Formulas for Stopped Diffusion Processes with Stopping Times Based on the Maximum

The joint Laplace transform of T and $X(T)$ is derived where X(•) is a time homogeneous diffusion process and T is the first time the process falls a specified amount below its current maximum. This generalizes the work of Taylor. The distribution of the maximum at T is shown to be exponential for Brownian motion. Formulas for more general stopping times based on the current maximum are given.