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