VISCOSITY SOLUTIONS OF FULLY NONLINEAR ELLIPTIC PATH DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: Ren, Zhenjie
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1178
• 发表日期: 2016
• 页码: 3381-3414
• 关键词: pdes; expectations

摘要

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.