### 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 is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most 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 Cramér-Lundberg 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 equations, 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. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

Original language | English |
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Pages (from-to) | 127-150 |

Journal | ASTIN Bulletin |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2015 |

### Keywords

- Dual risk model
- Erlang(n) interarrival times
- expected discounted dividends
- generalized Lundberg's equation
- Phase-Type distribution
- ruin probability
- time of ruin

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## Cite this

*ASTIN Bulletin*,

*45*(1), 127-150. https://doi.org/10.1017/asb.2014.19