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

Serena Della Corte, Fabian Fuchs, Richard C. Kraaij, Max Nendel

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 Lévy 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 Lévy processes.

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

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 Lévy 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 Lévy processes.

Paper Structure

This paper contains 23 sections, 21 theorems, 258 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and general setting
  3. Notation and Preliminaries
  4. Operator notions
  5. Viscosity solutions
  6. Notions for our framework
  7. Assumptions and main result
  8. Regularity and compatibility assumptions
  9. Application to partial integro-differential operators
  10. Deterministic Example: Drift terms and convex first-order Hamiltonians
  11. Stochastic Example: Diffusion operators
  12. Stochastic Example: Integral operators
  13. Construction of test functions
  14. Proof of the strict comparison principle
  15. Comparison in terms of estimating the difference of Hamiltonians
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $\mathbb{H} \subseteq C(E) \times C(E)$ be given by with $\Theta_1$ and $\Theta_2$ compact, metric spaces and $\mathcal{I} : E \times \Theta_1 \times \Theta_2 \rightarrow (-\infty,\infty]$ a cost functional. Furthermore, consider a containment function $V$ and penalization functions $\{\zeta_{z,p}\}_{z \in E, p \in \mathbb{R}^q}$, $\{\zeta_z\}_{z \in Let $H \coloneqq \left\{(f,g) \in \mathbb{

Figures (1)

  • Figure 1: Relation between the optimizing points with a note which parts of the propositions give us distance control.

Theorems & Definitions (68)

  • Definition 2.1: Sequential Denseness
  • Definition 2.2: Maximum principle
  • Definition 2.3: Viscosity sub- and supersolutions
  • Definition 2.4
  • Definition 2.5: Coupling
  • Definition 2.6: Controlled growth
  • Definition 2.7: Controlled growth coupling
  • Definition 2.8: Local first-order operator
  • Definition 2.9: Local semi-monotonicity
  • Definition 2.10: Convex semi-monotone operator
  • ...and 58 more