Levy copulas: Dynamics and transforms of Upsilon type

Levy processes and infinitely divisible distributions are increasingly defined in terms of their Levy measure. In order to describe the dependence structure of a multivariate Levy measure, Tankov (2003) introduced Levy copulas on R-+(m.) (For an extension to R-m,R- see Kallsen & Tankov, 2006.) Together with the marginal Levy measures they completely describe multivariate Levy measures on . In this article we show that any such Levy copula defines itself a Levy measure with one-stable margins, in a canonical way. A limit theorem is obtained, characterizing convergence of Levy measures with the aid of Levy copulas. Homogeneous Levy copulas are considered in detail. They correspond to Levy processes which have a time-constant Levy copula, and a complete description of homogeneous Levy 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 Levy 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.