Pseudo Markovian Viscosity Solutions of Fully Nonlinear Degenerate PPDEs

TL;DR

This paper introduces a new class of viscosity solutions for fully nonlinear path-dependent PDEs, relaxing previous non-degeneracy conditions and applying to stochastic control and game problems.

Contribution

It proposes pseudo Markovian viscosity solutions that remove the uniform non-degeneracy requirement and establishes a comparison principle for these solutions.

Findings

Comparison principle proven under mild conditions

Application to stochastic HJB equations

Application to path dependent Isaacs equations

Abstract

In this paper we propose a new type of viscosity solutions for fully nonlinear path dependent PDEs. By restricting to certain pseudo Markovian structure, we remove the uniform non- degeneracy condition imposed in our earlier works [9, 10]. We establish the comparison principle under natural and mild conditions. Moreover, as applications we apply our results to two important classes of PPDEs: the stochastic HJB equations and the path dependent Isaacs equations, induced from the stochastic optimization with random coefficients and the path dependent zero sum game problem, respectively.