Research Of The Stochastic Optimal Control Problems In The Management Of Pension Fund

The pension fund is one of the most important part of the social security system. It is very important to keep and increase the value of the pension fund by rational investment in the financial market. Since the accumulation value of the pension fund is not only related with the levels of life and happiness of each pension fund participant after retirement, but also effects the stability and unity of the society. In recent years, in the field of insurance actuary and mathematical finance, the research of the optimal investment decision models for DB and DC pension funds have become a hot issue for many scholars. It also has been widely paid attention by industry. This thesis combines the characteristics of the pension funds, focuses on the optimal portfolio choice problems in the pension funds under some stochastic environments such as the stochastic interest rate and stochastic inflation rate, and considers the risk of model uncertainty. First, we study the optimal investment problem for a DC pension fund under the Heston’s volatility model with inflation risk. Second, we consider a DC pension plan with premium return clause and obtain the optimal investment strategy. Then, a stochastic optimal control problem which is derived by a dynamic investment target of the DC pension plan member is discussed. Finally, the robust optimal control problem for a DC pension fund is studied based on the background of stochastic interest rate and stochastic income.In Chapter 1, we firstly introduce the background of the stochastic control prob-lems in the management of the pension plan. Then, we state the main work of this thesis. Finally, we introduce the general theory of stochastic optimal control, the method of dealing with the problem of time inconsistency and list some preliminary knowledge.Chapter 2 is devoted to studying a portfolio selection problem for a DC pension plan under the mean-variance criteria. We take into account the inflation risk and assume that the salary income process of the pension plan member is stochastic. Fur-thermore, the financial market consists of a risk-free asset, an inflation-linked bond and a risky asset with Heston’s stochastic volatility. Under the framework of game theory, we derive two extended Hamilton-Jacobi-Bellman (HJB) equations systems and give the corresponding verification theorems both in the periods of accumulation and distribution of the DC pension plan. The explicit expressions of the equilibri-um investment strategies, corresponding equilibrium value functions, and the efficient frontiers are also obtained. Finally, some numerical simulations and sensitivity anal-ysis are presented to verify our theoretical results.In Chapter 3, we study an optimal investment problem for a defined contribution pension plan during the accumulation phase under the mean-variance criterion. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump-diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) [38] and Bjork et al. (2014) [9]. Finally, some numerical simulations are provided to illustrate our derived results.In Chapter 4, based on the dynamic investment target, we consider the opti-mal investment problem of a DC pension plan at the phase of accumulation before retirement. We assume that the pension plan member sets an expected investment target via the income level of the member when she retires. And the pension fund is invested into the financial market consists of one risk-free asset and n risky assets. From the point of view of deficit and surplus, we analyze the deviation between the pension fund account and expected investment target, and formulate the continuous time portfolio problems under the quadratic loss and mean-variance criteria, respec-tively. The explicit expressions of the optimal investment strategies for the above problems are obtained, and the relationship between the expected terminal wealth under the two optimal strategies are also analysed. Finally, some numerical examples are presented to verify our results.Chapter 5 mainly considers the robust optimal portfolio problem for a DC pension fund with stochastic interest rate and stochastic income. In the reality, because of the difficulty of estimating the expected rate of return of the risk assets, we assume that the pension fund participant is an ambiguity aversion investor, she worries about the uncertainty risk of the model and it’s parameters. To make the expected utility of the pension fund account maximized at retirement time, we assumed that the ambiguity aversion pension plan investor is allowed to invest the pension fund into the financial market consists of a risk-free asset, zero coupon bond and a risky asset. Using the method of stochastic optimal control, we obtain the closed-form of the robust optimal investment strategy and the corresponding value function under the power utility. Some special cases are also provided. Finally, through some numerical examples, we discuss the sensitivity of the optimal investment strategy to the ambiguity aversion parameters, and compare the utility loss due to ignore the model uncertainty in the investment decision-making process.