supermartingales

A supermartingale is a concept in probability theory and stochastic processes used to describe a sequence of random variables whose future expected value, given the present, does not exceed the current value.

Formally, let (F_n) be a filtration and let (X_n) be an adapted sequence of integrable random variables. The process (X_n) is a supermartingale if for every n,

E[X_{n+1} | F_n] ≤ X_n almost surely.

If equality holds for all n, the process is a martingale; if the inequality is reversed, it is a submartingale. The condition implies that the unconditional expectations satisfy E[X_n] is nonincreasing in n.

Key properties include:

- E[X_n] ≤ E[X_0] for all n.

- If (X_n) is nonnegative, it converges almost surely to a limit X_∞, and E[X_∞] ≤ E[X_0] (Doob’s convergence theorem).

- Doob’s decomposition states that an integrable supermartingale can be written as X_n = M_n − A_n, where M_n is a martingale and A_n is a predictable, increasing process with A_0 = 0.

Examples:

- If a gambler participates in a fair game with a constant house edge, the gambler’s wealth sequence may form a supermartingale because its conditional expectation does not exceed the current wealth.

- More generally, any nonincreasing conditional expectation process derived from a fixed future payoff yields a supermartingale; every supermartingale can be decomposed into a decreasing predictable part and a martingale part.

Extensions and applications:

In continuous time, supermartingales are defined similarly with respect to a filtration, and they play a central role in finance (pricing bounds, hedging), optimal stopping, and stochastic calculus. Doob’s inequalities and related convergence results provide useful tools for analysis and modeling.