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

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.

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