On the first passage problem for correlated Brownian motion

Suppose that X=(X1,X2)X=(X1,X2) is two-dimensional correlated Brownian motion. Let τiτi denote the first passage time of XiXi to a fixed level, and ττ the minimum of τ1,τ2τ1,τ2. When XX has zero drift, several distributions of interest are available in closed form, including the joint density of the passage times and the distribution of X(τ)X(τ). Unfortunately these published formulae contain errors, and the corresponding distributions in the presence of drift are not expressible in closed form. The purpose of this paper is to address these issues by presenting corrected formulae and outlining a Monte Carlo algorithm for approximating quantities of interest in the presence of drift.