Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

Julien Randon-Furling; Satya N. Majumdar

doi:10.1088/1742-5468/2007/10/P10008

paper

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

arXiv:0708.2101 · doi:10.1088/1742-5468/2007/10/P10008

Abstract

We calculate analytically the probability density $P (t_{m})$ of the time $t_{m}$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P (M, t_{m})$ of the maximum $M$ and $t_{m}$ . In the driftless case, we find that $P (t_{m})$ has power-law tails: $P (t_{m}) \sim t_{m}^{- 3/2}$ for large $t_{m}$ and $P (t_{m}) \sim t_{m}^{- 1/2}$ for small $t_{m}$ . In presence of a drift towards the origin, $P (t_{m})$ decays exponentially for large $t_{m}$ . The results from numerical simulations are in excellent agreement with our analytical predictions.

arXiv comment13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008)

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

Abstract

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