On Patchwork Techniques for 2-Increasing Aggregation Functions and Copulas

In recent years, there has been a raise of interest in the determination of copulas with given values at some fixed points, or with given horizontal, vertical, affine, diagonal, or sub-diagonal sections and combinations thereof. Closely related to these investigations are the determination and characterization of increasing and 2-increasing functions with given margins whose domain is a subset of the unit square as well as necessary and sufficient conditions providing that the combination (patchwork) of such functions on sub-domains yields a (new) 2-increasing aggregation function on [0,1]2, in particular a copula. In the present contribution we provide a full characterization of increasing, 2-increasing functions with prescribed margins acting on a sub-rectangle of the unit square. The characterization allows to determine easily the greatest and smallest such functions and to look at the results on copulas with given horizontal and/or vertical sections and its boundaries from a more general and unified viewpoint. We further discuss necessary and sufficient conditions for a patchwork based on triangular sub-domains.