In this paper, an open problem is solved, for the stochastic optimal control problem with delay where the control domain is nonconvex and the diffusion term contains both control and its delayed term. Inspired by previous results about delayed stochastic control systems, Peng’s global stochastic maximum principle is generalized to the time delayed case. A special backward stochastic differential equation is introduced to deal with the cross terms, when applying the duality technique. Comparing with the classical result, the maximum condition contains an indicator function, which in fact is the characteristic of the stochastic optimal control problem with delay. Furthermore, to illustrate the applications of our theoretical results, three dynamic optimization problems are addressed based on the global maximum principle.

本文针对控制域为非凸且扩散项包含控制项及其延迟项的随机时滞随机最优控制问题，解决了一个开放性问题。受先前关于延迟随机控制系统的结果的启发，Peng的全局随机最大原理被推广到时滞情况。应用对偶技术时，引入了特殊的后向随机微分方程来处理交叉项。与经典结果相比，最大条件包含一个指标函数，这实际上是具有延迟的随机最优控制问题的特征。此外，为了说明我们的理论结果的应用，基于全局最大值原理解决了三个动态优化问题。