The Wald Martingale and the Maximum

Abstract

We introduce the first of our two key martingales and consider two immediate applications. In the first application, we will use the martingale to construct a change of measure and thereby consider the dynamics of X under the new law. In the second application, we shall use the martingale to study the law of the process $\overline{X}= \{\overline{X}_{t}\,\colon t\geq 0\}$, where

$$\overline{X}_t =\sup_{s\leq t}X_s, \quad t\geq 0. $$

In particular, we shall discover that the position of the trajectory of $\overline{X}$, when sampled at an independent and exponentially distributed time, is again exponentially distributed.