Let Xt be a geometric Brownian motion, i.e. dXt = rXtdt + XtdBt, X0 = x > 0 where Bt R; r, are constants. a) Find the generator A of Xt and compute Af(x) when f(x) = x; x > 0, constant. b) If r 1 22 then Xt as t, a.s. Qx. Put = inf{t > 0;Xt R} . Use Dynkin's formula with f(x) = lnx, x > 0 to prove that Ex[] = ln R x r 1 22 . (Hint: First consider exit times from (,R), > 0 and then let 0. You need estimates for (1 p()) ln , where p() = Qx[Xt reaches the value R before ] , which you can get from the calculations in a), b).)