Patchwork Methods for Multivariate Quasi-Copulas

In this contribution, we exhibit the work presented in a series of recent papers of the authors, members of Ljubljana Copula Group. These results appear to be new since the patchwork approach has been presented for copulas so far. In the bivariate case we can find actual constructions of desired patches. It seems that the multivariate case is much more involved, so that these constructions cannot be extended to higher dimensions. Our main result is to present the conditions both on the domain of the patch and the values on its border for the desired multivariate quasi-copula to exist. This work is somewhat related to patchwork constructions of aggregation functions with given upper and/or lower margins being modular or 2-increasing.