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 [math]\displaystyle{ (X_n)_{n \geq 1} }[/math] be a sequence of i.i.d. random variables whose moment generating function [math]\displaystyle{ M: t \mapsto \mathbb{E}(e^{t X_1}) }[/math] is finite for some [math]\displaystyle{ \theta \gt 0 }[/math], and let [math]\displaystyle{ S_n = X_1 + \cdots + X_n }[/math], with [math]\displaystyle{ S_0 = 1 }[/math]. Then, the process [math]\displaystyle{ (W_n)_{n \geq 0} }[/math] defined by

[math]\displaystyle{ W_n = \frac{e^{\theta S_n}}{M(\theta)^n} }[/math]

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

See also

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