Multi-dimensional Reflected FBSDEs With Quasi-monotonously Increasing Continuous Coefficients

In this paper,we study the existence of the multi-dimensional reflected FBSDEs with quasi-monotonously increasing continuous coefficients through existence and uniqueness result,comparison theorem of multi-dimensional re-flected BSDEs and four step scheme.In 1990,Pardoux and Peng[2]firstly introduced the nonlinear backward stochastic differential equations(BSDEs in short)as following and gave the existence and uniqueness of BSDEs:Over the past 20 years,BSDEs used as the tool of control system,engineering science,stochastic control and finance mathematics becoming more and more famous.In 1993,Antonelli[21]firstly introduced the FBSDE when he re-searched the control theory and gave the existence and uniqueness result of FBSDE.In 1994,Ma,Protter and Yong[22]using the study of partial differen-tial equation(PDE in short)system to give the corresponding result rely on a four step scheme,this method combined the stochastic control theory and the theory of the PDE very well.In 1997,El-Karoui et al.[13]firstly introduced the one dimensional reflected BSDEs and they draw the conclusion that under the Lipschitz condition the solution of the question was exist and unique and they gave the comparison theorem.In 2010,Huang,Lepeltier and Wu[27]made extension from the work of Antonelli and Hamadene[25],they added the obstacles process to constraint and derived the existence of one-dimensional reflected FBSDEs with continuous monotone coefficients.In 2010,Wu and Xiao[26]studied the multi-dimensional reflected BSDEs,proved the existence and uniqueness and the comparison theorem of the solution.In 2012,El.Asri[28]studied a system of multi-dimensional reflected FBSDEs and provided an application to optimal switching problem.In 2013,Aazizi and Fakhouri[29]studied the solvability of a class of multi-dimensional FBSDEs with oblique reflection and unbounded stopping time.We divide our paper into four chapters.In chapter 1,we introduce the some research of SDEs,BSDEs,reflected BSDEs,FBSDEs,reflected FBSDEs,put forward our problem formula and framework.In chapter 2,on the basis of the one-dimensional reflected FBSDEs which given from Huang,Lepeltier and Wu[27],we first give the following model for multi-dimensional reflected FBSDEs and we give some preliminary.In the aspect of preparation,we give the comparison theorem of multi-dimensional SDE,BSDEs and reflected BSDEs and some approximate knowledge.In chapter 3,we give the proof of the existence of multi-dimensional reflected FBSDEs with quasi-monotonously increasing continuous coefficients.We suppose that the coefficients and parameters in(1)satisfy the following assumptions.(i)b is monotonously increasing in y and is quasi-monotonously increasing in x;(?)f is monotonously increasing in x and is quasi-monotonously increasing in y;f_j only contains z_j,f_j is independent of every z_l,l?j;(?)There exists a constant C>0 such that In order to get the main results,we first define the following iteration sequences associated with(2.1)According to Ma,Protter and Yong[22],we will divide our proof into four steps,we draw our conclusion through some iteration algorithms and approx-imation techniques.In chapter 4,we summarize our research results and made a prospect for further research.