martingale

A martingale is any stochastic process that stays the same in expectation. Formally, $x_t$ is a martingale if

This condition is related to the Markov property, but is simultaneously

1. Stronger, in that we require the expectation to equal $x_t$ rather than being a function of $x_t$ as a Markov chain would allow.
2. Weaker, since we require only memorylessness of expectations rather than the full distribution. For example, $x_{t + 1} = x_t + \epsilon x_0$ is a martingale for any zero-mean random variable $\epsilon$ independent of $x_0$, but it is not a Markov chain.

The canonical example is a gambler's fortune, assuming that the gambler plays only fair games.

A submartingale or supermartingale is a process in which the conditional expectation at the next timestep is no less (sub) or no greater (super) than the current value.