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 σ in forward stochastic differential equations depends on control, but does not depend on z. In our paper, we generalize their work to the case where σ depends on both control and z, 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 W, we prove that the value function W is the minimum viscosity solution of the related nonlocal Hamilton–Jacobi–Bellman equation combined with an algebraic equation.

中文翻译：

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涉及价值函数的一般全耦合 FBSDES 和相关的非局部 HJB 方程与代数方程相结合

最近，郝和李[完全耦合的前向-后向SDEs涉及价值函数。非局部 Hamilton–Jacobi–Bellman 方程，ESAIM：控制优化，计算。变量。 22(2016) 519-538] 研究了一种新的前向后向随机微分方程 (FBSDEs)，即在扩散系数为σ在正向随机微分方程中，依赖于控制，但不依赖于z. 在我们的论文中，我们将他们的工作推广到以下情况σ取决于控制和z，称为涉及值函数的一般全耦合 FBSDE。证明了该类方程在适当假设下的存在唯一性定理。获得价值函数的动态规划原理后W，我们证明了价值函数W是相关的非局部 Hamilton–Jacobi–Bellman 方程与代数方程相结合的最小粘度解。