stopping-time

Intuitively, a stopping time is simply a random variable that determines when to stop a random process. Naturally, the decision to stop the process should be based on only information until that time, not any future information.

Mathematically, a stopping time $\tau$ is a random variable on a probability space $(\Omega ,\mathcal{F},P)$ with filtration $\{{\mathcal{F}}_t\}$ such that for all $t$, the event is $\{t\le t\}$ is ${\mathcal{F}}_t$ measurable. Eg suppose you have a stochastic process $(X_t)$ and ${\mathcal{F}}_t=\sigma (X_1,\dots ,X_t)$. Then the decision to stop the process at time $t$ should be based only on $X_1,\dots ,X_t$, not on $X_j$ for any $j>t$.