  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.
