The optimal uniform approximation of systems of stochastic differential equations

We analyze numerical methods for the pathwise approximation of a system of stochastic differential equations. As a measure of performance we consider the qth mean of the maximum distance between the solution and its approximation on the whole unit interval. We introduce an adaptive discretization that takes into account the local smoothness of every trajectory of the solution. The resulting adaptive Euler approximation performs asymptotically optimal in the class of all numerical methods that are based on a finite number of observations of the driving Brownian motion.