A comparison Principle Based on Couplings of Partial Integro-Differential Operators

This paper is concerned with a comparison principle for viscosity solu- tions to Hamilton–Jacobi (HJ), –Bellman (HJB), and –Isaacs (HJI) equations for gen- eral 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 opera- tors, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic no- tion 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 Lévy processes.