Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic control systems

TL;DR

This paper explores the complex relationship between the stochastic maximum principle and dynamic programming principle for fully coupled forward-backward stochastic control systems, introducing new techniques and establishing connections under various conditions.

Contribution

It develops a new decoupling technique to analyze the relationship between MP and DPP in fully coupled systems with nonconvex control domains.

Findings

Established the relationship between MP and DPP for fully coupled FBSCS.

Discovered connections between derivatives of algebra equation solutions and adjoint equations.

Extended results to the local case under monotonicity conditions.

Abstract

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) in [9] and the dynamic programming principle (DPP) in [10] for a fully coupled forward-backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman (HJB) equation combine an algebra equation respectively. So this relationship becomes more complicated and almost no work involves this issue. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as in [14,27] and obtain the relationship between the MP in [27] and the DPP in [14].