First passage times of two-dimensional correlated diffusion processes: analytical and numerical methods

Laura Sacerdote, Massimiliano Tamborrino, Cristina Zucca

Published 2012-12-20, updated 2015-03-18Version 4

Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that the these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for any bivariate process. We also derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting some previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.