An integral representation of dilatively stable processes with independent increments

TL;DR

This paper provides a new integral representation for dilatively stable processes with independent increments, linking them to time-changed Lévy processes and translatively stable Ornstein-Uhlenbeck-type processes.

Contribution

It introduces a novel integral representation for dilatively stable processes with independent increments, expanding the understanding of their structure and connections to other stochastic processes.

Findings

Representation via integrals with respect to time-changed Lévy processes

Connection to translatively stable Ornstein-Uhlenbeck-type processes

Extension of known results for selfsimilar processes

Abstract

Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Igl\'oi, we will show how dilatively stable processes with independent increments can be represented by integrals with respect to time-changed L\'evy processes. Via a Lamperti-type transformation these representations are shown to be closely connected to translatively stable processes of Ornstein-Uhlenbeck-type, where translative stability generalizes the notion of stationarity. The presented results complement corresponding representations for selfsimilar processes with independent increments known from the literature.