Doob's Optional Stopping Theorem
Jump to navigation
Jump to search
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 $T$ be a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.
Let:
- $\map {X_T} \omega = \map {X_{\map T \omega} } \omega \map {\chi_{\set {\omega \in \Omega : \map T \omega < \infty} } } \omega$
for each $\omega \in \Omega$.
Suppose one of the following conditions holds:
- $(1) \quad$ $T$ is bounded
- $(2) \quad$ $T$ is finite almost surely, and there exists an integrable random variable $Y$ with $\size {X_n} \le Y$ for $n \in \Z_{\ge 0}$
- $(3) \quad$ $T$ is integrable, and there exists a real number $M > 0$ such that for each $n \in \Z_{\ge 0}$ we have $\size {X_{n + 1} - X_n} \le M$ almost surely.
Then $X_T$ is integrable and in particular:
- $\expect {X_T} = \expect {X_0}$
Also see
- Integrable Adapted Stochastic Process is Supermartingale iff Optional Stopping Theorem Holds establishes the converse to this result.
- Doob's Optional Stopping Theorem for Stopped Sigma-Algebra of Bounded Stopping Time generalises case $(1)$.
Source of Name
This entry was named for Joseph Leo Doob.