Solutions And Properties Of Multidimensional Mean-Field Backward Stochastic Differential Equations

This paper studies mainly the following mean-field backward stochastic differ-ential equations (MFBSDEs for short) whose generator f satisfies two types of non-Lipschitz assumptions, respectively: ζ∈L2 (FT, Rk).Assumption 1:f is assumed to satisfy the following conditions:(H1){f(t,0,0,0,0)}t∈[0,T] satisfies the necessary square integrability;(H2) for all t∈[0,T], y1’,y2’,y1,y2∈Rk,z1’,z2’,z1,z2∈Rk×d, it holds that where C is a positive constant and κ:R+â†'R+ is a continuous nondecreasing concave function such thatWe prove the existence and the uniqueness of solutions for this type of MFBS-DEs (1) under Assumption 1 by using the iteration and the Bihari inequality. Assumption 2:f satisfies (H1) and(H3) a weak monotonicity condition in both y and y’;(H4) yâ†'f is continuous and y’â†'f is continuous;(H5) a general growth condition in both y and y’;(H6) f is Lipschitz continuous in both z and z’.We establish the existence and the uniqueness of solutions for this type of MFBSDEs (1) under Assumption 2 with the help of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Finally, we prove comparison theorems and multidimensional comparison the-orems for MFBSDEs.