This paper is concerned with a Pontryagin’s maximum principle for the stochastic optimal control problem with distributed delays given by integrals of not necessarily linear functions of state or control variables. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, we establish a necessary maximum principle and a sufficient verification theorem. In particular, we deal with the controlled stochastic system where the distributed delays enter both the state and the control. To explain the theoretical results, we apply them to a dynamic advertising problem.

中文翻译：

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分布时滞随机最优控制问题的最大原理

本文关注的是关于随机最优控制问题的Pontryagin极大原理，该最优控制问题的分布时延由状态或控制变量的线性函数的积分给出。借助于对偶方法和广义的预期后向随机微分方程，我们建立了必要的最大原理和充分的验证定理。特别是，我们处理的是受控随机系统，其中分布式延迟同时进入状态和控制。为了解释理论结果，我们将其应用于动态广告问题。