Principles of Copula Theory explores the state of the art on copulas and provides you with the foundation to use copulas in a variety of applications. Throughout the book, historical remarks and further readings highlight active research in the field, including new results, streamlined presentations, and new proofs of old results. After covering the essentials of copula theory, the book addresses the issue of modeling dependence among components of a random vector using copulas. It then presents copulas from the point of view of measure theory, compares methods for the approximation of copulas, and discusses the Markov product for 2-copulas. The authors also examine selected families of copulas that possess appealing features from both theoretical and applied viewpoints. The book concludes with in-depth discussions on two generalizations of copulas: quasi- and semi-copulas. Although copulas are not the solution to all stochastic problems, they are an indispensable tool for understanding several problems about stochastic dependence. This book gives you the solid and formal mathematical background to apply copulas to a range of mathematical areas, such as probability, real analysis, measure theory, and algebraic structures.
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 space of copulas Graphical representations Copulas and Stochastic Dependence Construction of multivariate stochastic models via copulas Sklar's theorem Proofs of Sklar's theorem Copulas and risk-invariant property Characterization of basic dependence structures via copulas Copulas and order statistics Copulas and Measures Copulas and d-fold stochastic measures Absolutely continuous and singular copulas Copulas with fractal support Copulas, conditional expectation, and Markov kernel Copulas and measure-preserving transformations Shuffles of a copula Sparse copulas Ordinal sums The Kendall distribution function Copulas and Approximation Uniform approximations of copulas Application to weak convergence of multivariate d.f.s Markov kernel representation and related distances Copulas and Markov operators Convergence in the sense of Markov operators The Markov Product of Copulas The Markov product Invertible and extremal elements in C2 Idempotent copulas, Markov operators, and conditional expectations The Markov product and Markov processes A generalization of the Markov product A Compendium of Families of Copulas What is a family of copulas? Frechet copulas EFGM copulas Marshall-Olkin copulas Archimedean copulas Extreme-value copulas Elliptical copulas Invariant copulas under truncation Generalizations of Copulas: Quasi-Copulas Definition and first properties Characterizations of quasi-copulas The space of quasi-copulas and its lattice structure 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 Transforms of semi-copulas Semi-copulas and level curves Multivariate aging notions of NBU and IFR Bibliography Index
