Robust utility maximization with limited downside risk in incomplete markets

In this article we consider the portfolio selection problem of an agent with robust preferences in the sense of Gilboa and Schmeidler Itzhak Gilboa, David Schmeidler, Maxmin expected utility with non-unique prior, Journal of Mathematical Economics 18 (1989) 141–153] in an incomplete market. Downside risk is constrained by a robust version of utility-based shortfall risk. We derive an explicit representation of the optimal terminal wealth in terms of certain worst case measures which can be characterized as minimizers of a dual problem. This dual problem involves a three-dimensional analogue of f$f$-divergences which generalize the notion of relative entropy.