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On the singular components of a copula

  • Fabrizio Durante (a1), Juan Fernández-Sánchez (a2) and Wolfgang Trutschnig (a3)

We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.

Corresponding author
Postal address: Faculty of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy. Email address:
∗∗ Postal address: Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain. Email address:
∗∗∗ Postal address: Department for Mathematics, University of Salzburg, Salzburg, Austria. Email address:
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[1] Ash, R. B. (2000). Probability and Measure Theory, 2nd edn. Harcourt/Academic Press, Burlington, MA.
[2] De Amo, E., Díaz Carrillo, M. and FernáNdez-Sánchez, J. (2010). On concordance measures and copulas with fractal support. In Combining Soft Computing and Statistical Methods in Data Analysis (Adv. Intelligent Soft Comput. 77), eds Borgelt, C. et al., Springer, Berlin, pp. 131138.
[3] Durante, F., FernáNdez Sánchez, J. and Sempi, C. (2013). A note on the notion of singular copula. Fuzzy Sets Systems 211, 120122.
[4] Durante, F., Foschi, R. and Spizzichino, F. (2010). Aging functions and multivariate notions of NBU and IFR. Prob. Eng. Inf. Sci. 24, 263278.
[5] Durante, F., Sarkoci, P. and Sempi, C. (2009). Shuffles of copulas. J. Math. Anal. Appl. 352, 914921.
[6] Fredricks, G. A., Nelsen, R. B. and RodríGuez-Lallena, J. A. (2005). Copulas with fractal supports. Insurance Math. Econom. 37, 4248.
[7] Jaworski, P., Durante, F. and Härdle, W. K., (eds) (2013). Copulae in Mathematical and Quantitative Finance (Lecture Notes Statist. 213). Springer, Berlin.
[8] Jaworski, P., Durante, F., Härdle, W. K. and Rychlik, T., (eds) (2010). Copula Theory and Its Applications (Lecture Notes Statist. 198). Springer, Berlin.
[9] Joe, H. (1997). Multivariate Models and Dependence Concepts (Monogr. Statist. Appl. Prob. 73). Chapman & Hall, London.
[10] Joe, H. (2015). Dependence Modeling with Copulas. CRC, Boca Raton, FL.
[11] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
[12] Lange, K. (1973). Decompositions of substochastic transition functions. Proc. Amer. Math. Soc. 37, 575580.
[13] Mai, J.-F. and Scherer, M. (2012). Simulating Copulas (Ser. Quant. Finance 4). Imperial College Press, London.
[14] Mai, J.-F. and Scherer, M. (2014). Simulating from the copula that generates the maximal probability for a joint default under given (inhomogeneous) marginals. In Topics in Statistical Simulation, eds Melas, V. B. et al., Springer, New York, pp. 333341.
[15] Navarro, J. and Spizzichino, F. (2010). Comparisons of series and parallel systems with components sharing the same copula. Appl. Stoch. Models Business Industry 26, 775791.
[16] Navarro, J., Del áGuila, Y., Sordo, M. A. and SuáRez-Llorens, A. (2013). Stochastic ordering properties for systems with dependent identically distributed components. Appl. Stoch. Models Business Industry 29, 264278.
[17] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
[18] Trutschnig, W. and FernáNdez Sánchez, J. (2013). Some results on shuffles of two-dimensional copulas. J. Statist. Planning Infer. 143, 251260.
[19] Trutschnig, W. and FernáNdez-Sánchez, J. (2014). Copulas with continuous, strictly increasing singular conditional distribution functions. J. Math. Anal. Appl. 410, 10141027.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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