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Published online by Cambridge University Press:  04 April 2019

Technische Universität München, Germany;
Technische Universität Berlin and WIAS Berlin, Germany;
MPI for Mathematics in the Sciences, Leipzig and UC Berkeley, Germany;


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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 examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths 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.

Research Article
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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