Approximate viscosity solutions of path-dependent PDEs and Dupire's vertical differentiability

TL;DR

This paper introduces a new concept of approximate viscosity solutions for nonlinear path-dependent PDEs, establishing foundational results and exploring the regularity of solutions in the sense of Dupire.

Contribution

It develops a novel framework for approximate viscosity solutions of PPDEs, including existence, comparison, stability, and regularity results, extending the theory to higher dimensions and non-linear cases.

Findings

Established existence, comparison, and stability of approximate viscosity solutions.

Demonstrated consistency with smooth solutions in low dimensions or concave cases.

Investigated Dupire's regularity of solutions to PPDEs.

Abstract

We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), including the Hamilton-Jacobi-Bellman type equations. Existence, comparaison and stability results are established under fairly general conditions. It is also consistent with smooth solutions when the dimension is less or equal to two, or the non-linearity is concave in the second order space derivative. We finally investigate the regularity (in the sense of Dupire) of the solution to the PPDE.