Pareto Lévy measures and multivariate regular variation

TLDR: In this paper, the dependence between the jumps is modeled by a so-called Pareto Levy measure, which is a natural standardization in the context of regular variation, and the dependence structure is analyzed in terms of spectral density and the tail integral.

Abstract: We consider regular variation of a Levy process X := ( X t ) t ≥0 in with Levy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Levy measures to those of a standard 1-stable Levy process, we decouple the marginal Levy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Levy measure , which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Levy measures and the Pareto Levy measure. Moreover, we define upper and lower tail dependence coefficients for the Levy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Levy measures.