Note on Viscosity Solution of Path-Dependent PDE and G-Martingales

TL;DR

This paper introduces a viscosity solution framework for fully nonlinear path-dependent PDEs, proves a comparison theorem using a novel left frozen maximization approach, and links solutions to backward SDEs and G-martingales.

Contribution

It develops a new viscosity solution concept for path-dependent PDEs and establishes a comparison principle using an innovative maximization method.

Findings

Established a comparison theorem for the new viscosity solutions.

Connected solutions of path-dependent PDEs to backward SDEs and G-martingales.

Extended the maximum principle to classical PDEs as a special case.

Abstract

In the 2nd version of this note we introduce the notion of viscosity solution for a type of fully nonlinear parabolic path-dependent partial differential equations (P-PDE). We then prove the comparison theorem (or maximum principle) of this new type of equation which is the key property of this framework. To overcome the well-known difficulty of non-compactness of the space of paths for the maximization, we have introduced a new approach, called left frozen maximization approach which permits us to obtain the comparison principle for smooth as well as viscosity solutions of path-dependent PDE. A solution of a backward stochastic differential equation and a G-martingale under a G-expectation are typical examples of such type of solutions of P-PDE. The maximum principle for viscosity solutions of classical PDE, called state dependent PDE, is a special case.