The Annals of Applied Probability

Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations

Zhenjie Ren

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3381-3414.

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

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Mathematical Reviews number (MathSciNet)

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.)

Viscosity solutions optimal stopping path-dependent PDEs comparison principle Perron’s approach


Ren, Zhenjie. Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations. Ann. Appl. Probab. 26 (2016), no. 6, 3381--3414. doi:10.1214/16-AAP1178.

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