A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEs

Bismut [1] first introduced linear backward stochastic differential equations (BSDEs in short) as the adjoint equation of the classical stochastic optimal control problem. In 1990, Pardoux and Peng firstly proved the existence and uniqueness of nonlinear BSDEs with Lipschitz condition [2]. Since then, the theory of BSDEs has been studied by many researchers and applied in a wide range of areas, such as in stochastic optimal control and mathematical finance [3, 4]. When a BSDE is coupled with a (forward) stochastic differential equation (SDE in short), the system is usually called a forward-backward stochastic differential equation (FBSDE in short). We can refer to the literatures in [5, 6, 7, 8, 9, 10] which studied the existence, uniqueness and the applications of coupled or fully-coupled FBSDEs.