A comparison principle based on couplings of partial integro-differential operators

TL;DR

This paper develops a new comparison principle for viscosity solutions of complex integro-differential equations, using couplings and a novel test function framework to handle non-local operators.

Contribution

It introduces a unified coupling-based approach for comparison principles applicable to a broad class of integro-differential operators, including those from Lévy processes.

Findings

Established a comparison principle for viscosity solutions involving non-local operators.

Unified the treatment of differential, difference, and integral operators through couplings.

Enhanced the contractivity property to include continuity in the strict topology.

Abstract

This paper is concerned with a comparison principle for viscosity solutions to Hamilton-Jacobi (HJ), -Bellman (HJB), and -Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with L\'evy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to differential, difference, and integral operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous L\'evy processes.