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:

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

$\blacksquare$

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): $3.3$: Continuous Time Martingales and Supermartingales