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 \( E[|X_{t+h} - X_t|^{\alpha}] ≤ c h^{1+\epsilon}\).