  We study pathwise approximation of scalar stochastic differential equations
at a single point. We provide the exact rate of convergence of the minimal
errors that can be achieved by arbitrary numerical methods that are based (in a
measurable way) on a finite number of sequential observations of the driving
Brownian motion. The resulting lower error bounds hold in particular for all
methods that are implementable on a computer and use a random number generator
to simulate the driving Brownian motion at finitely many points. Our analysis
shows that approximation at a single point is strongly connected to an
integration problem for the driving Brownian motion with a random weight.
Exploiting general ideas from estimation of weighted integrals of stochastic
processes, we introduce an adaptive scheme, which is easy to implement and
performs asymptotically optimally.
