Theories Of Forward-Backward Stochastic Difference Equations And Related Optimization Problems

This thesis studies the solvability problem of the forward-backward stochastic differ-ence equations(FBS AEs)and the related optimization problems.The forward-backward stochastic difference equations are the discrete time analog of the forward-backward s-tochastic differential equations(FBSDE).Started from the 1990s,the study on the F-BSDEs has achieved lots of research results.It can be seen that the theory of FBSDE has a wide range of applications In the area of optimal control,financial and economics.However,there are few results on the theory of FBS?Es,and most of work is in the nu-merical solutions problems,in which the research on the difference equations is mainly about the approximation of the FBSDE.In this paper,the study of the FBSAEs and the related optimization problems are in their own right,not just as the approximation to the continuous time case.In fact,the stochastic difference equations and the discrete time models also have broad wide applications.For example,the digital technology gets the sample data on discrete time,so the models related to this problem are necessarily discrete time models.As the development of the digital technology,the value of the study on discrete models becomes more and more important.Our thesis mainly contains four parts.In the first part we study the theories of backward stochastic difference equations(BSAEs).We propose the framework of the problems and provide some basic results,and these results can be used to proof the theories in the following chapters.In the second part we research the existence and uniqueness results of the solutions to fully coupled linear FBS?Es.We obtain the sufficient and necessary condition to the solvability problems of the linear FBS?Es,and provide two corollaries,which will be used in the proof of the nonlinear case in the next part.In the third part we study the solvability problems of fully coupled nonlinear FBSAE.We obtain the necessary and sufficient conditions for the solvability of FBS?Es under different formulations,which is the theoretical basis of part four.Besides,the product rule for the FBSAEs we provide here will also be used in the following part.At last,we discuss the optimal control problems with the systems described by partially coupled and fully coupled FBSAE,and obtain the Maximum Principle to the related problems.The main work of this thesis is detailed as follows:First,we begin our work by the study of the backward stochastic difference equations(BS?Es).In FBS?Es,the backward part is the more important part.Based on the form of the martingale representation theorem in discrete time,the research on BS?Es is usually under two probability space frameworks.One is the finite state probability space which is generated by a stochastic process.taking values from the basis vectors in Rd space.The martingale process generated by the stochastic process is used as the driving process of the equation.The other one is the general probability space generated by an independent increment martingale process,which is used to drive the stochastic difference equations.In addition,based on the specific form of the generator,BS?Es can also be divided into two types.In one case,the generator at time t depends on the solution at time t,what we call the implicitly depending generator.In the other case,the generator at time t depends on the solution at time t + 1,what we call the explicitly depending generator.The equations with two generators have different meanings and cannot be transformed to each other.Here we study the related theories of BS?E and FBS?Es systematically under the framework of two types of probability spaces and two types of generators.Note that the results of the martingale representation theorem in the finite state probability space are not unique under the common sense,and the uniqueness holds only under an equivalence relation.The equivalence relation makes the formulation of the stochastic difference equations and the definition of the norm for the variables more complicated in the finite state probability space.We discuss this problem in the second chapter,in this chapter,our main work is to propose the explicit characterization of the equivalence relation which is independent of the probability structure in the finite state probability space,and construct the equivalence class based on the result,and then we define the norm on the class.Next we show the relation between this norm and the semi-norm which is defined by the martingale process.In addition,we give the explicit expression of the martingale representation theorem.Finally,we present the existence and uniqueness theory for the solutions to several types of BSAEs.Related content can be found in the second chapter of the paper.Secondly,we study the solvability theory of the fully coupled linear FBSAE,which is a special class of the general FBS ?E.Under the finite state probability space framework,we investigate the random coefficients FBSAE with implicitly depending generator and explicitly depending generator separately.Under the general state probability space,we investigate the deterministic homogeneous coefficients FBSAE with implicitly depending generator and explicitly depending generator separately.Our main work is to transform the linear FBSAE solvability problem into the linear algebraic equation solvability prob-lem and give the necessary and sufficient condition for solvability of the linear FBSAE.It should be pointed out that compared with the sufficient condition which is the solv-ability of the Riccati equation obtained in the continuous time case,the necessary and sufficient condition here in the discrete time case is easier to verify.Related content can be found in the third chapter of the paper.Next,we study the solvability theory of the fully coupled nonlinear FBSAE.Under the finite state probability space framework,we investigate the one dimensional FBSAE with explicitly depending generator and multidimensional FBSAE with implicitly de-pending generator separately.Under the general state probability space,we investigate the explicitly depending generator FBSAE in which the forward solution and backward solution are in the same dimension and implicitly depending generator FBSAE with multidimensional solutions separately.Our main work is to provide the discrete time product rule for the FBSAE with explicit depending generator,which,to some extent,works like the Ito formula in the differential equations.Then through this technique,combined with the monotone condition,we obtain the existence and uniqueness theo-rems for the solutions to the FBS?Es with explicit depending generator under the finite state probability space framework and the general state probability space framework.Besides,for the FBS?Es with implicit depending generator under the finite state prob-ability space framework and the general state probability space framework,through the introduction of the A-norm and the estimation of the difference equations,we obtain the solvability results by the contraction mapping method.Related content can be found in the fourth chapter of the paper.At last,we study the FBSAE optimal control problem.Under the finite state probability space framework,we study the multidimensional partially coupled FBSAE optimal control problem and one dimensional fully coupled FBSAE optimal control problem.Under the general state probability space framework,we study the multidi-mensional partially coupled FBSAE optimal control problem and fully coupled FBS?E optimal control problem with the same dimension forward and backward variables.In this part,our main work includes the estimating of the solution to the variational equa-tion in the fully coupled case by using the FBSAE product rule technique,constructing the formulation of the adjoint equations and the Hamiltonian system by product rule in both the partially and fully coupled case,and finally deriving the Maximum Principle for the optimal control problems.Related content can be found in the fifth chapter of the paper.