VISCOSITY SOLUTIONS OF FULLY NONLINEAR PARABOLIC PATH DEPENDENT PDES: PART I

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. Pr...

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Bibliographic Details
Published in:The Annals of probability 2016-03, Vol.44 (2), p.1212-1253
Main Authors: Ekren, Ibrahim, Touzi, Nizar, Zhang, Jianfeng
Format: Article
Language:English
Subjects:
35D40
35K10
60H10
60H30
comparison principle
Control theory
Differential equations
Dynamic programming
Mathematical problems
Nonlinear equations
nonlinear expectation
Nonlinearity
Parabolas
Partial differential equations
Path dependence
Path dependent PDEs
second-order backward SDEs
Stochastic control theory
Stochastic processes
Stopping distances
Studies
Uniqueness
Viscosity
viscosity solutions
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Summary: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.
ISSN:0091-1798
2168-894X
DOI:10.1214/14-aop999