirectlycomesfromtheindependentincrementpropertyofSBM.BrownianMotionrestart"atanyxedtimes0.Thisresultcanbeextendedtorandomstopping

5、time,i.e.B(t+)B()isalsoaSBM.Butnotalwaystrueforanyrandomvariable.ForexampleM(t)=maxfB(s):0stgLet=infft:B(t)=M(1)g,thenX(t)=B(t+)B()isnotaSBM.H.LanFinancialEngineeringBrownianMotion(2)QuadraticVariationThequadraticvariationofBrownianMotionBtisdenedasXn[B](t):=limjB(tn)B(tn)jn!

6、1ii1i=1wherethelimitistakenoverallshrinkingpartitionsof[0;t].TheoremQuadraticvariationofaBrownianmotionover[0;t]ist.PProof:LetQn=jB(tn)B(tn)j2.Itiseasytoseethatiii1XXE(Q)=EjB(tn)B(tn)j2=(tntn)=t:nii1ii1iiByusingthefourthmomentofN(0;2)distributionis34,weobtainthevarianceofQnXX

7、Var(Q)=Var(jB(tn)B(tn)j2=Var(B(tn)B(tn))2nii1ii1iiXnn)23maxftntngt=3(titi1H.LanFinancialEngineeringii1iBrownianMotion(2)Ifthepartitionrenesaseachintervalisdividedbytwo,P1i=1Var(PQn)<1.Usingmonotoneconvergencetheorem,wendEn(QEQ)2<1.i=1nnThisimpliesthattheseriesinsidetheexpe

8、ctati