The Research On Optimal Portfolio Of Pension Base On HARA Utility

In recent years, in the field of pension fund management, Pension fund managers want to invest to realize the value of pension funds. Therefore, one of the core problem is to determine the optimal pension fund investment strategies so that the expected utility of pension reach the maximum at the end of the time. Many scholars are applying stochastic control theory to research the optimal investment problem under different model and utility function, it has great practical significance.In this paper, we apply the stochastic control theory, Legendre transforma-tion and duality theorem, get the optimal investment strategy of DCS system of pension under the HARA (Hyperbolic Absolute Risk Aversion) utility func-tion, and the risky asset price process is Heston model and O-U(Ornstein-Uhlehbeck) model respectively. In this paper, the main content is as follows.In chapter 1, we simply introduce the research background and the present situation of the pension funds. Then, we introduce the main work of this paper.In chapter 2, we introduce the necessary knowledge of Brown motion, Leg-endre transformation and duality theorem, Heston model and O-U model and so on.In chapter 3. we discuss the optimal investment problem of the DCS system of pension fund under the Heston market framework. Assume that fund man-agers put pension funds into the risk market and risk free market, the prices of risky assets process using Heston model to depict. Fund manager’s goal is to make the expectation of the terminal wealth of HARA utility maximum respec-tively in the two stages of life-retired employees before and after retirement. Using the stochastic control theory, we obtain the nonlinear HJB equation of the two phases. Using Legendre transformation, dual theorems and methods of variable transformation, we obtain the corresponding optimal investment strat-egy and the displayed expression of the optimal value function’s dual function in the end of each stage respectively.In chapter 4, we discuss the optimal investment problem of the DCS system of pension fund under the O-U market framework. Assume that fund managers put pension funds into the risk market and risk free market, the prices of risky assets process using O-U model to depict. Fund manager’s goal is to make the expectation of the terminal wealth of HARA utility maximum respectively in the two stages of life-retired employees before and after retirement. Using the stochastic control theory, we obtain the nonlinear HJB equation of the two phases. Then, we using the Legendre transformation, duality theorem and the method of variable transformation, transform it into linear partial differential equation in order to solve it facilitately. Finally, we obtain The corresponding optimal investment strategy in the end of each stage respectively.