Before revision: This paper is concerned with the utility-based risk of a financial position in a multi-asset market with frictions. Risk is quantified by set-valued risk measures, and market frictions are modeled by conical/convex random solvency regions representing proportional transaction costs or illiquidity effects, and convex random sets representing trading constraints. First, with a general set-valued risk measure, the effect of having trading opportunities on the risk measure is considered, and a corresponding dual representation theorem is given. Then, assuming individual utility functions for the assets, utility-based shortfall and divergence risk measures are defined , which form two classes of  set-valued convex risk measures .
After revision: This paper is concerned with the utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined , which form two classes of set-valued convex risk measures .