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

Ren, Zhenjie

Ann. Appl. Probab. 26, No. 6, 3381-3414 (2016).

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.

MSC:

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.)

Keywords:

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

Citations:

Zbl 1394.35228