# Some random times and martingales associated with BES0(δ) processes (0<δ<2)

Nikeghbali, A (2006). Some random times and martingales associated with BES0(δ) processes (0<δ<2). ALEA Latin American Journal of Probability and Mathematical Statistics, 2:67-89 (electronic).

## Abstract

In this paper, we study Bessel processes of dimension $\delta\equiv2(1-\mu)$, with $0<\delta<2$, 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 $0<\delta<1$, these processes are even not semimartingales. The last time before 1 when a Bessel process hits 0, called $g_{\mu}$, plays a key role in our study: we characterize its conditional distribution and extend Paul L\'{e}vy'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 Az\'{e}ma 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 $g_{\mu}$, a one parameter extension of Az\'{e}ma'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.

In this paper, we study Bessel processes of dimension $\delta\equiv2(1-\mu)$, with $0<\delta<2$, 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 $0<\delta<1$, these processes are even not semimartingales. The last time before 1 when a Bessel process hits 0, called $g_{\mu}$, plays a key role in our study: we characterize its conditional distribution and extend Paul L\'{e}vy'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 Az\'{e}ma 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 $g_{\mu}$, a one parameter extension of Az\'{e}ma'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.

## Citations

Detailed statistics

Other titles: Some random times and martingales associated with $BES_{0}(\delta)$ processes $(0<\delta<2)$ Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2006 20 Jan 2010 11:06 05 Apr 2016 13:24 Instituto Nacional de Matematica Pura e Aplicada, Brazil 1980-0436 http://alea.impa.br/english/index_v2.htm http://arxiv.org/abs/math/0505423
Permanent URL: http://doi.org/10.5167/uzh-21634

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