A Viscosity Framework for Dynamic Programming Principles and Applications

TL;DR

This paper introduces a viscosity solution framework for approximation schemes related to PDEs, notably dynamic programming principles, enabling stability, existence, and convergence results without measurability constraints.

Contribution

It develops a viscosity-based approach that bypasses measurability issues, establishing comparison, stability, and convergence results for DPPs and broad classes of PDEs.

Findings

Established a comparison principle for DPPs and viscosity solutions.

Proved existence and convergence of viscosity solutions for approximation schemes.

Unified treatment of various PDE types within the viscosity framework.

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.