Some advances on the Erlang(n) dual risk model

Abstract

The dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It has applications to companies with economical activities involved in research and development. This model is dual to the well known Cram´er-Lundberg risk model with applications to insurance. Existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cram´er–Lunderg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg’s equation, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase–Type, PH(m), distribution. Finally, we perform illustrations working some examples for some particular gain distributions and obtain numerical results