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

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Rough paths theory was developed by Terry Lyons in the early 1990 in a series of articles^[1]^[2]. There are now several monographs^[2]

Overview and history

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

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1006/jfan.1996.3088, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1006/jfan.1996.3088 instead.
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[LQ97-1] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1006/jfan.1996.3088, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1006/jfan.1996.3088 instead.

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[1]

[2]