Differential equations driven by rough signals

This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process. However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see 10, 1). We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations 8 have been constructed as a consequence of these results.