Bivariate distributions with ordered marginals

This paper provides a characterization of all possible dependency structures between two stochastically ordered random variables. The answer is given in terms of copulas that are compatible with the stochastic order and the marginal distributions. The extremal values for Kendall's $\tau$ and Spearman's $\rho$ for all these copulas are given in closed form. We also find an explicit form for the joint distribution with the maximal entropy. A multivariate extension and a generalization to random elements in partially ordered spaces are also provided.