An overview of Viscosity Solutions of Path-Dependent PDEs

TL;DR

This paper reviews the development of viscosity solutions for path-dependent PDEs, focusing on well-posedness, stability, and approximation methods, with emphasis on the semilinear case and the underlying optimal stopping theory.

Contribution

It provides a comprehensive overview of viscosity solutions for path-dependent PDEs, including existence, uniqueness, and stability results, and connects these to optimal stopping under nonlinear expectation.

Findings

Established well-posedness for semilinear path-dependent PDEs.

Reviewed stability and approximation schemes like Barles-Souganidis.

Connected viscosity solutions to nonlinear optimal stopping theory.

Abstract

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].