On the time to ruin for a dependent delayed capital injection risk model.

*(English)*Zbl 1429.91091Summary: In this paper, we propose a generalisation to the Cramér-Lundberg risk model, by allowing for a delayed receipt of the required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of an insurance firm. The delay time of the capital injection depends on a critical value of the deficit in the following way: if the deficit of the firm is less than the fixed critical value, then it can be covered by available funds and therefore the required capital injection is received instantaneously. On the other hand, if the deficit of the firm exceeds the fixed critical value, then the funds are provided by an alternative source and the required capital injection is received after some time delay. In this modified model, we derive a Fredholm integral equation of the second kind for the ultimate ruin probability and obtain an explicit expression in terms of ruin quantities for the Cramér-Lundberg risk model. In addition, we show that other risk quantities, namely the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, we extend the capital injection delayed risk model, such that the delay of the capital injections depends explicitly on the amount of the deficit. In this generalised risk model, we derive another Fredholm integral equation for the ultimate ruin probability, which is solved in terms of a Neumann series.

##### MSC:

91B05 | Risk models (general) |

45B05 | Fredholm integral equations |

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

91G05 | Actuarial mathematics |

##### Keywords:

ruin probability; deficit dependent delayed capital injections; Fredholm integral equation; Neumann series solution
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\textit{L. Ramsden} and \textit{A. D. Papaioannou}, Appl. Math. Comput. 352, 119--135 (2019; Zbl 1429.91091)

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##### References:

[1] | Asmussen, S.; Albrecher, H., Ruin Probabilities (2000), World Scientific |

[2] | Dickson, D. C.; Drekic, S., Optimal dividends under a ruin probability constraint, Ann. Actuarial Sci., 1, 2, 291-306 (2006) |

[3] | Dickson, D. C.; Qazvini, M., Gerber-Shiu analysis of a risk model with capital injections, Eur. Actuarial J., 6, 2, 409-440 (2016) · Zbl 1394.91209 |

[4] | Eisenberg, J.; Schmidli, H., Optimal control of capital injections by reinsurance in a diffusion approximation, Blätter DGVFM, 30, 1, 1-13 (2009) · Zbl 1183.91069 |

[5] | Eisenberg, J.; Schmidli, H., Minimising expected discounted capital injections by reinsurance in a classical risk model, Scand. Actuarial J., 2011, 3, 155-176 (2011) · Zbl 1277.60145 |

[6] | Gerber, H. U.; Goovaerts, M. J.; Kaas, R., On the probability and severity of ruin, ASTIN Bull., 17, 02, 151-163 (1987) |

[7] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press · Zbl 0704.15002 |

[8] | ING, Insurance Annual Report (2010) |

[9] | Jin, Z.; Yin, G., An optimal dividend policy with delayed capital injections, ANZIAM J., 55, 2, 129-150 (2013) · Zbl 1302.91189 |

[10] | Kulenko, N.; Schmidli, H., Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance Math. Econ., 43, 2, 270-278 (2008) · Zbl 1189.91075 |

[11] | Landriault, D.; Willmot, G. 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 Am. Actuarial J., 13, 2, 252-270 (2009) |

[12] | MOODY, MOODY’s Report (2016) |

[13] | Nie, C.; Dickson, D. C.; Li, S., Minimizing the ruin probability through capital injections, Ann. Actuarial Sci., 5, 02, 195-209 (2011) |

[14] | Nie, C.; Dickson, D. C.; Li, S., The finite time ruin probability in a risk model with capital injections, Scand. Actuarial J., 2015, 4, 301-318 (2015) · Zbl 1398.91350 |

[15] | Pafumi, G., “On the time value of ruin”, Hans U. Gerber and Elias S.W. Shiu, January 1998, North Am. Actuarial J., 2, 1, 75-76 (1998) |

[16] | Polyanin, A. D.; Manzhirov, A. V., Handbook of Integral Equations (2008), CRC Press · Zbl 1154.45001 |

[17] | Wu, Y., Optimal reinsurance and dividend strategies with capital injections in Cramér-Lundberg approximation model, Bull. Malaysian Math. Sci. Soc., 36, 1 (2013) · Zbl 1257.62098 |

[18] | Zemyan, S. M., The Classical Theory of Integral Equations: A Concise Treatment (2012), Springer Science & Business Media · Zbl 1270.45001 |

[19] | Zhou, M.; Yuen, K. C., Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle, Econ. Modell., 29, 2, 198-207 (2012) |

[20] | Zhou, M.; Yuen, K. C., Portfolio selection by minimizing the present value of capital injection costs, ASTIN Bull., 45, 01, 207-238 (2015) · Zbl 1390.91291 |

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