AI 中文总结
本文针对带律延迟的离散时间McKean-Vlasov最优控制问题，研究全耦合前向-后向随机差分方程，在合适单调性条件下证明其解的存在唯一性，引入新条件处理对应最优控制问题，并推导了带律延迟线性二次系统的唯一最优控制。
AI 中文摘要
受带律延迟的离散时间McKean-Vlasov最优控制问题的驱动，本文研究一类前向-后向随机差分方程。主要困难源于离散时间环境下缺少Ito公式，以及系数同时依赖律延迟项和律提前项的全耦合结构。在适当的单调性条件下，建立了解的存在性与唯一性。此外，引入新的单调性条件以处理相关最优控制问题。作为应用，推导了带律延迟的线性二次系统的唯一最优控制。
英文摘要
In this paper, motivated by discrete-time McKean-Vlasov optimal control problem with law-delay, a class of forward-backward stochastic difference equations is investigated. The main difficulties come from the absence of Ito's formula in the discrete-time setting, as well as the fully coupled structure in which the coefficients depend on both law-delayed and law-anticipated terms. Under suitable monotonicity conditions, the existence and uniqueness of solutions are established. Furthermore, a new monotonicity condition is introduced to address the associated optimal control problem. As an application, the unique optimal control for linear quadratic systems with law-delay is derived.
