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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.