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Recently, it has been shown in [Jentzen, A., M\"uller-Gronbach, T., andYaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system ofautonomous stochastic differential equations (SDE) on the time interval $[0,T]$with infinitely differentiable and bounded coefficients such that no strongapproximation method based on evaluation of the driving Brownian motion atfinitely many fixed times in $[0,T]$, e.g. on an equidistant grid, can convergein absolute mean to the solution at the final time with a polynomial rate interms of the number of Brownian motion values that are used. In the literatureon strong approximation of SDEs, polynomial error rate results are typicallyachieved under the assumption that the first order derivatives of thecoefficients of the equation satisfy a polynomial growth condition. Thisassumption is violated for the pathological SDEs from the above mentionednegative result. However, in the present article we construct an SDE withsmooth coefficients that have first order derivatives of at most linear growthsuch that the solution at the final time can not be approximated with apolynomial rate, whatever method based on observations of the driving Brownianmotion at finitely many fixed times is used. Most interestingly, it turns outthat using a method that adjusts the number of evaluations of the drivingBrownian motion to its actual path, the latter SDE can be approximated withrate 1 in terms of the average number of evaluations that are used. To the bestof our knowledge, this is only the second example in the literature of an SDEfor which there exist adaptive methods that perform superior to non-adaptiveones with respect to the convergence rate.

A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives