The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 26, Number 6 (2016), 3381-3414.
Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations
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.
Ann. Appl. Probab., Volume 26, Number 6 (2016), 3381-3414.
Received: October 2014
Revised: November 2015
First available in Project Euclid: 15 December 2016
Permanent link to this document
Digital Object Identifier
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.)
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. https://projecteuclid.org/euclid.aoap/1481792588