Exponential functionals of Markov additive processes

Anita Behme,Apostolos Sideris

doi:10.1214/20-ejp441

Abstract

We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Lévy 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.

Highlights

Given a bivariate Levy processt≥0 the corresponding exponential functional is defined as e−ξt− dηt, (0,∞)
As shown in [25] exponential functionals of Levy processes describe exactly the stationary distributions of generalized Ornstein-Uhlenbeck processes, a class of processes that stems from physics, and nowadays has numerous applications e.g. in finance and insurance, see e.g. [21, 28]
Where (Wt)t≥0 is a standard Brownian motion and Nη(j) are Poisson random measures with intensity measures ds νη(j)(dx), respectively. Using this decomposition it is straightforward to define integration with respect to the additive component of a Markov additive processes (MAPs) given its Markovian component. Another property of the additive components that carries over from Levy processes and which will be of importance in our results is the well-known fact that Levy processes in R either drift to ±∞ or oscillate

Summary

Introduction

Given a bivariate Levy process (ξt, ηt)t≥0 the corresponding exponential functional is defined as e−ξt− dηt,. Provided that the integral converges a.s. Necessary and sufficient conditions for this convergence in terms of the Levy characteristics of (ξt, ηt)t≥0 have been given in [15, Thm. 2]. We prove necessary and sufficient conditions for convergence of Eξ,η(t) as t → ∞ As it will turn out, by contrast with the classical Levy setting here we have to distinguish between almost sure convergence and convergence in probability. Note that exponential functionals of (Markov) additive processes have recently attracted the attention of other researchers as well.

Exponential functionals of Levy processes

Markov modulated perpetuities

Bivariate Markov additive processes

Main theorem

Examples

Sufficient conditions