Nonconcave Utility Maximization with Portfolio Bounds

The problems of nonconcave utility maximization appear in many areas of finance and economics, such as in behavioral economics, incentive schemes, aspiration utility, and goal-reaching problems. Existing literature solves these problems using the concavification principle. We provide a framework for solving nonconcave utility maximization problems, where the concavification principle may not hold, and the utility functions can be discontinuous. We find that adding portfolio bounds can offer distinct economic insights and implications consistent with existing empirical findings. Theoretically, by introducing a new definition of viscosity solution, we show that a monotone, stable, and consistent finite difference scheme converges to the value functions of the nonconcave utility maximization problems. This paper was accepted by Agostino Capponi, finance.