07 02 2025 E TS

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PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartA February7,2025 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. StateandprovetheItoisometryforaprocessX∈M2([0,T]). Question2. StateandprovetheFeynman-Kacformula. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB February7,2025 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetXbeagaussianrandomvariablewithmeanµandvarianceσ2. 1.Showthat E[cos(X)]=e−σ2 2cos(µ). (Hint:Recallthatcos(x)=eix+e−ix 2 ). LetB=(Ω,F,(Ft)t,(Bt)t,P)bearealstandardBrownianMotion.Letusconsidertherealstochastic process: Xt:=cos(Bt),t≥0. 2.ComputeE[Xt|Fs]forallt,s≥0. (Hint:cos(α+β)=cos(α)cos(β)−sin(α)sin(β)) 3.IsXtan(Ft)t-martingale? 4.ProvethatXtisanItˆoprocessandfinditsstochasticdifferential. FixT>0anddefinetheprocess Mt:=E[XT|Ft],t≥0. 5.IsMtamartingale?Justifyrigorouslytheanswer. 6.DiscusswhetherMtconvergesP-a.s.andinL1fort→+∞and,ifso,determinethelimit. 7.FindastochasticprocessG∈M2([0,T])suchthat: XT=E[XT]+ 󰁝T 0 GtdBt. Solution 1.Weevaluatetheexpectation: E[cos(X)]=E 󰀗eiX+e−iX 2 󰀘 =1 2E 󰀅 eiX󰀆 +1 2E 󰀅 e−iX󰀆 =1 2φX(1)+1 2φX(−1) whereφXdenotesthecharacteristicfunctionoftherandomvariableX.Now,recallingthatif X∼N(µ,σ2)then φX(t)=E 󰀅 eitX󰀆 =eiµt−σ2 2t2 , weobtain: E[cos(X)]=1 2eiµ−σ2 2 +1 2e−iµ−σ2 2 =e−σ2 2cos(µ). 2.For0≤s≤twehave E[Xt|Fs]=E[cos(Bt−Bs+Bs)|Fs] =E[cos(Bt−Bs)cos(Bs)|Fs]−E[sin(Bt−Bs)sin(Bs)|Fs] =E[cos(Bt−Bs)|Fs]cos(Bs)−E[sin(Bt−Bs)|Fs]sin(Bs) =E[cos(Bt−Bs)]cos(Bs)=e−(t−s) 2 cos(Bs). Infact,bythepropertiesoftheBrownianMotionandbythecontinuityofthecosineandsine…