We consider fully coupled forward–backward stochastic differential equations (FBSDEs), where all function parameters are Lipschitz continuous, the terminal condition is monotone and the diffusion coefficient of the forward part depends monotonically on z, the control process component of the backward part. We show that there exists a class of linear transformations turning the FBSDE into an auxiliary FBSDE for which the Lipschitz constant of the forward diffusion coefficient w.r.t. z is smaller than the inverse of the Lipschitz constant of the terminal condition w.r.t. the forward component x. The latter condition allows to verify existence of a global solution by analysing the spatial derivative of the decoupling field. This is useful since by applying the inverse linear transformation to a solution of the auxiliary FBSDE we obtain a solution to the original one. We illustrate with several examples how linear transformations, combined with an analysis of the decoupling field's gradient, can be used for proving global solvability of FBSDEs.

我们考虑完全耦合的前向-后向随机微分方程 (FBSDE)，其中所有函数参数都是 Lipschitz 连续的，终端条件是单调的，前向部分的扩散系数单调依赖于z，后向部分的控制过程分量。我们表明存在一类将 FBSDE 转换为辅助 FBSDE 的线性变换，其中正向扩散系数 wrt  z的 Lipschitz 常数小于正向分量x的终端条件的 Lipschitz 常数的倒数. 后一个条件允许通过分析去耦场的空间导数来验证全局解的存在。这很有用，因为通过将逆线性变换应用于辅助 FBSDE 的解，我们可以获得原始解的解。我们用几个例子来说明线性变换是如何结合去耦场梯度的分析来证明 FBSDE 的全局可解性的。