Stochastic Optimal Control Strategies For Pension Funds Sustainability

Recently,pension fund management has become a preferred and crucial topic to scholars due to its essential role in the financial market,and in the social security system.What is more,the demographic development that is threatening the sustainability of an adequate retirement income for the retirees and the evolution of the equity market are the additional motives for its popularity.To be precise,when life expectancy is relatively low,providing for old age is not a central economic problem.However,since the average age of people and the lifespan have sharply increased in the past decades(and is expected to continue),the approach to organizing a pension system providing for old age and maintaining economic efficiency and growth has become a significant challenge worldwide.In this thesis,we embrace and focus on solving optimal stochastic control problems for pension funds sustainability.Specifically,we introduce the following models in the pension fund management:the time-dependent interest rate model(the Hull-White)for riskless assets;the Hull and White stochastic volatility model for risky assets,and the hybrid stochastic volatility model that combines a stochastic volatility model for asset prices and a stochastic short rate process(the Heston-Hull-White).In the first part,we analyze the optimal investment strategies for a DC pension fund under the stochastic interest rate model.In the proposed model,the pension fund manager allocates capital in the bank account,stock index,and in the real estates,where the dynamics of inter-est rate follows the Hull-White interest rate model and a drifted Brownian motion drives the risky assets price.The pension fund manager goals are to seek optimal investment strategies to maximize the expected utility of terminal wealth.By applying the stochastic dynamic program-ming principle and variable change techniques,we obtain the explicit expressions for the optimal portfolio decisions in the power utility case.To analyze the influence of the parameters of the interest rate on the optimal portfolio choices,we provide a numerical illustration to demonstrate our results.The second part formulates the optimal contributions and investment strategies problem for a defined benefit pension fund.The pension fund manager in this model allocates the capital in riskless and risky assets.The dynamics of the risky asset price follows the Hull and White Stochastic Volatility model.The stochastic dynamic programming principle is used to derive a non-linear second-order partial differential equation,the Hamilton-Jacobi-Bellman equation.On the other hand,the Legendre transform method is applied to deal with the complexity of the Hamilton-Jacobi-Bellman equation.Moreover,we obtain closed-form solutions for optimal strategies for the logarithm utility functions by variable transformation technique.Finally,a numerical example is provided to analyze the effects of parameters on the model and provide some economic implications.In the third part,we consider the optimal investment and benefit payments policies for a DC pension fund under the hybrid stochastic volatility.In our model,the pension fund manager is allowed to invest in riskless and risky assets where the hybrid stochastic volatility model drives the price dynamics of the risk assets.The proposed hybrid model(HHW)is formulated by two systems of stochastic differential equations to analyze the price of a risky asset,and its volatility.The first system defines the price of a risky asset and its volatility and the second describes the short rate process.This hybrid model fits in the class of affine diffusion processes for which a closed-form solution of the characteristic function exists.By applying the stochastic dynamic programming principle and variable change techniques,we find the explicit expressions for the optimal control strategies in the power utility case.To analyze the influence of model parameters on the optimal strategies,we provide a numerical example.