Perron's method for viscosity solutions of semilinear path dependent PDEs

• Published in: Stochastics (Abingdon, Eng. : 2005) Vol. 89; no. 6-7; pp. 843 - 867
• Main Author: Ren, Zhenjie
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03-10-2017; Taylor & Francis Ltd; Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Subjects: Analysis of PDEs; Computer Science; Electric noise; Machine Learning; Mathematics; Nonlinear equations; Optimization and Control; Regularization; Viscosity; Viscosity solution; Perron's method; path-dependent PDE; optimal stopping
• ISSN: 1744-2508; 1744-2516
• DOI: 10.1080/17442508.2016.1215451

This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron's method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions introduced in the work of Ekren, Keller, Touzi and Zhang which considers as test functions all those smooth processes which are tangent in mean. We also provide a comparison result for semicontinuous viscosity solutions, by using a regularization technique. As an interesting byproduct, we give a new short proof for the optimal stopping problem with semicontinuous obstacles.