Stochastic optimal control and its application in pension schemes

Abstract

The aim of this study is to entertain the theory of stochastic optimal control at a mathematically rigorous level and to use a few related methods for solving problems arising in pension fund management. Notation used will be written carefully to avoid omitting important information and proofs of some results will be given in a considerable detail. A stochastic dynamical system may be influenced by various control processes. Stochastic optimal control deals with selecting the control process which optimizes some objective functional. In this thesis the dynamical system considered is represented by a Markov diffusion process that solves a controlled stochastic integral equation, while the objective functional is given in an additive form. The method of dynamic programming will be used as a tool to find an optimal control in closed form. This leads to solving a partial differential equation analytically. In many applications, however, the solution is hard to compute. So numerical techniques for finding an approximate optimal control process will also be studied. In particular, the Markov Chain approximation method will be discussed extensively. In funded pension schemes, a central problem is to select optimally the distribution of the wealth generated from the contributions over various investment opportunities. Since the capital depends also on the return from the investment, the dynamics of the pension fund are clearly stochastic. By considering the fraction of the wealth to invest in each asset as the control process, another objective of this study is to formulate this portfolio investment problem as a stochastic optimal control problem and solve numerically, three suitably specified problems