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A Viscosity Framework for Dynamic Programming Principles and Applications

Félix del Teso, Julio D. Rossi, Jorge Ruiz-Cases

TL;DR

This work develops a viscosity-based framework for dynamic programming principles (DPPs) that eliminates measurability obstacles and provides a robust link between DPPs and elliptic (and parabolic) PDEs. The authors establish existence via Perron's method, a comparison principle under mild structural hypotheses, and convergence of DPP solutions to the limiting PDE under weak stability and consistency; they further connect the DPP with asymptotic mean value properties, showing equivalence between AMVP viscosity solutions and PDE viscosity solutions under appropriate consistency. The framework applies to a broad class of operators, including linear, nonlinear, and nonlocal ones such as the Laplacian, $p$-Laplacian, fractional Laplacian, and infinity-Laplacian, and extends to time-dependent problems and general spaces. These results unify game-theoretic, control, and numerical analysis perspectives, providing new existence results, stability tools, and structural insights for complex PDEs arising in diverse applications.

Abstract

In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it bypasses any measurability requirement on solutions of the DPP, an assumption that is often difficult to verify and may even fail in relevant examples. We establish a comparison principle between classical strict supersolutions and viscosity subsolutions of the DPP, which yields stability results under minimal and natural hypotheses. As a consequence, we prove existence of viscosity solutions of the DPP and their convergence to viscosity solutions of a PDE that is consistent with the underlying approximation scheme. Moreover, we show that solutions of the limiting PDE admit an asymptotic expansion encoded by the approximation operator. Finally, we demonstrate that a broad class of local, nonlocal, and nonlinear partial differential equations fits into our framework, recovering known examples in the literature and completing gaps in the existing literature.

A Viscosity Framework for Dynamic Programming Principles and Applications

TL;DR

This work develops a viscosity-based framework for dynamic programming principles (DPPs) that eliminates measurability obstacles and provides a robust link between DPPs and elliptic (and parabolic) PDEs. The authors establish existence via Perron's method, a comparison principle under mild structural hypotheses, and convergence of DPP solutions to the limiting PDE under weak stability and consistency; they further connect the DPP with asymptotic mean value properties, showing equivalence between AMVP viscosity solutions and PDE viscosity solutions under appropriate consistency. The framework applies to a broad class of operators, including linear, nonlinear, and nonlocal ones such as the Laplacian, -Laplacian, fractional Laplacian, and infinity-Laplacian, and extends to time-dependent problems and general spaces. These results unify game-theoretic, control, and numerical analysis perspectives, providing new existence results, stability tools, and structural insights for complex PDEs arising in diverse applications.

Abstract

In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it bypasses any measurability requirement on solutions of the DPP, an assumption that is often difficult to verify and may even fail in relevant examples. We establish a comparison principle between classical strict supersolutions and viscosity subsolutions of the DPP, which yields stability results under minimal and natural hypotheses. As a consequence, we prove existence of viscosity solutions of the DPP and their convergence to viscosity solutions of a PDE that is consistent with the underlying approximation scheme. Moreover, we show that solutions of the limiting PDE admit an asymptotic expansion encoded by the approximation operator. Finally, we demonstrate that a broad class of local, nonlocal, and nonlinear partial differential equations fits into our framework, recovering known examples in the literature and completing gaps in the existing literature.
Paper Structure (23 sections, 8 theorems, 150 equations)

This paper contains 23 sections, 8 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Comments on related literature.
  3. Structure of the paper.
  4. Definitions and main results
  5. Existence of viscosity solutions of the DPP.
  6. Comparison results and stability.
  7. Convergence of viscosity solutions of the DPP.
  8. Asymptotic mean value formulas in the viscosity sense.
  9. Existence of viscosity solutions to the DPP
  10. Comparison result and weak stability for the DPP
  11. Convergence of the solutions to the DPP towards the solution to a PDE
  12. Asymptotic mean value formulas in the viscosity sense
  13. Extensions
  14. Parabolic problems.
  15. Fractional/nonlocal problems.
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\Omega \subset \mathbb{R}^N$ be an open domain, let $g :\mathbb{R}^N\setminus \Omega \to \mathbb{R}$ bounded, and let $\mathcal{A}\colon \Omega \times \mathcal{X} \times \mathbb{R} \to \mathbb{R}$ satisfy assumptions asA-it:a--asA-it:abis--asA-it:b. Assume that either or Then, there exists at least one viscosity solution of eq:BVPDPP.

Theorems & Definitions (38)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Definition 2: Classical solution of the DPP with boundary values
  • Remark 8
  • ...and 28 more