Bahchedjioglou O. Duality theory under model uncertainty for non-concave utility functions

In a financial market model, the optimal investment strategies depend on the preferences of the investor. However, these preferences can vary greatly, depending on various factors such as market completeness, assumptions on probability measures, properties of the utility function, payoff modeling, and budget constraints, among others. This thesis addresses the existence and construction of optimal investment strategies in a general setup under model uncertainty. Specifically, it considers both the standard and robust utility maximization functionals, assuming that the investor’s utility function is not necessarily concave. The analysis is conducted under a general set of prior probability measures in both complete and incomplete market models. Additionally, two cases of admissible final endowments are examined: standard budget constraints and an additional upper bound represented by a random variable. The main tools employed to achieve the desired results are the minimax identity, which allows for the interchangeability of maximization over strategies and minimization over measures, in the context of the complete market model, and the duality theory for the incomplete market model. In both cases, the concavification principle is utilized, which involves considering the concave envelope of the initial utility function.