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Consider a Markov Renewal Process $(X_n, T_n)$ on a countable state space $E$ where $T_n \in [0, \infty]$, $0 = T_0 < T_1 < \dots$ and $T_n \to \infty$ almost surely.

By the above properties of $T_n$ the counting process $N_t := \sup \{ n \in \mathbb{N} \mid T_n \leq t \}$ takes finite values. (Moreover, on $T_n = \infty$ the process $N_t$ has a finite number of jumps.) We can now define the continuous-time process $Y_t := X_{N_t}$ for all $t \geq 0$. Such a process is called semi-Markov in the literature.

Define the holding times $S_n := T_n - T_{n-1}$ on $T_{n-1} < \infty$ and $S_n := \infty$ on $T_{n-1} = \infty$. If all the $S_n$ are exponentially distributed (conditioned on $X_n$) with rate $\lambda(X_n) \in [0, \infty)$ then $Y_t$ is a continuous-time Markov chain.

Question: Is $Y_t$ always a Feller process?

Note that if all these rates are uniformly bounded then $Y_t$ is a uniformizable Markov chain.

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