Work

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

Year: 2017

Type: article

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, and Zhang [ Ann. Probab., 44 (2016), pp. 2507--2553], 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, and Souganidis [ Proc. Amer. Math. Soc., 102, (1988)] in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof of Ekren, Touzi, and Zhang but requires an $\mathbb{L}^p$-type of continuity (with respect to the path) for the viscosity semisolutions and for the nonlinearity defining the equation.

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/16m1090338

Authors: Zhenjie Ren, Nizar Touzi, Jianfeng Zhang

Institutions: Centre de Recherche en Mathématiques de la Décision, École Polytechnique, University of Southern California

Language: English

Topic: Stochastic processes and financial applications

Subfield: Finance

Field: Economics, Econometrics and Finance

Domain: Social Sciences

Open Access status: green

Funders: National Science Foundation, Agence Nationale de la Recherche, Fondation du Risque

Award: Some Topics on Path Dependent Partial Differential Equations and Stochastic Differential Equations