Wald's martingale

Let ${\displaystyle (X_{n})_{n\geq 1}}$ be a sequence of i.i.d. random variables whose moment generating function ${\displaystyle M:t\mapsto \mathbb {E} (e^{tX_{1}})}$ is finite for some ${\displaystyle \theta >0}$, and let ${\displaystyle S_{n}=X_{1}+\cdots +X_{n}}$, with ${\displaystyle S_{0}=1}$. Then, the process ${\displaystyle (W_{n})_{n\geq 0}}$ defined by

${\displaystyle W_{n}={\frac {e^{\theta S_{n}}}{M(\theta )^{n}}}}$

is a martingale known as Wald's martingale.[4] In particular, ${\displaystyle \mathbb {E} (W_{n})=1}$ for all ${\displaystyle n\geq 0}$.

• Martingale
• geometric Brownian motion
• Doléans-Dade exponential
• Wald's equation