Brownian Motion and Stochastic Calculus - 2018

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Nicholas Dykeman, 07.08.2018, 7:20-8:00

In the preparation room, I was given a piece of paper with two (handwritten) questions on it. The first told me to state and prove Blumenthal's 0-1-Law, and give it's consequences, in particular for regularity of boundary, etc.

The second question was a function of \(B_t\) and \(t\), use Ito to show that this is indeed a local martingale (straightforward, just calculate).

In the exam, I first answered these question (here Werner did not say very much, just listened to what I had to say). After this was done, he asked some other questions. These included Brownian Bridge (definition, what do we use it for, covariance function, independence from \(B_1\)), and then the Fourier decomposition of Brownian motion, and the law of the coefficients (I first spoke about the \(L^2\)-decomposition and how this relates to Brownian motion, and concluded in the special case (Fourier basis is an \(L^2\)-basis)).

Werner is as always very relaxed, and created a really nice atmosphere during the exam.


Marc, 07.08.2018, 10:10-10:50

My given task from the lecture was:

  • Recall the definition and the idea of the construction of the quadratic variation process for a bounded martingale \( (M_t)_{t\geq 0} \).

and the task from the exercises was:

  • Recall the idea of the proof of the fact that almost surely \( \limsup_{t \to 0+} \frac{B_t}{ \sqrt{2t \log \log (\frac{1}{t})} } \geq 1 \)

Afterwards, he asked me a version of a Dirichlet Problem in two dimensions, where:

  • \( D := B(0,R) \setminus B(0,1) \) for \( R>1 \), such that \( \Delta H = 0\) in \(D\), \( H=0 \) on \( \partial B(0,1) \) and \( H=1 \) on \( B(0,R) \)

He wanted a suitable harmonic function for such a problem in two dimensions and an explicit solution to this problem (Chapter 1, Section 5.3).

Then he changed topic and wanted to know how we showed continuity of Brownian motion. Finally, he wanted to hear the statement of Kolmogorov's continuity criterion.

Generally speaking, Prof. Werner is a very nice examiner. He gives you time to think about the new questions which were not given for preparation. He also gives some hints if you don't know how to proceed.