Answer all the questions well.

1 (1) Write down Ito's Lemma as it applies to a function f(X, ) of a stochastic process X, that satisfies the stochastic differential equation dX, =ordB, + 4dt , where By is a standard Brownian motion. (ii) Hence find the stochastic differential equations for each of the following processes: (a) G. = exp(X+) (b) Q = X- (c) V = (1+X,)1 (d) 4 =100+10X (e) J = InB (f) Kt = 5B +2B 2 Let B, (t 2 0 ) be a standard Brownian motion with 8, =0. By first writing down an expression for d(8; ), show that [ B,dB, = }(87 -1) (ii) What is the expected value at time 0 of [ B,dB, ? (iii) What is the expected value at time u (0