Definition:Supermartingale/Discrete Time

Definition
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \in \N}, \Pr}$ be a filtered probability space.

Let $\sequence {X_n}_{n \mathop \in \N}$ be an adapted stochastic process.

We say that $\sequence {X_n}_{n \mathop \in \N}$ is a supermartingale :


 * $(1): \quad$ $X_n$ is integrable for each $n \in \N$


 * $(2): \quad \forall n \in \N: \expect {X_{n + 1} \mid \FF_n} \le X_n$

Equation $(2)$ is understood as follows:


 * for any version $\expect {X_{n + 1} \mid \FF_n}$ of the conditional expectation of $X_{n + 1}$ given $\FF_n$, we have:


 * $\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost everywhere.