A Stopped Brownian Motion Formula

Abstract

We determine $E\lbrack \exp (\alpha X(T) - \beta T) \rbrack$ where $X(t)$ is a Brownian motion having arbitrary drift and variance and $T$ is the first time the process drops a specified amount below its maximum to date. From this result, the moments of $X(T)$ and $T$ and some asymptotic distributions may be found. Applications in process control and financial management are mentioned.