  In this article, we present a general methodology for stochastic control
problems driven by the Brownian motion filtration including non-Markovian and
non-semimartingale state processes controlled by mutually singular measures.
The main result of this paper is the development of a numerical scheme for
computing near-optimal controls associated with controlled Wiener functionals
via a finite-dimensional approximation procedure. The approach does not require
functional differentiability assumptions on the value process and ellipticity
conditions on the diffusion components. The general convergence of the method
is established under rather weak conditions for distinct types of non-Markovian
and non-semimartingale states. Explicit rates of convergence are provided in
case the control acts only on the drift component of the controlled system.
Near-closed/open-loop optimal controls are fully characterized by a dynamic
programming algorithm and they are classified according to the strength of the
possibly underlying non-Markovian memory. The theory is applied to stochastic
control problems based on path-dependent SDEs and rough stochastic volatility
models, where both drift and possibly degenerated diffusion components are
controlled. Optimal control of drifts for nonlinear path-dependent SDEs driven
by fractional Brownian motion with exponent $H\in (0,\frac{1}{2})$ is also
discussed. Finally, we present a simple numerical example to illustrate the
method.
