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

Abstract : Let B be a Brownian motion and T1 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 UT1/√T1, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.
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Romuald Elie, Mathieu Rosenbaum, Marc Yor. On the expectation of normalized Brownian functionals up to first hitting times. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2014, 19, pp.37. ⟨10.1214/EJP.v19-3049⟩. ⟨hal-01316640⟩

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