MPS-RR 2001-43
December 2001

Lévy processes and convolution semigroups with parameter in a cone and their subordination

Jan Pedersen, Ken-Iti Sato

Abstract

Convolution semigroups and Lévy processes with parameter in a cone $K$ are defined. Compared to ordinary convolution semigroups and Lévy processes (corresponding to $K=Rmathbb{R}_+$ the case of a general cone $K$ is more complicated in that there is generally not a one-to-one correspondence between semigroups and Lévy processes. Thus in particular we have to distinguish subordination of cone- parameter convolution semigroups and of cone-parameter Lévy processes. Several fundamental properties of cone-parameter convolution semigroups and Lévy processes are derived. In the study the destinction between cones with and without a strong basis is important. Conditions that a cone-parameter convolution semigroup is generative (that is, there is a cone-parameter Lévy process in law associated with it) are derived and examples of non-generative semigroups are given.

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