The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.

中文翻译：

---

具有控制和状态约束的受控随机演化方程的一阶和二阶必要最优条件

本文的目的是为具有控制和状态约束的随机演化方程的最优控制问题建立一阶和二阶必要最优性条件。控制在漂移和扩散项中都起作用，并且控制区域是可分希尔伯特空间的非空封闭子集。我们采用一些经典的集值分析工具和向量值向后随机演化方程的转置解和算子值向后随机演化方程的松弛转置解的理论来推导这些最优条件。二阶伴随方程的修正部分在一阶最优条件中没有出现，在二阶最优条件中起着基础性的作用。