Some extreme value problems in Lévy-Brownian motion

This talk will present recent results and open questions pertaining to extreme value problems in Brownian motion and more general Lévy processes. First, I will focus on the number of facets on the boundary of the convex hull of d-dimensional Brownian an Lévy motion. Then I will discuss two problems related to first passage times and lead changes for the supremum processes of one-dimensional Brownian and Lévy processes. The emphasis of the talk will be on the use of path-decomposition and path-transformation techniques, and on distribution-free results that may be obtained in this way.