Electronic Journal of Probability

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

Romuald Elie, Mathieu Rosenbaum, and Marc Yor

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 37, 23 pp.

Accepted: 29 March 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Brownian motion hitting times scaling random sampling Bessel process Brownian meander Ray-Knight theorem Feynman-Kac formula

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Elie, Romuald; Rosenbaum, Mathieu; Yor, Marc. On the expectation of normalized Brownian functionals up to first hitting times. Electron. J. Probab. 19 (2014), paper no. 37, 23 pp. doi:10.1214/EJP.v19-3049. https://projecteuclid.org/euclid.ejp/1465065679

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