Probability in Martingales

A martingale is a sequence of random variables where the conditional expectation of the next value equals the current value.

Definition

A sequence $(X_n)$ is a martingale with respect to filtration $(\mathcal{F}_n)$ if:

$E[X_{n+1} \mid \mathcal{F}_n] = X_n$

Optional Stopping Theorem

Under suitable conditions, if $(X_n)$ is a martingale and $\tau$ is a stopping time:

$E[X_\tau] = E[X_0]$

This has profound implications for gambling strategies — no betting system can turn a fair game into a winning one.