In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas. 1.1 Historical Introduction The history of copulas may be said to begin with Frechet [70]. He studied the following problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space (Ω ,F ,P), what can be said about the set Γ (F1,F2) of the bivariate d.f.’s whose marginals are F1 and F2? It is immediate to note that the set Γ (F1,F2), now called the Frechet class of F1 and F2, is not empty since, if X1 and X2 are independent, then the distribution function (x1,x2) → F(x1,x2) = F1(x1)F2(x2) always belongs to Γ (F1,F2). But, it was not clear which the other elements of Γ (F1,F2) were. Preliminary studies about this problem were conducted in [65, 71, 90] (see also [31, 182] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [192]. In his own words [194]: Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Linz Austria e-mail: fabrizio.durante@jku.at Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi”, Universita del Salento, Lecce, Italy e-mail: carlo.sempi@unisalento.it P. Jaworski et al. (eds.), Copula Theory and Its Applications, Lecture Notes in Statistics 198, DOI 10.1007/978-3-642-12465-5_1, c © Springer-Verlag Berlin Heidelberg 2010 4 Fabrizio Durante and Carlo Sempi [...] In the meantime, Bert (Schweizer) and I had been making progress in our work on statistical metric spaces, to the extent that Menger suggested it would be worthwhile for us to communicate our results to Frechet. We did: Frechet was interested, and asked us to write an announcement for the Comptes Rendus [184]. This began an exchange of letters with Frechet, in the course of which he sent me several packets of reprints, mainly dealing with the work he and his colleagues were doing on distributions with given marginals. These reprints, among the later arrivals of which I particularly single out that of Dall’Aglio [29], were important for much of our subsequent work. At the time, though, the most significant reprint for me was that of Feron [65]. Feron, in studying three-dimensional distributions had introduced auxiliary functions, defined on the unit cube, that connected such distributions with their one-dimensional margins. I saw that similar functions could be defined on the unit n-cube for all n≥ 2 and would similarly serve to link n-dimensional distributions to their one-dimensional margins. Having worked out the basic properties of these functions, I wrote about them to Frechet, in English. He asked me to write a note about them in French. While writing this, I decided I needed a name for these functions. Knowing the word “copula” as a grammatical term for a word or expression that links a subject and predicate, I felt that this would make an appropriate name for a function that links a multidimensional distribution to its one-dimensional margins, and used it as such. Frechet received my note, corrected one mathematical statement, made some minor corrections to my French, and had the note published by the Statistical Institute of the University of Paris as Sklar [192]. The proof of Sklar’s theorem was not given in [192], but a sketch of it was provided in [193] (see also [185]), so that for a few years practitioners in the field had to reconstruct it relying on the hand-written notes by Sklar himself; this was the case, for instance, of the second author. It should be also mentioned that some “indirect” proofs of Sklar’s theorem (without mentioning copula) were later discovered by Moore and Spruill [145] and Deheuvels [37] For about 15 years, all the results concerning copulas were obtained in the framework of the theory of Probabilistic Metric spaces [186]. The event that arose the interest of the statistical community in copulas occurred in the mid seventies, when Bert Schweizer, in his own words (see [183]), quite by accident, reread a paper by A. Renyi, entitled On measures of dependence and realized that [he] could easily construct such measures by using copulas. See [166] for Renyi’s paper. The first building blocks were the announcement by Schweizer and Wolff in the Comptes Rendus de l’Academie des Sciences [187] and Wolff’s Ph.D. Dissertation at the University of Massachusetts at Amherst [200]. These results were presented to the statistical community in the paper [188] (compare also with [201]). However, for several other years, Chapter 6 of the fundamental book [186] by Schweizer and Sklar, devoted to the theory of Probabilistic metric spaces and published in 1983, was the main source of basic information on copulas. Again in Schweizer’s words from [183], After the publication of these articles and of the book . . . the pace quickened as more . . . students and colleagues became involved. Moreover, since interest in questions of statistical dependence was increasing, others came to the subject from different directions. In 1986 the enticingly entitled article The joy of copulas by C. Genest and R.C MacKay [82], attracted more attention. 1 Copula Theory: An Introduction 5 In 1990, Dall’Aglio organized the first conference devoted to copulas, aptly called “Probability distributions with given marginals” [32]. This turned out to be the first in a series of conferences that greatly helped the development of the field, since each of them offered the chance of presenting one’s results and to learn those of other researchers; these conferences were held in Seattle in 1993 [176], in Prague in 1996 [11], in Barcelona in 2000 [26], in Quebec in 2004 [75, 76], and in Tartu in 2007 [119]; the next one is scheduled to be in Sao Paulo in 2010. At end of the nineties, the notion of copulas became increasingly popular. Two books about copulas appeared and were to become the standard references for the following decade. In 1997 Joe published his book on multivariate models [104], with a great part devoted to copulas and families of copulas. In 1999 Nelsen published the first edition of his introduction to copulas [150] (reprinted with some new results in [151]). But, the main reason of this increased interest has to be found in the discovery of the notion of copulas by researchers in several applied field, like finance. Here we should like briefly to describe this explosion by quoting Embrechts’s comments [57]: As we have seen so far, the notion of copula is both natural as well as easy for looking at multivariate d.f.’s. But why do we witness such an incredible growth in papers published starting the end of the nineties (recall, the concept goes back to the fifties and even earlier, but not under that name). Here I can give three reasons: finance, finance, finance. In the eighties and nineties we experienced an explosive development of quantitative risk management methodology within finance and insurance, a lot of which was driven by either new regulatory guidelines or the development of new products; see for instance Chapter 1 in [138] for the full story on the former. Two papers more than any others “put the fire to the fuse”: the [...] 1998 RiskLab report [58] and at around the same time, the Li credit portfolio model [121]. The advent of copulas in finance [79] originated a wealth of investigations about copulas and, especially, applications of copulas. See, for example, the books [19, 130, 138, 181]. At the same time, different fields like hydrology [77, 177] discovered the importance of this concept for constructing more flexible multivariate models. Nowadays, it is near to impossible to give a complete account of all the applications of copulas to the many fields where they have be used. As Schweizer wrote [183]: The “era of i.i.d.” is over: and when dependence is taken seriously, copulas naturally come into play. It remains for the statistical community at large to recognize this fact. And when every statistics text contains a section or chapter on copulas, the subject will have come of age. However, a word of caution is in order here. Several criticisms have been recently raised about copulas and their applications, and several people started to speak about “copula craze” [57]. See, for example, the very interesting discussion related to the paper by Mikosch [140, 141] (see also [56, 85, 94, 105, 126, 161, 189]). From our point of view, these criticisms were a quite natural reaction to such a wide diffusion of applications of copulas, not always in a well motivated way. It should be said that several people have wrongly interpreted copulas as the solution 6 Fabrizio Durante and Carlo Sempi to “all problems of stochastic dependence”. This is definitely not the case! Copulas are an indispensable tool for understanding several problems about stochastic dependence, but they are not the “panacea” for all stochastic models. Despite this broad range of interest about copulas, we still believe that this concept is still in “its infancy” [151] and several other investigations may (and should) be conducted in order to stress whether copulas, or related copula-based concepts, can be really considered as a “strong” mathematical concept worth of use in several applications (see Sect. 1.8).