Problem 2. Exponential of BM. Let (W+)to be a Brownian Motion and let a R. (a) Find an expression with integrals for eaWt using It formula. (b) Let r(t) = E[eW.]. Using part (a), find a differential equation satisfied by the function z(t). (Hint: take the expectation in part (a) and differentiate with respect to t). (c) Solve the differential equation in (b) and prove that the moment generating function of Brownian motion is given by EleWi] = e. Problem 2. Exponential of BM. Let (W+)to be a Brownian Motion and let a R. (a) Find an expression with integrals for eaWt using It formula. (b) Let r(t) = E[eW.]. Using part (a), find a differential equation satisfied by the function z(t). (Hint: take the expectation in part (a) and differentiate with respect to t). (c) Solve the differential equation in (b) and prove that the moment generating function of Brownian motion is given by EleWi] = e