27 06 2024 E TS

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PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartA June,272024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. 1.GivethedefinitionofrealBrownianMotion. 2.Showthatitisagaussianprocess. 3.Showthatithasacontinuousmodification. 4.HavethepathsofaBrownianMotionfinitevariation?Justifytheanswer. Question2. Givearesultofexistenceanduniquenessforthesolutionofastochasticdifferentialequation.Specify theassumptionsandgiveasketchoftheproof. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB June,272024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardBrownianMotion.Foranya>0,b>0and t≥0let Lt(a,b)=E 󰀗 exp 󰀕 −aB2 t−b2 2 󰁝t 0 B2 udu 󰀖󰀘 1.(1pt)Showthat,foranyZ∼N(0,1)andc>0,itholds E[exp(−cZ2)]= 1√1+2c. Then,computeLt(a,0)foranya>0. 2.(3pt)Findaprocessψ∈M1 locsothatthefollowingprocessZisamartingale: Zt=exp 󰀕 −b 󰁝t 0 BudBu−1 2 󰁝t 0 ψudu 󰀖 . 3.(2pt)ExpressZtintermsoftherandomvariablesBtand 󰁕t 0B2 uduonly,anddeducethat Lt(a,b)=E 󰀗 Ztexp 󰀝󰀕b 2−a 󰀖 B2 t 󰀞󰀘 exp 󰀕 −bt 2 󰀖 Hint:Computeexplicitely 󰁕t 0BudBu. 4.(2pt)Fixt≥0.ConstructaprobabilitymeasureQton(Ω,Ft)underwhichtheprocessW= (Ws)s∈[0,t]definedby Ws=Bs+b 󰁝s 0 Budu isaBrownianmotion. 5.(2pt)Showthat,forallt≥0, Bt=e−bt 󰁝t 0 ebudWu. 6.(2pt)DeducethelawofBtunderQt. Solutions 1.Bydirectcomputations,wehave E[e−cZ2 ]= 󰁕∞ −∞e−cz2 1√ 2πe−z2/2dz= 󰁕∞ −∞ 1√ 2πe−z2 2(1+2c)dz = 1√1+2c 󰁕∞ −∞ √1+2c√ 2πe−z2 2(1+2c)dz= 1√1+2c sincethefunction √1+2c√ 2πe−z2 2(1+2c)isthedensityfunctionofagaussianrandomvariableofmean0 andvariance 1√1+2canditsintegraloverRisequalto1.…