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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

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  • $\begingroup$ If $ X_0$ is fixed (not random) then I don't see the problem... $\endgroup$ – Ian Jan 31 '15 at 15:24
  • $\begingroup$ It is fixed, it's 100€. Still, I can't figure it out. Intuitively I would say that there's a chance I will never double the money because I run out of it before time n. Yet, given the definition the event of doubling the initial money when I've run out of money is impossible, so it's {} which belongs to any Fn.. $\endgroup$ – Tommaso Guerrini Jan 31 '15 at 15:33
  • $\begingroup$ That isn't a problem. If you ran out already then the doubling event doesn't happen for any n, which is fine. $\endgroup$ – Ian Jan 31 '15 at 15:36
  • $\begingroup$ You say it's a stopping time then? $\endgroup$ – Tommaso Guerrini Jan 31 '15 at 15:37
  • $\begingroup$ Looks like it to me. What exactly have you seen to the contrary? $\endgroup$ – Ian Jan 31 '15 at 15:47
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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.

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