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 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.…
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