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Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part A June, 6 2025 ©I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Question 1 . Show that the Brownian Motion is a gaussian process. State and prove the theorem of characterization of the Brownian Motion in terms of martingales. Question 2 . Give the definition of solution of the stochastic differential equation  dXt = b(t, Xt)dt + σ(t, Xt)dBt Xu = x ∈ Rn What is a strong solution? Give an example of SDE with strong solution. What is a weak solution? Give an example of SDE with weak solution. Are they equivalent? Justify the answer with a proof or an example. In which sense can we speak of uniqueness? Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part B June, 6 2025 ©I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Exercise 1 . Let B(t) = (B1(t), B2(t)) be a bidimensional continuous standard Ft-Brownian Motion on (Ω , F , P). We define the stochastic processes Xt = t(B2 1(t) + 2B2 2(t)), t ≥ 0. 1. Is Xt a continuous process? Why? 2. Is Xt a progressively measurable process? Why? 3. Compute E[Xt], V ar(Xt) and Cov(Xt, Xs), justifying all steps. 4. Determine the law of the random variable V = X1 − B2 2(1). 5. Compute E[Xt − 3t2|Fs] for all s, t ≥ 0. Is the process Yt = Xt − 3t2 a martingale? 6. Show that Xt is an Ito process. 7. Compute the stochastic differential of Xt. 8. Determine the P-a.s. lim t→+∞ Xt t3 . Solution 1. Xt is a continuous process since it is sum and product of continuous processes ( t, B1(t)2, B2(t)2) and B1(t) and B2(t)…