On the Characterization of Copulas by Differential Equations

Abstract

We study the semigroup action induced by univariate conditioning of copulas. Based on this, we give a new characterization of bivariate copulas in terms of flows generated by solutions of ordinary differential equations with not necessary continuous right side. Several applications, related to concordance ordering of copulas, illustrate the usefulness of this result.

Keywords:

• Copulas
• Discontinuous ordinary differential equations
• Univariate conditioning
• Archimedean copulas
• Left invariant copulas
• Positive quadrant dependence (PQD)
• Concordance ordering

Mathematics Subject Classification:

• 62H20
• 60E05
• 34A36