Rough path theory

Understand

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.