On exponential functionals of Levy processes

TL;DR

This paper studies exponential functionals of Lévy processes, their connection to generalized Ornstein-Uhlenbeck processes, and characterizes the mapping from Lévy processes to these functionals, including properties like injectivity and continuity.

Contribution

It derives the infinitesimal generator of GOU processes, proves they are Feller processes, and analyzes the properties of the mapping al, including its injectivity, inverse, and continuity.

Findings

GOU processes are Feller processes with explicitly derived generators.

The mapping al is often injective and its inverse is characterized by Lévy characteristics.

Continuity and range properties of al are established.

Abstract

Exponential functionals of L\'evy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping \Phi, which maps two independent L\'evy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (L\'evy) characteristics. Also, continuity of \Phi is treated and some results on its range are obtained.