Definition:Closed Martingale/Continuous Time

Definition

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

Let $\sequence {X_t}_{t \ge 0}$ be a $\sequence {\FF_t}_{t \ge 0}$-martingale.

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

$X_t = \expect {Z \mid \FF_t}$

for each $t \in \hointr 0 \infty$.

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

$X_t = \expect {Z \mid \FF_t}$ almost surely.

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): Definition $3.20$