Larisa Yaroslavtseva

Simon Ellinger, Thomas Müller-Gronbach, Larisa Yaroslavtseva

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)].

Thomas Müller-Gronbach, Larisa Yaroslavtseva

In recent work of Hairer, Hutzenthaler and Jentzen, see [9], a stochastic differential equation (SDE) with infinitely often differentiable and bounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In the present paper we show that this type of bad performance for quadrature of SDEs with infinitely often differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely often differentiable and bounded by one coefficients such that the first order derivatives of all diffusion coefficients are bounded by one as well and the first order derivatives of all drift coefficients are uniformly dominated by a single real-valued function and such that the corresponding sequence of mean absolute errors for approximation of the expected value of the first component of the solution at the final time can not converge to zero faster than the given speed.

Thomas Müller-Gronbach, Larisa Yaroslavtseva

Recently a lot of effort has been invested to analyze the $L_p$-error of the Euler-Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_p$-error rate of at least $1/(2p)-$ has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler-Maruyama scheme in fact achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$ as in the case of SDEs with Lipschitz coefficients.