Renewal series and square-root boundaries for Bessel processes

TL;DR

This paper links renewal series representations of Brownian functionals to the distribution of Bessel process hitting times of square-root boundaries, extending classical Brownian motion results to Bessel processes.

Contribution

It introduces a renewal series approach to analyze Bessel process hitting times, generalizing known Brownian motion results to Bessel processes.

Findings

Derived the law of Bessel process hitting times for square-root boundaries.

Extended classical Brownian motion results to Bessel processes.

Provided a new method using renewal series for these analyses.

Abstract

We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.