First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

TL;DR

This paper investigates boundary crossing probabilities for diffusion processes, establishing bounds and densities for first passage times, with extensions to time-inhomogeneous cases and numerical illustrations.

Contribution

It provides new bounds and existence results for first passage time densities for both homogeneous and certain inhomogeneous diffusions.

Findings

Bounds for approximation errors of crossing probabilities

Existence and bounds for first passage time densities

Extension to time-inhomogeneous diffusions

Abstract

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to whole real line this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.