Expected Value of Submartingale is Increasing in Time

Theorem

Continuous Time

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 $\sequence {\FF_t}_{t \ge 0}$-submartingale.

Let $t, s \in \hointr 0 \infty$ with $0 \le s < t$.

Then, we have:

$\expect {X_s} \le \expect {X_t}$