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.

Robin, 08.08.2018, 07:20-08:00

I got the following two tasks for preparation:

• State Girsanov's theorem and prove it.

• Let \(D\) be a bounded open domain in \(\mathbb R^2\) with regular boundary. What do you know about the solution of the problem

$$ \begin{cases} \Delta H = 2 \lambda H \\ H = 1 \ \mathrm{on} \ \partial D \\ H \ \mathrm{continuous \ on} \ \bar D \end{cases}$$ for a given \(\lambda \in \mathbb R\).

After discussing these two tasks he asked me some further questions:

• The statement of Cameron-Martin and why it is no longer valid if the derivative of \(h\) is not square integrable.

• The statement of Yamada-Watanabe.

• The statements about existence of weak solutions (first in general and then for the one-dimensional case).

• The statement of the Lévy characterisation of a Brownian motion.

So, the focus of the exam was mainly on the chapters about SDEs. If one cannot solve the tasks Werner helps with some hints and tries to guide you to the solution. For example, I could not figure out the eigenfunction problem, so he gave me the solution and I had to apply Ito's formula to show that this function must be the solution.

Beat, 09.08.2018, 07:00-07:40

I had two questions. First, I had to state Itô's Formula with the proof. The second question was an exercise to prove that for f continuous, \(P(\sup_{[0,1]}|B_t - f|<\varepsilon)>0\).

Janine, 09.08.2018, 08:20-09:00

I had to prepare the construction and definition of the stochastic integral for progressively measurable variables and the exercise was to show that \(B_t^3-3tB_t\) is a local martingale using Itô.

Then he asked me about Cameron-Marting space. I told him, I couldn't rememeber, so he skipped it and asked me more about harmonic functions: Existence and uniqueness of solutions to the Dirichlet problem. I also had to state Blumenthal's 0-1 law. Then I had to talk about 3-dimensional Brownian motion at infinity.

David, 10.08.2018, 7:20-8:00

I had the following tasks to prepare:

• Proof that a local martingale with finite variation is a.s. zero

• For \(T_a\) the hitting time of \(a\) of a one dimensional BM, calculate \(P[T_a < T_b]\) and \(E[T_a \wedge T_b]\) for \(a<0<b\)

• The semimartingale decomposition of \( B_t^2 e^{B_t + t} \)

Then he asked about the statement of Girsanov's theorem, and which existence theorems for solutions of SDEs it implies. Next I should give an example of a SDE which has random explosion time (apparently one can add a certain random term to \(y' = y^2 \) and then the explosion time is random). Afterwards, he wanted to know how mirror coupling works and where we used this. And also Kolmogorov's continuity criterion with idea of the proof.

Patrick, 11.08.2018, 8:40-9:20

For the 20 minutes of preparation I was given 3 questions that hadn't any particular correlation:

• The first question was a pretty simple application of the optional stopping theorem (simple version): It asked to consider an one dimensional Brownian motion started from 0, and compute \( E (e^{ \lambda B_T}) \), where \( \lambda >0 \) and \( T \) was the first time \(t\) for wich \( B_t = 1 \).
• The second question asked to prove the uniqueness of the quadratic variation for local martingales.
• The third question was a strange variation of the Dirichlet problem for a bounded domain with regular boundary.

After having discussed these 3 questions (we skipped question 3 since I had no idea..), he asked me to state Itô's formula, apply it to a concrete case, he asked me to state and prove Levy's characterization Theorem (for the case of an one dimensional Brownian motion) and finally to state the Kolmogorov's continuity criterion (with the idea of the proof).

Prof. Werner is generally relaxed and let you enough time to think without set you under pressure. Anyway, I had the impression that he was a little bit tired and unmotivated.