Boundary crossing identities for diffusions having the time inversion property

TL;DR

This paper introduces boundary crossing identities for diffusions with the time inversion property, providing explicit relations for crossing times and new examples for Brownian motion and Bessel processes.

Contribution

It develops a family of transformations linking boundary crossing times of diffusions with the time inversion property, including new explicit formulas and examples.

Findings

Derived explicit relations for boundary crossing times.

Provided new examples for Brownian motion and Bessel processes.

Connected results to the method of images and asymptotic behaviors.

Abstract

We review and study a one-parameter family of functional transformations, denoted by $(S^{(\beta)})_{\beta\in \R}$, which, in the case $\beta<0$, provides a path realization of bridges associated to the family of diffusion processes enjoying the time inversion property. This family includes the Brownian motion, Bessel processes with a positive dimension and their conservative $h$-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function $f$ to those over the image of $f$ by the mapping $S^{(\beta)}$, for some fixed $\beta\in \mathbb{R}$. We give some new examples of boundary crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.