Statistics of the first passage time of Brownian motion conditioned by maximum value or area
Michael J. Kearney, Satya N. Majumdar
TL;DR
This paper derives moments of the first passage time for Brownian motion conditioned on maximum value or area, providing new insights and formulas for these stochastic quantities with potential applications.
Contribution
It introduces a unified mathematical framework to derive moments of the first passage time conditioned on maximum or area for Brownian motion, including new formulas for Brownian excursions.
Findings
Derived moments of first passage time conditioned on maximum value
Derived moments of first passage time conditioned on area
Provided formulas for moments of Brownian excursions
Abstract
We derive the moments of the first passage time for Brownian motion conditioned by either the maximum value or the area swept out by the motion. These quantities are the natural counterparts to the moments of the maximum value and area of Brownian excursions of fixed duration, which we also derive for completeness within the same mathematical framework. Various applications are indicated.
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