Rough path theory

The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in stochastic analysis, where it has given a new perspective on It\^o calculus and a meaning to stochastic differential equations driven by irregular paths outside the setting of semi-martingales. In this survey, we present some of the main ideas that enter rough path theory. We discuss complementary notions of solutions for rough differential equations and the related notion of path signature, and give several applications and generalisations of the theory.