Computing Moments of the Exit Distribution for Diffusion Processes Using Linear Programming

Summary

We present a new approach to the numerical computation of (higher) moments of the exit time distribution of diffusion processes. The method relies on a linear programming (LP) formulation of a process exiting from a bounded domain. The LP formulation characterizes the evolution of the stopped process through the moments of the induced occupation measure and leads to an infinite dimensional LP problem on a space of measures. Our functional method yields finite dimensional linear programs which approximate the infinite dimensional problem. An important aspect of our approach is the fact that excellent software is readily available.

We shall illustrate the method by looking at some examples. We shall specifically report on numerical computations and results related to change-point detection methods for drifted Brownian motion. Procedures for detecting changes in these models, such us CUSUM, EWMA and the Roberts-Shiryaev procedure, can be described in terms of one-dimensional diffusion processes and lead to exit time problems. Important operating characteristics for all of these procedures are the mean exit time and higher moments of the exit distributions of the diffusion processes.