First Hitting Time Distributions for Brownian Motion and Regions with Piecewise Linear Boundaries

Abstract

Explicit formulas for the first hitting time distributions for a standard Brownian motion and different regions including rectangular, triangle, quadrilateral and a region with piecewise linear boundaries are derived. Moreover, approximations to the first hitting time distribution of a standard Brownian motion with respect to regions with general nonlinear continuous boundaries are also obtained. The rules for assessing the accuracies of the approximations are given. We generalize the results of one-sided boundaries which include the general nonlinear continuous boundaries, piecewise linear boundaries, linear boundaries and constant boundaries to a region and give the relationships among the first hitting time distributions. Some numerical examples are presented to illustrate the results obtained in the paper. These formulas can be further extended to compute the first hitting time distributions of a Brownian motion with linear drift with respect to regions.