Copula constructions using ultramodularity

We discuss some copula constructions by means of ultramodular bivariate copulas. In general, the ultramodularity of a real function is a stronger version of both its convexity and its supermodularity (the latter property being always satisfied in the case of a bivariate copula). In a statistical sense, ultramodular bivariate copulas are related to random vectors whose components are mutually stochastically decreasing with respect to each other. Analytically speaking, an ultramodular bivariate copula is characterized by the convexity of all of its horizontal and vertical sections. Among other results, we give a sufficient condition for the additive generators of Archimedean ultramodular bivariate copulas, and we propose two constructions for bivariate copulas: the first one being based on ultramodular aggregation functions, and the other one showing the special role of ultramodularity and Schur concavity for a product-like composition of bivariate copulas being again a bivariate copula.