Wald's martingale

Short description: Exponential martingale associated to sum of iid variables

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 $(X_n{)}_{n\ge 1}$ be a sequence of i.i.d. random variables whose moment generating function $M:t\mapsto \mathbb{𝔼}(e^{t{X}_1})$ is finite for some $\theta >0$, and let $S_n=X_1+\cdots +X_n$, with $S_0=1$. Then, the process $(W_n{)}_{n\ge 0}$ 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 $n\ge 0$.

See also

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