On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals

In this paper, we consider an insurance risk model governed by a Markovian arrival claim process and by phase-type distributed claim amounts, which also allows for claim sizes to be correlated with the inter-claim times. A defective renewal equation of matrix form is derived for the Gerber–Shiu discounted penalty function and solved using matrix analytic methods. The use of the busy period distribution for the canonical fluid flow model is a key factor in our analysis, allowing us to obtain an explicit form of the Gerber–Shiu discounted penalty function avoiding thus the use of Lundberg’s fundamental equation roots. As a special case, we derive the triple Laplace transform of the time to ruin, surplus prior to ruin, and deficit at ruin in explicit form, further obtaining the discounted joint and marginal moments of the surplus prior to ruin and the deficit at ruin.