Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I

TL;DR

This paper introduces a new framework for viscosity solutions of fully nonlinear parabolic path-dependent PDEs, extending previous work and establishing key properties like consistency, stability, and comparison, with applications to stochastic control.

Contribution

It develops a novel notion of viscosity solutions for fully nonlinear path-dependent PDEs, generalizing earlier semilinear cases and connecting to stochastic control problems.

Findings

Defined a consistent notion of viscosity solutions for path-dependent PDEs.

Proved stability and partial comparison properties for these solutions.

Showed that stochastic control value processes satisfy the path-dependent PDEs.

Abstract

The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in \cite{ETZ2}. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equation.