# Definition:Stopping Time

## 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 $t \in \hointr 0 \infty$.