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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.

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.

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Additional indexing

Other titles:Some random times and martingales associated with $BES_{0}(\delta)$ processes $(0<\delta<2)$
Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2006
Deposited On:20 Jan 2010 11:06
Last Modified:21 Nov 2017 14:21
Publisher:Instituto Nacional de Matematica Pura e Aplicada, Brazil
ISSN:1980-0436
Official URL:http://alea.impa.br/english/index_v2.htm
Related URLs:http://arxiv.org/abs/math/0505423

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