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.