On the moments of the surplus process perturbed by diffusion.

*(English)*Zbl 1063.91051The results of X. S. Lin and G. E. Willmot [Insur. Math. Econ. 27, 19–44 (2000; Zbl 0971.91031)] are extended to those based on the surplus process perturbed by diffusion. First the expression for the (discounted) moments of deficit at the time of ruin is derived. An upper bound is also given if the claim size distribution function satisfies a certain condition. Next, it is shown that the joint moment of the penalty function and the time of ruin due to a claim satisfies a defective renewal equation and has an explicit expression. Finally, the moments of the time of ruin due to oscillation and caused by a claim are studied.

Reviewer: Giacomo Bonanno (Davis)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

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\textit{C. C. L. Tsai} and \textit{G. E. Willmot}, Insur. Math. Econ. 31, No. 3, 327--350 (2002; Zbl 1063.91051)

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##### References:

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

[2] | Fagiuoli, E.; Pellerey, F., Preservation of certain classes of life distributions under Poisson shock models, Journal of applied probability, 31, 458-465, (1994) · Zbl 0806.60075 |

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

[4] | 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 |

[5] | Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550 |

[6] | Hesselager, O., Wang, S., Willmot, G.E., 1998. Exponential and scale mixtures and equilibrium distributions. Scandinavian Actuarial Journal, 125-142. · Zbl 1076.62559 |

[7] | Lin, X.; Willmot, G.E., Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics, 25, 63-84, (1999) · Zbl 1028.91556 |

[8] | Lin, X.; Willmot, G.E., The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: mathematics and economics, 27, 19-44, (2000) · Zbl 0971.91031 |

[9] | Picard, Ph.; Lefèvre, C., The moments of ruin time in the classical risk model with discrete claim size distribution, Insurance: mathematics and economics, 23, 157-172, (1998) · Zbl 0957.62089 |

[10] | Picard, Ph.; Lefèvre, C., Corrigendum to the moments of ruin time in the classical risk model with discrete claim size distribution, Insurance: mathematics and economics, 25, 105-107, (1999) · Zbl 1007.62529 |

[11] | Tsai, C.C.-L., 2002. On the expectation of the present value of the time of ruin perturbed by diffusion, submitted for publication. |

[12] | Tsai, C.C.-L.; Willmot, G.E., A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: mathematics and economics, 30, 51-66, (2002) · Zbl 1074.91563 |

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