### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### 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

- Mathematics(all)
- Statistics and Probability

### Cite this

**On a Generalization from Ruin to Default in a Lévy Insurance Risk Model.** / Feng, Runhuan; Shimizu, Yasutaka.

Research output: Contribution to journal › Article

*Methodology and Computing in Applied Probability*, vol. 15, no. 4, pp. 773-802. https://doi.org/10.1007/s11009-012-9282-y

}

TY - JOUR

T1 - On a Generalization from Ruin to Default in a Lévy Insurance Risk Model

AU - Feng, Runhuan

AU - Shimizu, Yasutaka

PY - 2013/12

Y1 - 2013/12

N2 - 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.

AB - 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.

KW - Compound geometric distribution

KW - Costs up to default

KW - Defective renewal equation

KW - Expected discounted penalty function

KW - Lévy risk model

KW - Operator calculus

KW - Potential measure

KW - Scale function

UR - http://www.scopus.com/inward/record.url?scp=84885834976&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885834976&partnerID=8YFLogxK

U2 - 10.1007/s11009-012-9282-y

DO - 10.1007/s11009-012-9282-y

M3 - Article

AN - SCOPUS:84885834976

VL - 15

SP - 773

EP - 802

JO - Methodology and Computing in Applied Probability

JF - Methodology and Computing in Applied Probability

SN - 1387-5841

IS - 4

ER -