Stopping time

Remarks

In discrete time, $T = \N$, the condition is equivalent to $\set{\tau = n} \in \mathcal{F}_n$ for every $n \in \N$, since $\set{\tau = n} = \set{\tau \le n} \setminus \set{\tau \le n - 1}$ and $\set{\tau \le n} = \bigcup_{k \le n} \set{\tau = k}$. The value $\infty$ means that the event never happens.

Example: let $(S_n)_{n \in \N}$ be a random walk with natural filtration $\mathcal{F}_n = \sigma(S_0, \dots, S_n)$, let $a \in \R$, and let $\tau = \min\set{n : S_n \ge a}$, with $\min \emptyset = \infty$. Then $\set{\tau \le n} = \bigcup_{k \le n} \set{S_k \ge a} \in \mathcal{F}_n$, so $\tau$ is a stopping time.

Non-example: with the same walk started at $S_0 = 0$ and with steps $\pm 1$, let $N \ge 2$ and $\sigma = \max\set{n \le N : S_n = 0}$, the last visit to $0$ before time $N$. The event $\set{\sigma \le 0}$ is the event that $S_n \ne 0$ for $1 \le n \le N$, which depends on steps after time $0$ and is not in $\mathcal{F}_0 = \set{\emptyset, \Omega}$. So $\sigma$ is not a stopping time.

Constant times are stopping times, and the minimum and the maximum of two stopping times are stopping times. The $\sigma$-algebra of the past at $\tau$ is $\mathcal{F}_\tau = \set{A \in \mathcal{F} : A \cap \set{\tau \le t} \in \mathcal{F}_t \text{ for all } t \in T}$.