A transformation method to study the solvability of fully coupled FBSDEs

We present a new method for checking global solvability of 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 one can reduce, via a linear transformation, the FBSDE to 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 analyzing the space derivative of the decoupling field. We illustrate with several examples how the transformation method can be used for proving global solvability of FBSDEs.