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