In this paper, we apply the stochastic optimal control theory to the pension fund management before and after retirement in the defined contribution and defined benefit pension schemes, where benefits are paid as investment returns for a period or duration of time. The goal of the management problem is to optimize the long-term growth of expected utility of returns. We consider different types of power law utility function of the form U(X)=γ-1Xγ, γ<1, γ≠0 to examine the different investment schemes. Our result shows the advantage of the defined contribution scheme over the defined benefit scheme before and after retirement.

The problem of A fund manager is to minimize the expected utility loss function, the noise generated in the dynamics of the wealth process are driven by fractional Brownian motions with long range dependence (if H>1/2). We replaced the classical Brownian motion by fractional Brownian motion with Hurst parameter more than 1/2. We finally use time-inversion of diffusions to obtain singular equations.

We investigate the optimal investment strategies of DC pension under stochastic volatility model using combined Heston-Hull-White (HHW) model with a constant income drawdown. The pension fund manager (PFM) aims to maximize the expected terminal utility of wealth in a complete market setting under constant relative risk aversion (CRRA). The goal of the PFM is to maintain the standard of living of the participants after retirement. We derive the HJB equation associated with the control problem and finally established the close form solution using stochastic dynamic programming principle (SDPP). The results show that the optimal investment and benefit payment strategies converge uniquely with time.

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