Recursive methods for a multi-dimensional risk process with common shocks.

*(English)*Zbl 1235.91090Summary: In this paper, a multi-dimensional risk model with common shocks is studied. Using a simple probabilistic approach via observing the risk processes at claim instants, recursive integral formulas are developed for the survival probabilities as well as for a class of Gerber-Shiu expected discounted penalty functions that include the surplus levels at ruin. Under the assumption of exponential or mixed Erlang claims, the recursive integrals can be simplified to give recursive sums which are computationally more tractable. Numerical examples including an optimal capital allocation problem are also given towards the end.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

##### Keywords:

common shock; deficit at ruin; Gerber-Shiu expected discounted penalty function; recursive methods; survival probability; multi-dimensional risk process; optimal capital allocation
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\textit{L. Gong} et al., Insur. Math. Econ. 50, No. 1, 109--120 (2012; Zbl 1235.91090)

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##### References:

[1] | Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific New Jersey · Zbl 1247.91080 |

[2] | Avram, F.; Palmowski, Z.; Pistoris, M., A two-dimensional ruin problem on the positive quadrant, Insurance: mathematics and economics, 42, 1, 227-234, (2008) · Zbl 1141.91482 |

[3] | Avram, F.; Palmowski, Z.; Pistoris, M., Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results, Annals of applied probability, 18, 6, 2421-2449, (2008) · Zbl 1163.60010 |

[4] | Badescu, A.L.; Cheung, E.C.K.; Rabehasaina, L., A two-dimensional risk model with proportional reinsurance, Journal of applied probability, 48, 3, 749-765, (2011) · Zbl 1239.91073 |

[5] | Boudreault, M.; Cossette, H.; Landriault, D.; Marceau, E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian actuarial journal, 5, 265-285, (2006) · Zbl 1145.91030 |

[6] | Cai, J.; Li, H., Multivariate risk model of phase type, Insurance: mathematics and economics, 36, 2, 137-152, (2005) · Zbl 1122.60049 |

[7] | Cai, J.; Li, H., Dependence properties and bounds for ruin probabilities in multivariate compound risk models, Journal of multivariate analysis, 98, 4, 757-773, (2007) · Zbl 1280.91090 |

[8] | Chan, W.-S.; Yang, H.; Zhang, L., Some results on the ruin probability in a two-dimensional risk model, Insurance: mathematics and economics, 32, 3, 345-358, (2003) · Zbl 1055.91041 |

[9] | Collamore, J.F., Hitting probabilities and large deviations, The annals of probability, 24, 4, 2065-2078, (1996) · Zbl 0879.60021 |

[10] | Collamore, J.F., First passage times of general sequences of random vectors: a large deviations approach, Stochastic processes and their applications, 78, 1, 97-130, (1998) · Zbl 0934.60025 |

[11] | Collamore, J.F., Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors, The annals of applied probability, 12, 1, 382-421, (2002) · Zbl 1021.65003 |

[12] | Cossette, H.; Marceau, E.; Marri, F., On the compound Poisson risk model with dependence based on a generalized farlie – gumbel – morgenstern copula, Insurance: mathematics and economics, 43, 3, 444-455, (2008) · Zbl 1151.91565 |

[13] | Czarna, I.; Palmowski, Z., De finetti’s dividend problem and impulse control for a two-dimensional insurance risk process, Stochastic models, 27, 2, 220-250, (2011) · Zbl 1214.91051 |

[14] | Dang, L.; Zhu, N.; Zhang, H., Survival probability for a two-dimensional risk model, Insurance: mathematics and economics, 44, 3, 491-496, (2009) · Zbl 1162.91405 |

[15] | Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-72, (1998) |

[16] | Lee, S.C.K.; Lin, X.S., Modeling and evaluating insurance losses via mixtures of Erlang distributions, North American actuarial journal, 14, 1, 107-130, (2010) |

[17] | Li, J.; Liu, Z.; Tang, Q., On the ruin probabilities of a bidimensional perturbed risk model, Insurance: mathematics and economics, 41, 1, 185-195, (2007) · Zbl 1119.91056 |

[18] | Rabehasaina, L., Risk processes with interest force in Markovian environment, Stochastic models, 25, 4, 580-613, (2009) · Zbl 1222.91025 |

[19] | Stanford, D.A.; Stroiński, K.J., Recursive methods for computing finite-time ruin probabilities for phase-type distributed claim sizes, Astin bulletin, 24, 2, 235-254, (1994) |

[20] | Stanford, D.A.; Stroiński, K.J.; Lee, K., Ruin probabilities based at claim instants for some non-Poisson claim processes, Insurance: mathematics and economics, 26, 2-3, 251-267, (2000) · Zbl 1013.91068 |

[21] | Willmot, G.E.; Woo, J.-K., On the class of Erlang mixtures with theoretic applications, North American actuarial journal, 11, 2, 99-115, (2007) |

[22] | Yuen, K.C.; Guo, J.; Wu, X., On the first time of ruin in the bivariate compund Poisson model, Insurance: mathematics and economics, 38, 2, 298-308, (2006) · Zbl 1095.62120 |

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