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Exponential functionals of Markov additive processes. (English) Zbl 07206374
Summary: We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Lévy case studied by K. B. Erickson and R. A. Maller [J. Theor. Probab. 18, No. 2, 359–375 (2005; Zbl 1075.60044)] 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 G. Alsmeyer and F. Buckmann [J. Difference Equ. Appl. 23, No. 4, 699–740 (2017; Zbl 1371.60174)].
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J74 Jump processes on discrete state spaces
60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI Euclid
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