<p>In controlled diffusions, a critical assumption underlying the mathematical analysis is the Brownian idealization of the driving noise, whereas in nearly all applications the noise is never precisely the Brownian. This gives rise to the problem of performance loss due to the mismatch between the actual system and the assumed idealized system. In this article, we present a robustness theorem for controlled stochastic differential equations driven by approximations of Brownian motion, where the approximations are those that converge to the Brownian under the rough paths topology along sample paths. These approximations include the Wong–Zakai, Karhunen–Loève, mollified Brownian and fractional Brownian processes, which can be interpreted to be more physical than the Brownian. We establish 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.</p>