T. (30 points] (a) Let X be a random variable and f be a function of two variables t, X) such that it is integrable for all t and continuously differentiable with respect to t. Assume that |"'\" (m s z E for some random variable Z E L1 then show that a _ a; (b) By taking derivatives with respect to the time parameter of the martingale Mt = exprBt tog/'2), show that i. Bf t, ii. 33 stat iii. a: 6:32 + 3:2 are also martingales. (c) Find E[T2] for T = min{t 3 t] : Bt g (o, M}, a. e t] c: a