# Toric geometry of path signature varieties

## TL;DR

This paper explores the algebraic geometry of path signature varieties, demonstrating that they are toric in certain classes of paths, thus connecting stochastic analysis with algebraic geometry.

## Contribution

It establishes that the signature varieties of rough paths and axis-parallel paths are toric, providing new geometric insights into path signatures in stochastic analysis.

## Key findings

- Signature varieties of rough paths are toric.
- Signature varieties of axis-parallel paths are toric in many cases.
- Bridges algebraic geometry and stochastic analysis through toric structures.

## 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.