## The Annals of Probability

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

#### Abstract

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

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

Dates
Revised: September 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960394

Digital Object Identifier
doi:10.1214/14-AOP999

Mathematical Reviews number (MathSciNet)
MR3474470

Zentralblatt MATH identifier
1375.35250

#### Citation

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. https://projecteuclid.org/euclid.aop/1457960394

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