The Annals of Probability

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

Ibrahim Ekren, Nizar Touzi, and Jianfeng Zhang

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The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204–236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277–3311]. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the well-posedness results established in [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint]. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 1212-1253.

Received: May 2013
Revised: September 2014
First available in Project Euclid: 14 March 2016

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Zentralblatt MATH identifier

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.)

Path dependent PDEs second-order backward SDEs nonlinear expectation viscosity solutions comparison principle


Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. Ann. Probab. 44 (2016), no. 2, 1212--1253. doi:10.1214/14-AOP999.

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