The Annals of Probability

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

Ibrahim Ekren, Nizar Touzi, and Jianfeng Zhang

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Abstract

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.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2507-2553.

Dates
Received: May 2013
Revised: September 2014
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139148

Digital Object Identifier
doi:10.1214/15-AOP1027

Mathematical Reviews number (MathSciNet)
MR3531674

Zentralblatt MATH identifier
06631778

Subjects
Primary: 35D40: Viscosity solutions 35K10: Second-order parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Path dependent PDEs nonlinear expectation viscosity solutions comparison principle Perron’s approach

Citation

Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 (2016), no. 4, 2507--2553. doi:10.1214/15-AOP1027. https://projecteuclid.org/euclid.aop/1470139148


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