An overview of viscosity solutions of path-dependent PDEs

Zhenjie Ren, Nizar Touzi, Jianfeng Zhang

Research output: Chapter in Book/Report/Conference proceedingConference contribution


This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial differential equations. We start by a quick review of the Crandall-Ishii notion of viscosity solutions, so as to motivate the relevance of our definition in the path-dependent case.We focus on thewellposedness theory of such equations. In particular, 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, including 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].

Original languageEnglish (US)
Title of host publicationStochastic Analysis and Applications 2014
EditorsDan Crisan, Ben Hambly, Thaleia Zariphopoulou
PublisherSpringer New York LLC
Number of pages57
ISBN (Electronic)9783319112916
StatePublished - 2014
EventConference on Stochastic Analysis and Applications, 2013 - Oxford, United Kingdom
Duration: Sep 23 2013Sep 27 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceConference on Stochastic Analysis and Applications, 2013
Country/TerritoryUnited Kingdom


  • Optimal stopping
  • Path-dependent PDEs
  • Viscosity solutions

ASJC Scopus subject areas

  • General Mathematics


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