Let $X$ be a continuous time Markov chain on a countable state space $S$, and let $\tau_n$ be the $n^{th}$ time at which the chain jumps out of a set $D$ (i.e. times $t$ at which, for some $\epsilon > 0$, $X_{t + a} \notin D$ for all $0 \leq a < \epsilon$ and $X_{t-a} \in D$ for all $0 < a \leq \epsilon$). The chain is assumed to be right continuous here.
The thing I'm curious about is whether $$ P[X_{\tau_n} = j \mid \tau_n] = P[X_{\tau_n} = j]. $$ Something weaker would do, but intuition seems to be that knowing the time at which the $n^{th}$ exit occurred doesn't inform about the point jumped to, and vice versa. If I'm wrong about this I may ask about a weaker condition, but I'd sure like a counter-example to this if one exists.
I've tried to justify the equality, but with no luck. Right continuity seems to ensure that $\tau_n$ is only a function of the "past" (one need not know about open intervals extending into the proximal future) but I'm having trouble formalizing this:
\begin{align} \int_{\tau_n^{-1}(a,b]} P[X_{\tau_n} = j | \tau_n] &= P[X_{\tau_n} = j, \tau_n \in (a,b]] \\ &= P\left[\left(\bigcap_{k \leq n-1} \bigcup_{ u_k \leq a}\{ X_{u_k}^- \in D, X_{u_k} \notin D\}\right) \bigcap \left( \bigcup_{u \in (a,b]}\{ X_u^- \notin D, X_u = j \}\right)\right] \end{align} and from here it isn't so easy to try the standard Markovian techniques, like dividing out by the probability of the past here and trying to reduce the resulting conditional probabilities (notice that these unions are not countable).
Thanks!
EDIT:
So, it looks like my original speculation was incorrect (thanks for the countexample Did). I did say I'd have a follow up: I wonder whether the following maneuver is allowable $$ \mathbb{E}[ \mathbb{E}[P_{X_{\tau_n}}[G] \mid \tau_n] I_{(0,1]}(\tau_n) ] = \mathbb{E}[ \mathbb{E}[P_{X_{\tau_0}}[G] \mid \tau_0] I_{(0,1]}(\tau_n) ]. $$ Something reminiscent but distinct from the strong Markov property is suggested in the equality, but my efforts to establish it led variously to dead ends such as the false conjecture above, or else equally frustrating conditions. If context is helpful the expression is arising from the Palm probability of the set $G$, where the point process considered is the one associated with exits from $D$.
Thanks again!