Question: Stochastic calculus, brownian motions, ito integral Problem 2. Finding Expectations (15 pts) In this problem, we consider a Brownian motion (Wt)tzo. We also consider the

Stochastic calculus, brownian motions, ito integral

Problem 2. Finding Expectations (15 pts) In this problem, we consider a Brownian motion (Wt)tzo. We also consider the process Xt := cos (Wt). (a) Find an expression for Xt as a sum of integrals (Ito and/ or Riemann). (b) Let x(t) := E[X ]. Using part (a), find a differential equation satisfied by the function x(t). (Hint: take the expectation in part (a) and differentiate with respect to t). (c) Find E[cos(Wt)]

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