On non-polynomial lower error bounds for adaptive strong approximation of SDEs

TL;DR

This paper investigates the limitations of adaptive strong approximation methods for SDEs, demonstrating that even with adaptive evaluation points, no polynomial error rate can be guaranteed for certain SDEs with smooth coefficients.

Contribution

It extends previous results by showing that adaptive evaluation strategies do not overcome the slow convergence rates for approximating solutions of specific SDEs.

Findings

Adaptive methods cannot guarantee polynomial error decay for certain SDEs.

There exist SDEs with smooth coefficients where approximation error decreases arbitrarily slowly.

Adaptive evaluation sites do not improve convergence rates beyond known bounds.

Abstract

Recently, it has been shown in [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43, 2 (2015), 468--527] that there exists a system of stochastic differential equations (SDE) on the time interval $[0,T]$ with infinitely often differentiable and bounded coefficients such that the Euler scheme with equidistant time steps converges to the solution of this SDE at the final time in the strong sense but with no polynomial rate. Even worse, in [Jentzen, A., M\"uller-Gronbach, T., and Yaroslavtseva, L. On stochastic differential equations with arbitrary slow convergence rates for strong approximation, Commun. Math. Sci. 14, 7 (2016), 1477-1500] it has been shown that for any sequence $(a_n)_{n\in\mathbb N}\subset (0,\infty)$, which may converge to zero arbitrary slowly, there exists an SDE on $[0,T]$ with infinitely often differentiable and bounded coefficients such that no approximation of the solution of this SDE at the final time based on $n$ evaluations of the driving Brownian motion at fixed time points can achieve a smaller absolute mean error than the given number $a_n$. In the present article we generalize the latter result to the case when the approximations may choose the location as well as the number of the evaluation sites of the driving Brownian motion in an adaptive way dependent on the values of the Brownian motion observed so far.