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July 2016 Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II
Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang
Ann. Probab. 44(4): 2507-2553 (July 2016). DOI: 10.1214/15-AOP1027


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.


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Ibrahim Ekren. Nizar Touzi. Jianfeng Zhang. "Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II." Ann. Probab. 44 (4) 2507 - 2553, July 2016.


Received: 1 May 2013; Revised: 1 September 2014; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 06631778
MathSciNet: MR3531674
Digital Object Identifier: 10.1214/15-AOP1027

Primary: 35D40 , 35K10 , 60H10 , 60H30

Keywords: Comparison principle , nonlinear expectation , path dependent PDEs , Perron’s approach , viscosity solutions

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 4 • July 2016
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