Some aspects of L\'evy copulas

ABSTRACT

L\'evy processes and infinitely divisible distributions are increasingly defined in terms of their L\'evy measure. In order to describe the dependence structure of a multivariate L\'evy measure, Tankov (2003) introduced positive L\'evy copulas. Together with the marginal L\'evy measures they completely describe multivariate L\'evy measures on $\mathbb{R}^m_+$. In this talk, it is shown that any L\'evy copula defines itself a L\'evy measure with 1-stable margins in a canonical way. Homogeneous L\'evy copulas are considered in detail. They correspond to L\'evy processes which have a time-constant L\'evy copula. Furthermore, we show how the L\'evy copula concept can be used to construct multivariate distributions in the Bondesson class with prescribed margins in the Bondesson class. The construction depends on a mapping $\Upsilon$, recently introduced by Barndorff-Nielsen and Thorbj\o rnsen (2004) and Barndorff-Nielsen, Maejima and Sato (2004). Similar results are obtained for self-decomposable distributions and for distributions in the Thorin class. The talk is based on joint work with Ole Barndorff-Nielsen.

Refreshments will be served at 3:45 p.m. in Room 008 of the Statistics Building

College of Natural Sciences