A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary

TL;DR

This paper establishes a comparison principle for viscosity solutions of boundary value problems involving integro-differential Hamilton-Jacobi-Bellman equations on bounded and unbounded domains, accommodating diffusive and jump processes.

Contribution

It introduces a novel test function framework and Lyapunov function estimates to handle unbounded solutions and boundary complexities in integro-differential equations.

Findings

Valid comparison principle for boundary value problems with jumps

Effective Lyapunov function estimates for unbounded solutions

Applicability to parabolic and elliptic equations on complex domains

Abstract

This work provides a comparison principle for viscosity solutions to boundary value problems on (partially) bounded, cylindrical spaces. The comparison principle is based on a test function framework, that allows for the simultaneous treatment of diffusive as well as jump terms. Estimates in the proof of the comparison principle incorporate the use of Lyapunov functions that act as growth bounds for the solutions, effectively yielding a theory for unbounded viscosity solutions. We apply the results to a wide class of parabolic equations and elliptic problems on a space with corners.