The Solution Of Fully-coupled Reflected Forward-backward Stochastic Differential Equations

In this paper, we study the solution of a kind of fully coupled reflected forward-backward stochastic differential equations (RFBSDE in short), includ-ing the existence and the uniqueness of the solutions and their connection with quasilinear parabolic partial differential equations (PDE in short).The following equation is so-called backward stochastic differential equations (BSDE in short). Par-doux and Peng[38] firstly introduced BSDE in 1990, and they proved the ex-istence and the uniqueness of the solution when the coefficient f is Lipschitz with respect to y and z, and the terminal condition is square integrable. BSDE theory develops very quickly because of its many applications, for example, in the mathematic finance, stochastic optimal control and PDE.In 1997, E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez firstly in-troduced the reflected backward stochastic differential equations (RBSDE in short). They introduced an increasing process to push the solution Y upward to stay above a given continuous stochastic process, which is called the obstacle with "minimal energy". This kind of equations is following They proved the existence and the uniqueness of the solution and the related comparison theorem. They also proved that the solution of RBSDE can be represented as a unique viscosity solution associated PDE with an obstacle within Markovian framework.On the other hand, there are many works about fully-coupled forward-backward stochastic differential equations (FBSDE in short). In 1993, An-tonelli[35] firstly investigated the fully coupled FBSDE and proved that there exists only one solution under Lipschitz condition when the time duration is sufficient small. When the coefficients involved are deterministic and the dif-fusion coefficient (?) of forward SDE is independent of z. in 1994, Ma and Yong introduced "four steps scheme" to obtain the of solvability of FBSDE. With-out the assumption that the drift coefficient of forward SDE is non-degenerate, in 1995, Hu and Peng proved the existence and the uniqueness of the solution of fully-coupled FBSDE under the Lipschitz condition and the monotonicity assumption while the solutions X and Y must have the same dimension. In 1999, Wu and Peng proved the existence and the uniqueness of the solution of fully coupled FBSDE when the solutions X and Y are not in the same dimensional under the weak monotone assumption. In 1998, Antonelli and Hamadene studied the existence of the solution for one kind of fully coupled FBSDE in the following form They constructed two increasing sequences to approximate respectively the so-lutions of SDE and BSDE. In 2008, using the same method, Huang[20] studied the existence of the solution of the following fully coupled RFBSDE under the condition that the coefficients are continuous. For the uniqueness of the solution, the problem is more difficult. In this paper, we investigated a special case and proved the uniqueness of the solution. In order to find the relation between the RFBSDE and the obstacle problem for PDE, we consider the obstacle Lt:= h(t, Xt) i.e.the obstacle Lt:= h(t, Xt). To be more precise, we consider the following fully-coupled reflected FBSDEWe construct two increasing sequences to approximate respectively the solutions of SDE and RBSDE to study the existence of the solution of the fully coupled RFBSDE. Assume that h.ψsatisfies:(i) h.ψare linear growing and increasing functions with respect to x;(ii)|h(s,X1)－h(X2)|< M|x1-x2|,for all s∈[O,T],ω∈Ω, and x1,x2∈R:Then, we give the condition:for (?)t∈[O,T],x1, x2, y1, y2, Z1, z2∈R, (B1) (?)C1> O,s.t.We study the uniqueness theory and the related comparison theorem for one class of RFBSDE as following: At last, using reduction to absurdity, we prove that the solution Y(?) is the viscosity solution u(t,x) for a kind of PDEThis paper is divided into five chapters.The first chapter is an introduction.In chapter 2, we will give the preliminary knowledge about SDE, BSDE, RBSDE, which will be used in the next chapters.Chapter 3 is devoted to prove the existence and the uniqueness of the solu-tion of fully-coupled RFBSDE. We divide it into three parts. In the first part, we prove the existence theorem similar to Antonelli, Hamadene and Huang. In the second part, we prove the uniqueness of the solution for one kind of RFBSDE. In the last part, we prove the related comparison theorem.In Chapter 4, we prove the solution of RFBSDE is the viscosity solution of associated PDE with an obstacle.Chapter 5 is about the further research.