Definition:Submartingale/Continuous Time

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

Let $\sequence {X_t}_{t \ge 0}$ be an adapted stochastic process.

We say that $\sequence {X_t}_{t \ge 0}$ is a $\sequence {\FF_t}_{t \ge 0}$-submartingale :


 * $(1) \quad$ $X_t$ is integrable for each $t \in \hointr 0 \infty$


 * $(2) \quad \forall s, t \in \hointr 0 \infty, \, 0 \le s < t: \expect {X_t \mid \FF_s} \ge X_s$

Equation $(2)$ is understood as follows:


 * for any version $\expect {X_t \mid \FF_s}$ of the conditional expectation of $X_t$ given $\FF_s$, we have:


 * $\expect {X_t \mid \FF_s} \ge X_s$ almost surely.