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Stochastic differential equations with noise perturbations and Wong-Zakai approximation of fractional Brownian motion

In this article we study effects that small perturbations in the noise have to the solution of differential equations driven by Hölder continuous functions of order $H>\frac12$. As an application, we consider stochastic differential equations driven by a fractional Brownian motion. We introduce a Wong--Zakai type stationary approximation to the fractional Brownian motions and prove that it converges in a suitable space. Moreover, we provide sharp results on the rate of convergence in the $p$-norm. Our stationary approximation is suitable for all values of $H\in (0,1)$.