L~P Solutions Of Mean-field Backward Stochastic Differential Equations

In this paper, we study the following mean-field backward stochastic differential equations (MFBSDEs for short):In2009, Buckdahn, Djehiche, Li and Peng [1] first introduced a new type of backward stochastic differential equations—mean-field backward stochastic differential equations (mean-field BSDEs). This new type of backward stochastic differential equations has been widely used. Many papers have studied mean-field BSDEs with square integrable terminal conditionand under the Lipschitz and the square integrability assumptions on the generator f.But in many cases, the generator f doesn’t satisfy the Lipschitz condition and the square integrability. Therefore, much attention has been paid to relax the Lipschitz hypothesis on the generator and find the existence and the uniqueness of Lp(1<p<1) solutions.With the help of the theory of BSDEs, in this paper we explore Lp (1<p<2) solutions for mean-field BSDEs(1). First, with Lipschitz hypothesis on the generators, we derive a priori estimates and prove the existence and the uniqueness of solutions in Lp(1<p<2),but also the comparison theorom.Then, under monotonicity condition, we obtain the existence and the uniquenes of solutions for mean-field BSDEs in Lp(1<p<2),as well as the comparison theorem. In the end, we consider to study the Lp solutions for mean-field BSDEs with continuous coefficients, we will prove the existence of the minimal Lp solution and the maximal Lp solution when f is continuous in(y’,y2z) by constructing approximation functions.