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

Romuald Elie (University Paris-Est Marne-La-Vallée)
Mathieu Rosenbaum (University Pierre et Marie Curie (Paris 6))
Marc Yor (University Pierre et Marie Curie)


Let $B$ be a Brownian motion and $T_1$ its first hitting time of the level $1$. For $U$ a uniform random variable independent of $B$, we study in depth the distribution of $B_{UT_1}/\sqrt{T_1}$, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.

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Pages: 1-23

Publication Date: March 29, 2014

DOI: 10.1214/EJP.v19-3049


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