# Prove that discrete first hitting time is a stopping time

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:

$\{\omega \in \Omega : \tau_A(\omega) \le n\} =\bigcup_{k=0}^n\{\omega \in \Omega : X_k(\omega) \in A\} \in \mathcal F_n$

• Go back wayyyy behind, to the first chapter of notational conventions in probability theory: there, they say that $\{\tau_A \le n\}$ is a shorthand for the set $$\{\omega\in\Omega\,|\,\tau_A(\omega) \le n\}.$$ If this point is clear to you, you should also be able to correct the definition of $\tau_A$ in your post, which is absurd at the moment since it confuses $\tau_A$ with $\{\tau_A\}$.
– Did
Mar 29 '15 at 13:07
• $\tau$ is random variable so it takes as argument $\omega$ and I agree, but in a practical example what could be an $\omega$ in this case? for r.v. which are like "n° of heads in 2 coins tosses" a possible $\omega=\{HT\}$. I imagine that also the notation $\{X_k\in A\}$ means $\{\omega\in\Omega : X_k(\omega)\in A\}$, but again now, which could be an $\omega$ for $X_k$? I have problems in visualizing this in my head since stochastic processes for me are r.v. indexed on the time, but I never asked myself which values they take from $\Omega$ to give back the output real number. Mar 29 '15 at 13:20
• The identity of $\Omega$ is irrelevant, all that counts is that there exists such suitable probability space. Try to solve the exercise and you will see that you never need to know what $\Omega$ is. I might even have explained this in details somewhere on the site.
– Did
Mar 29 '15 at 13:24
• If I had to explain with words the proof, I would say that the omegas for which $\tau_A(\omega)\le n$ are the same as the union of the omegas such that $X_k(\omega)\in A$ but since we are considering a sequence of $X_k$ until n, these omegas are in $\mathcal F_n$, is that correct? Mar 29 '15 at 13:28
• No, that some $\omega$ is or is not in $\mathcal F_n$ is far from being correct. What is the nature of the object $\mathcal F_n$, already?
– Did
Mar 29 '15 at 13:30

$\{\tau_A\le n\}=\{\bigcup_{k=0}^n X_k\in A\}$ 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.
• $\{\bigcup_{k=0}^n X_k\in A\}$: incorrect notation.
• do I have to write $\bigcup _{k=0}^n\{X_k\in A\}$ ? Mar 31 '15 at 17:58
• You have.