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December 2016 Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations
Zhenjie Ren
Ann. Appl. Probab. 26(6): 3381-3414 (December 2016). DOI: 10.1214/16-AAP1178


This paper extends the recent work on path-dependent PDEs to elliptic equations with Dirichlet boundary conditions. We propose a notion of viscosity solution in the same spirit as [Ann. Probab. 44 (2016) 1212–1253, Part 1; Ekren, Touzi and Zhang (2016), Part 2], relying on the theory of optimal stopping under nonlinear expectation. We prove a comparison result implying the uniqueness of viscosity solution, and the existence follows from a Perron-type construction using path-frozen PDEs. We also provide an application to a time homogeneous stochastic control problem motivated by an application in finance.


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Zhenjie Ren. "Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations." Ann. Appl. Probab. 26 (6) 3381 - 3414, December 2016.


Received: 1 October 2014; Revised: 1 November 2015; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1372.35386
MathSciNet: MR3582806
Digital Object Identifier: 10.1214/16-AAP1178

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

Keywords: Comparison principle , Optimal stopping , path-dependent PDEs , Perron’s approach , viscosity solutions

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 6 • December 2016
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