  We consider one-step methods for integrating stochastic differential
equations and prove pathwise convergence using ideas from rough path theory. In
contrast to alternative theories of pathwise convergence, no knowledge is
required of convergence in pth mean and the analysis starts from a pathwise
bound on the sum of the truncation errors. We show how the theory is applied to
the Euler-Maruyama method with fixed and adaptive time-stepping strategies. The
assumption on the truncation errors suggests an error-control strategy and we
implement this as an adaptive time-stepping Euler-Maruyama method using bounded
diffusions. We prove the adaptive method converges and show some computational
experiments.
