We study the pathwise approximation of stochastic differential equations with respect to the global error in the L₂-norm. For general scalar equations we establish sharp lower error bounds that hold for arbitrary methods that are based on a finite number of observations of a trajectory of the driving Brownian motion. As a consequence higher order methods do not exist if the global error is analyzed. We introduce a Milstein scheme with adaptive step-size control which performs asymptotically optimal. In particular, the new method is more efficient than an equidistant discretization. Moreover, we present results on pathwise approximation of stochastic differential systems with respect to the global error in the uniform norm.

MINISIMPOSIUM SESSION: 7. Numerical methods for Stochastic equations