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Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Lévy copulas

  • Benjamin Avanzi (a1) (a2), Jamie Tao (a1), Bernard Wong (a1) and Xinda Yang (a1)

Abstract

The class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e.g. between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

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Corresponding author

*Correspondence to: B. Wong, School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW, Sydney, NSW 2052, Australia. Tel: +61 2 9385 2837. Fax: +61 2 9385 1883. E-mail: bernard.wong@unsw.edu.au

References

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Aas, K. & Berg, D. (2009). Models for construction of multivariate dependence: a comparison study. The European Journal of Finance, 15(7–8), 639659.
Avanzi, B., Cassar, L.C. & Wong, B. (2011). Modelling dependence in insurance claims processes with Lévy copulas. ASTIN Bulletin, 41(2), 575609.
Barndorff-Nielsen, O.E. & Lindner, A.M. (2004). Some aspects of Lévy copulas, technical report no. 388, Sonderforschungsbereich 386, Munich.
Barndorff-Nielsen, O. & Lindner, A. (2007). Lévy copulas: dynamics and transforms of Upsilon type. Scandinavian Journal of Statistics, 34, 298316.
Bäuerle, N. & Blatter, A. (2011). Optimal control and dependence modeling of insurance portfolios with Lévy dynamics. Insurance: Mathematics and Economics, 48, 398405.
Bertoin, J. (1998). Lévy Processes. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK.
Biagini, F. & Ulmer, S. (2009). Asymptotics for operational risk quantified with expected shortfall. ASTIN Bulletin, 39(2), 735752.
Böcker, K. & Klüppelberg, C. (2008). Modeling and measuring multivariate operational risk with Lévy copulas. The Journal of Operational Risk, 3(2), 327.
Böcker, K. & Klüppelberg, C. (2010). Multivariate models for operational risk. Quantitative Finance, 1, 115.
Bregman, Y. & Klüppelberg, C. (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scandinavian Actuarial Journal, 2005(6), 462480.
Bücher, A. & Vetter, M. (2013). Nonparametric inference on Lévy measures and copulas. The Annals of Statistics, 41, 14851515.
Cassar, L.C. (2010). Dependence modelling in multivariate compound Poisson processes with Lévy copulas. Honours thesis. School of Risk and Actuarial Studies, UNSW Australia Business School.
Cont, R. & Tankov, P. (2004). Financial Modelling With Jump Processes. Chapman & Hall/CRC, London.
Eder, I. & Klüppelberg, C. (2009). The quintuple law for sums of dependent Lévy processes. The Annals of Applied Probability, 19(6), 20472079.
Esmaeili, H. & Klüppelberg, C. (2010). Parameter estimation of a bivariate compound Poisson process. Insurance: Mathematics and Economics, 47(2), 224233.
Esmaeili, H. & Klüppelberg, C. (2011). Parametric estimation of a bivariate stable Lévy process. Journal of Multivariate Analysis, 102, 918930.
Esmaeili, H. & Klüppelberg, C. (2013). Two-step estimation of a multi-variate Lévy process. Journal of Time Series Analysis, 34, 668690.
Farkas, W., Reich, N. & Schwab, C. (2006). Anisotropic stable Lévy copula processes – analytical and numerical aspects, technical report no. 2006-08, Eidgenössische Technische Hochschule, Zurich.
Frees, E.W. & Valdez, E.A. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), 125.
Grothe, O. & Hofert, M. (2015). Construction and sampling of Archimedean and nested Archimedean Lévy copulas. Journal of Multivariate Analysis, 138, 182198.
Grothe, O. & Nicklas, S. (2013). Vine constructions of Lévy copulas. Journal of Multivariate Analysis, 119, 115.
Hofert, M. (2008). Sampling Archimedean copulas. Computational Statistics and Data Analysis, 12, 51635174.
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.
Kallsen, J. & Tankov, P. (2006). Characterisation of dependence of multidimensional Lévy processes using Lévy copulas. Journal of Multivariate Analysis, 97(7), 15511572.
Kurowicka, D. & Joe, H. (2011). Dependence Modeling Vine Copula Ha ndbook. World Scientific, Singapore.
McNeil, A.J. (2008). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78, 567581.
McNeil, A.J., Frey, R. & Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, New Jersey.
McNeil, A.J. & Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l 1-norm symmetric distributions. Annals of Statistics, 37(5B), 30593097.
Mikosch, T. (2006). Non-Life Insurance Mathematics: An Introduction with Stochastic Processes. Springer, Berlin Heidelberg.
Nadarajah, S. & Bakar, S. (2014). New composite models for the Danish fire insurance data. Scandinavian Actuarial Journal, 2014, 180187.
Nelsen, R.B. (1999). An Introduction to Copulas. Springer, Springer-Verlag, New York.
Ross, S. (1983). Introduction to Stochastic Dynamic Programming. Academic Press, San Diego, CA.
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
Tankov, P. (2003). Dependence structure of spectrally positive multidimensional Lévy processes. Available online at the address http://www.proba.jussieu.fr/pageperso/tankov/.
Tao, J. (2011). Capturing non-exchangeable dependence in multivariate insurance claims processes with nested Lévy copulas, Honours thesis, School of Risk and Actuarial Studies, UNSW Australia Business School.

Keywords

Capturing non-exchangeable dependence in multivariate loss processes with nested Archimedean Lévy copulas

  • Benjamin Avanzi (a1) (a2), Jamie Tao (a1), Bernard Wong (a1) and Xinda Yang (a1)

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