## Abstract

In thi s article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek-Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.

Original language | English |
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Pages (from-to) | 2197-2208 |

Number of pages | 12 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 34 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2005 |

Externally published | Yes |

## Keywords

- Adjustment-coefficient
- Brownian motion with drift
- Diffusion
- Erlang(2) risk process
- Integral equation
- Lundberg's inequality
- Martingale
- Random walk