Wald's martingale

In probability theory, Wald's martingale is the name sometimes given to a martingale used to study sums of i.i.d. random variables. It is named after the mathematician Abraham Wald, who used these ideas in a series of influential publications.

Wald's martingale can be seen as discrete-time equivalent of the Doléans-Dade exponential.

Formal statement

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

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

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

See also

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