Stochastic Optimal Control With Applications To Pension Funds And Differential Game Theory

This work thrives to put forward the background and significance of stochastic optimal control in pension funds and nonlinear di?erential games. We intend to explore new ideas and direction in tackling pension fund investment strategies and optimal solutions of a nonlinear stochastic di?erential game. Our ideas laid in this document would be of much help to other researchers and in our view is of much importance in new directions of stochastic optimal control theory in pension funds and stochastic nonlinear di?erential games. Stochastic optimal control is a branch of control theory that deals with uncertainties in observations or decisions to be taken. Stochastic optimal control is mathematically defined as a process of how to act optimally in order to attain future rewards.The topic has been under discussion since the past decade. Though stochastic optimal control is mainly of two types: the discrete time and continuous time, here only the continuous time would be under discussion. The general aim of this dissertation is to find the best decisions to be undertaken by pension fund manager or an investor to obtain the best future rewards. How much wealth or such optimal controls yields would as well be under discussion in both pension funds and nonlinear stochastic di?erential game. Such a research would be pursed in the following manner.1. The research would briefly discuss the background of the stochastic di?erential equations. The existence and uniqueness of stochastic di?erential equations with controls is investigated. Also, the Dynamic Programming Principle and Hamiltonian Jacobi Bellman are presented as they are needed in this research. A strong unique solution of stochastic di?erential equations with controls has been found to exists.2. The core part of our research begins here. A stochastic optimal investment problem under inflationary market would be under discussion. The market is driven by two noise terms associated to the stock price and inflation modeled as geometric Brownian motion. An investor has three assets to trade upon, two risky assets(stock and inflation-linked bond) and one non-risky asset(cash/bank account).The best investment strategies are to be investigated under the following cases in defined contribution pension plan.(i) The expected minimum guarantee process acts as the solvency level.(ii) When the contribution process is in two forms: the fixed and supplementary contributions.Finally, a numerical example is further used to analyze our results. The relationship between the expected minimum guarantee process, contribution process and wealth process would be built. The constant of proportionality between the two discussed process, was found to have e?ect on the investment strategies that an investor has to purse. The e?ects of the constant relative risk aversion parameter plays an important role in choosing the asset of investment.3. The research extends to the study of an optimal strategy to be e?ected by the pension fund manager, before and after a client’s retirement. The benefits are to be paid as annuities in a defined contribution pension plan. Here, the manager has two assets to trade upon: the stock price and the cash account. A Vasicek interest rate is considered. The noble idea is to find which asset, when and how much the manager should invest under the more general new proposed Power Law Utility Function. This showed that the pension fund manager may purse with the same investment strategy before and after retirement, though more liabilities were shown to exits after retirement.4. Lastly, a brief background and literature review on game theory is given. As part of the work, a nonlinear stochastic di?erential game of two persons subjected to noisy measurements is studied. The controls or players decisions are drawn randomly. The existence of a solution to such a nonlinear stochastic di?erential game is investigated with the help of logarithmic transformation. Finally, the iterative optimal control path estimates for our minimization maximization problem are attained. The solution to such a nonlinear stochastic di?erential game was found to exist though provided certain assumptions are taken into consideration.