The surpluses immediately before and at ruin, and the amount of the claim causing ruin.

*(English)*Zbl 0674.62072Summary: In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u;x,y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place).

For an arbitrary claim amount distribution we find that \(f(0;x,y)=ap(x+y)\), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u;x,y) can be calculated explicitly, if the claim amount distribution is exponential or more generally, a combination of exponential distributions. We are also interested in \(X+Y\), the amount of the claim that causes ruin. Its density h(u;z) can be obtained from f(u;x,y). One finds, for example, that \(h(0;z)=azp(z)\).

For an arbitrary claim amount distribution we find that \(f(0;x,y)=ap(x+y)\), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u;x,y) can be calculated explicitly, if the claim amount distribution is exponential or more generally, a combination of exponential distributions. We are also interested in \(X+Y\), the amount of the claim that causes ruin. Its density h(u;z) can be obtained from f(u;x,y). One finds, for example, that \(h(0;z)=azp(z)\).

##### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

##### Keywords:

compound Poisson model; collective risk theory; surplus; ruin; claim amount distribution; rate of premium income; combination of exponential distributions
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\textit{F. Dufresne} and \textit{H. U. Gerber}, Insur. Math. Econ. 7, No. 3, 193--199 (1988; Zbl 0674.62072)

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##### References:

[1] | Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1987), Society of Actuaries Itasca, IL |

[2] | Dufresne, F.; Gerber, H.U., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: mathematics and economics, 7, 2, 75-80, (1988) · Zbl 0637.62101 |

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

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

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