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The Limiting Properties of Copulas Under Univariate Conditioning

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Copulae in Mathematical and Quantitative Finance

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 213))

Abstract

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.

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Notes

  1. 1.

    Sometimes in some special cases the conditional copula is called tail-dependence or threshold copula.

  2. 2.

    For basic facts concerning the univariate upper conditioning of random variables and copulas, see Section 3 of [24].

  3. 3.

    Copulas C f are known as Durante–Jaworski–Mesiar copulas[18, Section 5.4.4].

References

  1. Charpentier, A., Juri, A.: Limiting dependence structures for tail events, with applications to credit derivatives. J. Appl. Probab., 43(2), 563–586 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Charpentier, A., Segers, J.: Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100, 1521–1537 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chavez-Demoulin, V., Embrechts, P.: An EVT primer for credit risk. In: Lipton, A., Rennie, A. (eds.) Handbook of Credit Derivatives. Oxford University Press, Oxford (2010)

    Google Scholar 

  4. Cherubini, U., Luciano, E., Vecchiato, W.: Copula Methods in Finance. Wiley, Hoboken (2004)

    Book  MATH  Google Scholar 

  5. Czado, C.: Pair copula constructions of multivariate copulas. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and Its Applications. Lecture Notes in Statistics - Proceedings, vol. 198, pp. 93–109. Springer, Berlin (2010)

    Chapter  Google Scholar 

  6. Donnelly, C., Embrechts, P.: The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. Astin Bull. 40, 1–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Durante, F., Fernández-Sánchez, J.: Multivariate shuffles and approximation of copulas. Stat. Probab. Lett. 80, 1827–1834 (2010)

    Article  MATH  Google Scholar 

  8. Durante, F., Foscolo, E.: An analysis of the dependence among financial markets by spatial contagion. Int. J. Intell. Syst. 28(4), 319–331 (2013)

    Article  Google Scholar 

  9. Durante, F., Jaworski, P.: Spatial contagion between financial markets: a copula-based approach. Appl. Stoch. Models Bus. Ind. 26, 551–564 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Durante, F., Jaworski, P.: Invariant dependence structure under univariate truncation. Statistics 46, 263–277 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Durante, F., Sempi, C.: Copula theory: an introduction. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and Its Applications. Lecture Notes in Statistics - Proceedings, vol. 198, pp. 3–31. Springer, Berlin (2010)

    Chapter  Google Scholar 

  12. Durante, F., Klement, E.P., Quesada-Molina, J.J., Sarkoci, P.: Remarks on two product-like construction for copulas. Kybernetika 43, 235–244 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Durante, F., Klement, E.P., Quesada-Molina, J.J.: Bounds for trivariate copulas with given bivariate marginals. J. Inequal. Appl. 2008, 1–9, Article ID 161537 (2008)

    Google Scholar 

  14. Durante, F., Foschi, R., Spizzichino, F.: Threshold copulas and positive dependence. Stat. Probab. Lett. 78(17), 2902–2909 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Durante, F., Jaworski, P., Mesiar, R.: Invariant dependence structures and Archimedean copulas. Stat. Probab. Lett. 81, 1995–2003 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Durante, F., Foscolo, E., Sabo, M.: A spatial contagion test for financial markets. In: Kruse, R. et al. (eds.) Synergies of Soft Computing and Statistics for Intelligent Data Analysis. Advances in Intelligent and Soft Computing, vol. 190, pp. 321–329. Springer, Berlin (2013)

    Chapter  Google Scholar 

  17. Embrechts, P.: Copulas: a personal view. J. Risk Insur. 76, 639–650 (2009)

    Article  Google Scholar 

  18. Genest, C., Nešlehová, J.: Assessing and modeling asymmetry in bivariate continuous data. In: Jaworski, P., Durante, F., Härdle, W. (eds.) Copulae in Mathematical and Quantitative Finance, pp. 91–114. Springer, Berlin (2013)

    Google Scholar 

  19. Genest, C., Gendron, M., Bourdeau-Brien, M.: The advent of copulas in finance. Eur. J. Finance. 15, 609–618 (2009)

    Article  Google Scholar 

  20. Gidea, M., Niculescu, C.: Chaotic dynamical systems: An introduction. Universitaria Press, Craiova (2002)

    MATH  Google Scholar 

  21. de Haan, L.: Von Mises-type conditions in second order regular variation. J. Math. Anal. Appl. 197, 400–410 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Härdle, W., Okhrin, O.: De copulis non est disputandum - Copulae: an overview. AStA Adv. Stat. Anal. 94(1), 1–31 (2010)

    Article  MathSciNet  Google Scholar 

  23. Hashorva, E.: Gaussian approximation of conditional elliptical random vectors. Stoch. Model. 22, 441–457 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hashorva, E., Jaworski, P.: Gaussian approximation of conditional elliptical copulas. J. Multivariate Anal. 111, 397–407 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hofert, M.: Construction and sampling of nested Archimedean copulas. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and Its Applications. Lecture Notes in Statistics - Proceedings, vol. 198, pp. 147–160. Springer, Berlin (2010)

