On optimal error rates for strong approximation of SDEs with a Hölder continuous drift coefficient
Simon Ellinger, Thomas Müller-Gronbach, Larisa Yaroslavtseva
TL;DR
This paper addresses the problem of strong approximation for scalar SDEs with bounded, Hölder-continuous drift and constant diffusion, focusing on the final-time error when using finite evaluations of the driving Brownian motion. By transforming the SDE to a zero-drift form via a bi-Lipschitz mapping and employing a Milstein-type scheme, the authors derive lower bounds that match the known Euler scheme rate of $(1+\alpha)/2$ for $\alpha\in(0,1)$, up to logarithmic factors. They extend sharp lower bounds to drift coefficients with fractional Sobolev regularity, showing that the $(1+\alpha)/2$ rate cannot be improved by any method based on finitely many Brownian evaluations, up to a polylog term. The results rely on carefully constructed Weierstrass-type drift functions and a noise-coupling technique, providing a deeper understanding of the complexity of strong SDE approximation and the near-optimality of the Euler scheme in this setting.
Abstract
In the present article we study strong approximation of solutions of scalar stochastic differential equations (SDEs) with bounded and $α$-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 all $α\in(0, 1]$ 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)] 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 $α=1$. 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 $α\in (0,1)$ and constant diffusion coefficient at time point $1$. We prove that for every $α\in (0,1)$, the $L^p$-error rate $ (1 + α)/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 $α\in (1/2,1)$ 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)].