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Toric geometry of path signature varieties

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2020

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Colmenarejo, Laura
Galuppi, Francesco

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https://doi.org/10.1016/j.aam.2020.102102
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http://arxiv.org/abs/1903.03779v2

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Advances in Applied Mathematics. Elsevier. 2020, 121, 102102. ISSN 0196-8858. eISSN 1090-2074. Available under: doi: 10.1016/j.aam.2020.102102

Zusammenfassung

In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions.

In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavor, and we prove that it is toric in many cases.

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510 Mathematik

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ISO 690COLMENAREJO, Laura, Francesco GALUPPI, Mateusz MICHALEK, 2020. Toric geometry of path signature varieties. In: Advances in Applied Mathematics. Elsevier. 2020, 121, 102102. ISSN 0196-8858. eISSN 1090-2074. Available under: doi: 10.1016/j.aam.2020.102102
BibTex
@article{Colmenarejo2020Toric-52316,
  year={2020},
  doi={10.1016/j.aam.2020.102102},
  title={Toric geometry of path signature varieties},
  volume={121},
  issn={0196-8858},
  journal={Advances in Applied Mathematics},
  author={Colmenarejo, Laura and Galuppi, Francesco and Michalek, Mateusz},
  note={Article Number: 102102}
}
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