The maximum of Brownian motion with parabolic drift
Svante Janson, Guy Louchard, Anders Martin-L\"of
TL;DR
This paper analyzes the distribution and moments of the maximum of a Brownian motion with parabolic drift, providing series expansions, integral formulas, and high-precision numerical values.
Contribution
It introduces new series expansions and integral formulas for the distribution and moments of the maximum of Brownian motion with parabolic drift.
Findings
Derived series expansions for the distribution
Provided integral formulas for moments
Achieved high-precision numerical evaluations
Abstract
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
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