Linear and Non-linear Boundary Crossing Prob-abilities for Brownian Motion and Related Processes

Abstract

We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear or non-linear boundaries for the Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundaries studied in the literature and a simple application to pricing corporate debt are provided in order to illustrate our method.