I have the following problem
Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and $\tau^{n+1}=\inf\{k > \tau^n: X_k \in A\}$. A classic result is that in some conditions $(X_{\tau^n})$ is a Markov chain, but my question is if the natural filtration of $(X_{\tau^n})$ $\mathcal{F}_n= \sigma(X_0,X_{\tau^1},...,X_{\tau^n})$ and the $\sigma-$algebra of the stopping time $\mathcal{F}_{\tau^n}$ are equal?
Is easy to see that $\mathcal{F}_n \subseteq \mathcal{F}_{\tau^n}$ but I don't know if they are equal.
Any help will be appreciated