Confusion regarding stopping times

Question

I am having a hard time with "stopping time" definitions, notation and reasoning behind it.

$\tau \in \{0,1,2,...;+\infty\}$

1. What formaly does $\{ \tau \leq n \}$ mean? Does it represent an event? Particularly in $ \{ \tau \leq n \} = \{ w:\tau(w) \leq n \} \in \mathscr{F_n} $. What is $\tau(w)$ then?

2. Can you please explain meaning of the $ \{ \tau = n \} = \{ \tau \leq n \} \setminus \{ \tau \leq n - 1 \} \in \mathscr{F_n} $? I unfortunately have no intuitive understandring of this statement.

3. Can you please refer to clear construction examples of sigma algebra / filtration needed for stopping time to be measurable / adapted.

4. Can you please recommend good book/article with clear and intuituve guidance to grasp a motivation and understanding of stopping times?

Answer 1

To start $\tau$ is a random variable. Therefore many authors use a capital $T$ to denote stopping times. So, everything you know about random raviables applies. Now,

1. Yes, $\{\tau\le n\}$ is an event. It is the event, that your process will stop before time $n$.
2. The event, that the stopping criterion will occur at point $n$ is the event, that the stopping criterion occured before $n$ but after $n-1$. The information is obtained only at point $n$ not earlier, hence $\mathcal F_n$.
3. The most common example, is an infinite sequence of coin tosses which induces a random walk. Formally, let $$P(X_i=1)=P(X_i=-1)=\frac12$$ for any $i\in\mathbb N$ and let $S_n=\sum_{i=1}^n X_i$ with $S_0=0$ (this is an arbitrary starting condition). You can define the stopping time $$\tau=\inf{\{n\in N: S_n\ge 5\}}$$ or in words, the first point in time that your random walk, equals or exceeds $5$. Now, e.g. $P(\tau=1)=0$, because there is no chance to reach $5$ starting from $0$ and adding at most $1$ at each step. But $P(\tau=5)>0$.
4. There are many, many, so hmm no, I cannot pick one.