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,72024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Question1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardBrownianMotion.Letusconsiderthestochastic processIt= t 0XsdBsfort∈[0,T]andX∈M2[0,T]. 1.ShowthatItisboundedinL2. 2.IsItanFt-martingale?Justifytheanswer. 3.ProvethatthequadraticvariationoftheprocessItis <I>t= t 0 X2 sds. Question2. 1.Givethedefinitionofstrongandweaksolutionofastochasticdifferentialequation. 2.Givethedefinitionofpathwiseuniquenessanduniquenessinlawforthesolutionofastochastic differentialequation. 3.Stateandprovethetheoremofexistenceofaweaksolutionforastochasticdifferentialequation. PolitecnicodiMilano-ScuoladiIngegneriaIndustrialeedell’Informazione StochasticDifferentialEquations-PartB June,72024 c©Idirittid’autoresonoriservati.Ognisfruttamentocommercialenonautorizzatosar`aperseguito. Surname,Name,Matricola Exercise1. LetB=(Ω,F,(Ft)t,(Bt)t,P)beacontinuousstandardBrownianMotion.Foranyx>0,letZ=(Zt)t≥0 bedefinedby Zt=xe−t+e−tMt,Mt= √ 2 t 0 esdBs. 1.(2pt)ShowthatZtisanItoprocessandcomputeitsstochasticdifferential. 2.(1pt)ComputethevariationV0 t(Z)andthequadraticvariation〈Z〉t. 3.(1pt)Isitamartingale? 4.(2pt)IsitGaussian?IncaseitisGaussian,finditsmeanandcovariance. 5.(2pt)WedefinetheprocessV=(Vt)t≥0bysetting Vt=xe−t+e−tBe2t−1. ShowthatVisnotanItoprocesswithrespecttotheBrownianmotionB=(Ω,F,(Ft)t,(Bt)t,P). 6.(2pt)Showforeveryt≥0thattheprocessesZandVareequivalent. 7.(2pt)AretheprocessesZandValsoindistinguishable? Solutions. 1.WecanwritetheprocessZtas Zt=e−t x+ √ 2 t 0 esdBs Thus,Zistheproductofthesmoothdeterministicfunctione−tandtheprocessx+ √ 2 t 0esdBs,…
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