Expected Value of Martingale is Constant in Time/Discrete Time

Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a discrete-time filtered probability space.

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a martingale.

Then:


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

for each $n \in \Z_{\ge 0}$.

Proof
From Definition 2 of a discrete time martingale, we have:


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

So:


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

From Expectation of Conditional Expectation, we have:


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

So:


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

for each $n \in \Z_{\ge 0}$.