This book presents a much-needed synthesis of recent research on copulas, a statistical tool widely used in applications in financial risk assessment and actuarial analysis. The text provides an overview of the mathematical properties of copulas with an emphasis on recent developments. It contains numerous examples using real data, especially from finance and insurance. This unique resource not only provides up-to-date information on this growing field, but it also covers the multidimensional case.

Copulas: Basic Definitions and Properties Notations Preliminaries on random variables and distribution functions Definition and first examples Characterization in terms of properties of d.f.s Continuity and absolutely continuity The derivatives of a copula The family of copulas Cd Classification of copulas Copulas and Random Variables Construction of multivariate stochastic models via copulas Sklar’s theorem Copulas and conditional probabilities Proofs of Sklar’s theorem Characterization of dependence structures via copulas Extensions of Sklar’s theorem Copulas and Measure Copulas and d-fold stochastic measures Absolutely continuous and singular copulas Copulas and measure–preserving transformations Shuffles of a copula Sparse copulas The Kendall’s measure Copulas and Markov Operators Markov operators Copulas and Markov operators Copulas and Approximation Approximations of copulas Application to weak convergence of multivariate d.f.s Approximation in M- and ?-convergence Approximation in the Sobolev sense The TFS metrics Generalizations of Copulas: Quasi-Copulas Quasi-copulas Characterizations of quasi–copulas The space Qd Mass distribution associated with a quasi-copula Generalizations of Copulas: Semi-Copulas Definition and basic properties Bivariate semi-copulas, triangular norms and fuzzy logic Relationships among capacities and semi-copulas Semi-copulas and level curves Multivariate ageing notions of NBU and IFR Copula-Based Measures of Association The bivariate case The multivariate case Multivariate extensions of bivariate measures Copulas and Stochastic Processes General considerations The *-product Markov operators and *-product The Sobolev geometry of C2 Copulas and conditional expectations The *-product and Markov processes Copulas and Brownian motion Other Markov processes Coupled Brownian motion Exchangeable sequences The *C-product The compatibility question The special case d = 3 and k = 2 Properties of D(A, B) Appendix A: Preliminaries Appendix B: A Compendium of Families of Copulas Bibliography Index