In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions’ sense. We also provide a sufficient condition for the optimality of the control.

中文翻译：

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广义平均场时滞控制问题的随机极大值原理

在本文中，我们首先推导了一类广义平均场时滞随机微分方程和平均场预期后向随机微分方程（MFABSDE）解的存在唯一性定理。然后我们研究了广义平均场时滞控制问题的随机极大值原理。由于状态方程是分布相关的，我们将伴随方程定义为 MFABSDE，其中系数的所有导数均符合 Lions 意义。我们还为控制的最优性提供了充分条件。