Multivariate subordination, self-decomposability and stability

Abstract

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class L<sub>m</sub> property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.