On the Lamperti stable processes

TL;DR

This paper introduces the Lamperti stable processes, a new family of Lévy processes with explicit functional laws, series representations for simulation, and properties similar to tempered and layered stable processes, enriching the understanding of their behavior.

Contribution

The paper defines the Lamperti stable processes, explores their properties, and provides examples and series representations for simulation, expanding the class of Lévy processes with explicit functional laws.

Findings

Explicit laws for related functionals are computable.

Series representation enables sample path simulation.

Shares properties with tempered and layered stable processes.

Abstract

We consider a new family of $\R^d$-valued L\'{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.