  We formulate a path-dependent stochastic optimal control problem under
general conditions, for which weprove rigorously the dynamic programming
principle and that the value function is the unique Crandall-Lions viscosity
solution of the corresponding Hamilton-Jacobi-Bellman equation. Compared to the
literature, the proof of our core result, that is the comparison theorem, is
based on the fact that the valuefunction is bigger than any viscosity
subsolution and smaller than any viscosity supersolution. It alsorelies on the
approximation of the value function in terms of functions defined on
finite-dimensionalspaces as well as on regularity results for parabolic partial
differential equations.
