On the Markov property of the occupation time for continuous-time inhomogeneous Markov chains

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.