First passage times of two-correlated processes: analytical results for the Wiener process and a numerical method for diffusion processes

TL;DR

This paper derives analytical results for the joint density of first passage times in correlated Wiener processes and introduces a numerical method for general diffusion processes, with convergence proof and practical examples.

Contribution

It provides explicit formulas for Wiener processes and a new numerical algorithm with convergence proof for general diffusion processes.

Findings

Explicit formulas for Wiener process passage times

A convergent numerical method for diffusion processes

Illustrative examples demonstrating the method's effectiveness

Abstract

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 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 bivariate diffusion processes. We derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.