Brownian Motion and Stochastic Calculus - 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. \( (M)^4+a*(M)*(\langle M \rangle)^2+b*(\langle M \rangle)^2\) (or at least something similar to this) is a local Martingale. (Rmk: for a=-6 and b=3 or something this is actually the fourth Hermite Polynomial So answering with the exercise sheet instead of Itô might be neat but also awfully pretentious)

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...?

Tobias, 04.08, 9:00-9:40

Questions to prepare:

- Give some condition such that there exists a solution to the Dirichlet problem for all continuous boundary functions \(f\).

- Exercise 11.4: Explain why the ODE \(dX_t=dB_t-X_t dt, X_0=0\) has a unique solution given by the Ornstein-Uhlenbeck process \(X_t=\exp (-t)\int_0^t \exp(s) dB_s\).

He didn’t want me to write down the details for the mean value property part. So I told everything orally. After answering the first question, he wanted to know if the regular boundary is necessary, so I gave a counterexample.

For the second question I said that \( \sigma(x)=1\) and \(b(x)=-x\) are Lipschitz and it follows by the Theorem. Then I proved that \(X\) solves the SDE in differential notation (he did not want further details).

He then wanted to know if we instead start from a standardnormal r.v. \(N\) (instead of 0), if \(X_t\) has normal distribution. He told me that the solution is given by \(X_t=\exp(-t)(\int_0^t \exp(s)dB_s+N).\)

I had no idea so he asked me if \(X_t\) is a time changed BM. I told him it is as a stochastic integral is a continuous martingale. However it I forgot the \( \exp(-t)\) (which makes it to a supermartingale). However it still has a normal distribution. He then explained that this can for example be seen by the approximation of the stochastic integral choosing a nested sequence of subdivisions.

He then changed topic and wanted to hear the statement of Girsanov and Cameron-Martin (orally as there was no time left).

Emanuele, 07.08, 8:00-8:40

Questions to prepare:

1) Statement and proof of Donsker's Theorem

2) Prove some properties of \( \sup B_t -t \), very similar to exercise 2 i),ii) of exercise sheet 6

I used all the time of the exam to present the solution to the above problems and so there was no follow up questions.

Selim, 07.08, 8:20-9:00

Questions to prepare (same as Emanuele)

1) Statement and proof of Donsker's Theorem

2) Prove some properties of \( \sup B_t -t \), very similar to exercise 2 i),ii) of exercise sheet 6

These questions took a lot of time so after that, I only had to state Girsanov's theorem and explain what happens when M is a Brownian Motion.

Kirill, 07.08, 09:00-9:40

Two questions to prepare:

1) Explain how do we construct the stochastic integral of a progressively measurable process with respect to Brownian Motion

2) calculate the semimartingale decomposition of \(t^2 \int_0^t B_s dB_s + B_t \exp (B_t)\)

In 1) we discussed the construction without proving the density of elementary processes and completeness of the space of L2 martingales, just mentioned it.

In 2) the trick is to define \(Z_t = \int_0^t B_s dB_s\) and use Ito for \(f(Z,B,t)=t^2Z_t + B_t \exp (B_t)\) and \(d Z_s = B_s d B_s\)

Then we talked about Donkers Theorem without proving it and also about Tanaka's SDE without calculations

Thea, 07.08, 10:30-11:10

Questions to prepare:

1) Main steps of proof of existence of solution to an SDE with globally Lipschitz coefficients

2) Calculation of semimatingale decomposition of \( (t^2+B_t)\exp(B_tx) \)

I explained 1), he also wanted to see the calculation of the proof.

Then I wrote part of the solution to 2), but both of us agreed that it was rather boring.

Then he asked me about Lévy's characterization theorem and its proof.