Comparison of viscosity solutions of fully nonlinear degenerate parabolic Path-dependent PDEs

2015/11/18 by Zhenjie Ren, Ren, Zhenjie, Nizar Touzi +3
Mathematics · #34G20 #35B51 #35D40 #93E20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR #msc:34G20 #msc:35B51 #msc:35D40 #msc:93E20

paper · pdf · doi:10.48550/arxiv.1511.05910

24 pages

arxiv created 2015/11/18 · arxiv updated 2015/11/19

Abstract

We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang, our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Souganidis in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof in Ekren, Touzi & Zhang, but requires an L^p-type of continuity (with respect to the path) for the viscosity semi-solutions and for the nonlinearity defining the equation.

Related

• Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I
• Note on Viscosity Solution of Path-Dependent PDE and G-Martingales
• Monotone Schemes for Fully Nonlinear Parabolic Path Dependent PDEs
• Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs
• Strong-viscosity Solutions: Semilinear Parabolic PDEs and Path-dependent\n PDEs