I am having a hard time for understanding stopping time, especially on notations.
My book says:
Def. A stopping time for the filtration $(g_n)$ is a random variable
$T:\Omega \rightarrow \mathbb{N} \cup \infty \ \text{such that} $ { $T=n$ } $ \in g_n \ \text{for all} \ n \ge 0$
It is difficult for me to understand what $T$ is. Suppose $\Omega = $ { 1,2,3 } .
Then $T(1) =1 , T(2) = 3 ...$ works that way?
Also, $g_n$ is a $\sigma$ - algebra on $\Omega$, It gets more confusing.
Could you help me what the $T$ represents and, what is stopping time?
Thanks in advance:)
The information that goes with this is ... $\mathcal G_n$ is information known time $n$. That is, for an event $E$, we have $E \in \mathcal G_n$ if and only if the outcome of $E$ is known at time $n$. Then a stopping time $T$ is a random variable such that we know when it happens ... that is, for any time $n$, the event $\{T = n\}$ is known at time $n$.
Example. Suppose we toss a fair coin repeatedly. Let $X_n$ be the outcome of the $n$th toss. Then take $\mathcal G_n$ to be the events determined by $X_1,\dots, X_n$. So an event $E$ belongs to $\mathcal G_3$ iff we can tell at time $3$ whether event $E$ occurs.
Let $T$ be "the first time when the coin shows heads". Convince yourself this is a stopping time.
Now let $T$ be defined by $T=n$ iff $X_{n+1} = H$. This is not a stopping time: at time $n$ we do not know whether the next toss will be H.