基于随机波动下不同效用函数的集体最优投资问题研究

Research on Collective Optimal Investment Problem with Different Utility Functions under Stochastic Volatility

摘要: 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。

Abstract: 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.