08 06 2023 E TS

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Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part A June, 8 2023 c⃝I diritti d’autore sono riservati. Ogni sfruttamento commerciale non autorizzato sar` a perseguito. Surname, Name, Matricola Question 1 . Give the definition of continuous time martingale. Is the Brownian Motion a continuous time martingale? Giustify the answer. Give a characterization of Brownian Motion in terms of martingales. Prove the result. Question 2 . State and proof the Ito Isometry for a process X∈M2[0,T ]. Politecnico di Milano - Scuola di Ingegneria Industriale e dell’Informazione Stochastic Differential Equations - Part B June, 8 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 ) be a standard continuous real Brownian Motion and let St a geometric Brownian Motion with dynamic dSt =St(bdt +σdBt) S0 = 1 where b and σ are constants. 1. Show that for every o≤to≤t lnSt = lnSt0 + (b− σ2 2 )(t−t0) +σ(Bt−Bt0), 0≤t0≤t. (1) Let us define G(t,T ) := 1 T ∫T t (Br−Bt)dr 2. Show in rigorous and detailed way that, for every fixed t and T , G(t,T ) is a gaussian random variable independent ofFt. 3. Compute the E[G(t,T )|Ft] and Var (G(t,T )|Ft). Fort> 0 let At = 1 t ∫t 0 lnSsds. 4. Is At a gaussian process? 5. Show that the following equality holds AT = t TAt + (1− t T )[lnSt + 1 2(b− σ2 2 )(T−t)] +σG(t,T ). 6. Deduce that AT = Zt +U where Zt is an Ft-measurable random variable and U is a gaussian random variable independent ofFt. 7. Compute E(eAT|Ft) and show that E(eAT|Ft) =α(t)eZt. Determine α(t). Solution. 1. The process S is a geometric Brownian Motion. So St =S0e(b− 1 2σ2)t+σBt and for s>t Ss St = S0e(b− 1…