Could you check if my solution is correct? I find the filtrations quite tricky.
Here is the problem: Let $\{X_n\}_{n \in \mathbb{N}}$ be a stochastic process and $B$ a borel set in $\mathbb{R}^N$.
Prove that $ \tau_k = \inf \{n> \tau_{k-1}: \ X_n \in B\}$ is a stopping time w.r. to the natural filtration generated by the process.
Here is what I've done:
I assume that we know that the first hitting time is a stopping time and I want to prove the statement by induction.
Assume $\tau_k$ is a stopping time, $k \ge 1$.
We want to check if $\{ \tau_{k+1} \le n \} \in \mathcal{F}_n$ .
We have that $$\{ \tau_{k+1} \le n \} = \{ \tau_{k+1} =0 \} \cup \dots \cup \{ \tau_{k+1} =n \} $$
$\{ \tau_{k+1} =0 \} = \emptyset$, because if $k \ge 1$ and $\tau_{k+1} =0$, then $\tau_{k} <0$.
Generally, $$\{ \tau_{k+1} =j \} = \{ \tau_{k+1} < j \} \cap \{X_{j} \in B\} \in \mathcal{F}_j$$
And using the inductive assumption for the first set, measurability and the fact that $\{X_n\}$ is adapted to the filtration with the property that $\forall n: \ \mathcal{F}_n \subset \mathcal{F}_{n+1}$, we get that $\{ \tau_{k+1} \le n \} \in \mathcal{F}_n$
Is everything correct here?