Varieties of Signature Tensors

Carlos Amendola, Bernd Sturmfels, and Peter Friz

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Submission date: 24. Apr. 2018
Pages: 46
published in: Forum of mathematics / Sigma, 7 (2019), art-no. e10 
DOI number (of the published article): 10.1017/fms.2019.3
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Abstract:
The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is here examined through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.