STAT4207/5207 Homework assignment 4 Due by April 15, 2021. Write down the complete derivations for full credits. Question 3 and 4 are bonus questions (worth 12 and 13 points respectively). Exercise 1 (55 points) For a standard Brownian motion (Bt : t ≥ 0) a) Find P(B2 − B1 > 1|B0.5 = 2). b) Consider the process Xt = Bt − tB1 0 ≤ t ≤ 1. i) Derive E[Xt] and Cov(Xt, Xs) for some fixed t, s ∈ [0, 1] and specify the joint distribution of Xt and Xs. ii) Calculate P(X0.15 > 0.2). iii) Find E[Xs|Xt = 1] for some fixed s, t ∈ [0, 1] such that 0 ∈ R, σ > 0. Derive E[Xt] and Cov(Xt, Xs) for some fixed t, s ∈ [0, 1] and specify the joint distribution of Xt and Xs. d) Consider the process Xt = e µt+σBt t ≥ 0. for constants µ ∈ R, σ > 0. i) Prove that, for Z ∼ N (0, 1), E[e λZ] = e λ 2/2 ∀λ ∈ R. ii) Use your previous answer and the self-similarity property of Brownian motion to show that EXt = e (µ+0.5σ 2 )t . iii) Derive Cov(Xt, Xs) for some fixed t, s ∈ [0, 1]. iv) Give the distribution of Xt for some fixed t > 0. e) Find the distribution of the random variable (Bt − Bs) 2 (Bv − Bt) 2 for 0
