Scalarized utility-based multi-asset risk measures

We introduce a risk measure that simultaneously allows to minimize hedging costs and maximize expected utility in the presence of a risk constraint. We call it the scalarized utility-based multi-asset (SUBMA) risk measure. For the SUBMA risk measure we state the following results: If the utility function has constant relative risk aversion and the risk constraint is coherent, then the SUBMA risk measure is coherent. In a one-period financial market setup, we present a sufficient condition for the SUBMA risk measure to be finite-valued. We derive results about the existence of optimal payoffs. Finally, we present a dual representation for a map that generalizes classical risk measures and has not been analyzed so far. In particular, it gives us the dual representation for the SUBMA risk measure.