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VARIETIES OF SIGNATURE TENSORS

Part of: Computational aspects in algebraic geometry Stochastic analysis

Published online by Cambridge University Press:  04 April 2019

CARLOS AMÉNDOLA ,
PETER FRIZ  and
BERND STURMFELS
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.

MSC classification

Primary: 14Q15: Higher-dimensional varieties 60H99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e10
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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