Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in games and control theory. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent master equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in stochastic control theory and games. We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupires vertical derivative, and applying stochastic forward-backward system argument. Moreover, we consider a general non-smooth case with a functional mollifying method.
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