Exercise: Identifying Stopping Times — Six Cases

Prerequisites: Stopping Times, Filtrations and Information

Problem

Let $(S_n)_{n \ge 0}$ be a random walk with $\mathcal{F}_n = \sigma(S_0, \ldots, S_n)$. For each of the following random times, decide whether it is a stopping time and give a one-sentence justification.

1. $\tau_1 = 7$.
2. $\tau_2 = \inf\{n : S_n \ge 10\}$, the first time the walk reaches $10$.
3. $\tau_3 = \sup\{n \le 100 : S_n = 0\}$, the last time the walk hits $0$ before time $100$.
4. $\tau_4 = \inf\{n : S_{n-1} \ge 10\}$ (one step after the first hitting time).
5. $\tau_5 = \inf\{n : \max_{k \le n} S_k - \min_{k \le n} S_k \ge 5\}$, the first time the running range exceeds $5$.
6. $\tau_6 = \inf\{n : S_n \ge S_{n+1}\}$, the first step immediately before a drop.

For the ones that are stopping times, write down the measurability condition explicitly for a concrete $n$ (e.g. $n = 5$).

Hint

The test is always: "can I decide $\{\tau \le n\}$ using only $S_0, S_1, \ldots, S_n$?"