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.
Citation
Howard M. Taylor. "A Stopped Brownian Motion Formula." Ann. Probab. 3 (2) 234 - 246, April, 1975. https://doi.org/10.1214/aop/1176996395
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