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 ≥ 1 {\displaystyle (X_{n})_{n\geq 1}} be a sequence of i.i.d. random variables whose moment generating function M : θ ↦ E ( e θ X 1 ) {\displaystyle M:\theta \mapsto \mathbb {E} (e^{\theta X_{1}})} is finite for some θ > 0 {\displaystyle \theta >0} , and let S n = X 1 + ⋯ + X n {\displaystyle S_{n}=X_{1}+\cdots +X_{n}} , with S 0 = 0 {\displaystyle S_{0}=0} . Then, the process ( W n ) n ≥ 0 {\displaystyle (W_{n})_{n\geq 0}} defined by
W n = e θ S n M ( θ ) n {\displaystyle W_{n}={\frac {e^{\theta S_{n}}}{M(\theta )^{n}}}}
is a martingale known as Wald's martingale. In particular, E ( W n ) = 1 {\displaystyle \mathbb {E} (W_{n})=1} for all n ≥ 0 {\displaystyle n\geq 0} .
See also Martingale geometric Brownian motion Doléans-Dade exponential Wald's equation
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