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.
"This book presents an overview of the theory of copulas. The research on copulas and its applications has grown very rapidly from Sklar's seminal paper in 1959. Recent applications in finance have boosted new research, and several books have been written on this subject during the last two decades.
… In this book, the authors' focus is on the mathematical foundations of the theory of copulas. In fact, the first chapters are devoted to put the theory of copulas into the frame of measure theory. Moreover, the theory is presented for general d-copulas (d > 2), whenever possible. Pointers to applications are given in special paragraphs named \Further readings"…. The presentation is always clear and the mathematical language is precise. The book can be used as a reference book for mathematicians and statisticians interested in the theory of copulas, and especially in the mathematical foundations of that theory. Moreover, the book can also be useful to those researchers in applied fields who need a strong reference manual on the mathematical aspects of the theory of copulas."
—Fabio Rapallo (Alessandria) in Zentralblatt Mathematik, April 2018
"This book represents a rigourous introduction to the theory of copula models, the biggest and the most thorough yet at that. The level of detail and rigour targets mathematicians working in probability theory. The exposition starts with an overview of the history of the subject followed by eight chapters laying out the theory of Copula models. The idea of the book was to present modern theoretical foundations for Copula models now that the field has seen over 50 years of research that has greatly accelerated recently with the development of applications in Finance, Operations Research, Statistics and Biostatistics. … For a mathematically-oriented researcher in Statistics, Biostatistics and Applied Probability, and further in the specific applied fields of science, the book will serve as a reference for theoretical ideas potentially inspiring applied theory development."
— Alex Tsodikov, University of Michigan, in International Statistical Review, December 2017
Copulas: Basic Definitions and Properties
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
Copulas and Stochastic Dependence
Construction of multivariate stochastic models via copulas
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
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?
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