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We study pathwise approximation of scalar stochastic differential equationsat a single time point or globally in time by means of methods that are basedon finitely many observations of the driving Brownian motion. We prove lowererror bounds in terms of the average number of evaluations of the drivingBrownian motion that hold for every such method under rather mild assumptionson the coefficients of the equation. The underlying simple idea of our analysisis as follows: the lower error bounds known for equations with coefficientsthat have sufficient regularity globally in space should still apply in thecase of coefficients that have this regularity in space only locally, in asmall neighborhood of the initial value. Our results apply to a huge variety ofequations with coefficients that are not globally Lipschitz continuous in spaceincluding Cox-Ingersoll-Ross processes, equations with superlinearly growingcoefficients, and equations with discontinuous coefficients. In many of thesecases the resulting lower error bounds even turn out to be sharp.

Lower Error Bounds for Strong Approximation of Scalar SDEs with non-Lipschitzian Coefficients