We determine E exp (α X(T) - β T) 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.
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Howard M. Taylor (1975) studied this question.
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