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Copulae became an extremely popular tool in different areas of research. Since the first applications in risk management in the late 90th, they attracted many other quantitatively oriented sciences like biostatistics, hydrology and finance. The main reason originates in the  Sklar (1959) theorem, which allows for separation of the marginal distributions from the dependency structure between the random variables. This editorial is organized as follows. In the first section we define the copulae and state the Sklar theorem. Some literature suggestions are given in the second section. The last section presents the content of this special issue.

We present a list of challenges one faces when given the task of modeling dependence between stochastic objects, with a special focus on financial applications. Our aim is to draw the readers' attention to common (and not so common) pitfalls and fallacies, and we particularly address readers who are new to dependence modeling. The presented list of challenges is clearly not complete, but it gives a flavor of how difficult and subtle the task of dependence modeling can be. Moreover, the readers shall get some intuition about what challenges are structural and cannot be overcome, and what challenges allow for a better solution than common practice might suggest.

The demand for an accurate financial risk management involving larger numbers of assets is strong not only in view of the financial crisis of 2007–2009. Especially dependencies among assets have not been captured adequately. While standard multivariate copulas have added some flexibility, this flexibility is insufficient in higher dimensional applications. Vine copulas can fill this gap by benefiting from the rich class of existing bivariate parametric copula families. Exploiting this in combination with GARCH models for the margins, we develop a regular vine copula based factor model for asset returns, the Regular Vine Market Sector model, which is motivated by the classical CAPM and shown to be superior to the CAVA model proposed by Heinen and Valdesogo (2009). The model can also be used to separate the systematic and idiosyncratic risk of specific stocks, and we explicitly discuss how vine copula models can be employed for active and passive portfolio management. In particular, Value-at-Risk forecasting and asset allocation are treated in detail. All developed models and methods are used to analyze the Euro Stoxx 50 index, a major market indicator for the Eurozone. Relevant benchmark models such as the popular DCC model and the common Student's t copula are taken into account.

Copulas are widely used for modeling the dependence structure of multivariate data. Many methods for estimating the copula density functions are investigated. In this paper, we study the asymptotic properties of the Bernstein estimator for unbounded copula density functions. We show that the estimator converges to infinity at the corner and we establish its relative convergence when the copula density is unbounded. Also, we provide the uniform strong consistency of the estimator on every compact in the interior region. We investigate the finite sample performance of the estimator via an extensive simulation study and we compare the Bernstein copula density estimator with other nonparametric methods. Finally, we consider an empirical application where the asymmetric dependence between international equity markets (US, Canada, UK, and France) is examined.

There is an increasing demand for models of multivariate time-series with time-varying and non-Gaussian dependencies. The available models suffer from the curse of dimensionality or from restrictive assumptions on the parameters and distributions. A promising class of models is that of hierarchical Archimedean copulae (HAC), which allows for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change-point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. Both examples reveal periods with constant and turmoil dependencies. The economic significance of the suggested modelling is evaluated using the Value-at-Risk of a portfolio.