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

In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness FBSDEs with totally different coefficients. A family is proved be equivalent respect the Unified Approach. Thus whole once a member exists unique solution. Another are investigated by introducing transformation method. Owing coupling structure between forward backward equations, it leads highly interdependence in solutions. We able decouple into partial coupling, virtue transformation, without losing existence uniqueness Moreover, owing non-degeneracy matrix, solution original determined solutions after transformation. addition, an example LQ problem presented illustrate.