Bivariate copula decomposition in terms of comonotonicity, countermonotonicity and independence

Copulas are statistical tools for modelling the multivariate dependence structure among variables in a distribution free way. This paper investigates bivariate copula structure; the existence and uniqueness of a bivariate copula decomposition into a comonotonic, an independent, a countermonotonic, and an indecomposable part are proved, while the coefficients are determined from partial derivatives of the corresponding copula. Moreover, for the indecomposable part, an optimal convex approximation is provided and analyzed on the basis of the usual criterion. Some applications of the decomposition in finance and insurance are mentioned.