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

arXiv: https://arxiv.org/abs/1803.06804 · HTML: https://mlsanity.com/paper/1803.06804/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Mar 2018.

## Built on by (most cited first, 1 of 1)

- [A deep learning method for solving stochastic optimal control problems driven by fully-coupled FBSDEs](https://mlsanity.com/paper/2204.05796/) (2022)

## Abstract (arXiv)

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].