Existence And Uniqueness Of Solution For The (Anticipated) BSDEs Driven By Brown Motion And Time-Changed Lévy Noises

Since Pardoux-Peng(1990)first proposed nonlinear form of backward stochastic differential equation theory,which has aroused the interest of many scholars at home and abroad.BSDE theory has important applications in many fields,such as stochastic analysis,partial differential equation,random control,financial mathematics,etc.At present,BSDE theory has become one of the hot research fields in the field of stochastic analysis and probability theory.This paper makes a thorough and systematic discussion on the two aspects of the BSDE theory research,and has made some progress.On the one hand,the anticipated BSDE is derived from BSDE.The main problem it solves is how to make today's decisions when faced with uncertain goals for a period of time.On the other hand,it is a BSDE driven by time-changed Lévy noise.The structure of this noise is related to the Brown motion of time-changed and the Poisson random measure of time-changed.The first aspect of this paper study the existence,uniqueness and stability of solu-tion to the anticipated BSDE.This part includes chapter 2 and chapter 3.The detailed description are as follows:in Chapter 2,first of all,when the generator g satisfies p-order weak monotonicity condition in y and Lipschitz continuous condition in(z,?,v),we prove the stability theorems of Lp(1<p? 2)solutions for anticipated BSDEs(see Theorem 2.1-2.2).Then,if g further satisfies general growth condition and the con-tinuity condition in y,we establish the existence and uniqueness of L2 solutions for anticipated BSDE,which generalizes the corresponding results in Peng-Yang(2009)and Hu-Chen(2016).Finally,with the help of existence and uniqueness of L2 solu-tions and stability results,we also get the existence and uniqueness of Lp(1<p<2)solutions for anticipated BSDE.In chapter 3,first of all,we study the uniqueness of L1 solution for anticipated BSDE(see Theorem 3.1),when g satisfies a one-sided Osgood and general growth conditions in y,and Lipschitz continuous and the sublinear growth condition in z and anticipated term of y,z.Moreover,if ? belongs to[1/2,1)in sublin-ear growth condition of assumption(3H3),the one-sided Osgood condition is replaced by the stronger p(p>1)-order one-sided Constantin's condition in assumption(3H1),we obtain the existence of L1 solutions for anticipated BSDE(see Theorem 3.2)and then get the existence and uniqueness of L1 solution for anticipated BSDE under mono-tonicity condition in this chapter,and furthermore establish the solution for anticipated BSDE in the space of SF1(0,T + K;Rn)x MF1(0,T + K;Rn×d).The second aspect of this paper include the existence and uniqueness of solution and generator representation theorem etc.to BSDE driven by time-changed Lévy noise.This part includes chapter 4 and chapter 5.The details of the chapters are as follows:Chapter 4 contains three parts.Firstly,when the generator g satisfies non-Lipschitz con-tinuous condition,we establish the existence and uniqueness of L2 solution for BSDE driven by time-changed Lévy noise(see Theorem 4.1).The results improve and include the corresponding results in Di Nunno-Sjursen(2014).Secondly,we prove the exis-tence and uniqueness of Lp(p>2)solutions for BSDE driven by time-changed Lévy noise(see Theorem 4.2).Finally,we study the generator representation theorem and converse comparison theorem for this kind of BSDE(see Theorem 4.3-4.4).In chapter 5,when the generator g satisfies the monotonicity,continuity and general growth con-ditions in y,we establish the existence and uniqueness of solutions for BSDE driven by time-changed Lévy noise(see Theorem 5.1),which also generalize the corresponding results in Di Nunno-Sjursen(2014).In addition,this chapter first introduce and obtain the stability result for this kind of BSDE(see Theorem 5.2).