This book presents the theory of bivariate copulas and shows their potential for quantifying stochastic dependence between two random variables through measures of concordance. It emphasizes the importance of copula measures, patchwork copulas and transformations of copulas.

Opening with a prologue leading to the famous theorem of Sklar, which can be considered the backbone of copula theory, the book is organized into three parts. The first part presents the foundations of bivariate copulas and copula measures, as well as the construction of patchwork copulas. The second part is devoted to transformations and measures of concordance. It introduces a group of transformations of copulas, studies invariance properties and the integral of a copula with respect to a copula measure, and then develops the theory of measures of concordance, including their comparison and estimation. The third part covers a range of special topics, including Archimedean and max-stable copulas, diagonals, distortions and shuffles of copulas, measures of dependence and asymmetry, the Markov product of copulas and measures of functional dependence, as well as order statistics and quasicopulas. The text concludes with a glance at multivariate copulas.

Including problems for self-study, the book provides a solid basis for graduate students and researchers new to the field who are interested in copula theory.

Sebastian Fuchs is a Professor at the Department of Artificial Intelligence and Human Interfaces and the Head of the IDA Lab Team Dependence Modeling at the Paris Lodron University of Salzburg, Austria. His research focuses on dependence modeling and copula theory, multivariate and nonparametric statistics, statistical learning theory, nearest-neighbor statistics, and quantitative risk management. He develops statistical and data-analytic methods for quantifying and estimating dependence in multivariate data, aiming to deepen our understanding of complex, high-dimensional dependence structures.

Klaus D. Schmidt is a Senior Professor of Mathematics at the Institute of Mathematics, University of Mannheim, Germany, and Professor Emeritus at the Institute of Mathematical Stochastics, TU Dresden, Germany. His research is devoted to various areas in functional analysis, measure theory, probability theory, and actuarial mathematics. He is the author or co-author of several monographs and textbooks.