Copulas and Preserver Problems

TL;DR

This paper explores preserver problems within the context of copulas, demonstrating that the copula property is uniquely preserved under increasing transformations, contributing to the mathematical understanding of invariance in this field.

Contribution

It introduces the concept of copula preserver problems and proves the uniqueness of copula property preservation under increasing transformations.

Findings

Copula property is preserved only under increasing transformations.

The paper extends preserver problem theory to the field of copulas.

Provides a mathematical characterization of invariance in copula transformations.

Abstract

Preserver problems concern the characterization of operators on general spaces that leave invariant some categories of subsets or ratios. The most known in the mathematical literature are those of linear preserver problems (LPP) which date back to the ninth century. Here, we treat the preserver problem in the recent and emerging field of copulas. Precisely, we prove that \emph{copula property} is preserved uniquely under increasing transformations.