Martingales

We introduce martingales, sub- and super-martingales. We define stopping times, which is fundamental in martingale theory, and prove Doob’s Optional Sampling Theorem. This result essentially says that no legitimate strategy can alter the nature (that is, from fair to one sided) of a two-person game. The famous identities of Wald follow from this theorem. Doob’s maximal inequality shows how the maximum of a sub-martingale is controlled by the last point of the sequence probabilistically.