On the distribution of surplus immediately after ruin under interest force.

*(English)*Zbl 1012.91027A compound Poisson model is considrered \(dU_\delta(t)= pdt+ U_\delta(t) \delta dt-dX(t)\). Here \(U_\delta(t)\) is the value of the reserve at time \(t\), \(p\) is the premium rate that the insurance company receives, \(\delta\) is the interest force, and \(X(t)= \sum^{N(t)}_{j=1} Y_j\), where \(N(t)\) is a homogeneous Poisson process counting the number of claims in the time interval \((0,t]\), and \(Y_j\) is the amount of the \(j\)th claim. Using the techniques of B. Sundt and J. L. Tengels [Isur. Math. Econ. 16, 7-22 (1995; Zbl 0838.62098)], the equations are obtained for the probability of ruin \(G_\delta (u,y)\) beginning with initial reserve \(u\) and the deficit at most \(y> 0\) immediately after the claim causing ruin. The bounds for \(G_\delta (0,y)\) are derived. In the case \(\delta=0\), the asymptotic result for \(G_\delta (u,y)\), as \(u\to +\infty\), is obtained, which generalizes the result in J. Grandell [Aspects of Risk Theory. Springer, Berlin (1991; Zbl 0717.62100)] to the case where the severity of ruin is taken into account.

Reviewer: Oleksandr Kukush (Kiev)

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

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\textit{H. Yang} and \textit{L. Zhang}, Insur. Math. Econ. 29, No. 2, 247--255 (2001; Zbl 1012.91027)

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##### References:

[1] | Boogaert, P.; Crijns, V., Upper bound on ruin probabilities in case of negative loadings and positive interest rates, Insurance: mathematics and economics, 6, 221-232, (1987) · Zbl 0642.62058 |

[2] | Di Lorenzo, E.; Tessitore, G., Approximation solutions of severity of ruin, Blatter deutsche gesellschaft fur versicherungsmathematik, XXII, 705-709, (1996) · Zbl 0860.62078 |

[3] | 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 |

[4] | 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, 75-80, (1988) · Zbl 0637.62101 |

[5] | Gerber, H.V.; 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 |

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

[7] | Gerber, H.V.; Goovaerts, M.J.; Kaas, R., On the probability and severity of ruin, Astin bulletin, 17, 2, 151-153, (1987) |

[8] | Grandell, J., 1991. Aspects of Risk Theory. Springer, Berlin. · Zbl 0717.62100 |

[9] | Paulsen, J., Ruin theory with compounding assets: a survey, Insurance: mathematics and economics, 22, 3-16, (1998) · Zbl 0909.90115 |

[10] | Paulsen, J.; Gjessing, H.K., Ruin theory with stochastic return on investments, Advances in applied probability, 29, 965-985, (1997) · Zbl 0892.90046 |

[11] | Sundt, B.; Teugels, J.L., Ruin estimates under interest force, Insurance: mathematics and economics, 16, 7-22, (1995) · Zbl 0838.62098 |

[12] | Willmot, G.E.; Lin, X., Exact and approximate properties of the distribution of surplus before and after ruin, Insurance: mathematics and economics, 23, 91-110, (1997) · Zbl 0914.90074 |

[13] | Yang, H., 1999. Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal, 66-79. · Zbl 0922.62113 |

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