Wald's martingale explained

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

(X _{n)} _{n} be a sequence of i.i.d. random variables whose moment generating function M: \theta \mapsto E (e ^{\theta X _{1}} ) is finite for some \theta > 0 , and let S _{n} = X _{1} + … + X _{n} , with S _{0} = 0 . Then, the process (W _{n)} _{n} defined byW _{n} = (e ^{\theta S _{n}})/(M(\theta) ^{n}) is a martingale known as Wald's martingale.^[4] In particular, E (W _{n)} = 1 for all n \geq 0 .

See also

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

Notes and References

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