 Solution

For the natural filtration ${\mathcal{F}}_t=\sigma (X_s:0\le s\le t)$, a nonnegative random time $\tau$ is a stopping time when $\{\tau \le t\}\in {\mathcal{F}}_t$ for every $t\ge 0$: whether it has occurred can be decided from the observations already made.

For a time-homogeneous finite-state continuous-time Markov chain, the Strong Markov property says that, on $\{\tau <\infty \}$, conditional on ${\mathcal{F}}_{\tau }$, the process $(X_{\tau +s}{)}_{s\ge 0}$ has the law of a fresh chain started at $X_{\tau }$ and is independent of the earlier history given that state. In particular, with transition matrix $P_s=e^{sQ}$,The property applies also to bounded functions of several future times or of the entire future path.