Expected Value of Martingale is Constant in Time
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Theorem
Discrete Time
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 $n \in \Z_{\ge 0}$.
Continuous Time
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 $t \in \hointr 0 \infty$.