Geodesic
Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling
Rosenbaum, Mathieu1 ; Yor, Marc1 
1 LPMA, University Pierre et Marie Curie - Paris 6, Paris, France.
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 578-589

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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 U T 1 / T 1 , where , where B is a Brownian motion, T 1 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.

DOI : 10.1051/ps/2015009
Classification : 60G40, 60J55, 60J65
Keywords: Brownian motion, Brownian bridge, Brownian meander, pseudo-Brownian bridge, Bessel process, uniform sampling, local times, hitting times, enlargement of filtration

Rosenbaum, Mathieu 1 ; Yor, Marc 1

1 LPMA, University Pierre et Marie Curie - Paris 6, Paris, France. 
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     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},
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     year = {2015},
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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

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