I have problems with the proof that a first hitting time is a stopping time:
Let $\tau$ be the first hitting time into the set A, for a process $\{ X_n \}$ adapted to a filtration $\mathcal F_n$.
I know that a random time is a stopping time if the set $ \{\tau\le n\}\in\mathcal F_n \, \, \forall n \in[0,\infty)$
Define now the first hitting time as $\tau_A= \mathrm{inf}(n\ge0 : X_n\in A)$
I find everywhere the same proof:
$ \{\omega \in \Omega : \tau_A(\omega) \le n\} =\bigcup_{k=0}^n\{\omega \in \Omega : X_k(\omega) \in A\} \in \mathcal F_n$
Best Answer
So then I got the answer:
$\{\tau_A\le n\}=\{\bigcup_{k=0}^n X_k\in A\}$ because if the first hitting time of A has occurred before n, the omegas for which this occur are the same omegas for $X_k$ for which "At least a $X_k$ at some point before n has entered A", then these omegas belong to $\mathcal F_n$ because $X_k$ is $\mathcal F_n$-measurable.