Connection between the Adjoint Variables and Value Function for Controlled Fully Coupled FBSDEs: The Local Case

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].

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General fully coupled FBSDES involving the value function and related nonlocal HJB equations combined with algebraic equations

Stochastics and Dynamics ◽

10.1142/s0219493721500325 ◽

2020 ◽

pp. 2150032

Author(s):

Tao Hao ◽

Qingfeng Zhu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Value Function ◽

Dynamic Programming Principle ◽

Algebraic Equations ◽

Hjb Equations ◽

Bellman Equations ◽

Hamilton Jacobi Bellman ◽

Fully Coupled ◽

The Value Function

Recently, Hao and Li [Fully coupled forward-backward SDEs involving the value function. Nonlocal Hamilton–Jacobi–Bellman equations, ESAIM: Control Optim, Calc. Var. 22(2016) 519–538] studied a new kind of forward-backward stochastic differential equations (FBSDEs), namely the fully coupled FBSDEs involving the value function in the case where the diffusion coefficient [Formula: see text] in forward stochastic differential equations depends on control, but does not depend on [Formula: see text]. In our paper, we generalize their work to the case where [Formula: see text] depends on both control and [Formula: see text], which is called the general fully coupled FBSDEs involving the value function. The existence and uniqueness theorem of this kind of equations under suitable assumptions is proved. After obtaining the dynamic programming principle for the value function [Formula: see text], we prove that the value function [Formula: see text] is the minimum viscosity solution of the related nonlocal Hamilton–Jacobi–Bellman equation combined with an algebraic equation.

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