  In this article we show a robustness theorem for controlled stochastic
differential equations driven by approximations of Brownian motion. Often,
Brownian motion is used as an idealized model of a diffusion where
approximations such as Wong-Zakai, Karhnen-Lo\`eve or fractional Brownian
motion are often seen as more physical. However, there has been extensive
literature on solving control problems driven by Brownian motion and little on
control problems driven by more realistic models that are only approximately
Brownian. The question of robustness naturally arises from such approximations.
We show robustness using rough paths theory, which allows for a pathwise theory
of stochastic differential equations. To this end, in particular, we show that
within the class of Lipschitz continuous control policies, an optimal solution
for the Brownian idealized model is near optimal for a true system driven by a
non-Brownian (but near-Brownian) noise.
