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

Abstract

The joint Laplace transform of $T$ and $X(T)$ is derived where $X(\bullet)$ 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.