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

Abstract

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 56 (2018) 4309–4335] and the dynamic programming principle (DPP) from M. Hu, S. Ji and X. Xue [SIAM J. Control Optim. 57 (2019) 3911–3938] 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. 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-order and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662] and Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259], and obtain the relationship between the MP from Z. Wu [Syst. Sci. Math. Sci. 11 (1998) 249–259] and the DPP from J. Li and Q. Wei [SIAM J. Control Optim. 52 (2014) 1622–1662].