The Mean, Variance And Stochastic Differential Game Problem Is Studied

Portfolio selection is a classic and important problem in financial economics. Together with asset pricing and risk management, portfolio selection are coined as three pillars in modern finance. An early scientific contribution to portfolio selection was made by the work of Markowitz, where an elegant mathematical model for portfolio selection was introduced. There are many uncertain events, such as inflation, disaster, wars and so on, which may influence the return rate and the volatility of the asset greatly. Therefore, it is necessary to introduce the more realistic model to represent the random variation of the environmental or economical condition. It has very important significance in theory and practial to study the strategy selection problem under the more realistic asset price model.In this paper we first study the dynamic mean-variance portfolio selection problem under which the stock price follows a CEV model, a Heston stochastic variance model and a jump-diffusion risk model, respectively. By invoking the use of the stochastic linear-quadratic approach, we obtain the closed-form solutions of the efficient portfolio strategy and the efficient frontier. Then we study the stochastic differential portfolio games between the insurance company and the market in a jump-diffusion model, and study the zero-sum stochastic differential games problem in a Vasicek stochastic interest rate model. By using the linear-quadratic method, we obtain the closed-form soultions of the optimal portfolio strategy, the optimal market strategy and the optimal value function. Finally, we study the Robust utility maximization portfolio selection problem under which the stock price followed a CEV model. We also obtain the closed-form solutions of the optimal portfolio strategy, the optimal equivalent probability scenario and the optimal value function using the linear-quadratic technique.