Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II

TL;DR

This paper establishes the well-posedness of viscosity solutions for fully nonlinear path-dependent PDEs by extending previous work and assuming comparison principles for associated path-frozen PDEs.

Contribution

It introduces a full well-posedness result for nonlinear path-dependent PDEs under assumptions on local PDE comparison and Perron existence methods.

Findings

Proves uniqueness of viscosity solutions under new assumptions.

Extends the theory of viscosity solutions to fully nonlinear path-dependent PDEs.

Provides conditions for existence and comparison principles in this context.

Abstract

In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence, we prove that the nonlinear path-dependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.