This paper focuses on the study of forward–backward stochastic evolution equations (FBSEEs), which are a class of nonlinear fully coupled forward–backward stochastic differential equations (FBSDEs), in infinite dimensions. Drawing inspiration from various linear-quadratic (LQ) optimal control problems, we apply a set of domination-monotonicity conditions that are more relaxed compared to general conditions. Within this framework, we employ the method of continuation to establish the unique solvability result and provide a pair of solution estimates. Notably, to address the challenges posed by the infinite-dimensional setting, we introduce a class of approximating equations, as Itô’s formula is not directly applicable. Conversely, these results find application in various LQ problems, where the stochastic Hamiltonian systems precisely correspond to the FBSEEs satisfying the aforementioned domination-monotonicity conditions. Consequently, by solving the corresponding stochastic Hamiltonian systems, we can obtain explicit expressions for the optimal controls.

中文翻译：

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无限维前向-后向随机演化方程及其在LQ最优控制问题中的应用

本文重点研究前向-后向随机演化方程（FBSEE），它是一类无限维非线性全耦合前向-后向随机微分方程（FBSDE）。从各种线性二次（LQ）最优控制问题中汲取灵感，我们应用了一组比一般条件更宽松的支配单调性条件。在此框架内，我们采用连续方法来建立唯一的可解性结果并提供一对解估计。值得注意的是，为了解决无限维设置带来的挑战，我们引入了一类近似方程，因为伊藤公式不能直接适用。相反，这些结果适用于各种 LQ 问题，其中随机哈密顿系统精确对应于满足上述支配单调性条件的 FBSEE。因此，通过求解相应的随机哈密顿系统，我们可以获得最优控制的显式表达式。