Principles of Copula Theory

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 functionsDefinition and first examplesCharacterization in terms of properties of d.f.s Continuity and absolutely continuity The derivatives of a copulaThe space of copulasGraphical representations Copulas and Stochastic Dependence Construction of multivariate stochastic models via copulas Sklar’s theorem Proofs of Sklar’s theoremCopulas 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 copulasCopulas with fractal supportCopulas, conditional expectation, and Markov kernelCopulas and measure-preserving transformations Shuffles of a copula Sparse copulas Ordinal sumsThe Kendall distribution function Copulas and Approximation Uniform approximations of copulasApplication to weak convergence of multivariate d.f.s Markov kernel representation and related distancesCopulas and Markov operatorsConvergence 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 expectationsThe Markov product and Markov processes A generalization of the Markov product A Compendium of Families of Copulas What is a family of copulas? Fréchet 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 propertiesCharacterizations of quasi-copulas The space of quasi-copulas and its lattice structureMass distribution associated with a quasi-copula Generalizations of Copulas: Semi-Copulas Definition and basic properties Bivariate semi-copulas, triangular norms, and fuzzy logicRelationships among capacities and semi-copulas Transforms of semi-copulasSemi-copulas and level curves Multivariate aging notions of NBU and IFR Bibliography Index

Fabrizio Durante is a professor in the Faculty of Economics and Management at the Free University of Bozen–Bolzano. He is an associate editor of Computational Statistics & Data Analysis and Dependence Modeling. His research focuses on multivariate dependence models with copulas, reliability theory and survival analysis, and quantitative risk management. He earned a PhD in mathematics from the University of Lecce and habilitation in mathematics from the Johannes Kepler University Linz. Carlo Sempi is a professor in the Department of Mathematics and Physics at the University of Salento. He has published nearly 100 articles in many journals. His research interests include copulas, quasi-copulas, semi-copulas, weak convergence, metric spaces, and normed spaces. He earned a PhD in applied mathematics from the University of Waterloo.