User:Sebastian.riedel/draft article on rough paths theory

Rough paths theory was developed by Terry Lyons in the early 1990 in a series of articles.

Rough paths theory and stochastic analysis
Allows for pathwise stochastic calculus in several dimensions (even in Banach spaces), example: Stratonovich SDE, advantage over Ito's theory: strong regularity of the Ito-Lyons map

Brownian motion sample paths seen as rough paths
Lift of a Brownian motion, connection to Stratonovich integration, continuity of Ito-Lyons map may be used to prove support theorem, Freidlin-Wentzel large deviations, Wong-Zakai theorem

Other Stochastic processes
Pathwise stochastic calculus possible for:

Gaussian processes
prime example: fractional Brownian motion, Hoermander theory, application in SPDE theory

Rough paths spaces
path plus some extra information defines rough path, extra information: iterated integrals, levy-area

Geometric rough paths
rough paths as paths in a Lie-group

Controlled paths
Gubinelli