Differential Equations Driven By Rough Paths: An Approach Via Discrete Approximation

A theory of systems of differential equations of the form dy(i) = Sigma(j)f(j)(i) (y) dx(i), 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.