On viscosity solutions of path dependent PDEs

TL;DR

This paper introduces a new notion of viscosity solutions for path dependent semi-linear PDEs, extending the nonlinear Feynman-Kac formula to non-Markovian cases using functional Itô calculus, and proves fundamental properties like existence and uniqueness.

Contribution

It develops a framework for viscosity solutions of non-Markovian PDEs using functional Itô calculus, extending classical results to path-dependent cases.

Findings

Proves existence and uniqueness of viscosity solutions.

Establishes a comparison principle for these solutions.

Demonstrates stability of the solutions.

Abstract

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman-Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional It\^{o} calculus recently introduced by Dupire [Functional It\^{o} calculus (2009) Preprint].