Geodesic
On the Markov property of the occupation time for continuous-time inhomogeneous Markov chains
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 23-49

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It is known that the occupation time random field for a homogeneous Markov chain is Markovian. One investigates the possibility of generalizing this result for inhomogeneous chains. Consider a process which is a homogeneous Markov chain with the transition probability density $Q_1$ up to time $T$ and with the density $Q_2$ after $T$ ($Q_1\ne Q_2$). It turns out that even in this simplest case the occupation time is not Markovian. 
@article{ZNSL_2013_420_a1,
     author = {A. A. Vorotov},
     title = {On the {Markov} property of the occupation time for continuous-time inhomogeneous {Markov} chains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {23--49},
     publisher = {mathdoc},
     volume = {420},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a1/}
}
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A. A. Vorotov. On the Markov property of the occupation time for continuous-time inhomogeneous Markov chains. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 23-49. http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a1/