Confusion regarding stopping times

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

τ∈{0,1,2,…;+∞}\tau \in \{0,1,2,…;+\infty\}

What formaly does {τ≤n}\{ \tau \leq n \} mean? Does it represent an event? Particularly in {τ≤n}={w:τ(w)≤n}∈Fn \{ \tau \leq n \} = \{ w:\tau(w) \leq n \} \in \mathscr{F_n} . What is τ(w)\tau(w) then?
Can you please explain meaning of the {τ=n}={τ≤n}∖{τ≤n−1}∈Fn
\{ \tau = n \} =
\{ \tau \leq n \} \setminus \{ \tau \leq n – 1 \} \in \mathscr{F_n}
? I unfortunately have no intuitive understandring of this statement.
Can you please refer to clear construction examples of sigma algebra / filtration needed for stopping time to be measurable / adapted.
Can you please recommend good book/article with clear and intuituve guidance to grasp a motivation and understanding of stopping times?



1 Answer


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

Yes, {τ≤n}\{\tau\le n\} is an event. It is the event, that your process will stop before time nn.
The event, that the stopping criterion will occur at point nn is the event, that the stopping criterion occured before nn but after n−1n-1. The information is obtained only at point nn not earlier, hence Fn\mathcal F_n.
The most common example, is an infinite sequence of coin tosses which induces a random walk. Formally, let P(Xi=1)=P(Xi=−1)=12P(X_i=1)=P(X_i=-1)=\frac12 for any i∈Ni\in\mathbb N and let Sn=∑ni=1XiS_n=\sum_{i=1}^n X_i with S0=0S_0=0 (this is an arbitrary starting condition). You can define the stopping time τ=inf{n∈N:Sn≥5}\tau=\inf{\{n\in N: S_n\ge 5\}} or in words, the first point in time that your random walk, equals or exceeds 55. Now, e.g. P(\tau=1)=0P(\tau=1)=0, because there is no chance to reach 55 starting from 00 and adding at most 11 at each step. But P(\tau=5)>0P(\tau=5)>0.
There are many, many, so hmm no, I cannot pick one.