supermartingale

A stochastic process $(S_t)$ is a supermartingale with respect to a filtration $({\mathcal{F}}_t)$ if

$$\mathbb{E}[S_t|{\mathcal{F}}_{t-1}]\le S_{t-1},\forall t.$$

Supermartingales are thus decreasing over time in expectation. If $\le$ is replaced by an equality then we have a martingale and, if in addition $\mathbb{E}[S_0]=1$ then we have a test-martingale. If $\le$ is swapped with $\ge$ then we have a submartingale.

Supermartingales obey Ville’s inequality which makes them useful tools in game-theoretic statistics. They also define betting strategies in game-theoretic probability.

Supermartingales obey the optional stopping theorem, which states that $\mathbb{E}[S_{\tau }]\le \mathbb{E}{S}_0$ for stopping times $\tau$, under certain conditions on the process and the stopping time. They also obey the martingale convergence theorem, which states that if $S_t$ is nonnegative, then $S_t\to S$ a.s. to some random variable $S$ and $\mathbb{E}S\le \mathbb{E}{S}_0$.