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 January,102024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. LetX∈M2([0,T]). 1.Givethedefinitionof T 0XsdBs. 2.Showthatthedefinitioniswell-posed. 3.Howcanwedeterminethevalueof T 0XsdBs? Question2. 1.GivethedefinitionofrealstandardBrownianMotion. 2.StateandprovethetheoremofcharacterizationofBrownianMotionintermsofmartingales. 3.StateothercharacterizationsofBrownianMotion. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB January,102024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardBrownianMotion.Considerforsomeλ>0 theprocess Xt:=e−λtBe2λt. 1.ShowthatXtisagaussianprocess. 2.IstheprocessXtacontinuousprocess? 3.AretheincrementsoftheprocessXtindependent? 4.ShowthatXtisastationaryprocess:thismeansthatthelawof(Xt1+h,...,Xtn+h)isequaltothe lawof(Xt1,...,Xtn),foreverychoiceofn∈N,0≤t1<t2<....<tnandh≥0. 5.IstheprocessXtaBrownianMotion?Justifyrigorouslytheanswer. 6.IstheprocessXtboundedinL2? 7.Let(Gt)tthefiltrationdefinedbysettingforallt≥0 Gt=Fe2λt. IstheprocessXtaGt-martingale? 8.Computethea.s.−limt→∞e−λtXt. Solutions. 1.Forevery(t1,...,tm)thevector(Be2λt1,...,Be2λtm)isagaussianvectorsincetheprocess{Bt}tisa BrownianMotion,henceitisagaussianprocess.(Xt1,...,Xtm)isgaussian,beingalinearfunction of(Be2λt1,...,Be2λtm): Xt1 Xt2 ... Xtm = e−λt1 0 ...... 0 0 e−λt2 0... 0 ... ... ...... 0 0...e−λtn Be2λt1 Be2λt2 ... Be2λtn . 2.Forallω∈Ωthetrajectorys→Bs(ω)isacontinuousfunction.Itfollowsthatt→Xt(ω)=…
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