Exercise 5.5. Let B, be a standard Brownian motion with B = 1. Let T= min{t:...

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Exercise 5.5. Let B, be a standard Brownian motion with B = 1. Let T= min{t: B =0} Let r > 0 and let X = B. 1. Find a function g such that M =X, exp exp{9(x) ds} is a local martingale for t < T. (Do not worry about what happens after time T.) 2. What SDE does M, satisfy? 3. Let Q be the probability measure obtained by tilting by M. Find the SDE for B, in terms of a Q-Brownian motion. Exercise 5.5. Let B, be a standard Brownian motion with B = 1. Let T= min{t: B =0} Let r > 0 and let X = B. 1. Find a function g such that M =X, exp exp{9(x) ds} is a local martingale for t < T. (Do not worry about what happens after time T.) 2. What SDE does M, satisfy? 3. Let Q be the probability measure obtained by tilting by M. Find the SDE for B, in terms of a Q-Brownian motion.