Effective algorithms for inverting the signature of a path

Thesis

The signature of a path is an important concept in rough paths theory. It has been proved in the literature that the terms in the signature of a path are bounded above, and it would then be interesting to consider whether a lower bound exists for the signature of a bounded-variation path. It has also been proved that the signature of a bounded-variation path is unique up to some modifications, then a natural question is reconstructing the path from its signature, i.e. inverting the signature of a path.

We show the connection between the two questions above, and provide practical methods for signature inversion. First we prove a result about the super-multiplicativity and the decay of the signature of a path with bounded variation, then we describe the method of symmetrisation, which was first introduced by Lyons and Xu, and demonstrate an algorithm to invert the signature of a monotone path. Moreover we introduce the method of inverting the signature by insertion, and provide examples using the insertion method to invert the signature of a path. We compare these two methods of signature inversion, and illustrate the differences with computational results.

Authors: Chang, J

• DOI: 10.5287/ora-gw55yepdj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Keywords: Rough paths
• Subjects: Mathematics