Differential equations driven by rough paths: An approach via discrete approximation

A theory of systems of differential equations of the form dyⁱ = Σ_j f_jⁱ (y)dxⁱ , where the driving path x(t) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I also give some other examples showing that the main results are reasonably sharp.