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,82025 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beaBrownianMotion. 1.ShowthatBadmitsacontinuousmodification. 2.ShowthatthepathsofaBrownianMotiondonothavefinitevariationinanytimeintervala.s. 3.Explaintheconsequenceoftheresultatpoint2.intheconstructionofthestochasticintegral. Question2. Stateandprovetheexistenceanduniquenesstheoremforstrongsolutionsofastochasticdifferential equationunderlocally-Lipschitzassumptionsonthecoefficients. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB January,82025 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardBrownianMotion.Wedefinethestochastic processes Xt=4t−4B2 t−32 t 0 B2 sds, Yt=eXt. 1.ShowthatXtisanItoprocess. 2.CalculatethestochasticdifferentialofXt. 3.DeterminethequadraticvariationandthefirstvariationofXt. 4.CalculatethestochasticdifferentialofYt. 5.ShowthatYtisalocalmartingale. 6.ShowthatYtisamartingale. 7.FixedT>0,findaprobabilityQequivalenttoP|FT anddefineintermsofBtanewprocess (Bt)t∈[0,T]suchthatitisanFt-standardcontinuousBrownianMotionunderQ. Solution 1.WecanwriteXt=Ut+VtwhereUt=−4B2 tandVt= t 0(4−32B2 s)ds.UtisanItoprocesssince Ut=f(Bt)andthefunctionf(x)=−4x2belongstothespaceC2(R).VtisanItoprocesswith Gt=0andFt=4−32B2 t∈M1 loc[0,T]sincethepathsoftheprocessB2 tarecontinuous.Hence theprocessXtisanItoprocessassumoftwoItoprocesses. 2. dXt =−8BtdBt−4dt+4dt−32B2 tdt =−32B2 tdt−8BtdBt, X0=0 3.Wehavethat <X>t= t 0 64B2 tds.…
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