New Classes of Infinitely Divisible Distributions Related to the Goldie-Steutel-Bondesson Class

TL;DR

This paper introduces new classes of infinitely divisible distributions on R^d using stochastic integral mappings, focusing on a subclass related to the Goldie-Steutel-Bondesson class, and explores their properties.

Contribution

It develops a novel class of infinitely divisible distributions via stochastic integral mappings, extending the Goldie-Steutel-Bondesson class in a multidimensional setting.

Findings

New classes of distributions constructed using stochastic integrals.

Characterization of distributions without Gaussian parts.

Connections established with the Goldie-Steutel-Bondesson class.

Abstract

Recently, many classes of infinitely divisible distributions on R^d have been characterized in several ways. Among others, the first way is to use Levy measures, the second one is to use transformations of Levy measures, and the third one is to use mappings of infinitely divisible distributions defined by stochastic integrals with respect to Levy processes. In this paper, we are concerned with a class of mappings, by which we construct new classes of infinitely divisible distributions on R^d. Then we study a special case in R^1, which is the class of infinitely divisible distributions without Gaussian parts generated by stochastic integrals with respect to a fixed compound Poisson processes on R^1. This is closely related to the Goldie-Steutel-Bondesson class.