time-homogeneous continuous time Markov chain I have a question about the continuous time Markov chain. In the Poisson process we have independent and stationary increments. Do we have this in a continuous time Markov chain that is time-homogeneous?
I know that a Poisson process is a time-homogeneous continuous time Markov chain, so some of these has both independent and stationary increments. But do all of them have it?
 A: No. For a counterexample, consider a continuous time Markov chain $X_t$ whose state-space is $\{0,1\}$ and both states are recurrent. Let $t_3>t_2>t_1$. If the increment $X_{t_2}-X_{t_1}$ equals $1$, this clearly implies that $X_{t_2} = 1$, hence increment $X_{t_3}-X_{t_2}$ is either $0$ or $-1$. On the other hand, if the increment $X_{t_2}-X_{t_1}$ is $-1$,  $X_{t_2} = -1$, hence increment $X_{t_3}-X_{t_2}$ is either $0$ or $1$. Thus, as the recurring states assumption also implies that $P(\textrm{increment}=0)<1$, the increments $X_{t_3}-X_{t_2}$ and $X_{t_2}-X_{t_1}$ are not independent.
Furthermore, the state-space of a continuous time Markov chain could be something where subtraction (and hence) increments are not even defined. 
Concerning stationarity of increments
The question is about independent and stationary increments, and the previous counterexample showed that the increments can be not independent, thus not (stationary and independent). In addition, the increments can be not stationary, too. Let $X$ be a Poisson process. Let $Y_t=X_t^2$ for all $t$. Now, $Y$ is also a time-homogenous continuous-time Markov chain, but the expected increment of $Y$ increases over time, thus the increments are not stationary.
