On optimal error rates for strong approximation of SDEs with a drift coefficient of fractional Sobolev regularity
Simon Ellinger, Thomas Müller-Gronbach, Larisa Yaroslavtseva
TL;DR
The paper establishes a near-optimal lower bound for the strong approximation of SDEs with drift in fractional Sobolev spaces $W^{s,2}$, showing that for $s\in(1/2,1)$, no method based on finitely many fixed-time evaluations of the driving Brownian motion can beat the Euler-rate $(1+s)/2$ in $L^2$ up to a logarithmic factor. The authors deploy a coupling-of-noise framework, coupling the Brownian motion at discretization points with an independent copy otherwise, and use a bi-Lipschitz drift-removal transform $G_\mu$ to reduce the problem to driftless SDEs with Lipschitz diffusion. A specially constructed drift $\mu_s$ with $\mu_s=\mathscr{F}h_s$ (where $h_s(z)=1/((e+|z|)^{1/2+s}\ln(e+|z|))$) enables precise lower bounds via occupation-time functionals and Fourier-analytic techniques, yielding $\inf_{\pi} e_2(\pi) \ge c/(\ln(n+1)\,n^{(1+s)/2})$ for fixed-time evaluations. The work closes the gap between known upper bounds and matching lower bounds in this non-Lipschitz setting and highlights the potential gap between adaptive and nonadaptive strategies, leaving open whether similar lower bounds extend to adaptive schemes.
Abstract
We study strong approximation of scalar additive noise driven stochastic differential equations (SDEs) at time point $1$ in the case that the drift coefficient is bounded and has Sobolev regularity $s\in(0,1)$. Recently, it has been shown in [arXiv:2101.12185v2 (2022)] that for such SDEs the equidistant Euler approximation achieves an $L^2$-error rate of at least $(1+s)/2$, up to an arbitrary small $\varepsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In the present article we prove a matching lower error bound for $s\in(1/2,1)$. More precisely we show that, for every $s\in(1/2,1)$, the $L^2$-error rate $(1+s)/2$ 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. Up to now, this result was known in the literature only for the cases $s=1/2-$ and $s=1-$. For the proof we employ the coupling of noise technique recently introduced in [arXiv:2010.00915 (2020)] to bound the $L^2$-error of an arbitrary approximation from below by the $L^2$-distance of two occupation time functionals provided by a specifically chosen drift coefficient with Sobolev regularity $s$ and two solutions of the corresponding SDE with coupled driving Brownian motions. For the analysis of the latter distance we employ a transformation of the original SDE to overcome the problem of correlated increments of the difference of the two coupled solutions, occupation time estimates to cope with the lack of regularity of the chosen drift coefficient around the point $0$ and scaling properties of the drift coefficient.