Areas of areas generate the shuffle algebra

TL;DR

This paper introduces the 'area' operator, an anti-symmetrization of the half-shuffle, showing it can recover the iterated-integral signature of a path and characterizing the algebraic structure of the shuffle algebra.

Contribution

It defines the 'area' operator and demonstrates its sufficiency in reconstructing path signatures, providing new insights into the algebraic structure of iterated integrals.

Findings

The area operator corresponds to signed areas of iterated integrals.

Iterated application of the area operator recovers the path signature.

Characterization of generating sets of the shuffle algebra.

Abstract

We consider the anti-symmetrization of the half-shuffle on words, which we call the 'area' operator, since it corresponds to taking the signed area of elements of the iterated-integral signature. The tensor algebra is a so-called Tortkara algebra under this operator. We show that the iterated application of the area operator is sufficient to recover the iterated-integral signature of a path. Just as the "information" the second level adds to the first one is known to be equivalent to the area between components of the path, this means that all the information added by subsequent levels is equivalent to iterated areas. On the way to this main result, we characterize (homogeneous) generating sets of the shuffle algebra. We finally discuss compatibility between the area operator and discrete integration and stochastic integration and conclude with some results on the linear span of the areas of areas.