The probability law of the Brownian motion normalized by its range

Florin Spinu (OMERS Capital Markets)


In the present paper we deduce explicit formulas for the probability laws of the quotients $X_t/R_t$ and $m_t/R_t$, where $X_t$ is the standard Brownian motion and $m_t$, $M_t$, $R_t$ are its running minimum, maximum and range, respectively.The computation makes use of standard techniques from analytic number theory and the theory of the Hurwitz zeta function.

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Pages: 1-8

Publication Date: June 13, 2013

DOI: 10.1214/ECP.v18-2568


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