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March 2016 Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I
Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang
Ann. Probab. 44(2): 1212-1253 (March 2016). DOI: 10.1214/14-AOP999

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.

Citation

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Ibrahim Ekren. Nizar Touzi. Jianfeng Zhang. "Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I." Ann. Probab. 44 (2) 1212 - 1253, March 2016. https://doi.org/10.1214/14-AOP999

Information

Received: 1 May 2013; Revised: 1 September 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1375.35250
MathSciNet: MR3474470
Digital Object Identifier: 10.1214/14-AOP999

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 2 • March 2016
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