Difference between revisions of "Brownian Motion and Stochastic Calculus - 2020"
| Line 10: | Line 10: | ||
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 \(E[\sigma] = \infty\). | 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. | + | He's a very pleasant examiner and doesn't get too hung up on small details. He doesn't say much when you are presenting the questions you prepared. He will ask the follow-up questions once you're done. When you don't know how to proceed at some point he will provide some useful hints. |
== Carlo, 03.08., 14:40 == | == Carlo, 03.08., 14:40 == | ||
Revision as of 22:55, 3 August 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. He doesn't say much when you are presenting the questions you prepared. He will ask the follow-up questions once you're done. 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.
