User:Sebastian.riedel/draft article on rough paths theory
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Some introductory words.
Overview and history
Deterministic theory designed for solving controlled differential equations, established by T. Lyons, (generalization of Young's theory?), universal limit theorem
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