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MiS Preprint

Toric geometry of path signature varieties

Laura Colmenarejo, Francesco Galuppi and Mateusz Michałek


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 flavour, and we prove that it is toric in many cases.

Mar 12, 2019
Mar 14, 2019

Related publications

2020 Repository Open Access
Laura Colmenarejo, Francesco Galuppi and Mateusz Michałek

Toric geometry of path signature varieties

In: Advances in applied mathematics, 121 (2020), p. 102102