Lumpability and marginalisability for continuous-time Markov chains

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let { X { t )} = {( X 1 ( t ), X 2 ( t ), · ··, X m ( t ))} be a continuous-time Markov chain. Under what conditions are the marginal processes { X 1 ( t )}, { X 2 ( t )}, · ··, { X m ( t )} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.