  This paper proposes to parameterize open loop controls in stochastic optimal
control problems via suitable classes of functionals depending on the driver's
path signature, a concept adopted from rough path integration theory. We
rigorously prove that these controls are dense in the class of progressively
measurable controls and use rough path methods to establish suitable conditions
for stability of the controlled dynamics and target functional. These results
pave the way for Monte Carlo methods to stochastic optimal control for generic
target functionals and dynamics. We discuss the rather versatile numerical
algorithms for computing approximately optimal controls and verify their
accurateness in benchmark problems from Mathematical Finance.
