一类非Lipschitz条件的BSDE解的存在唯一性

Existence and Uniqueness of Solutions on a Class of BSDEs with Non-Lipschitz Coefficient

本文讨论了倒向随机微分方程在f(t,y,z)满足:(?)N>0,(?)CN0,LN>0,使得对任意y1,y2∈Rn,z1,z2∈Rn×d 当0≤|y1|,|y2|,|z1|,|z2|≤N时,有 |f,(s,y1,z1)-f(s,y2,z2)|2≤CNK(t,|y1-y2|2)+LN|z1-z2|2 的非Lipschitz条件时解的存在性和唯一性。2003年,王赢、王向荣证明了一类倒向随机微分方程解的存在唯一性,我们使用函数逼近法,得到一列满足王赢,王向荣文中条件的倒向随机微分方程,因而每个方程均有唯一解,然后通过取极限的方法证明我们所讨论的方程有唯一解(Y Z),从而推广了他们的结果。

We study the existence and uniqueness of solutions on the following BSDEs： Yt=ξ＋1∫tf（s,Ys,Zs）ds-1∫tZsdWs where f（t,y,z） satisfies non-Lipschitz condition： νN〉0,ЭCN〉0,LN〉0,when yl,y2 ∈R^n,z1,z2∈R^n×d and 0≤｜y1｜,｜y2｜,｜z1｜,｜z2｜≤N, ｜f（s,y1,z1）-f（s,y2,z2）｜^2≤CNK（t,｜y1-y2｜^2）＋LN｜z1-z2｜^2 In 2003, Wang Ying and Wang Xiangrong proved existence and uniqueness of the locally and globally adapted solutions on a class of BSDEs. We extend their results. Firstly, we construct a serial of BSDEs satisfying those conditions in their paper, hence each equation has a unique solution. Finally, we prove that our equation has a unique solution by taking a limit on the equations serial.