Remarks on Viscosity Super-Solutions of Quasi-Variational Inequalities

TL;DR

This paper proposes a modified definition of viscosity super-solutions for HJB type quasi-variational inequalities to ensure the comparison principle, which is crucial for uniqueness and stability of solutions.

Contribution

It introduces a new definition for viscosity super-solutions of HJB type QVIs that guarantees the comparison theorem, addressing a gap in existing formulations.

Findings

Established a modified definition ensuring comparison between super- and sub-solutions.

Proved the comparison theorem under the new definition.

Enhanced the theoretical foundation for uniqueness of solutions in impulse control problems.

Abstract

For Hamilton-Jacobi-Bellman (HJB) equations, with the standard definitions of viscosity super-solution and sub-solution, it is known that there is a comparison between any (viscosity) super-solutions and sub-solutions. This should be the same for HJB type quasi-variational inequalities (QVIs) arising from optimal impulse control problems. However, according to a natural adoption of the definition found in Barles 1985, Barles 1985b, the uniqueness of the viscosity solution could be guaranteed, but the comparison between viscosity super- and sub-solutions could not be guaranteed. This paper introduces a modification of the definition for the viscosity super-solution of HJB type QVIs so that the desired comparison theorem will hold.