  In the past decade, an intensive study of strong approximation of stochastic
differential equations (SDEs) with a drift coefficient that has discontinuities
in space has begun. In the majority of these results it is assumed that the
drift coefficient satisfies piecewise regularity conditions and that the
diffusion coefficient is globally Lipschitz continuous and non-degenerate at
the discontinuities of the drift coefficient. Under this type of assumptions
the best $L_p$-error rate obtained so far for approximation of scalar SDEs at
the final time is $3/4$ in terms of the number of evaluations of the driving
Brownian motion. In the present article we prove for the first time in the
literature sharp lower error bounds for such SDEs. We show that for a huge
class of additive noise driven SDEs of this type the $L_p$-error rate $3/4$ can
not be improved.
  For the proof of this result we employ a novel technique by studying
equations with coupled noise: we reduce the analysis of the $L_p$-error of an
arbitrary approximation based on evaluation of the driving Brownian motion at
finitely many times to the analysis of the $L_p$-distance of two solutions of
the same equation that are driven by Brownian motions that are coupled at the
given time-points and independent, conditioned on their values at these points.
To obtain lower bounds for the latter quantity, we prove a new quantitative
version of positive association for bivariate normal random variables $(Y,Z)$
by providing explict lower bounds for the covariance $\text{Cov}(f(Y),g(Z))$ in
case of piecewise Lipschitz continuous functions $f$ and $g$. In addition it
turns out that our proof technique also leads to lower error bounds for
estimating occupation time functionals $\int_0^1 f(W_t)\, dt$ of a Brownian
motion $W$, which substantially extends known results for the case of $f$ being
an indicator function.
