Study material for STOCHASTIC DIFFERENTIAL EQUATIONS, shared by the Studwiz community and reviewed by moderators.
Study material for STOCHASTIC DIFFERENTIAL EQUATIONS, shared by the Studwiz community and reviewed by moderators.
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PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartA September,22024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. GivethedefinitionofrealBrownianmotionanddescribeitsmainproperties. WhatisanaturalBrownianmotion?andastandardBrownianmotion? LetnowB=(Ω,F,(Ft)t,(Bt)t,P)beaBrownianmotion. Let(F′ t)tafiltrationlargerthan(Ft)t,i.esuchthat Ft⊂F′ tforeveryt≥0. IsBaBrownianmotionwithrespectto(F′ t)t? Denote,asusual,(¯Ft)ttheaugmentedfiltrationof(Ft)t. IsBaBrownianmotionwithrespectto(¯Ft)t? Question2. StateIto’sformulaTheoremandgiveasketchoftheproof,underliningthemainpoints. Showatleastanapplication. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB September,22024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardrealBrownianMotion.Letusconsiderthe process bt=Bt−tB1, t∈[0,1]. 1.Showthat(bt)t∈[0,1]isaGaussianprocess.Showthattheprocess(bt)t∈[0,1]isindependentofthe randomvariableB1. 2.Isbacontinuousprocess? 3.ComputeE(bt),Var(bt). 4.ComputeE(bt·bs). 5.ConsidertheprocessX=(Xt)t∈[0,1]definedas Xt=b1−t,t∈[0,1]. ShowthatXisequivalenttob. 6.IstheprocessXamodificationofb?Aretheyindistinguishable? 7.ShowthattheprocessY=(Yt)t≥0definedbyYt=(1+t)bt/(1+t),t≥0,isBrownianmotion.With respecttowhichfiltration? Solutions 1.Foreverychoiceof(t1,...,tn)thevector(bt1,...,btn)isgaussian,beingalinearfunctionof(Bt1,...,Btn,B1). {(bt,t≥0)∪B1}isagaussianfamilyofrandomvariablesand∀t∈[0,1] E[btB1]=E[(Bt−tB1)B1]=t∧1−t=0. Thetwor.v.’sbtandB1,beingjointlygaussiananduncorrelated,areindependentforeveryt≤1.…
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