Rough path

Summary

A rough path is a path together with extra data that play the role of its iterated integrals. For paths with Hölder regularity of exponent at most $1/2$, such as Brownian paths, these iterated integrals are not determined by the path itself. Once they are given, differential equations driven by the path can be solved path by path, and the solution depends continuously on the rough path. Rough path theory, introduced by Terry Lyons, thus separates the probabilistic and the analytic parts of stochastic calculus.