Viscosity Solutions of Second Order Path-Dependent Partial Differential Equations and Applications

TL;DR

This paper introduces a framework for viscosity solutions of fully nonlinear second order path-dependent PDEs, establishing fundamental properties and applying the theory to stochastic differential games.

Contribution

It extends the concept of viscosity solutions to complex path-dependent PDEs and proves key properties like existence, comparison, and stability, with applications to stochastic games.

Findings

Established existence and uniqueness of viscosity solutions

Proved comparison principle and stability results

Applied theory to path-dependent stochastic differential games

Abstract

In this article, a notion of viscosity solutions is introduced for fully nonlinear second order path-dependent partial differential equations in the spirit of [Zhou, Ann. Appl. Probab., 33 (2023), 5564-5612]. We prove the existence, comparison principle, consistency and stability for the viscosity solutions. Application to path-dependent stochastic differential games is given.