Some Researches On Backward Stochastic Differential Equations

Backward stochastic differential equations?BSDE for short?has become an important field of probability theory,it has many important applications in fi-nance,stochastic control and partial differential equations,etc.This dissertation studies BSDE theory in the following three aspects.Firstly,we establishes the representation theorems for F-expectations and F-evaluations.In 2000,Coquet et al.^[79]proved that any F-expectations can be represented by g-expectations under a Lipschitz type control condition.This means that in some way,any asset pricing and risk measures can be carried out by solving BSDEs.Furthermore,an interesting problem was given by Peng^[82]:are the notions of g-expectations and g-evaluations general enough to representall�enough regular�F-expectations and F-evaluations?Our paper further answers Peng's problem.We introduce an uniformly continu-ous control condition,under which we show any F-expectations and F-evaluations are g-expectations and g-evaluations,respectively.Specifically speaking,our rep-resentation theorem shows that under uniformly continuous control condition,any F-expectations and F-evaluations are the solutions of BSDEs whose generators satisfy an uniformly continuous condition.Our study has to deal with the diffi-culties rising from the lack of Lipschitz continuous.Boundedness and localization play important roles in our proof.Secondly,we obtain the representation theorems of generators for quadratic BSDEs?generators have a quadratic growth in z?and local representation theo-rems of generators for reflected BSDEs?RBSDE for short?.Representation the-orem of generator plays a important role in the study of the properties of BSDE and nonlinear expectation.We obtain these theorems for quadratic BSDEs in two cases,one is that the generator has a linear growth in y,the other is that the generator is monotonic and has a convex growth in y.Using the representation theorem,we obtain some properties of quadratic g-expectations,which generalize some results in[66].In order to study the properties of RBSDE,we establish a local representation theorem of generator for RBSDE.Different from the BSDE case,this theorem contains the solution K of RBSDE,and it is established in a local space.Using the representation theorem,we obtain a generalized converse comparison theorem of RSBDE.Finally,we introduces a new kind of BSDE,called functional BSDE.Such B-SDE can be considered an extension of anticipated BSDE introduced by Peng and Yang^[34]and delayed BSDE introduced by Delong and Imkeller^[41].In more details,the generator g?t,�,�?of functional BSDE depends on the solution on[t-l,t+u],where l?0 and u?0 are the time delayed parameter and the time advanced parameter,respectively.For a sufficiently small time delay l or a sufficiently small Lipschitz constant,the existence and uniqueness of solution of such BSDE are ob-tained.Our result is independent of the terminal time T,which is different from the corresponding result in Delong and Imkeller^[41].As an adjoint equation of func-tional BSDE,we introduce a type of functional SDE.Such SDE can be considered an extension of classic functional SDE?see[5]?and advanced SDE introduced by Chen and Huang^[95].The coefficients b?t,�?and??t,�?of such SDE depend on the solution on[t-l,t+u],where l?0 and u?0 are the time delayed parameter and the time advanced parameter,respectively.For a sufficiently small time advance u or a sufficiently small Lipschitz constant,the existence and uniqueness of solution of such SDE are proved.We also obtain comparison theorems,continuous depen-dence properties for such BSDEs and SDEs,and a duality between such BSDEs and SDEs.