This paper studies a general class of time-inconsistent stochastic control problems under ambiguous covariance matrix. The time inconsistency is caused in various ways by a general objective functional and thus the associated control problem does not admit Bellman’s principle of optimality. Moreover, we model the state by a McKean–Vlasov dynamics under a set of non-dominated probability measures induced by the ambiguous covariance matrix of the noises. We apply a game-theoretic concept of subgame perfect Nash equilibrium to develop a robust equilibrium control approach, which can yield robust time-consistent decisions. We characterize the robust equilibrium control and equilibrium value function by an extended optimality principle and then we further deduce a system of Bellman–Isaacs equations to determine the equilibrium solution on the Wasserstein space of probability measures. The proposed analytical framework is illustrated with its applications to robust continuous-time mean-variance portfolio selection problems with risk aversion coefficient being constant or state-dependent, under the ambiguity stemming from ambiguous volatilities of multiple assets or ambiguous correlation between two risky assets. The explicit equilibrium portfolio solutions are represented in terms of the probability law.

本文研究了模糊协方差矩阵下的一类一般时间不一致随机控制问题。时间不一致是由一般目标函数以各种方式引起的，因此相关的控制问题不承认贝尔曼的最优性原理。此外，我们在由噪声的模糊协方差矩阵引起的一组非支配概率测量下，通过 McKean-Vlasov 动力学对状态进行建模。我们应用子博弈完美纳什均衡的博弈论概念来开发稳健的均衡控制方法，该方法可以产生稳健的时间一致决策。我们通过扩展的最优性原理来表征鲁棒平衡控制和平衡值函数，然后进一步推导贝尔曼-艾萨克斯方程组以确定概率测度的 Wasserstein 空间上的平衡解。所提出的分析框架通过其在多种资产的模糊波动性或两种风险资产之间的模糊相关性引起的模糊性下，在风险厌恶系数恒定或依赖于状态的鲁棒连续时间均值方差投资组合选择问题中的应用进行了说明。显式均衡投资组合解用概率定律表示。