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

Ibrahim Ekren ^[1] ; Nizar Touzi ^[3] ; Jianfeng Zhang ^[2]
1. [1] Swiss Federal Institute of Technology in Zurich
  
  Swiss Federal Institute of Technology in Zurich
  
  Zürich, Suiza
2. [2] University of Southern California
  
  University of Southern California
  
  Estados Unidos
3. [3] Ecole Polytecnique Paris
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 4, 2016, págs. 2507-2553
Idioma: inglés
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Resumen
- 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.