Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time

Theorem

Discrete Time

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

Let $\sequence {X_n}_{n \ge 0}$ be an $\sequence {\FF_n}_{n \ge 0}$-martingale.

Let $S$ and $T$ be bounded stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$ and $S \le T$.

Let $\FF_S$ be the stopped $\sigma$-algebra associated with $S$.

Let $X_T$ and $X_S$ be $X$ at the stopping times $T$ and $S$.

Then:

$\expect {X_T \mid \FF_S} = X_S$ almost surely.

Also see

• Doob's Optional Stopping Theorem