Lévy systems and the time value of ruin for Markov additive processes

In this paper we study the ruin problem for an insurance risk process driven by a spectrally-positive Markov additive process. Particular attention is given to the family of spectrally-positive Markov-modulated Lévy processes. We give an expression for the expected discounted penalty function by extending results available in the literature. In particular, we generalize some results in Biffis and Kyprianou (Insur Math Econ 46:85–91, 2010) to a more general setting provided by the theory of Markov additive processes. This natural extension is possible thanks to the concept of Lévy systems that allows us to generalize well-known results for Lévy processes to a larger family of Markov additive processes. We also discuss how more compact expressions for the expected discounted penalty function can be obtained using the notion of scale matrix of a Markov additive process.