Abstract: We propose to parameterize open loop controls in stochastic optimal control problems via suitable classes of functionals depending on the driver’s path signature, a feature map on path space adopted from rough path theory. We prove that these controls are dense in the class of admissible controls and establish suitable stability conditions. 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.