Classes of Infinitely Divisible Distributions on $\mathbf{R}^d$ Related to the Class of Selfdecomposable Distributions

Abstract

This paper studies new classes of infinitely divisible distributions on $\mathbf{R}^d$. Firstly, the connecting classes with a continuous parameter between the Jurek class and the class of selfdecomposable distributions are revisited. Secondly, the range of the parameter is extended to construct new classes and characterizations in terms of stochastic integrals with respect to Lévy processes are given. Finally, the nested subclasses of those classes are discussed and characterized in two ways: One is by stochastic integral representations and another is in terms of Lévy measures.