Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-18T05:56:15.495Z Has data issue: false hasContentIssue false
  • English
  • Français

On the singular components of a copula

Part of: Mathematical finance Distribution theory - Probability Classical measure theory

Published online by Cambridge University Press:  30 March 2016

Fabrizio Durante ,
Juan Fernández-Sánchez  and
Wolfgang Trutschnig
Fabrizio Durante*
Affiliation:
Free University of Bozen-Bolzano
Juan Fernández-Sánchez*
Affiliation:
Universidad de Almería
Wolfgang Trutschnig*
Affiliation:
University of Salzburg
*
∗Postal address: Faculty of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy. Email address: fabrizio.durante@unibz.it
∗∗Postal address: Grupo de Investigación de Análisis Matemático, Universidad de Almería, La Cañada de San Urbano, Almería, Spain. Email address: juan.fernandez@ual.es
∗∗∗Postal address: Department for Mathematics, University of Salzburg, Salzburg, Austria. Email address: wolfgang@trutschnig.net
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Copulacouplingdisintegration theoremMarkov kernelsingular measure

MSC classification

Primary: 60E05: Distributions
Secondary: 28A50: Integration and disintegration of measures 91G70: Statistical methods, econometrics
Type
Short Communications
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1175 - 1182
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Ash, R. B. (2000). Probability and Measure Theory, 2nd edn. Harcourt/Academic Press, Burlington, MA.Google Scholar
[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. 131–138.Google Scholar
[3] Durante, F., FernáNdez Sánchez, J. and Sempi, C. (2013). A note on the notion of singular copula. Fuzzy Sets Systems 211, 120–122.Google Scholar
[4] Durante, F., Foschi, R. and Spizzichino, F. (2010). Aging functions and multivariate notions of NBU and IFR. Prob. Eng. Inf. Sci. 24, 263–278.Google Scholar
[5] Durante, F., Sarkoci, P. and Sempi, C. (2009). Shuffles of copulas. J. Math. Anal. Appl. 352, 914–921.Google Scholar
[6] Fredricks, G. A., Nelsen, R. B. and RodríGuez-Lallena, J. A. (2005). Copulas with fractal supports. Insurance Math. Econom. 37, 42–48.Google Scholar
[7] Jaworski, P., Durante, F. and Härdle, W. K., (eds) (2013). Copulae in Mathematical and Quantitative Finance (Lecture Notes Statist. 213). Springer, Berlin.Google Scholar
[8] Jaworski, P., Durante, F., Härdle, W. K. and Rychlik, T., (eds) (2010). Copula Theory and Its Applications (Lecture Notes Statist. 198). Springer, Berlin.Google Scholar
[9] Joe, H. (1997). Multivariate Models and Dependence Concepts (Monogr. Statist. Appl. Prob. 73). Chapman & Hall, London.Google Scholar
[10] Joe, H. (2015). Dependence Modeling with Copulas. CRC, Boca Raton, FL.Google Scholar
[11] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[12] Lange, K. (1973). Decompositions of substochastic transition functions. Proc. Amer. Math. Soc. 37, 575–580.Google Scholar
[13] Mai, J.-F. and Scherer, M. (2012). Simulating Copulas (Ser. Quant. Finance 4). Imperial College Press, London.Google Scholar
[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. 333–341.Google Scholar
[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, 775–791.Google Scholar
[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, 264–278.Google Scholar
[17] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.Google Scholar
[18] Trutschnig, W. and FernáNdez Sánchez, J. (2013). Some results on shuffles of two-dimensional copulas. J. Statist. Planning Infer. 143, 251–260.Google Scholar
[19] Trutschnig, W. and FernáNdez-Sánchez, J. (2014). Copulas with continuous, strictly increasing singular conditional distribution functions. J. Math. Anal. Appl. 410, 1014–1027.Google Scholar