Ruin probabilities for the phase-type dual model perturbed by diffusion

In risk theory, ruin probabilities and dividend strategies have drawn lots of attentions. The dual risk model assumes that the surplus process decreases at a constant rate over time and gains by means of random jumps at random times, which is in accordance with real-life scenes such as life insurance. The dual model is widely used in describing security portfolio, pension funds, profits of enterprises whose income-generating depends on inventions or found, etc. In this paper, we consider a phase-type dual model perturbed by diffusion, whose inter-claim times are continuously distributed in phase-type distribution. We derive integro-differential equations, boundary conditions for ruin probability and obtain an explicit expression of the ruin probability when the phase-type distribution degenerates into the generalized Erlang (n) distribution. Finally, we obtain the integro-differential equations for the expected discounted dividend function under dividend payment with a threshold strategy.