Brownian Motion and Stochastic Calculus - 2019

Mirja, 21.08.2019, 08:30-09:10

I was given two question to prepare.

The first one was to show that if a solution to the Dirichlet Problem with boundary value f exist then it is equal to $$U(x) = E_{x}[f(B_{\tau})]$$ where \( B \) is a BM started from \( x \) and \( \tau \) is the first time that the BM \( B \) hits de boudary of D.

The second question was an application of the optional stopping theorem. Let \( B \) be a one dimensional BM started from 0 an \( \lambda > 0 \). Consider the exit time \( \tau \) of \( [-1,1] \) by B and let \( \sigma \) be the first time such that \( B_t =1 \). Calculate: $$ E[e^{ - \lambda \tau}] \text{ and } E[ e^{- \lambda \sigma}].$$

In the exam I first presentet my solutions to the two question. Then he asked me about Cameron-Martin spaces and I had to state Girsanov's Theorem. After this I had to state Kolmogorov’s continuity criterion and he asked me where we need that we have a power of \(1 + \epsilon \) in the assumption \( E[|X_{t+h} - X_t|^{\alpha}] ≤ c h^{1+\epsilon}\).

Patrick, 21.08.2019, 09:10-09:50

My exam was more or less identical to the one of Mirja. So the two question to prepare were:

- (Proposition 5.3)

Show that if a solution to the Dirichlet Problem with boundary value \(f \) exist then it is equal to $$U(x) = E_{x}[f(B_{T})]$$ where \( B \) is a BM started from \( x \) and \( T \) is the first time that the BM \( B \) hits de boundary of \(D\).

- (Application of the optional stopping theorem)

Let \( B \) be a one dimensional BM started from 0 an \( \lambda > 0 \). Consider the exit time \( \tau \) of \( [-1,1] \) by B and let \( T \) be the first time such that \( B_t =1 \). Calculate: $$ E[e^{ - \lambda \tau}] \text{ and } E[ e^{- \lambda T}].$$

When the exam started I first presented the detailed proof of Proposition 5.3. In the preparation time, I didn't manage to prepare the solution of the second question. That didn't seem to be a problem to Prof. Werner, so first I just told him by voice what tools I would use to compute these two expectations (optional stopping theorem and exponential martingales..). Then he asked me if I could use these ideas to compute concretely \( E [e^{- \lambda T}] \). I did the computations and he was happy.

After that, he asked me to talk about the Cameron-Martin space. I introduced the topic as it is done in the Lecture Notes (...for which \( h : [0,1] \to \mathbb{R} \) do we have that the law of \( (B_t)_{t \in [0,1]} \) is absolutely continuous with respect to the law of \( (B_t + h(t))_{t \in [0,1]} \)? It is not the case for \( h(t) = t ^{1/3} \) ). Then I stated Definition/Proposition 6.4. He asked me about the Radon-Nikodym derivative and to state Girsanov's theorem.

Then we changed topic and he asked me to state Kolmogorov's continuity theorem. He asked me to explain what a continuous modification is, and why we need to have the power \( 1 + \varepsilon \) in the assumption of Kolmogorov's theorem. As last question he asked me if i knew the analogues for random scalar fields (see Remark 2.2), in particular he wanted to know how this power would look like (\(d+\varepsilon \)).

Werner is as always very relaxed and creates a nice atmosphere during the exam. He gives time to think and useful hints if one is stuck.

Skander, 27.08.2019 08:30-9:10

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

$$ \begin{cases} \Delta H = 2 \\ H = 0 \ \mathrm{on} \ \partial D \\ \end{cases}$$ Can one recover in one dimension the formula for \(E_x[T]\) where \(T=inf{t>0:B_t is -1 or 1}\)