Viscosity properties with singularities in a state-constrained expected utility maximization problem

TL;DR

This paper investigates the value function in a constrained utility maximization problem, linking it to a nonlinear degenerate HJB equation with singularities, and establishes conditions for classical and viscosity solutions.

Contribution

It introduces a verification method and a comparison principle to connect the value function with solutions of a singular HJB equation in a constrained setting.

Findings

Established conditions for classical solutions to the HJB equation.

Proved the value function is the unique viscosity solution.

Linked the value function to a nonlinear degenerate PDE.

Abstract

We consider the value function originating from an expected utility maximization problem with finite fuel constraint and show its close relation to a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. On one hand, we give a so-called verification argument based on the dynamic programming principle, which allows us to derive conditions under which a classical solution of the HJB equation coincides with our value function (provided that it is smooth enough). On the other hand, we establish a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.