J 1

D. Suppose that X is a generalized Weiner process dX = di + dW(t), where W(t) is a Brownian motion. What is the probability that X ever reaches -1? Solution: To solve this problem, we again can use the equation E exp(-2mX) =1 from the previous problem with m =1. It may not be obvious since we only have one apparent boundary, -1. To apply the stopping time, we also need a corresponding positive boundary. To address this problem, we can simply use too as the positive boundary and the equation becomes