A Strict Comparison Principle for Integro-Differential Hamilton-Jacobi-Bellman Equations on Domains with Boundary
Serena Della Corte, Fabian Fuchs, Richard C. Kraaij, Max Nendel
TL;DR
This work develops a strict comparison principle for viscosity solutions of integro-differential HJB-type boundary value problems on cylindrical domains, accommodating both diffusion and jump terms and handling unbounded solutions via Lyapunov growth bounds and doubling penalties. It introduces a cohesive framework that decomposes operators into a coupled stochastic part and a deterministic part, with Phi-controlled growth and Jensen perturbations guiding optimizer and test-function construction. The main result provides a sharp comparison bound that reduces to boundary data, with concrete applications to parabolic problems on [0,T]×R^q and elliptic problems on spaces with corners, under explicit regularity and compatibility conditions. The paper also furnishes detailed optimizer/test-function construction and demonstrates how Lyapunov functions ensure convex semi-monotonicity and controlled operator action in bounded and corner 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.