Mean viability theorems and second-order Hamilton-Jacobi equations

TL;DR

This paper introduces mean viability for stochastic differential equations, establishing viability theorems, and provides a probabilistic approach to second-order path-dependent Hamilton-Jacobi equations, avoiding traditional compactness methods.

Contribution

It develops a new notion of mean viability and proves viability theorems, offering a novel probabilistic proof for solutions of complex path-dependent PDEs.

Findings

Established necessary and sufficient conditions for mean viability.

Provided a probabilistic proof of comparison principle.

Proved existence of viscosity solutions without compactness arguments.

Abstract

We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo's classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton-Jacobi-Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs.