In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forward-backward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance.

中文翻译：

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时间不一致的随机最优控制问题：一种后向随机偏微分方程方法

在本文中，我们研究了具有确定性系数的随机微分方程的一类时间不一致的随机控制问题。我们在博弈论的框架内研究这些问题，并寻找开环纳什均衡控制。在适当的条件下，我们通过前向后向随机偏微分方程组的流量推导了平衡控制的验证定理。为了说明我们的结果，我们讨论均值方差问题，并在均值和方差之间进行状态依赖的权衡。