Exercise 5.5. Let Bt be a standard Brownian motion with Bo T = min{t : B = 0} Let r > 0 and let X = B. = 1. Let 1. Find a function such that M = Xt exp t { I(X,) ds} is a local martingale for t < T. (Do not worry about what happens after time T.) 2. What SDE does Mt satisfy? 3. Let Q be the probability measure obtained by tilting by Mt. Find the SDE for Bt in terms of a Q-Brownian motion.