Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 26J / f

At a state with positive departure rate ${\lambda }_i=R-r_i$, the next-jump probabilities areIf at least two rates are positive, every state has positive departure rate and the normalizing constant $Z={\sum }_i{r}_i(R-r_i)=R^2-{\sum }_i{r}_i^2$ is positive. The jump-chain equilibrium law isIndeed ${\hat{\pi }}_i{\hat{p}}_{ij}=r_i{r}_j/Z$ is symmetric, proving both stationarity and reversibility. Jump observations weight the continuous-time equilibrium by the departure rate, so $\hat{\pi }$ is generally different from $\pi$.

If exactly one rate, $r_k$, is positive, state $k$ is absorbing and the denominator $R-r_k$ vanishes. There is no next jump from that state: the genuine embedded jump sequence terminates there. Under the usual extended-chain convention set ${\hat{p}}_{kk}=1$; every other state jumps to $k$ with probability one. Its equilibrium law is ${\delta }_k$ and it is trivially reversible there. The normalized $r_i(R-r_i)$ formula cannot be used in that absorbing case.