On Boundary Crossing Probabilities for Diffusion Processes

TL;DR

This paper links boundary crossing probabilities with first passage time densities for diffusion processes, providing a product-form expression and differentiability results under broad conditions.

Contribution

It introduces a novel relationship between asymptotic crossing probabilities and first passage time densities, with explicit formulas and differentiability analysis.

Findings

Derived a product-form for first crossing time density.

Established Gateaux differentiability of boundary crossing probability.

Provided explicit representation of the derivative of crossing probability.

Abstract

In this paper, we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and first passage time densities for diffusion processes. Namely, we show that, under broad assumptions, the first crossing time density of a general curvilinear boundary by a general time-homogeneous diffusion process has a product-form, the factors being the transition density of the process and the coefficient of the leading term in the asymptotic representation of the non-crossing probability of the boundary by the respective diffusion bridge (as the end-point of the bridge approaches the boundary). Using a similar technique, we also demonstrate that the boundary crossing probability is a Gateaux differentiable function of the boundary and give an explicit representation of its derivative.