This paper focuses on indefinite stochastic mean-field linear–quadratic (MF-LQ, for short) optimal control problems, which allow the weighting matrices for state and control in the cost functional to be indefinite. The solvability of stochastic Hamiltonian system and Riccati equations is presented under indefinite case. The optimal controls in open-loop form and closed-loop form are derived, respectively. In particular, dynamic mean–variance portfolio selection problem can be formulated as an indefinite MF-LQ problem to tackle directly. Another example also sheds light on the theoretical results established.

中文翻译：

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不确定平均场型线性-二次随机最优控制问题

本文关注不确定的随机平均场线性二次方程（MF-LQ）最优控制问题，这使得状态和成本函数中的控制权矩阵不确定。给出了不确定情况下的随机哈密顿系统和Riccati方程的可解性。分别推导了开环形式和闭环形式的最优控制。特别是，动态均方差投资组合选择问题可以表述为不确定的MF-LQ问题，直接解决。另一个例子也阐明了所建立的理论结果。