Construction of Order-Preserving Coupling for One-Dimensional Markov Chains

This paper is devoted to construct an order-preserving coupling for two regular stochastically comparable Markov chains with state space E = 0, 1,… endowed with the ordinary order. Meanwhile, for a specific type of distance p, the p-optimal Markovian couplings are studied. It is proved that the order-preserving coupling constructed in the paper is indeed p-optimal for two stochastically comparable marginal Markov chains.