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In this paper, we investigate two families of fully coupled linearForward-Backward Stochastic Differential Equations (FBSDE). Within thesefamilies, one could get the same well-posedness of FBSDEs with totallydifferent structures. The first family of FBSDEs are proved to be equivalentwith respect to the Unified Approach. Thus one could get the well-posedness ofthe whole family if one member exists a unique solution. Another equivalentfamily of FBSDEs are investigated by introducing a linear transformationmethod. By reason of the fully coupling structure between the forward andbackward equations, it leads to a highly interdependence in solutions. We areable to lower the coupling of FBSDEs, by virtue of the idea of transformation,without losing the well-posedness. Moreover, owing to the non-degeneracy of thetransformation matrix, the solution to original FBSDE is totally determined bysolutions of FBSDE after transformation. In addition, an example of optimalLinear Quadratic (LQ) problem is presented to illustrate.

Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations