A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model.

*(English)*Zbl 1231.91156Summary: A risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber-Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber-Shiu function can be expressed in terms of the original Gerber-Shiu function (see e.g. [H. U. Gerber and E. S.W. Shiu [N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550)]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber-Shiu function is also shown to be closely related to the original Gerber-Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber-Shiu function. Numerical illustrations are then examined.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

##### Keywords:

generalized penalty function; maximum surplus level before ruin; Markovian arrival process; discounted joint distribution
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\textit{E. C. K. Cheung} and \textit{D. Landriault}, Insur. Math. Econ. 46, No. 1, 127--134 (2010; Zbl 1231.91156)

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##### References:

[1] | Ahn, Soohan; Badescu, Andrei L., On the analysis of the gerber – shiu discounted penalty function for risk processes with Markovian arrivals, Insurance: mathematics and economics, 41, 2, 234-249, (2007) · Zbl 1193.60103 |

[2] | Asmussen, Søren; Avram, Florin; Usabel, Miguel, The Erlang approximation of finite time ruin probabilities, Astin bulletin, 32, 2, 267-281, (2002) · Zbl 1081.60028 |

[3] | Badescu, Andrei L.; Breuer, Lothar; Da Silva Soares, Ana; Latouche, Guy; Remiche, Marie-Ange; Stanford, David, Risk processes analyzed as fluid queues, Scandinavian actuarial journal, 2005, 2, 127-141, (2005) · Zbl 1092.91037 |

[4] | Biffis, Enrico, Morales, Manuel, 2009. On a new generalization of the expected discounted penalty function. Preprint · Zbl 1231.91146 |

[5] | Breuer, Lothar, 2009. A quintuple law for Markov-additive processes with phase-type jumps. Preprint · Zbl 1205.60095 |

[6] | Bühlmann, Hans, Mathematical methods in risk theory, (1970), Springer New York · Zbl 0209.23302 |

[7] | Cheung, Eric C.K., 2009. Analysis of some risk models involving dependence. Ph.D. Thesis, University of Waterloo (in preparation) |

[8] | Cheung, Eric C.K.; Landriault, David, Perturbed MAP risk models with dividend barrier strategies, Journal of applied probability, 46, 2, 521-541, (2009) · Zbl 1180.60071 |

[9] | Cheung, Eric C.K., Landriault, David, 2009b. Analysis of a generalized penalty function in a semi-Markovian risk model. North American Actuarial Journal (in press) · Zbl 1180.60071 |

[10] | Cheung, Eric C.K., Landriault, David, Willmot, Gordon E., Woo, Jae-Kyung, 2009. Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models. Insurance: Mathematics and Economics, in press (doi:10.1016/j.insmatheco.2009.05.009) · Zbl 1231.91157 |

[11] | Gerber, Hans U.; Goovaerts, Marc J.; Kaas, Rob, On the probability and severity of ruin, Astin bulletin, 17, 2, 151-163, (1987) |

[12] | Gerber, Hans U.; Lin, Sheldon X.; Yang, Hailiang, A note on the dividends-penalty identity and the optimal dividend barrier, Astin bulletin, 36, 2, 489-503, (2006) · Zbl 1162.91374 |

[13] | Gerber, Hans U.; Shiu, Elias 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, 2, 129-137, (1997) · Zbl 0894.90047 |

[14] | Gerber, Hans U.; Shiu, Elias S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-72, (1998) · Zbl 1081.60550 |

[15] | Landriault, David; Willmot, Gordon E., On the joint distributions of the time to ruin, the surplus prior to ruin and the deficit at ruin in the classical risk model, North American actuarial journal, 13, 2, 252-270, (2009) |

[16] | Latouche, Guy; Ramaswami, Vaidyanathan, Introduction to matrix analytic methods in stochastic modeling, (1999), ASA SIAM Philadelphia · Zbl 0922.60001 |

[17] | Li, Shuanming; Dickson, David C.M., The maximum surplus before ruin in an Erlang(n) risk process and related problems, Insurance: mathematics and economics, 38, 3, 529-539, (2006) · Zbl 1168.60363 |

[18] | Li, Shuanming; Garrido, Jose, On a general class of renewal risk process: analysis of the gerber – shiu function, Advances in applied probabilities, 37, 3, 836-856, (2005) · Zbl 1077.60063 |

[19] | Li, Shuanming; Lu, Yi, Moments of the dividend payments and related problems in a Markov-modulated risk model, North American actuarial journal, 11, 2, 65-76, (2007) |

[20] | Li, Shuanming; Lu, Yi, The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model, Astin bulletin, 38, 1, 53-71, (2008) · Zbl 1169.91390 |

[21] | Lin, Sheldon X.; Willmot, Gordon E., Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics, 25, 1, 63-84, (1999) · Zbl 1028.91556 |

[22] | Lin, Sheldon X.; Willmot, Gordon E.; Drekic, Steve, The compound Poisson risk model with a constant dividend barrier: analysis of the gerber – shiu discounted penalty function, Insurance: mathematics and economics, 33, 3, 551-566, (2003) · Zbl 1103.91369 |

[23] | Lu, Yi; Tsai, Cary Chi-Liang, The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion, North American actuarial journal, 11, 2, 136-152, (2007) |

[24] | Neuts, Marcel F., Structured stochastic matrices of M/G/1 type and their applications, (1989), Marcel Dekker New York · Zbl 0695.60088 |

[25] | Ramaswami, Vaidyanathan, Passage times in fluid models with application to risk processes, Methodology and computing in applied probability, 8, 4, 497-515, (2006) · Zbl 1110.60067 |

[26] | Ren, Jiandong, The discounted joint probability of the surplus prior to ruin and the deficit at ruin, North American actuarial journal, 11, 3, 128-136, (2007) |

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