Differential Equations Driven by Π-Rough Paths
By Lajos Gergely Gyurkó. Proceedings of the Edinburgh Mathematical Society, 2012
223 citations recorded by the source corpus.
Abstract This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lip γ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.
Subfields: Computational Mechanics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/826bf0ce5e3d4193a1aedb702a0d7e12fad32adc
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In Computational Mechanics
- Huilin Zhang
- Martina Hofmanov'a
- Massimiliano Gubinelli
- Christa Cuchiero
- Martin Hairer
- Samy Tindel
- Peter K. Friz
- Franco Flandoli
In Computational Theory and Mathematics
- Massimiliano Gubinelli
- Peter K. Friz
- Samy Tindel
- Martin Hairer
- H. Boedihardjo
- Terry Lyons
- T. Cass
- Martina Hofmanov'a
Citations in the corpus
- Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “view of existence and uniqueness results, the appropriate way to measure the regularity of the Vi’s turns out to be the notion of Lipschitz-γ (short: Lip-γ) in the sense of Stein (see [12] and [26]).”
- A priori estimates for rough PDEs with application to rough conservation laws (2016, background): “Since its invention, Rough path theory (RPT) has been developed very intensively to provide robust analysis of ODEs and a novel way to define solutions of stochastic differential equations driven by non semimartingale signals (for a comprehensive review see the book of Friz and Victoir [25] and the ”
- Differential equations driven by rough paths with jumps (2017, background): “[35, 12]), one has the following Taylor’s expansion, for any k < [p]− 1, s < t,”
- Joint calibration to SPX and VIX options with signature‐based models (2023, background): “The proof of the paths version of the next result for can be found for instance in Ree (1958) or Lyons et al. (2007).”
- The Signature of a Rough Path: Uniqueness (2014, background): “, jn) as the set of ordered shu es (see p72 [22]).”