On Utility Maximization Under Convex Portfolio Constraints
The authors consider a utility-maximization problem in a general semimartingale financial market, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin, i.e., no risky investment at all may be inadmissible. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present. The main result establishes existence of the optimal trading strategies in such markets under no smoothness requirements on the utility function, and relates them to the corresponding dual objects.