The time to ruin for a class of Markov additive risk processes

Risk processes are considered, which locally behave as a Brownian motion with some drift and variance, both depending on an underlying Markov chain that is used also to generate the claims arrival process. Thus claims arrive according to a renewal process with waiting times of phase-type. The claims are assumed to form an iid sequence, independent of everything else, and with a distribution with a Laplace transform that is a rational function. In the main results of the paper, the joint Laplace transform of the time to ruin and the undershoot at ruin as well as the probability of ruin is determined explicitly. Furthermore, both the Laplace transform and the ruin probability is decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. Results from complex function theory are used as an important tool. Keyword and phrases. Probability of ruin; time to ruin; undershoot; passage time; martingales; optional sampling; additive processes; Rouché’s theorem. AMS subject classification: 60J25, 60K15, 60J35, 60J60, 60G40, 60G44. ∗ MaPhySto, Network in Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation.