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MiS Preprint
28/2019

Toric geometry of path signature varieties

Laura Colmenarejo, Francesco Galuppi and Mateusz Michałek

Abstract

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.

Received:
Mar 12, 2019
Published:
Mar 14, 2019

Related publications

inJournal
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