ABSTRACT This paper considers the stochastic optimal control of a multi-period mean-variance trade-off performance criterion with and without constraints for discrete-time linear systems subject to multiplicative noises. We adopt a mean-field approach to tackle the problem and obtain a solution for the unconstrained case in terms of a Riccati-like difference equation. From this general result, we obtain a sufficient condition for a closed-form solution for one of the constrained problems considered in the paper. When particularised to the portfolio selection problem, we show that our results retrieve some of the results available in the literature. We conclude the paper by illustrating the obtained optimal controls with a multi-period portfolio selection problem where we minimise the sum of the mean-variance trade-off costs of a portfolio against a benchmark along the time.

中文翻译：

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具有乘性噪声的离散时间线性系统的平均方差控制的平均场公式

摘要 本文考虑了受乘性噪声影响的离散时间线性系统的多周期均值-方差权衡性能标准的随机最优控制和无约束。我们采用平均场方法来解决这个问题，并根据类 Riccati 的差分方程获得无约束情况的解决方案。从这个一般结果，我们获得了论文中考虑的约束问题之一的封闭形式解的充分条件。当特定于投资组合选择问题时，我们表明我们的结果检索了文献中可用的一些结果。