Solution: Identifying Stopping Times — Six Cases

Exercise: Identifying Stopping Times: Six Cases

#	$\tau$	Stopping time?	Reason
1	$\tau_1 = 7$	Yes	$\{\tau_1 \le n\}$ is either $\emptyset$ (for $n < 7$ ) or $\Omega$ (for $n \ge 7$ ), both in $\mathcal{F}_n$ .
2	$\tau_2 = \inf\{n : S_n \ge 10\}$	Yes	$\{\tau_2 \le n\} = \bigcup_{k=0}^n \{S_k \ge 10\}$ , a union of events in $\mathcal{F}_k \subseteq \mathcal{F}_n$ .
3	$\tau_3 = \sup\{n \le 100 : S_n = 0\}$	No	At $n = 5$ , you need to know if $S_k = 0$ for some $k \in (5, 100]$ to decide if $\tau_3 \le 5$ — that requires the future.
4	$\tau_4 = \inf\{n : S_{n-1} \ge 10\}$	Yes	Equivalently $\tau_4 = \tau_2 + 1$ , and a stopping time plus a constant is a stopping time. Explicitly $\{\tau_4 \le n\} = \{\tau_2 \le n - 1\} \in \mathcal{F}_{n-1} \subseteq \mathcal{F}_n$ .
5	$\tau_5 = \inf\{n : \max_{k \le n} S_k - \min_{k \le n} S_k \ge 5\}$	Yes	The running max and min over $\{0, \ldots, n\}$ are in $\mathcal{F}_n$ .
6	$\tau_6 = \inf\{n : S_n \ge S_{n+1}\}$	No	At time $n$ , you don't know $S_{n+1}$ , so you can't decide if $\{\tau_6 = n\}$ has occurred. Classic "peeking-one-step-ahead" failure.

Explicit check for $\tau_2$ at $n = 5$ :

\{\tau_2 \le 5\} = \{S_0 \ge 10\} \cup \{S_1 \ge 10\} \cup \cdots \cup \{S_5 \ge 10\}.

Each $\{S_k \ge 10\}$ is $\mathcal{F}_k$ -measurable, hence $\mathcal{F}_5$ -measurable. The union is in $\mathcal{F}_5$ .

Takeaways

First hitting times are stopping times; last hitting times are not. The difference is peeking — the last hit requires knowing no future hit occurs.
Running extrema are adapted, so first-exit times based on them are stopping times — this is why drawdown triggers and range-break triggers are legal trading rules.
Any rule that references $S_{n+k}$ for $k > 0$ at time $n$ is not a stopping time. The test is purely syntactic: does the description mention future indices? If yes, stop — pun intended.
Stopping times plus constants are stopping times (provided the constant is non-negative integer; the shift indexes only backwards into known information).