### Abstract

A Cox risk process with a piecewise constant intensity is considered where the sequence (L_{i}) of successive levels of the intensity forms a Markov chain. The duration s, of the level L_{i} is assumed to be only dependent via L_{i}. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of L_{i} contains a discrete component, a Cramér-Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

Original language | English |
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Pages (from-to) | 196-210 |

Number of pages | 15 |

Journal | Journal of Applied Probability |

Volume | 33 |

Issue number | 1 |

Publication status | Published - Mar 1996 |

### Keywords

- Change of measure
- Cox model
- Cramér-Lundberg approximation
- Lundberg inequality
- Martingale methods
- Risk theory
- Ruin probability

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## Cite this

*Journal of Applied Probability*,

*33*(1), 196-210.