On the discounted penalty at ruin in a jump-diffusion and the perpetual put option.

*(English)*Zbl 0924.60075The paper deals with a stochastic model of ruin theory which is obtained by adding a Wiener process to the right side term of the classical non-random model. The corresponding expected discounted value of a penalty at ruin satisfies a renewal equation, which is obtained via a probabilistic approach. Pricing perpetual put options is examined, and the new equations so obtained extend classical known results already established by Merton.

Reviewer: G.Jumarie (Montréal)

##### MSC:

60J75 | Jump processes (MSC2010) |

91B28 | Finance etc. (MSC2000) |

91B24 | Microeconomic theory (price theory and economic markets) |

91B30 | Risk theory, insurance (MSC2010) |

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\textit{H. U. Gerber} and \textit{B. Landry}, Insur. Math. Econ. 22, No. 3, 263--276 (1998; Zbl 0924.60075)

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##### References:

[1] | Bowers, N.; Gerber, H.U.; Hickman, J.; Jones, D.; Nesbitt, C., Actuarial mathematics, (1997), Society of Actuaries Schaumburg, IL |

[2] | Dickson, D.C.M., On the distribution of surplus prior to ruin, Insurance: mathematics and economics, 11, 191-207, (1992) · Zbl 0770.62090 |

[3] | Dickson, D.C.M., On the distribution of the claim causing ruin, Insurance: mathematics and economics, 12, 143-154, (1993) · Zbl 0783.62083 |

[4] | Dickson, D.C.M.; Waters, H.R., The probability and severity of ruin in finite and infinite time, ASTIN bulletin, 23, 177-190, (1992) |

[5] | Dufresne, F.; Gerber, H.U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: mathematics and economics, 7, 193-199, (1988) · Zbl 0674.62072 |

[6] | 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 |

[7] | Egídio dos Reis, A.D., How long is the surplus below zero?, Insurance: mathematics and economics, 12, 23-38, (1993) · Zbl 0777.62096 |

[8] | Feller, W., () |

[9] | Gerber, H.U.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, ASTIN bulletin, 17, 151-163, (1987) |

[10] | Gerber, H.U.; Shiu, E.S.W., From ruin theory to option pricing, (), 157-176 |

[11] | Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047 |

[12] | Gerber, H.U.; Shiu, E.S.W., From ruin theory to pricing reset guarantees and perpetual put options, Insurance: mathematics and economics, (1998), to appear |

[13] | 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 |

[14] | Gerber, H.U.; Shiu, E.S.W., Pricing perpetual options for jump processes, North American actuarial journal, (1998), to appear |

[15] | Lamberton, D.; Lapeyre, B., Introduction to stochastic calculus applied to finance, (1996), Chapman and Hall London · Zbl 0898.60002 |

[16] | Lin, X.; Willmot, G.E., Analysis of a defective renewal equation arising in ruin theory, () · Zbl 1028.91556 |

[17] | Merton, R.C., Theory of rational option pricing, Bell journal of economics and management science, 4, 1, 141-183, (1973) · Zbl 1257.91043 |

[18] | Picard, P., On some measures of the severity of ruin in the classical Poisson model, Insurance: mathematics and economics, 14, 107-115, (1994) · Zbl 0813.62093 |

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