## Abstract

Two risk models with a constant dividend barrier are considered. In the two models claims arrive according to a Poisson process. In the first model the claim size has a phase type distribution. In the second model the claim size is exponentially distributed, but the arrival rate, the mean claim size, and the premium rate are governed by a random environment, which changes according to a Markov process. Kella and Whitt [Kella, O., Whitt, W., 1992. Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probability 29, 396-403] martingale is applied in the first model. Asmussen and Kella [Asmussen, S., Kella, O., 2000. A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probability 32, 376-393] multi-dimensional martingale is applied in the second model. The expected time to ruin and the amount of dividends paid until ruin occurs are obtained for both models.

Original language | English |
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Pages (from-to) | 216-228 |

Number of pages | 13 |

Journal | Insurance: Mathematics and Economics |

Volume | 37 |

Issue number | 2 SPEC. ISS. |

DOIs | |

State | Published - 18 Oct 2005 |

### Bibliographical note

Funding Information:I want to thank the financial support of the Zimerman Foundation for the Study of Banking and Finance. I want to thank the referee for the careful reading, and for pointing out to me the paper by Dickson and Waters (2004) .

## Keywords

- Exponential distribution
- Laplace transform
- Lévy process
- Markov additive process
- Martingales
- Phase type distribution
- Reflected process
- Time to ruin

## ASJC Scopus subject areas

- Statistics and Probability
- Economics and Econometrics
- Statistics, Probability and Uncertainty