BSDE, Path-dependent PDE and Nonlinear Feynman-Kac Formula

In this paper, we introduce a type of path-dependent quasilinear (parabolic) partial differential equations in which the (continuous) paths on an interval [0,t] becomes the basic variables in the place of classical variables (t,x). This new type of PDE are formulated through a classical backward stochastic differential equation (BSDEs, for short) in which the terminal values and the generators are allowed to be general functions of Brownian paths. In this way we have established a new type of nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of regularities for this new PDE was obtained.