Some Problems Of Optimal Control And The Application In Financial Mathematics

With the growing of economic activities and the development of science and technology, as an important branch of control theory, stochastic optimal control has penetrated into various fields. Grasping the basic knowledge of stochastic control theory and using the "stochastic" perspective to analyze and solve practical problems, is gradually becoming an indispensable knowledge quality of scientific workers. Financial Mathematics is a new discipline to solve financial problems by the application of stochastic optimal control and stochastic analysis.Modern Portfolio Theory has been a key concern of the economists in the world upon theoretical research. In the financial market, the traditional investment includes stocks and bonds. The investment in bounds is of both the advantage of low investment risk and a relatively low rate of return. In contrast, the investment in stocks is of high risk and high return. To reduce the risks and improve returns, investors will allocate assets in various markets, such as part of the stock rnarket and part of investment bonds. In this thesis, we study this portfolio problem by optimal control theory.The thesis is of four Chapters.We introduce some background information about optimal portfolio in Chapter 1. Also we put forward the problem we research in optimal portfolio.We convert a stochastic finance model to a confirm control problem in Chapter 3, and find the optimal portfolio process in the set of the confirm portfolio process by using the method of working viscosity solution. The static optimal portfolio process we obtain can be used to forecast the market by the information we have now. The we can operate the trade by the portfolio process in a relatively short time.We use a traditional feedback means to work out the problem in Chapter 3. We solve the optimization problem inside the HJB-Equation to obtain the stochastic optimal portfolio process and the continuous optimal value function depending on the yet unknown value function. The optimal portfolio process we obtain is stochastic, it can be changed correspond to the stochastic change of the wealth process at any time.Chapter 4 is independent of the other three chapters of the thesis. We research another optimal control problem in this Chapter. That is to say, we study a linear-quadratic optimal control problem for the dynamic systems in thc bchavioral setting. A kind problem of positive-semidefinite quadratic cost function in the system can convert to a linear-quadratic optimization. Then we can use a linear program to work out the linear-quadratic optimization, and obtain the optimal trajectory and the optimal value of the optimal control problem.