Dividend Payments in a Risk Model Perturbed by Diffusion with Multiple Thresholds

We consider a perturbed renewal risk model where the inter-claim times are phase-type distributed and the dividend payment is a step function depending on the current surplus level. We obtain the integro-differential equations with boundary conditions for the moment-generating functions and the moments of arbitrary order of the present value of all dividend payments until ruin. Explicit expressions for the expected discounted dividend payments are derived if the claim amount distribution belongs to the rational family. Last, we present an example to show how the results are applied