A Geometrical Characterization of Multidimensional Hausdorff Polytopes with Applications to Exit Time Problems

Abstract

We present a formula for the corner points of the multidimensional Hausdorff polytopes and show how this result can be used to improve linear programming models for computing, e.g., moments of exit time distributions of diffusion processes. Specifically, we compute the mean exit time of two-dimensional Brownian motion from the unit square, as well as higher moments of the exit time of time-space Brownian motion, i.e., the two-dimensional process composed of a one-dimensional Wiener process and the time component, from a rectangle. The corner point formula is complemented by a convergence result, which provides the analytical underpinning of the numerical method that we use.