Characteristic functions of measures on geometric rough paths

TL;DR

This paper introduces a characteristic function for probability measures on geometric rough path signatures, establishing conditions for unique determination and convergence, with applications to Lévy, Gaussian, and Markovian rough paths.

Contribution

It defines a new characteristic function for measures on rough path signatures and analyzes its properties, advancing the understanding of measure determination and convergence in rough path theory.

Findings

Unique determination of measures by expected signature

Analytic properties of the characteristic function

Method of moments for weak convergence

Abstract

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from L\'evy, Gaussian and Markovian rough paths.