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

TL;DR

This paper proves a central limit theorem for the Euler scheme approximating multidimensional SDEs with irregular drifts, showing the limiting distribution can be characterized by hybrid Young-Itô equations or transformed SDEs, extending classical results.

Contribution

It extends the central limit theorem for Euler schemes to SDEs with irregular, non-differentiable drifts, using noise regularization effects.

Findings

Limiting law characterized by hybrid Young-Itô equations for Hölder continuous drifts.

Limiting law characterized by transformed SDEs for Sobolev regular drifts.

Extension of classical results to less regular drift coefficients.

Abstract

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 H\"older 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-It\^o differential equation. When the drift has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our result is an extension 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 utilize the regularisation effect from the non-degenerate noise.