Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I

Abstract

The main objective of this paper and the accompanying one ETZ2 is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work EKTZ, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in ETZ0. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in ETZ2. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equation.