This paper studies a class of robust mean–variance portfolio selection problems with state-dependent risk aversion. Model uncertainty, in the sense of considering alternative dominated models, is introduced to the problem to reflect the investor’s uncertainty-averse preference. To characterise the robust portfolios, we consider closed-loop equilibrium control and spike variation approaches. Moreover, we show that a closed-loop equilibrium strategy exists and is unique under some technical conditions. This partially addresses open problems left in Björk et al. (Finance Stoch. 21:331–360, 2017) and Pun (Automatica 94:249–257, 2018). By using a necessary and sufficient condition for the equilibrium, we manage to derive the analytical form of the equilibrium strategy via the unique solution to a nonlinear ordinary differential equation system. To validate the proposed closed-loop control framework, we show that when there is no uncertainty, our equilibrium strategy is reduced to the strategy in Björk et al. (Math. Finance 24:1–24, 2014), which cannot be deduced under the open-loop control framework.

本文研究了一类具有状态依赖风险规避的稳健均值-方差投资组合选择问题。在考虑替代主导模型的意义上，模型不确定性被引入该问题以反映投资者的不确定性厌恶偏好。为了表征稳健的投资组合，我们考虑了闭环平衡控制和尖峰变化方法。此外，我们表明闭环均衡策略存在并且在某些技术条件下是独一无二的。这部分解决了 Björk 等人留下的开放性问题。(Finance Stoch. 21:331–360, 2017) 和 Pun (Automatica 94:249–257, 2018)。通过使用均衡的充分必要条件，我们设法通过非线性常微分方程系统的唯一解导出均衡策略的解析形式。为了验证所提出的闭环控制框架，我们表明当没有不确定性时，我们的均衡策略被简化为 Björk 等人的策略。(Math. Finance 24:1–24, 2014)，在开环控制框架下无法推导出。