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

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This category contains pages concerning **Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time**:

### 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.

## Pages in category "Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time"

The following 5 pages are in this category, out of 5 total.

### D

- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time
- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time
- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Martingale
- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Submartingale
- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time/Discrete Time/Supermartingale