12 01 2023 E TS

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Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part A January, 12 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Question 1 . State and prove the Ito isometry for an elementary process in M2[0,T ]. This isometry is crucial to define the stochastic integral for an integrand process in M2[0,T ]. Why? Question 2 . State an existence and uniqueness result for the solution of the stochastic differential equation { dXt = b(t,Xt)dt +σ(t,Xt)dBt Xu = η Give an example of stochastic differential equation whose coefficients satisfy the assumptions of the stated result. Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part B January, 12 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Exercise 1 . Let B = (Ω,F, (Ft)t, (Bt)t,P ) be a real standard continuous Brownian Motion and U an N(µ,σ 2) distributed random variable independent of (Bt)t. For α∈ R, let Xt =U·t +αBt andGt =σ(Xs,s≤t). 1. Is (Xt)t≥0 a martingale with respect to (Ft)t? 2. Compute Cov(U,Xs),Cov (Xs,Xt). 3. Show that (Xt)t is a gaussian process. 4. Prove that, for every t≥ 0, there exists a number λ (depending on t) and a random variable Y , independent ofGt, such that U =λXt +Y . 5. Compute E[U|Gt] 6. Is Xt aGt-martingale ? 7. Show that lim t→+∞ E[U|Gt] =U a.s. Solution. 1. The process ( Xt)t≥0 is not anFt-martingale since it is not adapted to (Ft)t. Otherwise, we can observe that is not an Ft-martingale since E[Xt] = µt is not constant if µ̸= 0. If µ = 0 we can compute E[Xt|Fs] =E[Ut +αBt|Fs] = 0·t +αBs̸=Xs. Also in this case Xt…