Nonexplosive continuous-time Markov chain

A continuous-time Markov chain with finite holding rates is nonexplosive when its jump times have no finite accumulation almost surely. It then remains in its specified state space at every finite time, rather than being lost to a cemetery state after infinitely many jumps. Bounded rates are sufficient for nonexplosion, but finite rates at individual states alone are not.

For a nonexplosive continuous-time Markov chain on the nonnegative integers, suppose zero is absorbing and each positive state has positive rate to zero. On every finite positive set the conditional chance of absorption at the next jump has a positive minimum. By the Strong Markov property, the chance of surviving $r$ departures from that set is bounded by $(1-\delta {)}^r$. Without absorption only finitely many such departures occur, and finite positive holding rates prohibit staying forever at a positive state. Thus eventually the path stays outside each finite set: it escapes to infinity. This conclusion needs no upward skip-free assumption.