Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (1): 97-124.

### THE OPTIMAL REINSURANCE-INVESTMENT PROBLEM CONSIDERING THE JOINT INTERESTS OF AN INSURER AND A REINSURER UNDER HARA UTILITY*

Yan Zhang1, Peibiao Zhao2,†, Huaren Zhou1

1. 1. Department of General Education, Army Engineering University of PLA, Nanjing 211101, China;
2. School of Mathematics & Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
• Received:2021-08-27 Revised:2022-06-26 Published:2023-03-01
• Contact: †Peibiao ZHAO. E-mail: pbzhao@njust.edu.cn
• About author:Yan Zhang, E-mail: sdzyjyw@126.com; Huaren Zhou,zhouzhu123123123@126.com
• Supported by:
*Natural Science Foundation of China (11871275; 11371194).

Abstract: This paper focuses on an optimal reinsurance and investment problem for an insurance corporation which holds the shares of an insurer and a reinsurer. Assume that the insurer can purchase reinsurance from the reinsurer, and that both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset which are governed by the Heston model and are distinct from one another. We aim to find the optimal reinsurance-investment strategy by maximizing the expected Hyperbolic Absolute Risk Aversion (HARA) utility of the insurance corporation's terminal wealth, which is the weighted sum of the insurer's and the reinsurer's terminal wealth. The Hamilton-Jacobi-Bellman (HJB) equation is first established. However, this equation is non-linear and is difficult to solve directly by any ordinary method found in the existing literature, because the structure of this HJB equation is more complex under HARA utility. In the present paper, the Legendre transform is applied to change this HJB equation into a linear dual one such that the explicit expressions of optimal investment-reinsurance strategies for $-1\le \rho_i \le 1$ are obtained. We also discuss some special cases in a little bit more detail. Finally, numerical analyses are provided.

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