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Rough paths theory was developed by Terry Lyons in the early 1990 in a series of articles[1][2].
Overview and history
editRough paths theory and stochastic analysis
editAllows 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
editLift 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
editPathwise stochastic calculus possible for:
Gaussian processes
editprime example: fractional Brownian motion, Hoermander theory, application in SPDE theory
Markov processes
editSemimartingales
editLevy processes
editRough paths spaces
editpath plus some extra information defines rough path, extra information: iterated integrals, levy-area
Geometric rough paths
editrough paths as paths in a Lie-group
Controlled paths
editGubinelli
References
edit- ^ Lyons, T.; Qian, Z. (1997). "Flow Equations on Spaces of Rough Paths". Journal of Functional Analysis. 149: 135. doi:10.1006/jfan.1996.3088.
- ^ Lyons, T. (1998). "Differential equations driven by rough signals". Revista Matemática Iberoamericana: 215–310. doi:10.4171/RMI/240.