On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model with Phase-Type Gains

In this paper, we consider the dual of the classical Cramér-Lundberg model when gains follow a phase-type distribution. By using the property of phase-type distribution, two pairs of upcrossing and downcrossing barrier probabilities are derived. Explicit formulas for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin under a variety of dividend strategies can then be obtained without the use of Laplace transforms.

Keywords: Ruin theory; dual risk model; phase-type distribution; dividend strategy; upcrossing and downcrossing probabilities.