# Definition:Supermartingale/Discrete Time

## Definition

### Definition 1

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ 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 $\sequence {\FF_n}_{n \ge 0}$-supermartingale if and only if:

$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
$(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \le X_n$

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 surely.

### Definition 2

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ 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 $\sequence {\FF_n}_{n \ge 0}$-supermartingale if and only if:

$(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
$(2): \quad \forall n \in \Z_{\ge 0}, \, \forall m \ge n: \expect {X_m \mid \FF_n} \le X_n$.

Equation $(2)$ is understood as follows:

for any version $\expect {X_m \mid \FF_n}$ of the conditional expectation of $X_m$ given $\FF_n$, we have:
$\expect {X_m \mid \FF_n} \le X_n$ almost surely.