Extended Dynamic Programming Principle and Applications to Time-Inconsistent Control

Since Peng (1993) established a local maximum principle for a general stochastic control problem governed by forward-backward stochastic differential equations (FBSDEs), the corresponding partial differential equation (PDE) characterization has not been developed yet. The main difficulty stems from the potential time inconsistency inherent in this class of control problems. In a dimension-augmented space, we first establish an extended dynamic programming principle (DPP). Consequently, an extended Hamilton-Jacobi-Bellman (HJB) equation is derived. The existence and uniqueness of a new type of viscosity solution is also investigated for this extended HJB equation. Compared to extant research on the stochastic maximum principle, the present paper is the first normal work on the PDE method for a control system with states evolving in both forward and backward manners. Interestingly, our extended DPP provides an equilibrium solution for general time-inconsistent control problems associated with the traditional mean-variance model, risk-sensitive control and utility optimization for narrow framing investors, among others.