Stochastic calculus

We let B- be the standard Brownian motion. (a) Prove the following facts using Ito's isometry (i) (5 marks) Show that for any stochastic processes Xt E II2 and Y E I12, [. x, aB. ) (J Y,dB. ) = [ E[X, Y,]ds. (ii) (5 marks) We let Xt E I12 be a stochastic process and let X, (s), X2(s), ... be a sequence of stochastic process such that lim E IXn(s) - Xs/2ds = 0. n-+00 Show that So Xn(s)dBs - So XsdBs in L2-norm. (b) (5 marks) Compute the expectation Express your answer in terms of t. (c) (Harder, 10 marks) We let Yt = JesdB, for t 2 0 and define Zt = YsdBs Calculate IE[Zt, Ztz] for any 0 S t, S t2. (Hint: You can first think the special case when t, = t2.)