This article is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super‐ and sub‐jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.

中文翻译：

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黏性溶液框架下具有混合时滞的随机递归最优控制问题

本文涉及混合时滞的随机递推最优控制问题。讨论了庞特里亚金最大原理和贝尔曼动态规划原理之间的联系。在不包含值函数的任何导数的情况下，通过使用在定义粘度解中引入的超级射流和子射流的概念来研究伴随过程与值函数之间的关系。还给出了随机验证定理，以验证给定的可允许控制是否真的是最优的。