Extended Dynamic Programming Principle and Applications to Time-Inconsistent Control

TL;DR

This paper develops an extended dynamic programming principle and HJB equation for time-inconsistent stochastic control problems governed by FBSDEs, providing a PDE approach to equilibrium solutions in complex control scenarios.

Contribution

It introduces an extended DPP and HJB equation for FBSDE-based control problems, addressing the challenge of time inconsistency with a novel PDE framework.

Findings

Established an extended DPP in augmented space.

Derived and analyzed an extended HJB equation.

Provided equilibrium solutions for various time-inconsistent models.

Abstract

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.