Category:Expected Value of Martingale is Constant in Time

This category contains pages concerning Expected Value of Martingale is Constant in Time:

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 $t \in \hointr 0 \infty$.