Exact upper bound for copulas with a given diagonal section

TL;DR

This paper provides an explicit formula for the exact upper bound of bivariate copulas with a specified diagonal section, resolving a long-standing open problem and enabling practical applications in measuring copula asymmetry.

Contribution

It introduces a precise formula for the upper bound of copulas with a given diagonal section, addressing a 15-year-old open question and deriving a practical measure of maximal asymmetry.

Findings

Explicit formula for the upper bound of copulas with a given diagonal section

Determination of the maximal asymmetry for such copulas

Construction of a copula that attains maximal asymmetry

Abstract

We answer a 15-year-old open question about the exact upper bound for bivariate copulas with a given diagonal section by giving an explicit formula for this bound. As an application, we determine the maximal asymmetry of bivariate copulas with a given diagonal section and construct a copula that attains it. We derive a formula for the maximal asymmetry that is simple enough to be used by practitioners.