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.

Funding: M. Dai acknowledges financial support from the National Natural Science Foundation of China [Grants 12071333 and 11671292] and the SingaporeMinistry of Education [Grants R-146-000-243-114, R-146-000-306-114, R-146-000-311-114, and R-703-000-032-112]. X. Wan is supported by the National Natural Science Foundation of China [Grants 71850010, 72171109, 71972131, and 72271157].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2021.4228.