Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

TL;DR

This paper demonstrates the existence of smooth solutions to the Hamilton-Jacobi-Bellman equation in a class of constrained utility maximization problems with nonsmooth utility functions, using duality methods.

Contribution

It introduces a novel approach to obtain smooth solutions for nonsmooth utility maximization problems via dual control and HJB equations.

Findings

Existence of smooth classical solutions to the HJB equation for nonsmooth utilities.

Duality approach effectively constructs solutions satisfying boundary conditions.

Results extend the class of utility functions for which smooth solutions are known.

Abstract

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.