  We study stochastic differential equations (SDEs) whose drift and diffusion
coefficients are path-dependent and controlled. We construct a value process on
the canonical path space, considered simultaneously under a family of singular
measures, rather than the usual family of processes indexed by the controls.
This value process is characterized by a second order backward SDE, which can
be seen as a non-Markovian analogue of the Hamilton-Jacobi-Bellman partial
differential equation. Moreover, our value process yields a generalization of
the G-expectation to the context of SDEs.
