28 06 2023 E TS

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Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part A June, 28 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Question 1 . Discuss the regularity properties of Brownian paths. Are they in some sense related to the definition of the stochastic integral? Justify rigorously the answer. Question 2 . State the existence and uniqueness theorem for a solution of a stochastic differential equation under Assumption A and give a sketch of the proof. Specify in which sense existence and uniqueness hold. Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part B June, 28 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Exercise 1 . LetB = (Ω,F, (Ft)t, (Bt)t,P ) andβ = (Ω,F, (Ft)t, (βt)t,P ) be two real continuous and independent Brownian Motions (in particular, the random variables Bt and βs are independent for every s,t≥ 0). Let us consider the real process X = (Xt)t∈[0,∞) defined by Xt :=B 2 3 t−β 1 3 t. 1. Is the process X continuous? 2. Show that the process X is gaussian. 3. Show that the process X is a real natural Brownian Motion. 4. Is the process X a martingale? With respect to which filtration? Let us define now the real process Y = (Yt)t∈[0,∞) by setting Yt :=β 2 3 t +B 1 3 t. 5. Show that for every t∈ [0,∞), the random variablesXt andYt are uncorrelated, i.e. Cov(Xt,Y t) = 0. Are they independent? 6. Show that for every 0 <s<t< ∞, the random variables Xs and Yt are not independent. [Hint.: consider separately the cases s< t 2 and t 2≤s<t .] 7. Is the process {(Xt,Y t)}t∈[0,∞) with…