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Pure Mathematics 2024
基于随机波动下不同效用函数的集体最优投资问题研究
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Abstract:
效用函数是养老基金管理过程中最直接最重要的影响因素之一。本文利用Heston提出的随机波动率模型模拟金融市场，对基金代理人提供最低担保的混合养老金计划进行研究并给出最优投资策略问题，以最大化终端财富过程的期望效用为目标函数。假设投资者对风险的偏好程度满足HARA效用函数的一般框架，运用随机最优控制理论和静态鞅方法确定不同效用函数下的最优终端财富，并结合伊藤积分给出了不同的最优投资策略的显式表达式。最后通过数值模拟说明在不同效用函数下的参数对最优财富的影响不同，为决策者提供相应的参考。
Utility function is one of the most direct and important factors in the process of pension fund management. This paper uses the stochastic volatility model proposed by Heston to simulate the financial market, studies the hybrid pension plan with minimum guarantee provided by the fund agent, and presents the optimal investment strategy problem, taking maximizing the expected utility of the terminal wealth process as the objective function. Assuming that the degree of in-vestor's preference for risk meets the general framework of HARA utility function, the stochastic optimal control theory and static martingale method are used to determine the optimal terminal wealth under different utility functions, and the explicit expressions of different optimal investment strategies are given by combining ITO integral. Finally, numerical simulation is used to illustrate the different effects of parameters under different utility functions on the optimal wealth, which provides the corresponding reference for decision makers.
| [1] | 陈凯, 何银深. 混合型养老金产品的定价及风险分析[J]. 保险研究, 2011(12): 80-87. |
| [2] | Zhang, X.Y. and Guo, J.Y. (2020) Optimal Defined Contribution Pension Management with Salary and Risky Assets Following Jump Diffusion Processes. East Asian Journal on Applied Mathematics, 10, 22-39.
https://doi.org/10.4208/eajam.301218.170419 |
| [3] | 柴忠芃, 荣喜民, 赵慧. 具有保费退还条款的DC养老金最优投资研究[J]. 系统工程理论与实践, 2017, 37(7): 1688-1696. |
| [4] | 陈佳辰, 荣喜民, 赵慧. 带有死亡和意外返还条款的DC型养老金的最优投资问题[J]. 工程数学学报, 2020, 37(6): 651-663. |
| [5] | He, L. and Liang, Z.X. (2013) Optimal Investment Strategy for the DC Plan with the Return of Premiums Clauses in a Mean-Variance Framework. Insurance: Mathematics and Economics, 53, 643-649.
https://doi.org/10.1016/j.insmatheco.2013.09.002 |
| [6] | He, L., Liang, Z.X., Liu, Y. and Ma, M. (2019) Optimal Control of DC Pension Plan Management under Two Incentive Schemes. North American Actuarial Journal, 23, 120-141. https://doi.org/10.1080/10920277.2018.1513371 |
| [7] | Dong, Y.H., Lv, W.X., Wei, S.Y. and Gong, Y.Y. (2020) Optimal Investment of DC Pension Plan under Incentive Schemes and Loss Aversion. Mathematical Problems in Engineering, 2020, Article ID: 5145848.
https://doi.org/10.1155/2020/5145848 |
| [8] | 陈峥, 李仲飞. 目标日期基金应对养老金投资的有效性及最优策略选择[J]. 系统工程理论与实践, 2022, 42(12): 3231-3246. |
| [9] | 常浩, 王春峰, 房振明. 通胀风险下基于HARA效用的DC型养老金计划[J]. 运筹学学报, 2016, 20(4): 39-51.
https://doi.org/10.15960/j.cnki.issn.1007-6093.2016.04.005 |
| [10] | Wang, S.X. and Lu, Y. (2019) Optimal Invest-ment Strategies and Risk-Sharing Arrangements for a Hybrid Pension Plan. Insurance: Mathematics and Economics, 89, 46-62. https://doi.org/10.1016/j.insmatheco.2019.09.005 |
| [11] | Deelstra, G., Grasselli, M. and Koehl, P.F. (2004) Optimal Design of the Guarantee for Defined Contribution Funds. Journal of Economic Dynamics and Control, 28, 2239-2260. https://doi.org/10.1016/j.jedc.2003.10.003 |
| [12] | Deelstra, G., Grasselli, M. and Koehl, P.F. (2003) Optimal Investment Strategies in the Presence of a Minimum Guarantee. Insurance: Mathematics and Economics, 33, 189-207. https://doi.org/10.1016/S0167-6687(03)00153-7 |
| [13] | An, C., Nguyen, T. and Rach, M. (2021) A Collec-tive Investment Problem in a Stochastic Volatility Environment: The Impact of Sharing Rules. Annals of Operations Research, 302, 85-109. https://doi.org/10.1007/s10479-021-03983-8 |
| [14] | Lichtenstern, A., Shevchenko, P.V. and Zagst, R. (2021) Optimal Life-Cycle Consumption and Investment Decisions under Age-Dependent Risk Preferences. Mathematics and Financial Economics, 15, 275-313.
https://doi.org/10.1007/s11579-020-00276-9 |
| [15] | 黄冬冬, 陈传钟. 基于HARA效用函数最优投资问题的显示解[J]. 统计与决策, 2017(5): 81-84. |
