# Markov renewal process and Feller property

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.