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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (5): 915-930.doi: 10.3969/j.issn.1005-3085.2024.05.009

• • 上一篇    下一篇

O-U模型下基于HARA效用的最优投资–再保险策略问题

张  燕1,   王正艳2   

  1. 1. 陆军工程大学基础部，南京  211101
    2. 盐城工学院经济管理学院，盐城  224056
  • 收稿日期:2021-12-01 接受日期:2022-06-21 出版日期:2024-10-15
  • 基金资助:
    陆军工程大学基础前沿创新项目 (KYJBJKQTZQ23001).

Optimal Reinsurance-investment Problem with HARA Utility under O-U Model

ZHANG Yan1,   WANG Zhengyan2   

  1. 1. Department of General Education, Army Engineering University of PLA, Nanjing 211101
    2. School of Economics and Management, Yancheng Institute of Technology, Yancheng 224056
  • Received:2021-12-01 Accepted:2022-06-21 Online:2024-10-15
  • Supported by:
    The Fundamental Frontier Science Innovation Project of Army Engineering University of PLA (KYJBJKQTZQ23001).

摘要：

研究了O-U(Ornstein-Uhlenbeck)风险模型下最大化双曲绝对风险(Hyperbolic Absolute Risk Aversion, HARA)效用的最优投资–再保险问题。允许保险人购买比例再保险，且可投资于一种无风险资产和一种风险资产，其瞬间收益率由能够反映市场的牛市和熊市特征的O-U过程刻画。在保险人终端财富的HARA效用期望最大化的目标下，利用随机动态规划原理，首先建立了Hamilton-Jacobi-Bellman(HJB)方程。其次，由于HARA效用函数的复杂结构导致常规方法难以求解HJB方程，利用勒让德对偶变换将HJB方程转化为易于求解的对偶HJB方程。通过构造对偶HJB方程解的形式及变量变换，得到了最优再保险–投资策略的解析式。最后通过数值计算分析了参数对最优结果的影响。

关键词: HARA效用函数, 投资, 再保险, O-U风险模型, 勒让德变换

Abstract:

This paper investigates an optimal reinsurance-investment problem with HARA utility. In order to avoid claim risks, the insurer is allowed to purchase reinsurance, and is assumed to invest in one risk-free asset and one risky asset whose instantaneous rate is governed by an Ornstein-Uhlenbeck (O-U) process, which could describe the features of bull and bear markets. Firstly, under the criterion of maximizing the expected HARA utility of the insurer's terminal wealth, the HJB equation for the value function is obtained by applying dynamic programming principle. Secondly, due to the complexity of the structure of HARA utility, we use Legendre transform to change the original HJB equation into its dual one, whose solution is easy to conjecture. Closed-form solution of optimal investment-reinsurance strategy is obtained by constructing the solution form of the dual equation and the variable change technique. Finally, some numerical simulations are presented to illustrate the impacts of model parameters on the optimal reinsurance-investment strategy.

Key words: HARA utility function, investment, reinsurance, O-U risk model, Legendre transform

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