Affiliation:
1. Institut für Mathematik Technische Universität Berlin and WIAS Berlin Berlin Germany
2. Mathematical Institute University of Oxford Oxford UK
3. John A. Paulson School of Engineering and Applied Sciences Harvard University Cambridge Massachusetts USA
Abstract
AbstractThe signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log‐signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log‐signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log‐signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization.
Funder
Deutsche Forschungsgemeinschaft
Engineering and Physical Sciences Research Council
Lloyd's Register Foundation
Government of the United Kingdom
Office for National Statistics
National Science Foundation