On the expectation of normalized Brownian functionals up to first hitting times

Romuald Elie (1, 2) , Mathieu Rosenbaum (3) , Marc Yor (3, 4)

1 CREST - Centre de Recherche en Économie et Statistique
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
4 IUF - Institut universitaire de France

Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

Mots clés: Feynman-Kac formula; Brownian motion; hitting times; scaling; random sampling; Bessel process; Brownian meander; Ray-Knight theorem; Feynman-Kac formula.

Domaines: Probabilités [math.PR]

Romuald Elie, Mathieu Rosenbaum, Marc Yor. On the expectation of normalized Brownian functionals up to first hitting times. 2013. ⟨hal-00869298⟩