User:Sebastian.riedel/draft article on rough paths theory
 | This is not a Wikipedia article: It is an individual user's work-in-progress page, and may be incomplete and/or unreliable. For guidance on developing this draft, see Wikipedia:So you made a userspace draft. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Rough paths theory was developed by Terry Lyons in the early 1990 in a series of articles[1][2].
Overview and history
[edit]Rough paths theory and stochastic analysis
[edit]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
[edit]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
[edit]Pathwise stochastic calculus possible for:
Gaussian processes
[edit]prime example: fractional Brownian motion, Hoermander theory, application in SPDE theory
Markov processes
[edit]Semimartingales
[edit]Levy processes
[edit]Rough paths spaces
[edit]path plus some extra information defines rough path, extra information: iterated integrals, levy-area
Geometric rough paths
[edit]rough paths as paths in a Lie-group
Controlled paths
[edit]Gubinelli
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.