An overview of Viscosity Solutions of Path-Dependent PDEs

Zhenjie Ren (1) , Nizar Touzi (1) , Jianfeng Zhang (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 USC - University of Southern California

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].

Mots clés: Path-dependent PDEs; viscosity solutions; optimal stopping

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Zhenjie Ren, Nizar Touzi, Jianfeng Zhang. An overview of Viscosity Solutions of Path-Dependent PDEs. 2014. ⟨hal-01057300⟩