Definition:Stopping Time
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Discrete Time
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $T : \Omega \to \Z_{\ge 0} \cup \set {\infty}$ be a random variable.
Definition 1
We say that $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$ if and only if:
- $\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for all $t \in \Z_{\ge 0}$.
Definition 2
We say that $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$ if and only if:
- $\set {\omega \in \Omega : \map T \omega = t} \in \FF_t$
for all $t \in \Z_{\ge 0}$.
Continuous Time
Let $\struct {\Omega, \Sigma, \sequence {\FF_t}_{t \ge 0}, \Pr}$ be a continuous-time filtered probability space.
Let $T : \Omega \to \closedint 0 \infty$ be a random variable.
We say that $T$ is a stopping time with respect to $\sequence {\FF_t}_{t \ge 0}$ if and only if:
- $\set {\omega \in \Omega : \map T \omega \le t} \in \FF_t$
for each $t \in \hointr 0 \infty$.