Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations. (English) Zbl 1372.35386

Summary: 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 [I. Ekren et al., Ann. Probab. 44, No. 4, 2507–2553 (2016; Zbl 1394.35228)], 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.


35R60 PDEs with randomness, stochastic partial differential equations
35D40 Viscosity solutions to PDEs
35J60 Nonlinear elliptic equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)


Zbl 1394.35228
Full Text: DOI Euclid