On last passage times of linear diffusions to curved boundaries

TL;DR

This paper investigates the distribution of the last passage time of linear diffusions to curved boundaries, providing general formulas, computational methods, and examples within Brownian and Bessel processes.

Contribution

It introduces a general expression for the density of last passage times and connects it with first hitting times using martingale techniques and time inversion.

Findings

Derived a general density formula for last passage times.

Applied martingale methods to compute densities for implicit boundaries.

Provided explicit examples in Brownian and Bessel processes.

Abstract

The aim of this paper is to study the law of the last passage time of a linear diffusion to a curved boundary. We start by giving a general expression for the density of such a random variable under some regularity assumptions. Following Robbins & Siegmund, we then show that this expression may be computed for some implicit boundaries via a martingale method. Finally, we discuss some links between first hitting times and last passage times via time inversion, and present an integral equation (which we solve in some particular cases) satisfied by the density of the last passage time. Many examples are given in the Brownian and Bessel frameworks.