Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs

TL;DR

This paper introduces a probabilistic approach to compare viscosity solutions of semi-linear path-dependent PDEs, expanding the set of test functions and simplifying proofs, especially for the linear path-dependent heat equation.

Contribution

It provides a new, more general comparison proof for viscosity solutions of path-dependent PDEs using an enlarged set of test functions and introduces punctual differentiation for semimartingales.

Findings

Enlarged test functions simplify comparison proofs.

Semimartingales are punctually differentiable almost everywhere.

The approach does not rely on solution representation.

Abstract

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in \cite{EKTZ} which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions, and reduces to the notion of stochastic viscosity solutions analyzed in \cite{BayraktarSirbu1,BayraktarSirbu2}. Our main result takes advantage of this enlargement of the test functions, and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions \cite{CaffarelliCabre}, and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in \cite{EKTZ}. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.