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.