Robust state-dependent mean-variance portfolio selection: a closed-loop approach

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 Bjork 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 Bjork et al. (Math. Finance 24:1-24, 2014), which cannot be deduced under the open-loop control framework.