Definition:Closed Martingale/Discrete Time

Definition

Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{n \mathop \ge 0}, \Pr}$ be a discrete-time filtered probability space.

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a $\sequence {\FF_n}_{n \mathop \ge 0}$-martingale.

We say that $\sequence {X_n}_{n \mathop \ge 0}$ is a closed martingale if and only if there exists an integrable random variable $Z$ such that:

$X_n = \expect {Z \mid \FF_n}$

for each $n \in \Z_{\ge 0}$.

That is, for each version $\expect {Z \mid \FF_n}$ of the conditional expectation of $Z$ given $\FF_n$, we have:

$X_n = \expect {Z \mid \FF_n}$ almost surely.