    Chapter  Google Scholar 

  26. Jágr, V., Komorníková, M., Mesiar, R.: Conditioning stable copulas. Neural Netw. World 20(1), 69–79 (2010)

    Google Scholar 

  27. Jaworski, P.: On uniform tail expansions of bivariate copulas. Appl. Math. 31, 397–415 (2004)

    MathSciNet  MATH  Google Scholar 

  28. Jaworski, P.: On uniform tail expansions of multivariate copulas and wide convergence of measures. Appl. Math. 33, 159–184 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Jaworski, P.: On copulas and their diagonals. Inform. Sci. 179, 2863–2871 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jaworski, P.: Tail Behaviour of Copulas. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and Its Applications. Proceedings of the Workshop Held in Warsaw, 25–26 Sept 2009. Lecture Notes in Statistics - Proceedings, vol. 198, pp. 161–186. Springer, Berlin (2010)

    Google Scholar 

  31. Jaworski, P.: Invariant dependence structure under univariate truncation: the high-dimensional case. Statistics (2012) online. doi:10.1080/02331888.2012.664143

    Google Scholar 

  32. Jaworski, P.: On copulas and differential inclusions. In: Kruse, R. et al. (eds.) Synergies of Soft Computing and Statistics for Intelligent Data Analysis. Advances in Intelligent and Soft Computing, vol. 190, pp. 321–329. Springer, Berlin (2013)

    Chapter  Google Scholar 

  33. Jaworski, P.: On the characterization of copulas by differential equations. In: Communications in Statistics - Theory and Methods (2013, in press)

    Google Scholar 

  34. Jaworski, P., Pitera, M.: On spatial contagion and multivariate GARCH models. Appl. Stoch. Models Bus. Ind. (2013, in press)

    Google Scholar 

  35. Jaworski, P., Rychlik, T.: On distributions of order statistics for absolutely continuous copulas, with applications to reliability. Kybernetika 44, 757–776 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds.) Copula Theory and its Applications. Lecture Notes in Statistics - Proceedings, vol. 198. Springer, Berlin (2010)

    Google Scholar 

  37. Joe, H.: Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997)

    Google Scholar 

  38. Joe, H., Li, H., Nikoloulopoulos, A.: Tail dependence functions and vine copulas. J. Multivariate Anal. 101, 252–270 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Juri, A., Wüthrich, M.V.: Copula convergence theorems for tail events. Insur. Math. Econ. 30(3), 405–420 (2002)

    Article  MATH  Google Scholar 

  40. Juri, A., Wüthrich, M.V.: Tail dependence from a distributional point of view. Extremes 6(3), 213–246 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Larsson, M., Nešlehová, J.: Extremal behavior of Archimedean copulas. Adv. Appl. Probab. 43, 195–216 (2011)

    Article  MATH  Google Scholar 

  42. Łojasiewicz, S.: An Introduction to the Theory of Real Functions, 3rd edn. Wiley, Chichester (1988)

    MATH  Google Scholar 

  43. McNeil, A.J.: Sampling nested Archimedean copulas. J. Stat. Comput. Simul. 78, 567–581 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. McNeil, A.J., Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l 1-norm symmetric distributions. Ann. Stat. 37(5b), 3059–3097 (2009)

    Article  MATH  Google Scholar 

  45. McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton Series in Finance. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  46. Mesiar, R., Pekárová, M.: DUCS copulas. Kybernetika 46, 1069–1077 (2010)

    MATH  Google Scholar 

  47. Mesiar, R., Sempi, C.: Ordinal sums and idempotents of copulas. Aequationaes Math. 79, 39–52 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Mesiar, R., Jágr, V., Juráňová, M., Komorníková, M.: Univariate conditioning of copulas. Kybernetika 44, 807–816 (2008)

    MathSciNet  MATH  Google Scholar 

  49. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer Series in Statistics. Springer, New York (2006)

    MATH  Google Scholar 

  50. Nikoloulopoulos, A., Joe, H., Li, H.: Extreme value properties of multivariate t copulas. Extremes 12, 129–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Okhrin, O., Okhrin, Y., Schmid, W.: Properties of hierarchical Archimedean copulas. Stat. Risk Model. 30(1), 21–54 (2013)

    Article  MATH  Google Scholar 

  52. Patton, A.J.: Modelling asymmetric exchange rate dependence. Int. Econ. Rev. 47, 527–556 (2006)

    Article  MathSciNet  Google Scholar 

  53. Quesada-Molina, J.J., Rodriguez-Lallena, J.A.: Some advances in the study of the compatibility of three bivariate copulas. J. Ital. Stat. Soc. 3, 397–417 (1994)

    Article  MATH  Google Scholar 

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Acknowledgements

The author acknowledges the partial support by Polish Ministry of Science and Higher Education, via the grant N N201 547838.

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Authors and Affiliations

  1. Institute of Mathematics, Warsaw University, Warsaw, Poland

    Piotr Jaworski

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  1. Piotr Jaworski
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Correspondence to Piotr Jaworski .

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Editors and Affiliations

  1. Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, Warszawa, 02097, Poland

    Piotr Jaworski

  2. School of Economics and Management, Free University of Bozen-Bolzano, Piazza Università 1, Bozen, 39100, Italy

    Fabrizio Durante

  3. L.v.Bortkiewicz Chair of Statistics, C.A.S.E. Centre f. Appl. Stat. & Econ., Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, 10099, Germany

    Wolfgang Karl Härdle

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Jaworski, P. (2013). The Limiting Properties of Copulas Under Univariate Conditioning. In: Jaworski, P., Durante, F., Härdle, W. (eds) Copulae in Mathematical and Quantitative Finance. Lecture Notes in Statistics(), vol 213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35407-6_7

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