O-U模型下基于HARA效用的最优投资–再保险策略问题  

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

作　　者：张燕 王正艳 ZHANG Yan;WANG Zhengyan(Department of General Education,Army Engineering University of PLA,Nanjing 211101;School of Economics and Management,Yancheng Institute of Technology,Yancheng 224056)

机构地区：[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

出　　处：《工程数学学报》2024年第5期915-930,共16页Chinese Journal of Engineering Mathematics

基　　金：陆军工程大学基础前沿创新项目(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 Abso-lute 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。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 as-sumed 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 pro-gramming 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.

关 键 词：HARA效用函数 投资 再保险 O-U风险模型 勒让德变换 

分 类 号：O211.67[理学—概率论与数理统计]