Difference between revisions of "Brownian Motion and Stochastic Calculus - 2020"

Revision as of 07:25, 4 August 2020

Maran, 03.08., 14:00 - 14:40

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.

Josua, 04.08, 8:20-9:00

The two questions on the sheet were:

(1)State and prove Kolmogorov's Continuity Criterion

(2) Given a local Martingale M, is there a,b real numbers s.t. some function of M and its QV (sorry Mac-Keyboard) is a Local Martingale

After these two he asked (in no specific order):

(3) State Girsanov's Thm

(4) State sufficient cond s.t. the Exponential Local Martingale is an U.I. Martingale (Novikov & Kazamaki)

(5) Can you prove Novikov's Condition? (uhm... no?)

(6) Def. and sufficient (and necessary) criteria for Cameron Martin space for B.M.

(7) Proof of sufficient criterion for C.M.-space for B.M.

(8) How does one use Girsanov's Thm to find Weak sol for certain SDE's

(9) Does Girsanov imply uniqueness in Distr. for some SDE's

(10) How can one apply Stoch. Calculus to find Sol to ODE's which do not satisfy usual regularity cond.

Rmk: As was stated numerous times, Prof. Werner is always relaxed and very kind. Also if you are considering bringing a mask, although he wears one to go from one room to the other, you don't need one in the actual exam room. I brought one and the assistant told me that i can take it off. So if you wear one anyways - nice, else... still wear one on the way to the exam...?