One of the central problems in mathematical finance is the construction of dynamic investment strategies that maximize the utility functional of a risk-averse investor. In recent years, considerable progress has been made in the case where the utility functional corresponds to investor preferences under von Neumann-Morgenstern hypotheses. This assumption, however, relies on the precise knowledge of the future dynamics of the underlying, which rarely can be achieved in practice. For instance, it may be impossible to obtain a precise statistical forecast of the future drift of the price process. In such a situation, an investor will be subject to model uncertainty, also known as Knightian uncertainty. Knightian uncertainty should thus be regarded as a principal ingredient of financial modelling and in turn be taken into account when formulating optimal investment problems.
The aim of this project is to analyze dynamic optimization problems for investor preferences that take into account both risk and uncertainty aversion. Modern mathematical economics has shown that such preference structures typically admit numerical representations in terms of robust utility functionals, i.e., via functionals that arise by applying a non-additive expectation operator to the accumulated utility. A typical example could be the Choquet integral of the utility with respect to a capacity. More precisely, the aim of this project is to understand the influence of Knightian uncertainty on the qualitative and quantitative behavior of optimal consumption streams. New phenomena can already be expected in the case of a complete model. General incomplete models will be analyzed, e.g., by duality methods and Bellmann-type PDE techniques.