I'm starting the study of stochastic processes.
What are some of the applications of the Wald martingale? By Wald martingale I mean the following (defined by my professor).
Let $\{X_n\}_{n\in\mathbb{N}}$ be i.i.d. random variables defined on a common probability space $(\Omega, \mathscr{F}, \mathbb{P})$, with moment generating function $\mathbb{E}(e^{\alpha X_n})=M(\alpha)$. Then $$Z_n=M(\alpha)^{-n}e^{\alpha S_n}$$
where $S_n=\sum_{j=1}^{n}X_j$. $Z_n$ is then a martingale with respect to the natural filtration induced by the random variables.