Ruin theory for a Markov regime-switching model under a threshold dividend strategy.

*(English)*Zbl 1141.91558Summary: We study a Markov regime-switching risk model where dividends are paid out according to a certain threshold strategy depending on the underlying Markovian environment process. We are interested in these quantities: ruin probabilities, deficit at ruin and expected ruin time. To study them, we introduce functions involving the deficit at ruin and the indicator of the event that ruin occurs. We show that the above functions and the expectations of the time to ruin as functions of the initial capital satisfy systems of integro-differential equations. Closed form solutions are derived when the underlying Markovian environment process has only two states and the claim size distributions are exponential.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B28 | Finance etc. (MSC2000) |

60G40 | Stopping times; optimal stopping problems; gambling theory |

##### Keywords:

Markov regime-switching; dividend; ruin probability; deficit at ruin; integro-differential equation
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\textit{J. Zhu} and \textit{H. Yang}, Insur. Math. Econ. 42, No. 1, 311--318 (2008; Zbl 1141.91558)

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##### References:

[1] | Asmussen, S., Risk theory in a Markovian environment, Scandinavian actuarial journal, 69-100, (1989) · Zbl 0684.62073 |

[2] | Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore |

[3] | Baeuerle, N., Some results about the expected ruin time in Markov-modulated risk models, Insurance: mathematics and economics, 18, 119-127, (1996) |

[4] | Gerber, H.U., Martingales in risk theory, (), 205-216 · Zbl 0278.60047 |

[5] | Gerber, H.U., An introduction to mathematical risk theory, Monograph series, vol. 8, (1979), Huebner Foundation Philadelphia · Zbl 0431.62066 |

[6] | Gerber, H.U.; Shiu, E., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550 |

[7] | Gerber, H.U.; Shiu, E., Optimal dividends: analysis with Brownian motion, North American actuarial journal, 8, 1, 1-20, (2004) · Zbl 1085.62122 |

[8] | Lin, X.S.; Pavlova, K., The compound Poisson risk model with a threshold dividend strategy, Insurance: mathematics and economics, 38, 57-80, (2006) · Zbl 1157.91383 |

[9] | Lin, X.S.; Willmot, G.E.; Drekic, S., The classical risk model with a constant dividend barrier: analysis of the gerber – shiu discounted penalty function, Insurance: mathematics and economics, 33, 551-566, (2003) · Zbl 1103.91369 |

[10] | Lu, Y.; Li, S., On the probability of ruin in a Markov-modulated risk model, Insurance: mathematics and economics, 37, 522-532, (2005) · Zbl 1129.60066 |

[11] | Ng, A.; Yang, H., Lundberg-type bounds for the joint distribution of surplus immediately before and after ruin under a Markov-modulated risk model, Astin bulletin, 35, 351-361, (2005) · Zbl 1101.62102 |

[12] | Ng, A.; Yang, H., On the joint distribution of surplus prior and immediately after ruin under a Markovian regime switching model, Stochastic processes and their applications, 116, 244-266, (2006) · Zbl 1093.60051 |

[13] | Reinhard, J.M., On a class of semi-Markov risk models obtained as classical risk models in a Markovian enviroment, Astin bulletin, 14, 23-43, (1984) |

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