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

Serena Della Corte, Fabian Fuchs, Richard C. Kraaij and Max Nendel
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Serena Della Corte: Center for Mathematical Economics, Bielefeld University
Fabian Fuchs: Center for Mathematical Economics, Bielefeld University
Richard C. Kraaij: Center for Mathematical Economics, Bielefeld University
Max Nendel: Center for Mathematical Economics, Bielefeld University

No 696, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: 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.

Keywords: Comparison principle; viscosity solution; Hamilton–Jacobi-Bellman–Isaacs equation; coupling of operators; Lyapunov function; Jensen perturbation; mixed topology (search for similar items in EconPapers)
Pages: 47
Date: 2024-11-11
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