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Lévy Copulas: Dynamics and Transforms of Upsilon Type

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  • OLE E. BARNDORFF‐NIELSEN
  • ALEXANDER M. LINDNER

Abstract

. Lévy processes and infinitely divisible distributions are increasingly defined in terms of their Lévy measure. In order to describe the dependence structure of a multivariate Lévy measure, Tankov (2003) introduced Lévy copulas on . (For an extension to ℝm, see Kallsen & Tankov, 2006.) Together with the marginal Lévy measures they completely describe multivariate Lévy measures on . In this article we show that any such Lévy copula defines itself a Lévy measure with one‐stable margins, in a canonical way. A limit theorem is obtained, characterizing convergence of Lévy measures with the aid of Lévy copulas. Homogeneous Lévy copulas are considered in detail. They correspond to Lévy processes which have a time‐constant Lévy copula, and a complete description of homogeneous Lévy copulas is obtained. A general scheme to construct multivariate distributions having special properties is outlined, for distributions with prescribed margins having the same properties. This makes use of Lévy copulas and of certain mappings of Upsilon type. The construction is then exemplified for distributions in the Goldie–Steutel–Bondesson class, the Thorin class and for self‐decomposable distributions.

Suggested Citation

  • Ole E. Barndorff‐Nielsen & Alexander M. Lindner, 2007. "Lévy Copulas: Dynamics and Transforms of Upsilon Type," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(2), pages 298-316, June.
    • Handle: RePEc:bla:scjsta:v:34:y:2007:i:2:p:298-316
    DOI: 10.1111/j.1467-9469.2006.00527.x
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    Cited by:

    1. Esmaeili, Habib & Klüppelberg, Claudia, 2011. "Parametric estimation of a bivariate stable Lévy process," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 918-930, May.
    2. Barndorff-Nielsen, Ole E. & Maejima, Makoto, 2008. "Semigroups of Upsilon transformations," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2334-2343, December.

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