A central limit theorem for the Euler method for SDEs with irregular drifts

The goal of this article is to establish a central limit theorem for the Euler-Maruyama scheme approximating multidimensional SDEs with elliptic Brownian diffusion, under very mild regularity requirements on the drift coefficients. When the drift is Hölder continuous, we show that the limiting law of the rescaled fluctuations around the true solution is characterised as the unique solution of a hybrid Young-Itô differential equation. When the drift is bounded and has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our results are extensions of the results of Jacod-Kurtz-Protter (1991, 1998) in which SDEs with differentiable coefficients were considered. To compensate for the lack of regularity of the drifts, we utilise the regularisation effect from the non-degenerate noise.