Limit Theorem for a Rough Differential Equation with a Negative Long-Range Random Coefficient

We consider an ordinary differential equation driven by rough paths, in the T. Lyons sense (Rev Mat Iberoamer. 1998;14(2)215–310), depending on a small parameter and with a negative long-range random coefficient. We establish sufficient conditions under which the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter \(H\in ( \frac{1}{4},\frac{1}{2}) \) $H\in (\frac{1}{4},\frac{1}{2})$ , depends on the asymptotic behavior of the covariance function of the random coefficient.