Stochastic verification theorem of forward–backward controlled systems for viscosity solutions

In this paper, we investigate the controlled systems described by forward–backward stochastic differential equations with the control contained in drift, diffusion and generator of BSDEs. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. It is worth pointing out that this theorem has wider applicability than the restrictive classical verification theorems. As a relevant problem, the optimal stochastic feedback controls for forward–backward systems are discussed as well.