Stochastic domination and Markovian couplings

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P less than or equal to (st) P', we show t he existence of an order-preserving Markovian coupling and give a way to co nstruct it. From our proof, we also obtain the conditions of Brandt and Las t for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the ex istence of Markovian couplings between two Jackson networks.