Expected Value of Martingale is Constant 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 $\sequence {\FF_t}_{t \ge 0}$-martingale.

Then:


 * $\expect {X_t} = \expect {X_0}$

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

Proof
From the definition of a continuous-time martingale, we have:


 * $\expect {X_t \mid \FF_0} = X_0$ almost surely

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

So:


 * $\expect {\expect {X_t \mid \FF_0} } = \expect {X_0}$

From Expectation of Conditional Expectation, we have:


 * $\expect {\expect {X_t \mid \FF_0} } = \expect {X_t}$

So we have:


 * $\expect {X_t} = \expect {X_0}$

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