## Abstract

In this work, we propose a capital injection strategy which is periodically implemented based on the number of claims in the classical Poisson risk model. Especially, capital injection decisions are made at a predetermined accumulated number of claim instants, if the surplus is lower than a minimum required level. There appears to be a similar problem found in reliability theory such that preventive maintenance policies are performed at certain shock numbers. Assuming a combination of exponentials for the claim severities, we first derive an explicit expression for the discounted density of the surplus level after a certain number of claims if ruin has not yet occurred. Utilising this result, we study the expected total discounted capital injection until the first ruin time. To solve the differential equation associated with this quantity, we analyse an extended Lundberg's fundamental equation. Similarly, an expression for the Laplace transform of the time to ruin is also explicitly found. Finally, we illustrate the applicability of the present capital injection strategy and methodologies through various numerical examples. In particular, for exponential claim severities, some optimal capital injection strategy which minimises the expected capital spending per unit time is numerically studied.

Original language | English |
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Pages (from-to) | 296-325 |

Number of pages | 30 |

Journal | Annals of Actuarial Science |

Volume | 12 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Sept 2018 |

Externally published | Yes |

## Keywords

- Classical Poisson risk model
- Expected total discounted capital injection
- Laplace transform of time to ruin
- Number of claims
- Preventive maintenance