The Maximum Principle For Stochastic Control Problems With Jumps In Progressive Structure

In this thesis,we introduce a new structure named progressive structure to deal with stochastic control problems with random jumps.It can be summarized into five chapters as follows.In Chapter 1,we give some backgrounds on stochastic integrals of progressive processes with respect to martingales,backward stochastic differential equations and the Maximum Principles.In Chapter 2,we define the stochastic integrals of progressive processes with respect to Poisson random measure by extending the theories of stochastic integrals of progressive processes with respect to martingales.Furthermore,we also give some useful properties of the new integral.Then the related theories of stochastic differential equations and backward stochastic differential equations are given subsequently.In Chapter 3,first an example is given to show the difference between our new model and the classical one.Then we obtain the Maximum Principles of a stochastic control problem in which the control domain is convex.The Maximum Principle contains two parts;one is called the continuous part and the other is called the discontinuous part.In Chapter 4,a general Maximum Principle is given.The control domain is not necessarily convex compared with the system in Chapter 3,and the control is allowed to enter the diffusion term.We fixed the deficiency in the work of Tang and Li by introducing progressive structure and a new form of variation.In Chapter 5,we consider the stochastic control system driven by Markov Chains.First we give some preliminaries of Markov Chains.Then we obtain the Maximum Principle by following the way of Chapter 4.In Chapter 6,we consider a stochastic control system with singular control.The control domain of the singular control is unnecessarily convex.Similar to Chapter 3,there are two parts of the Maximum principle.One part is a necessary condition that the absolutely continuous control satisfies;the other is a necessary condition that the singular control satisfies.