Existence and Uniqueness of the Nonlinear BSDEs with a Small Parameter under Locally Lipschitz Condition

In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.