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VARIETIES OF SIGNATURE TENSORS

Published online by Cambridge University Press:  04 April 2019

CARLOS AMÉNDOLA
Affiliation:
Technische Universität München, Germany; carlos.amendola@tum.de
PETER FRIZ
Affiliation:
Technische Universität Berlin and WIAS Berlin, Germany; friz@math.tu-berlin.de
BERND STURMFELS
Affiliation:
MPI for Mathematics in the Sciences, Leipzig and UC Berkeley, Germany; bernd@mis.mpg.de

Abstract

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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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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