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In this paper, we consider a system of forward-backward stochasticdifferential equations (FBSDEs) with monotone functionals. We show theexistence and uniqueness of such a system by the method of continuationsimilarly to Peng and Wu (1999) for classical FBSDEs and obtain estimates underconditional probability. As applications, we prove the well-posedness resultfor a mean field FBSDE with conditional law and show the existence of adecoupling function. In addition, we show that mean field games with commonnoise are uniquely solvable under a linear controlled process with convex andweak-monotone cost functions and prove that the optimal control is in afeedback form depending only on the current state and conditional law.

Forward-backward stochastic differential equations with monotone functionals and mean field games with common noise