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

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 contributes to the literature in three ways: (1) We cast the Crandall–Ishii Lemma into a test function framework to tackle a wide class of second-order integro-differential operators in the spirit of the classical doubling of variables method. (2) We provide a unified approach to estimate the difference of Hamiltonians by adapting the probabilistic notion of couplings to an analytic setting. (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.