On optimal error rates for strong approximation of SDEs with a Hölder continuous drift coefficient

Abstract: In the present article we study strong approximation of solutions of scalar stochastic differential equations (SDEs) with bounded and $\alpha$-H\"older continuous drift coefficient and constant diffusion coefficient at time point $1$. Recently, it was shown in [arXiv:1909.07961v4 (2021)] that for such SDEs the equidistant Euler scheme achieves an $L^p$-error rate of at least $(1+\alpha)/2$, up to an arbitrary small $\varepsilon$, for all $p\geq 1$ and all $\alpha\in(0, 1]$ in terms of the number of evaluations of the driving Brownian motion $W$. In this article we prove a matching lower error bound for $\alpha\in(0, 1)$. More precisely, we show that for every $\alpha\in(0, 1)$, the $L^p$-error rate $(1+\alpha)/2$ of the Euler scheme in [arXiv:1909.07961v4 (2021)] can not be improved in general by no numerical method based on finitely many evaluations of $W$ at fixed time points. Up to now, this result was known in the literature only for $\alpha=1$. Additionally, we extend a result from [arXiv:2402.13732v2 (2024)] on sharp lower errror bounds for strong approximation of SDEs with a bounded drift coefficient of fractional Sobolev regularity $\alpha\in (0,1)$ and constant diffusion coefficient at time point $1$. We prove that for every $\alpha\in (0,1)$, the $L^p$-error rate $ (1 + \alpha)/2$ that was shown in [arXiv:2101.12185v2 (2022)] for the equidistant Euler scheme can, up to a logarithmic term, not be improved in general by no numerical method based on finitely many evaluations of W at fixed time points. This result was known from [arXiv:2402.13732v2 (2024)] only for $\alpha\in (1/2,1)$ and $p=2$. For the proof of these lower bounds we use variants of the Weierstrass function as a drift coefficient and we employ the coupling of noise technique recently introduced in [arXiv:2010.00915v1 (2020)].