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 , where , where is a Brownian motion, its first hitting time of level one and a uniform random variable independent of . 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.
Mots-clés : Brownian motion, Brownian bridge, Brownian meander, pseudo-Brownian bridge, Bessel process, uniform sampling, local times, hitting times, enlargement of filtration
@article{PS_2015__19__578_0,
author = {Rosenbaum, Mathieu and Yor, Marc},
title = {Some explicit formulas for the {Brownian} bridge, {Brownian} meander and {Bessel} process under uniform sampling},
journal = {ESAIM: Probability and Statistics},
pages = {578--589},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015009},
mrnumber = {3433427},
zbl = {1333.60181},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2015009/}
}
TY - JOUR AU - Rosenbaum, Mathieu AU - Yor, Marc TI - Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling JO - ESAIM: Probability and Statistics PY - 2015 SP - 578 EP - 589 VL - 19 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2015009/ DO - 10.1051/ps/2015009 LA - en ID - PS_2015__19__578_0 ER -
%0 Journal Article %A Rosenbaum, Mathieu %A Yor, Marc %T Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling %J ESAIM: Probability and Statistics %D 2015 %P 578-589 %V 19 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2015009/ %R 10.1051/ps/2015009 %G en %F PS_2015__19__578_0
Rosenbaum, Mathieu; Yor, Marc. Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 578-589. doi : 10.1051/ps/2015009. http://archive.numdam.org/articles/10.1051/ps/2015009/
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