User:Sebastian.riedel/draft article on rough paths theory
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Some introductory words.
Overview and history
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
Markov processes
Semimartingales
Levy processes
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