Independent factorization of the last zero arcsine law for Bessel processes with drift

TL;DR

This paper extends the last zero distribution law for Bessel processes with drift, showing it as a product of independent random variables, and introduces a new additive decomposition for Bessel process squares.

Contribution

It generalizes the last zero distribution law from Brownian motion to Bessel processes with drift and proposes a novel additive decomposition method.

Findings

Last zero distribution matches product of independent variables.

Extension from Brownian motion to Bessel processes with drift.

Introduces a new additive decomposition for Bessel process squares.

Abstract

We show that the last zero before time $t$ of a recurrent Bessel process with drift starting at $0$ has the same distribution as the product of an independent right censored exponential random variable and a beta random variable. This extends a recent result of Schulte-Geers and Stadje (2017) from Brownian motion with drift to recurrent Bessel processes with drift. Our proof is intuitive and direct while avoiding heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.