I'm attending a stocahstic processes course. I have some trouble with the intuition behind a stopping time. I will consider the discrete case to make it simpler.
a stopping time is given by Wikipedia as:
A stopping time with respect to a sequence of random variables
$X_1,X_2,X_3,…$ is a random variable $\tau$ with the property that for each $n$,
the occurrence or non-occurrence of the event $\{\tau \le n\}$ depends only on
the values of $X_1,X_2,X_3,…,X_n$.
Being more general, $\tau$ is a stopping time if $\{\tau \le n\}$ is $\mathcal{F}_n$-measurable for any $n$, where $\mathcal{F}_n$ is the natural filtration of the process up to time $n$.
Wikipedia makes the example of the Gambler's ruin, starting from 100€ and winning or losing 1€ at each $n$. $X_n$= amount of money at time n.
Let's consider $\tau=\inf \{n \ge 0: X_n=0\}$.
Is $\tau$ a stopping time for any $n$?
I would think that for any $n<100$ $\{\tau \le n\}$ is equal to $\emptyset$ so it belongs to $ \mathcal{F}_n$ for any $n<100$, and for any $n>100$ $\{\tau \le n\}$ can be written as the intersection of {losing 1€ at time $m$} for any $m$ from $1$ to $100$, so it still belongs to $\mathcal{F}_n$.
How do I concile this with the definition of WIkipedia say for $n<100$ (It depends on the future outcomes)? Is this the proper way to procede?
If so why $\tau=\inf\{ n\ge 0: X_n=2*X_0$ i.e. the $n$ at which I doubled my initial amount of money$\}$
is not a stopping time? Proceding as before I could construct an union of events belonging to $\mathcal{F}_n$ for any $n$. But ipse dixit it's not a stopping time.
I guess there's something big I'm missing here, I beg for mercy.
Thank you
Best Answer
Looks like the article on Wikipedia is requiring $\mathbb{P}(\tau < \infty) =1$. It is stated on the definition that sometimes it is required.
So, if you add this requirement, then of course your variable is not a stopping time, as you may also lose forever. If you don't (and I actually didn't add this requirement during my stochastic processes course) then it is a stopping time.