VISCOSITY SOLUTIONS OF FULLY NONLINEAR PARABOLIC PATH DEPENDENT PDES: PART I

• 成果类型: Article
• 署名作者: Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng
• 署名单位: University of Southern California; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP999
• 发表日期: 2016
• 页码: 1212-1253
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; backward sdes; formulation

摘要

The main objective of this paper and the accompanying one [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [Ann. Probab. (2014) 42 204-236], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [Stochastic Process. Appl. (2014) 124 3277-3311]. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the well-posedness results established in [Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012) Preprint]. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.