Viscosity solutions of fully nonlinear parabolic path dependent PDES: Part I

The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204-236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277-3311]. 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 well-posedness results established in [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint].We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.