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October, 2002 On independent times and positions for Brownian motions
Bernard Roynette, Pierre Vallois, Marc Yor, Bernard de Meyer
Rev. Mat. Iberoamericana 18(3): 541-586 (October, 2002).

Abstract

Let $(B_t ; t \ge 0)$, $\big(\mbox{resp. }((X_t, Y_t) ; t \ge 0)\big)$ be a one (resp. two) dimensional Brownian motion started at 0. Let $T$ be a stopping time such that $(B_{t \wedge T} ; t \ge 0)$ \big(resp. $(X_{t \wedge T} ; t \ge 0) ; (Y_{t \wedge T} ; t \ge 0)\big)$ is uniformly integrable. The main results obtained in the paper are: \begin{itemize} \item[1)] if $T$ and $B_T$ are independent and $T$ has all exponential moments, then $T$ is constant. \item[2)] If $X_T$ and $Y_T$ are independent and have all exponential moments, then $X_T$ and $Y_T$ are Gaussian. \end{itemize} We also give a number of examples of stopping times $T$, with only some exponential moments, such that $T$ and $B_T$ are independent, and similarly for $X_T$ and $Y_T$. We also exhibit bounded non-constant stopping times $T$ such that $X_T$ and $Y_T$ are independent and Gaussian.

Citation

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Bernard Roynette. Pierre Vallois. Marc Yor. Bernard de Meyer. "On independent times and positions for Brownian motions." Rev. Mat. Iberoamericana 18 (3) 541 - 586, October, 2002.

Information

Published: October, 2002
First available in Project Euclid: 28 April 2003

zbMATH: 1055.60078
MathSciNet: MR1954864

Subjects:
Primary: 60E10, 60G40, 60G44, 60J25, 60J65

Rights: Copyright © 2002 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.18 • No. 3 • October, 2002
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