Appears in collection : Schlumberger workshop on Topics in Applied Probability

We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We present a randomization approach of the control, and prove that the value function can be characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton-Jacobi-Bellman (HJB) equation, and an extension to G-expectation. This includes in particular equations in finance arising from model uncertainty. In the Markovian case, our BSDE representation provides a Feynman-Kac type formula to fully nonlinear HJB equation, and leads to a new probabilistic numerical scheme for solving this equation, taking advantage of high dimensional features of Monte-Carlo methods.