Brownian Motion and Stochastic Calculus - 2020
Maran, 03.08., 14:00
My exam was identical to the one from Carlo a few minutes later. The two questions, as well as the follow up questions, were the same, namely:
(1) State the uniqueness part in the statement about the solutions to SDE's with globally Lipschitz coefficients and recall the main steps of the proof.
(2) Given a real-valued Brownian motion \(B\) started from the origin, show that there exists \(\lambda < 1\) such that \(P( \max_{t \leq n} \lvert B_t \rvert < 1) \leq \lambda^n\). Then, given a \(d\)-dimensional Brownian motion \(X\) started from the origin, a bounded domain \(D \subseteq \mathbb{R}^d\) containing the origin, show that \(E[T^p] < \infty\) for all \(p\), where \(T\) denotes the exit time of \(X\) from \(D\).
After answering both questions, he asked me how the probability \(P(\sigma < t)\) decays when \(t \to \infty\), where \(\sigma\) denotes the hitting time of 1 and how to show that \(E[\sigma] = \infty\).
He's a very pleasant examiner and doesn't get too hung up on small details. When you don't know how to proceed at some point he will provide some useful hints.
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 containing 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.
