4. (5 marks) Ito calculus. (a) Let X(t) and Y (t) be two stochastic processes, such that dx = ux (X(t), t)at + ox (X(t), t )dZx dy = my (Y (t), t )di +oy (Y (t), t )dZy with Zx(t), Zy(t) being Brownian motions. Let X(t;) = X, and Y(t;) = Y;. Show that (Xitl - Xi)(Viti - Yi) = XitIViti - XiVi - Xi(YitI - Yi) - Vi(Xit1 - Xi) . Now, using the definition of the Ito integral which is the limit of a discrete sum, show that X(s)dY (s) = [XYh - Y(s)dX(s) - dX(s)dY(s). (b) Let Z(t) be a Brownian motion. Using the result in part (a), show that (assuming Ito calculus) Z(8) dZ(s) = t91 Z (t)2