Viscosity solutions of obstacle problems for Fully nonlinear path-dependent PDEs

TL;DR

This paper extends the concept of viscosity solutions to obstacle problems for fully nonlinear path-dependent PDEs, establishing existence, uniqueness, and stability results, and linking solutions to reflected backward stochastic differential equations.

Contribution

It introduces a new definition of viscosity solutions for obstacle problems in path-dependent PDEs and proves their consistency, stability, and uniqueness.

Findings

Viscosity solutions are consistent with classical solutions.

The value functional from reflected backward SDEs is the unique viscosity solution.

The new framework handles fully nonlinear, path-dependent obstacle problems.

Abstract

In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,\omega)$, and generator Lipschitz continuous in $(y,z,\gamma)$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.