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

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.

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