Construction methods for absolutely continuous copulas with specified curvilinear sections

This paper is devoted to the constructions of absolutely continuous copulas having a specified curvilinear section. To this end, we first characterize a generalization of curvilinear copula with a specified curvilinear section, and establish a necessary and sufficient condition for this generalization to be a bivariate copula. Subsequently, utilizing this generalization of curvilinear copula, we construct a family of absolutely continuous copulas, for a suitable function generated by a given curve, ensuring that this function as curvilinear section. We further verify that under an assumption on a curve, any curvilinear section that remains strictly below both the given curve and the diagonal of the unit square serves as the curvilinear section of an absolutely continuous copula. Finally, employing techniques based on the ordinal sum and the patchwork construction, we provide explicit construction methods for absolutely continuous copulas with a general curvilinear section.