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Principles of Copula Theory
Fabrizio Durante, Carlo Sempi
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Últimas novedades estadística matemáticas
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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. |
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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 |
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