Exponential functionals of Markov additive processes

TL;DR

This paper establishes criteria for the convergence of exponential integrals of Markov additive processes, highlighting differences from classical Lévy processes and utilizing recent Markovian perpetuity results.

Contribution

It provides necessary and sufficient conditions for convergence, distinguishing between almost sure and in-probability convergence in the Markov additive context.

Findings

Criteria for convergence of exponential integrals established

Differentiates between almost sure and in-probability convergence

Utilizes recent Markovian perpetuity results for proofs

Abstract

We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. Other than in the classical L\'evy case studied by Erickson and Maller we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by Alsmeyer and Buckmann.