×

zbMATH — the first resource for mathematics

A class of multivariate copulas with bivariate Fréchet marginal copulas. (English) Zbl 1231.91253
Summary: We present a class of multivariate copulas whose two-dimensional marginals belong to the family of bivariate Fréchet copulas. The coordinates of a random vector distributed as one of these copulas are conditionally independent. We prove that these multivariate copulas are uniquely determined by their two-dimensional marginal copulas. Some other properties for these multivariate copulas are discussed as well. Two applications of these copulas in actuarial science are given.

MSC:
91B30 Risk theory, insurance (MSC2010)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
Software:
CreditRisk+
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1997), The Society of Actuaries Schaumburg, Illinois · Zbl 0634.62107
[2] Credit Suisse First Boston, 1997. CreditRisk^+-A credit Risk Management Framework. Available on http://www.defaultrisk.com/pp_model_21.htm
[3] Cherubini, M.; Luciano, E.; Vecchiato, W., Copula methods in finance, (2004), Wiley and Sons, Inc England · Zbl 1163.62081
[4] CSO Task Force Report, 2002. 2001 Valuation Basic Table and 2001 CSO table. http://www.actuary.org/life/cso_0702.asp
[5] Denneberg, D., Non-additive measure and integral, (1994), Kluwer Academic Publishers Boston · Zbl 0826.28002
[6] Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vynche, D., The concept of comonotonicity in actuarial science and finance: theory, Insurance: mathematics and economics, 31, 1, 3-33, (2002) · Zbl 1051.62107
[7] Dhaene, J.; Denuit, M.; Goovaerts, M.J.; Kaas, R.; Vynche, D., The concept of comonotonicity in actuarial science and finance: application, Insurance: mathematics and economics, 31, 2, 133-161, (2002) · Zbl 1037.62107
[8] Embrechets, P., Lindskog, F., McNeil, A., 2001. Modelling dependence with copulas and applications to risk management. http://www.math.ethz.ch/finance
[9] Hull, J.; White, A., Valuation of a CDO and an \(n\)-th to default CDS without Monte Carlo simulation, Journal of derivatives, 12, 2, 8-23, (2004)
[10] Hürlimann, W., Multivariate Fréchet copulas and conditional value-at-risk, International journal of mathematics and mathematical sciences, 7, 345-364, (2004) · Zbl 1075.62043
[11] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[12] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2001), Kluwer Academic Publishers Boston
[13] Klugman, S.A.; Panjer, H.H.; Willmot, G.E., Loss models: from data to decision, (2004), Wiley and Sons, Inc New York · Zbl 1141.62343
[14] Mari, D.D.; Kotz, S., Correlation and dependence, (2001), Imperial College Press Singapore · Zbl 0977.62004
[15] Mikusinski, P.; Sherwood, H.; Taylor, M.D., Probabilistic interpretations of copulas and their convex sums, (), 95-112 · Zbl 0733.60023
[16] Nelsen, R.B., An introduction to copulas, (2006), Springer-Verlag New York · Zbl 1152.62030
[17] Salvadori, G.; De Michele, C.; Kottegoda, N.T.; Rosso, R., Extremes in nature: an approach using copulas, (2007), Springer The Netherlands
[18] Tiit, E-M., Mixtures of multivariate quasi-extremal distributions having given marginals, (), 337-357
[19] Yang, J.P.; Cheng, S.H.; Zhang, L.H., Bivariate copula decomposition in terms of comonotonicity, countermonotonicity and independence, Insurance: mathematics and economics, 39, 267-284, (2006) · Zbl 1098.62070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.