Step size control for the uniform approximation of systems of stochastic differential equations with additive noise

We analyze the pathwise approximation for systems of stochastic differential equations.The pathwise distance between the solution and its approximation is measured globally on the unit interval in the L.-norm, and we study the expectation of this distance. For systems with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary methods which use discrete observations of a Brownian path. The optimal order is achieved by an Euler scheme with adaptive step-size control. We illustrate the superiority of the adaptive method compared to equidistant discretization by a simulation experiment.