Some random times and martingales associated with $BES_{0}(δ)$ processes $(0<δ<2)$

Abstract

In this paper, we study Bessel processes of dimension $δ\equiv2(1-μ)$, with $0<δ<2$, and some related martingales and random times. Our approach is based on martingale techniques and the general theory of stochastic processes (unlike the usual approach based on excursion theory), although for $0<δ<1$, these processes are even not semimartingales. The last time before 1 when a Bessel process hits 0, called $g_μ$, plays a key role in our study: we characterize its conditional distribution and extend Paul Lévy's arc sine law and a related result of Jeulin about the standard Brownian Motion. We also introduce some remarkable families of martingales related to the Bessel process, thus obtaining in some cases a one parameter extension of some results of Azéma and Yor in the Brownian setting: martingales which have the same set of zeros as the Bessel process and which satisfy the stopping theorem for $g_μ$, a one parameter extension of Azéma's second martingale, etc. Throughout our study, the local time of the Bessel process also plays a central role and we shall establish some of its elementary properties.