31 01 2023 E TS

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Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part A January, 31 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Question 1 . Give the definition of quadratic variation of a stochastic process. What is the quadratic variation of a Brownian Motion? Show rigorously how we obtain it. Question 2 . State and proof the stochastic Leibniz rule. Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part B January, 31 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Exercise 1 . GivenB = (Ω,F,Ft,B 1(t),B 2(t),P ), a standard continuous 2-dimensional Brownian Motion, we consider the real stochastic processes Y1(t) := ∫t 0B1(s)dB1(s), t ≥ 0, Y2(t) := ∫t 0 √sdB 2(s), t ≥ 0, Xt := Y1(t)·Y2(t), t ≥ 0. 1. Prove that X is an Itˆ o process and find its stochastic differential. 2. Compute the quadratic variation <X >t. Is it decreasing, constant or increasing? 3. Prove that X is a continuousFt-martingale in L2(Ω). 4. Compute the mean E[Xt]. Is it decreasing, constant or increasing? 5. Compute the variance Var (Xt). Is it decreasing, constant or increasing? 6. Compute the sup 0≤t≤TE[X2 t ]. 7. Find a bound for the E[sup0≤t≤TX2 t ]. Solution. 1. We can observe that Y1 = ∫t 0B1(s)dB1(s) and Y2 = ∫t 0 √sdB 2(s) are two Ito processes. Endeed B1 is a process in M2[0,T ], since E ∫t 0|B1(s)|2ds = ∫t 0E|B1(s)|2ds = ∫t 0sds = t2 2 <∞. So it belongs to M2 loc[0,T ]. Moreover √ t is a square integrable function and it belongs to M2 loc[0,T ] ∫ t 0 |√s|2ds = t2 2. It follows that X is an Ito process being the…