Mathematics – Probability
Scientific paper
2005-05-19
Alea, 2, p. 67-89 (2006)
Mathematics
Probability
Typos corrected. Close to the published version
Scientific paper
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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