Optimal Investment Strategy for α-Robust Utility Maximization Problem

In reality, investors are uncertain about the dynamics of risky asset returns. Therefore, investors prefer to make robust investment decisions. In this paper, we propose an α-robust utility maximization problem under uncertain parameters. The investor is allowed to invest in a financial market consisting of a risk-free asset and a risky asset. The uncertainty about the expected return rate is parameterized by a nonempty set. Different from most existing literature on robust utility maximization problems where investors are generally assumed to be extremely ambiguity averse because they tend to consider only expected utility in the worst-case scenario, we pay attention to the investors who are not only ambiguity averse but also ambiguity seeking. Under power utility, we provide the implicit function representations for the precommitted strategy, equilibrium strategy of the open-loop type, and equilibrium strategy of the closed-loop type. Some properties about the optimal trading strategies, the best-case and worst-case parameters under three different kinds of strategies, are provided.Funding: This work was supported by National Natural Science Foundation of China [Grants 12071147, 12171169, 12271171, 12371470, 71721001, 71931004, 72371256], the Shanghai Philosophy Social Science Planning Office Project [Grant 2022ZJB005], Fundamental Research Funds for the Central Universities [Grant 2022QKT001], the Excellent Young Team Project Natural Science Foundation of Guangdong Province of China [Grant 2023B1515040001], the Philosophy and Social Science Programming Foundation of Guangdong Province [Grant GD22CYJ17], the Nature Science Foundation of Guangdong Province of China [Grant 2022A1515011472], and the 111 Project [Grant B14019].