Partially Observed Stochastic Control Systems And Their Applications

In this paper, we mainly study partially observed stochastic control systems and their applications to linear quadratic optimal control, differential game and optimal portfolio choice problems. This paper consists of five chapters.Filtering theory is very important for us to get explicit solutions for partially observed stochastic control problems. Therefore, in chapter 1, we first give a brief review of filtering theory and introduce an important lemma. We also introduce the main results obtained in our paper.The objective of Chapter 2 is to deal with the Kalman-Bucy filtering problem of forward and backward stochastic system which arising from one kind of classical stochastic optimal control problem. We also study the stability of filtering equation corresponding to this kind of forward and backward stochastic system. As an application of filtering theory, we study one kind of partially observed linear quadratic recursive optimal control problem. Combining the separation principle with a direct construction method, we get the optimal control which is the linear feedback of the state filtering estimation. At last, we give the value of the information for the partially observed recursive optimal control problem.In Chapter 3, we concern one kind of partially observed recursive optimal control problem. Combining Girsanov's theory with the classical method used in fully observed optimal control problems, we derive the maximum principle for this kind of partially observed recursive optimal control problem. Under the framework of state constraints, we also get the maximum principle for this kind of control problem by the similar method used in this chapter. As an application of the maximum principle, one kind of partially observed linear quadratic recursive optimal control problem is also studied.Risk-sensitive optimal control problem can be used to describe the risk-attitude of one investor, therefore it is very useful to solve some financial and economic problems. In chapter 4, we discuss one kind of partially observed risk-sensitive optimal control problem. By the similar method used in chapter 3, we get the general stochastic maximum for this kind of partially observed risk-sensitive optimal control problem. Although the form of the maximum principle is similar to its risk-neutral counterpart, the variational inequality and the adjoint equation heavily depend on the risk-sensitive parameterγ. This is one of the main difference from the risk-neutral case. As a special case of partial information, the general stochastic maximum principle for this kind of fully observed risk-sensitive optimal control problem is also obtained. At the last in this chapter, we give three interesting examples to show the applications of our risk-sensitive maximum principle. The optimal portfolio choice problem further illustrates the meaning of the word risk-sensitive.In the last chapter, one kind of partially observed linear quadratic non-zero sum differential game problem is studied. Combining the separation principle with the theory of forward and backward stochastic differential equations, we obtain the explicit observable Nash equilibrium point of this kind of game problem.