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**Optimal reinsurance-investment problem under mean-variance criterion with \(n\) risky assets.**
*(English)*
Zbl 1459.91165

Summary: Based on the mean-variance criterion, this paper investigates the continuous-time reinsurance and investment problem. The insurer’s surplus process is assumed to follow Cramér-Lundberg model. The insurer is allowed to purchase reinsurance for reducing claim risk. The reinsurance pattern that the insurer adopts is combining proportional and excess of loss reinsurance. In addition, the insurer can invest in financial market to increase his wealth. The financial market consists of one risk-free asset and \(n\) correlated risky assets. The objective is to minimize the variance of the terminal wealth under the given expected value of the terminal wealth. By applying the principle of dynamic programming, we establish a Hamilton-Jacobi-Bellman (HJB) equation. Furthermore, we derive the explicit solutions for the optimal reinsurance-investment strategy and the corresponding efficient frontier by solving the HJB equation. Finally, numerical examples are provided to illustrate how the optimal reinsurance-investment strategy changes with model parameters.

### MSC:

91G05 | Actuarial mathematics |

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\textit{P. Yang}, Discrete Dyn. Nat. Soc. 2020, Article ID 6489532, 16 p. (2020; Zbl 1459.91165)

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### References:

[1] | Browne, S., Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3, 3, 275-294 (1999) · Zbl 1047.91025 |

[2] | Chen, Z.; Yang, P., Robust optimal reinsurance-investment strategy with price jumps and correlated claims, Insurance: Mathematics and Economics, 92, 27-46 (2020) |

[3] | Yang, H.; Zhang, L., Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37, 3, 615-634 (2005) · Zbl 1129.91020 |

[4] | Zhao, H.; Rong, X.; Zhao, Y., Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, 53, 3, 504-514 (2013) · Zbl 1290.91106 |

[5] | Asmussen, S.; Taksar, M., Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20, 1, 1-15 (1997) · Zbl 1065.91529 |

[6] | Chen, S.; Zeng, Y.; Hao, Z., Optimal dividend strategies with time-inconsistent preferences and transaction costs in the Cramér-Lundberg model, Insurance: Mathematics and Economics, 74, 31-45 (2017) · Zbl 1394.91202 |

[7] | Belkina, T.; Luo, S., Asymptotic investment behaviors under a jump-diffusion risk process, North American Actuarial Journal, 21, 1, 36-62 (2017) · Zbl 1414.91164 |

[8] | Sun, Z., Upper bounds for ruin probabilities under model uncertainty, Communications in Statistics-Theory and Methods, 48, 18, 4511-4527 (2019) |

[9] | Markowitz, H., Portfolio selection, The Journal of Finance, 7, 1, 77-91 (1952) |

[10] | Zhou, X. Y.; Li, D., Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Applied Mathematics and Optimization, 42, 1, 19-33 (2000) · Zbl 0998.91023 |

[11] | Li, D.; Ng, W.-L., Optimal dynamic portfolio selection: multiperiod mean-variance formulation, Mathematical Finance, 10, 3, 387-406 (2000) · Zbl 0997.91027 |

[12] | Bielecki, T. R.; Jin, H.; Pliska, S. R.; Zhou, X. Y., Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15, 2, 231-244 (2005) · Zbl 1153.91466 |

[13] | Bäuerle, N., Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62, 1, 159-165 (2005) · Zbl 1101.93081 |

[14] | Yang, P., Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market, Optimization, 66, 5, 737-758 (2017) · Zbl 1369.91102 |

[15] | Wang, S.; Rong, X.; Zhao, H., Mean-variance problem for an insurer with default risk under a jump-diffusion risk model, Communications in Statistics-Theory and Methods, 48, 17, 4221-4249 (2019) |

[16] | Sun, Z.; Yuen, K. C.; Guo, J., A BSDE approach to a class of dependent risk model of mean-variance insurers with stochastic volatility and no-short selling, Journal of Computational and Applied Mathematics, 366, 112413 (2020) · Zbl 1427.91243 |

[17] | Yang, P.; Chen, Z.; Xu, Y., Time-consistent equilibrium reinsurance-investment strategy for n competitive insurers under a new interaction mechanism and a general investment framework, Journal of Computational and Applied Mathematics, 374, 112769 (2020) · Zbl 1435.62375 |

[18] | Liang, Z.; Guo, J., Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, Journal of Applied Mathematics and Computing, 36, 1-2, 11-25 (2011) · Zbl 1232.93100 |

[19] | Hu, X.; Duan, B.; Zhang, L., De Vylder approximation to the optimal retention for a combination of quota-share and excess of loss reinsurance with partial information, Insurance: Mathematics and Economics, 76, 48-55 (2017) · Zbl 1395.91253 |

[20] | Zhao, H.; Shen, Y.; Zeng, Y.; Zhang, W., Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean-variance insurer with ambiguity aversion, Insurance: Mathematics and Economics, 88, 159-180 (2019) · Zbl 1425.91238 |

[21] | Huang, Y.; Ouyang, Y.; Tang, L.; Zhou, J., Robust optimal investment and reinsurance problem for the product of the insurer’s and the reinsurer’s utilities, Journal of Computational and Applied Mathematics, 344, 532-552 (2018) · Zbl 1458.91184 |

[22] | Luenberger, D. G., Optimization by Vector Space Methods (1968), New York, NY, USA: John Wiley, New York, NY, USA · Zbl 0184.44502 |

[23] | Fleming, W. H.; Soner, H. M., Controlled Markov Processes and Viscosity Solution (1993), Berlin, Germany: Springer, Berlin, Germany · Zbl 0773.60070 |

[24] | Bi, J.; Guo, J., Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157, 1, 252-275 (2013) · Zbl 1266.91093 |

[25] | Zhang, X., Application of markov-modulated processes in insurance and finance (2009), Tianjin, China: Nankai University, Tianjin, China, Ph. D. thesis |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.