Brownian Motion

Michael, 05.02.2020, 09:40-10:20

I was given three question to prepare.

The first one was to show that the following is a martingale $$B^3^_t -3t B_t$$

The second question was to prove Levy characterisation of Brownian motion.

The third question was to prove uniqunes strong solution in the globally Lipschitz case for an SDE.

In the exam I first presentet my solutions to the three question. Then he asked me about one of the related problems to Dirichlet proble. We havee H=0 on the boundary and the Laplacian equal to 1. What can I say about the solution to this problem. Finally he wanted to hear the statement of Donsker's Theorem and wanted to know how we can argue that the max of S_epsilon goes to the max of Brownian motion. He wanted to hear that since it's continous it follows directly by weak convergence.