Differential Equations Driven by Rough Paths: an Approach via Discrete Approximation

A theory of systems of differential equations of the form dy = ∑ j f i j(y)dx , where the driving path x(t) is non-differentiable, has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp.