Piotr Jaworski

Copula Theory and Its Applications

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

On copulas and their diagonals

We formulate the necessary and sufficient conditions for a function @d:[0,1]->[0,1] to be the diagonal section of a multivariate absolutely continuous copula. Moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics.

Absolutely Continuous Copulas with Given Diagonal Sections

Necessary and sufficient conditions are given in order to ensure that a function δ:[0, 1] → [0, 1] is the diagonal section of an absolutely continuous copula. Explicit constructions are provided.

Invariant dependence structure under univariate truncation

The class of all bivariate copulas that are invariant under univariate truncation is characterized. To this end, a family of bivariate copulas generated by a real-valued function is introduced. The obtained results are also used in order to show that the Clayton family of copulas (including its limiting elements) coincides with the class of copulas that are invariant under bivariate truncation and contains all exchangeable copulas which are invariant under univariate truncation.

A spatial contagion measure for financial time series

A novel spatial contagion measure is proposed that is based on the calculation of suitable conditional Spearmans correlations extracted from the financial time series of interest. Algorithms for the numerical estimation of this measure are illustrated, together with a simulation study showing its features in relations with popular families of copulas. Finally, two applications are presented about the use of spatial contagion measure for determining (asymmetric) linkages in the financial systems, and creating clusters of financial time series. In particular, contrarily to previous approaches in the literature, such clusters identify which time series increase their (positive) association when the market is under distress. The presented methodology is also expected to be useful to select a diversified portfolio of asset returns.

A New Characterization of Bivariate Copulas

A new characterization of bivariate copulas is given by using the notion of Dini derivatives. Several examples illustrate the usefulness of this result.

Invariant dependence structure under univariate truncation: the high-dimensional case

In this paper, the class of all multivariate copulas that are invariant under univariate truncation is characterized.

The Limiting Properties of Copulas Under Univariate Conditioning

The dynamics of univariate conditioning of copulas with respect to the first variable is studied. Special attention is paid to the limiting properties when the first variable is attaining extreme values. We describe the copulas which are invariant with respect to the conditioning and study their sets of attraction. Furthermore we provide examples of the limit sets consisting of more than one element and discuss the chaotic nature of univariate conditioning.

Conditional risk based on multivariate hazard scenarios

We present a novel methodology to compute conditional risk measures when the conditioning event depends on a number of random variables. Specifically, given a random vector