Toric geometry of path signature varieties

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COLMENAREJO, Laura, Francesco GALUPPI, Mateusz MICHALEK, 2020. Toric geometry of path signature varieties. In: Advances in Applied Mathematics. Elsevier. 121, 102102. ISSN 0196-8858. eISSN 1090-2074. Available under: doi: 10.1016/j.aam.2020.102102

@article{Colmenarejo2020Toric-52316, title={Toric geometry of path signature varieties}, year={2020}, doi={10.1016/j.aam.2020.102102}, volume={121}, issn={0196-8858}, journal={Advances in Applied Mathematics}, author={Colmenarejo, Laura and Galuppi, Francesco and Michalek, Mateusz}, note={Article Number: 102102} }

eng 2021-01-08T10:11:25Z Colmenarejo, Laura Michalek, Mateusz 2021-01-08T10:11:25Z Galuppi, Francesco Galuppi, Francesco Colmenarejo, Laura 2020 terms-of-use Toric geometry of path signature varieties 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.<br /><br />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. Michalek, Mateusz

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