Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

TL;DR

This paper addresses a utility maximization problem involving current utility dependent on wealth, establishing the regularity and uniqueness of solutions to the associated HJB equation using a dual approach, with applications in financial portfolio optimization.

Contribution

It introduces a dual approach to prove the regularity and uniqueness of solutions to the HJB equation in utility maximization with current utility dependence.

Findings

Viscosity solutions are shown to be smooth functions.

Unique smooth solutions coincide with the value function.

Applications in portfolio optimization are demonstrated.

Abstract

We consider a utility maximization problem for an investment-consumption portfolio when the current utility depends also on the wealth process. Such kind of problems arise, e.g., in portfolio optimization with random horizon or with random trading times. To overcome the difficulties of the problem we use the dual approach. We define a dual problem and treat it by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth functions. This allows to define a smooth solution of the primal Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique in a suitable class and coincides with the value function of the primal problem. Some financial applications of the results are provided.