 Stopping time

Relative to a filtration $({\mathcal{F}}_t)$, a stopping time is a random variable $T$ with values in $[0,\infty ]$ such thatIn discrete time, use integer-valued $T$ and integers $t=n$. The event that stopping has occurred must be decidable from the information available now. The stopping-time sigma-algebra records the information available at $T$.

For a continuous adapted process, its first hit of a closed upper level is a stopping time, as shown by closed-level hitting times of continuous adapted processes. A fixed deterministic time also qualifies. The time of the maximum of a Brownian motion over a prescribed future interval generally does not: deciding whether the final maximum has already occurred requires future observations. This distinction determines whether the Strong Markov property or an optional stopping theorem applies.