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

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 [SGS17] 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.