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.
"Exponential functionals of Markov additive processes." Electron. J. Probab. 25 1 - 25, 2020. https://doi.org/10.1214/20-EJP441