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.[1][2][3]

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

Formal statement

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

See also

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

Notes

1. Wald, Abraham (1944). "On cumulative sums of random variables". Ann. Math. Stat. 15 (3): 283–296. doi:10.1214/aoms/1177731235.
2. Wald, Abraham (1945). "Sequential tests of statistical hypotheses". Ann. Math. Stat. 16 (2): 117–186. doi:10.1214/aoms/1177731118.
3. Wald, Abraham (1945). Sequential analysis (1st ed.). John Wiley and Sons.
4. Gamarnik, David (2013). "Advanced Stochastic Processes, Lecture 10". MIT OpenCourseWare. Retrieved 24 June 2023.