### Abstract

In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative Lévy risk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution.

Original language | English |
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Pages (from-to) | 773-802 |

Number of pages | 30 |

Journal | Methodology and Computing in Applied Probability |

Volume | 15 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 Dec 1 |

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### Keywords

- Compound geometric distribution
- Costs up to default
- Defective renewal equation
- Expected discounted penalty function
- Lévy risk model
- Operator calculus
- Potential measure
- Scale function

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)