Stochastic domination and Markovian couplings

For continuous-time Markov chains with semigroups P, P′ taking values in a partially ordered set, such that P ≤st P′, we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.