= Solution
For the natural <filtration> $\mathcal F_t=\sigma(X_s:0\leq s\leq t)$, a nonnegative random time $\tau$ is a <stopping time> when $\{\tau\leq t\}\in\mathcal F_t$ for every $t\geq0$: 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 $(X_{\tau+s})_{s\geq0}$ 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 $P_s=e^{sQ}$,
$$
\boxed{\mathbb E[f(X_{\tau+s})\mid\mathcal F_\tau]=(P_sf)(X_\tau)\quad\text{on }\{\tau<\infty\}.}
$$
The property applies also to bounded functions of several future times or of the entire future path.
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