Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

TL;DR

This paper characterizes semi-selfdecomposable distributions using stochastic integrals, introduces related Ornstein-Uhlenbeck processes, and explores the structure of subclasses, revealing their limits relate to semi-stable distributions.

Contribution

It provides a new stochastic integral characterization of semi-selfdecomposable distributions and studies associated Ornstein-Uhlenbeck processes and their nested subclasses.

Findings

Characterization of semi-selfdecomposable distributions via stochastic integrals.

Construction of Ornstein-Uhlenbeck type processes with semi-selfdecomposable limits.

Nested subclasses converge to the closure of semi-stable distributions.

Abstract

In this paper, three topics on semi-selfdecomposable distributions are studied. The first one is to characterize semi-selfdecomposable distributions by stochastic integrals with respect to Levy processes. This characterization defines a mapping from an infinitely divisible distribution with finite log-moment to a semi-selfdecomposable distribution. The second one is to introduce and study a Langevin type equation and the corresponding Ornstein-Uhlenbecktype process whose limiting distribution is semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes with semi-selfdecomposable distributions are constructed. The third one is to study the iteration of the mapping above. The iterated mapping is expressed as a single mapping with a different integrand. Also, nested subclasses of the class of semi-selfdecomposable distributions are considered, andit is shown that the limit of these nested subclasses is the closure of the class of semi-stable distributions.