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

Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2507-2553
Number of pages47
JournalAnnals of Probability
Issue number4
StatePublished - 2016


  • Comparison principle
  • Nonlinear expectation
  • Path dependent PDEs
  • Perron's approach
  • Viscosity solutions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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