Multivariate subordination, self-decomposability and stability

Multivariate subordinators are multivariate Levy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Levy processes with independent components by multivariate subordinators is defined. Multiparameter Levy processes and their subordination are introduced so that the subordinated processes are multivariate Levy 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, property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.