Definition:Supermartingale

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

Let $\sequence {X_n}_{n \ge 0}$ be an adapted stochastic process.

We say that $\sequence {X_n}_{n \ge 0}$ is a supermartingale if:


 * $(1) \quad$ $X_n$ is integrable for each $n \ge 0$
 * $(2) \quad$ $\expect {X_{n + 1} \mid \FF_n} \le X_n$ for each $n \ge 0$.

The 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.