The first crossing-time density for Brownian motion with a perturbed linear boundary

Abstract

An expansion is derived for the density of the first time a Brownian path crosses a perturbed linear boundary α+εf(t). When the perturbation f(t) is a finite mixture of negative exponentials of either sign the expansion is shown to converge for all values of the perturbation parameter ε. Numerical examples suggest that the technique works well for a wider choice of f(t), including cases where f(t) is periodic.