Question 6 [12 marks]. Consider the process Xt = (1 + t)~Bt. t 20, where Be is a standard Brownian motion. (a) Evaluate E[Xtth - Xt(Xt = x] a(t, x) = lim h-0 h to find the drift coefficient of X, [3 marks]. (b) Find the Ito representation of exp(X,) [3 marks]. Now consider the standard geometric OU process Zt = expect + e-t [ eudBu). t20, where Br is a standard Brownian motion. (c) Show that the transition density function of Z, is given by 1 f (y, t|x, s) = (log(y) - e-(t-s)log (x)) = exp yvn(1 - e-2(t-s) ) 1 - e-2(t-s) [3 marks] (d) Show that the stochastic differential equation of the process Z, is given by dZt = (5-log(Zz) ) Zedt + Z,dBy. [3 marks]