Brownian Motion and Stochastic Calculus - 2020

Carlo, 03.08., 14:40 The two questions were the following: (1) State the uniqueness part in the statement about the solutions to SDE's in the globally Lipschitz case and recall the main steps in the proof. (2) Given a BM $$B$$ in 1d started from the origin, show that there exists $$\lambda <1$$ so that $$P( \max_{t \leq n} \lvert B_t \rvert <1) \leq \lambda^n$$. Then, given a BM $$X$$ started from the origin in d dimensions, and $$D$$ a bounded open domain started from the origin, $$T$$ its exit time from $$D$$, show that for all integers $$p$$, the expectation of $$T^p$$ is finite.

After having solved the first two questions, he asked me if I know how to provide an asymptotic on $$P(\sigma < t)$$ where $$\sigma$$ is the first hitting time of 1, and how to show that the expectation of $$\sigma$$ is infinite. I didn't know how to do it so he gave me some hints, but I needed all the time left anyway.