##
**Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion.**
*(English)*
Zbl 1183.91077

Summary: In the absence of dividends, the surplus of an insurance company is modeled by a compound Poisson process perturbed by diffusion. Dividends are paid at a constant rate whenever the modified surplus is above the threshold, otherwise no dividends are paid. Two integro-differential equations for the expected discounted dividend payments prior to ruin are derived and closed-form solutions are given. Accordingly, the Gerber-Shiu expected discounted penalty function and some ruin related functionals, the probability of ultimate ruin, the time of ruin and the surplus before ruin and the deficit at ruin, are considered and their analytic expressions are given by general solution formulas. Finally the moment-generating function of the total discounted dividends until ruin is discussed.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

45J05 | Integro-ordinary differential equations |

60K05 | Renewal theory |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

### Keywords:

compound Poisson model; diffusion; threshold strategy; dividend payments; Gerber-Shiu discounted penalty function; ruin related functionals
Full Text:
DOI

### References:

[1] | Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20, 1-15 (1997) · Zbl 1065.91529 |

[2] | Chiu, S. N.; Yin, C. C., The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion, Insurance: Mathematics and Economics, 33, 1, 59-66 (2003) · Zbl 1055.91042 |

[3] | De Finetti, B., Su un’impostazione alternativa dell teoria colletiva del rischio, Transactions of the XV International Congress of Actuaries, 2, 433-443 (1957) |

[4] | Dufresne, F.; Gerber, H. U., Risk theory for the command Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10, 51-59 (1991) · Zbl 0723.62065 |

[5] | Gerber, H. U., An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, 205-210 (1970) · Zbl 0229.60062 |

[6] | Gerber, H. U.; Landry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22, 263-276 (1998) · Zbl 0924.60075 |

[7] | Gerber, H. U.; Shiu, E. S.W., Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8, 1, 1-20 (2004) · Zbl 1085.62122 |

[8] | Gerber, H. U.; Shiu, E. S.W., On optimal dividend strategy in the compound Poisson model, North American Actuarial Journal, 10, 2, 76-93 (2005) · Zbl 1479.91323 |

[9] | Gerber, H. U.; Shiu, E. S.W., On optimal dividends: From reflection to refraction, Journal of Computational and Applied Mathematics, 186, 4-22 (2006) · Zbl 1089.91023 |

[11] | Lin, X. S.; Willmot, G. E.; Drekic, S., The classical risk model with a constant dividend barrier, Insurance: Mathematics and Economics, 33, 551-566 (2003) · Zbl 1103.91369 |

[12] | Lin, X. S.; Pavlova, K. P., The compound Poisson risk model with a threshold dividend strategy, Insurance: Mathematics and Economics, 38, 57-80 (2006) · Zbl 1157.91383 |

[14] | Jeanblanc-Picqué, M.; Shiryaev, A. N., Optimization of the flow of dividends, Russian Mathematical Surveys, 20, 257-277 (1995) · Zbl 0878.90014 |

[15] | Wang, G.; Wu, R., Some distributions for classical risk processes that is perturbed by diffusion, Insurance: Mathematics and Economics, 26, 15-24 (2000) · Zbl 0961.62095 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.