Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of Hölder or Sobolev regularity using a Weierstraß scale

Abstract

We study strong approximation of solutions of SDEs with bounded $α$-Hölder 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 $(1+α)/2$, up to an arbitrary small $\varepsilon$, for all $p\geq 1$ and $α\in (0,1]$, in terms of the number of evaluations of the driving Brownian motion $W$. In this article, we prove a matching lower error bound for $α\in (0,1)$. More precisely, we show that for every $α\in (0,1)$, the $L^p$-error rate $(1+α)/2$ of the Euler scheme in [arXiv:1909.07961v4 (2021)] cannot be improved in general by any numerical method based on finitely many evaluations of $W$ in $[0,1]$. Up to now, this result was known only for $α=1$. Even stronger, an $L^p$-error rate better than $(1+α)/2$ cannot be achieved, even if algorithms additionally use a finite number of time integrals of $W$. Thus, Wagner-Platen type schemes are not superior to the Euler scheme. Additionally, we extend a result from [arXiv:2402.13732v2 (2024)] on final time approximation of SDEs with a bounded drift coefficient of fractional Sobolev regularity $α\in (0,1)$. We prove that for every $α\in (0,1)$, the $L^p$-error rate $(1+ α)/2$ shown in [arXiv:2101.12185v2 (2022)] for the equidistant Euler scheme can essentially not be improved by any numerical method based on finitely many evaluations and time integrals of $W$ in $[0,1]$. This lower bound was known from [arXiv:2402.13732v2 (2024)] only for $α\in (1/2,1)$, $p=2$ and numerical methods based on finitely many evaluations of $W$. For the proof of our results we use variants of the Weierstrass function as a drift coefficient and we extend the coupling of noise technique introduced in [arXiv:2010.00915v1 (2020)].