General fully coupled FBSDES involving the value function and related nonlocal HJB equations combined with algebraic equations

Abstract

Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var. 22 (2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient [Formula: see text] in forward stochastic differential equations depends on control, but does not depend on [Formula: see text]. In our paper, we generalize their work to the case where [Formula: see text] depends on both control and [Formula: see text], which is called the general fully coupled FBSDEs involving the value function. The existence and uniqueness theorem of this kind of equations under suitable assumptions is proved. After obtaining the dynamic programming principle for the value function [Formula: see text], we prove that the value function [Formula: see text] is the minimum viscosity solution of the related nonlocal Hamilton–Jacobi–Bellman equation combined with an algebraic equation.