Expected Value of Submartingale is Increasing in Time/Continuous Time

Theorem
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:


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

Proof
From the definition of a submartingale, we have:


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

From Expectation is Monotone, we have:


 * $\expect {\expect {X_t \mid \FF_s} } \ge \expect {X_s}$

From Expectation of Conditional Expectation, we have:


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