Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling

LPMA, University Pierre et Marie Curie - Paris 6, Paris, France.
mathieu.rosenbaum@upmc.fr

Received: 24 November 2014

Abstract

We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge given in [M. Rosenbaum and M. Yor, Séminaire de Probabilités XLVI (2014) 359–375], together with various relationships between the laws of these four processes. Finally, we consider the variable B_UT1/√T₁ , where B is a Brownian motion, T₁ its first hitting time of level one and U a uniform random variable independent of B. This variable is shown to be centered in [R. Elie, M. Rosenbaum and M. Yor, Electron. J. Probab. 37 (2014) 1–23; M. Rosenbaum and M. Yor, Séminaire de Probabilités XLVI (2014) 359–375]. The results obtained here enable us to revisit this intriguing property through an enlargement of filtration formula.