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 $t \in \hointr 0 \infty$.

$\blacksquare$

Sources

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