A Markov additive risk process with a dividend barrier

We study a risk process with dividend barrier b where the claims arrive according to a Markovian additive process (MAP). For spectrally negative MAPs, we present linear equations for the expected discounted dividends and the expected discounted penalty function. We apply results for the first exit times of spectrally negative Lévy processes and change-of-measure techniques. Explicit expressions are given when there are positive and negative claims, with phase-type distribution.