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 \(T\) is the first time where a Brownian motion hits -1 or 1.

• What can you say about the SDE:

$$ \begin{cases} dX_t = X_t^2 dB_t + sin(X_t)dt \\ X_0=0 \end{cases}$$ For the first question, write down the assumption of the second problem we saw that is similar to the Dirichlet problem. The second part of the question follows instantly and given that it is in one dimension, one can add a few specifications (exercise sheets). For the second question, use the theorem about existence of strong solutions with Lipschitz and note that the square function used in the first term is only locally so. The fact that it doesn't explode (martingale) allows us to deduce that there is no problem.

Afterwards, he asked me to state Donsker's theorem (the one in the later chapters, not the first one), say that we consider the space of random functions, write down the metric used. He also wanted me to prove the theorem which I did not.

He then asked me to state Girsanov with context and say how it relates to the second exercise (we can switch between spaces using the local martingale equivalence).

Finally, he asked me to state multidimensional Itô and apply it to \(B_t^3-3tB_t\) or something like that.

Prof. Werner and Bálint seemed to be pretty relaxed, although I got the feeling that Prof. Werner was slowly but surely getting awfully bored.