On a perturbed MAP risk model under a threshold dividend strategy

In this paper, we consider a perturbed risk model where the claims arrive according to a Markovian arrival process (MAP) under a threshold dividend strategy. We derive the integro-differential equations for the Gerber–Shiu expected discounted penalty function and the moments of total dividend payments until ruin, obtain the analytical solutions to these equations, and give numerical examples to illustrate our main results. We also get a matrix renewal equation for the Gerber–Shiu function, and present some asymptotic formulas for the Gerber–Shiu function when the claim size distributions are heavy-tailed.