## The Annals of Probability

### Characteristic functions of measures on geometric rough paths

#### 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 Lévy, Gaussian and Markovian rough paths.

#### Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 4049-4082.

Dates
Revised: October 2015
First available in Project Euclid: 14 November 2016

https://projecteuclid.org/euclid.aop/1479114270

Digital Object Identifier
doi:10.1214/15-AOP1068

Mathematical Reviews number (MathSciNet)
MR3572331

Zentralblatt MATH identifier
06674845

#### Citation

Chevyrev, Ilya; Lyons, Terry. Characteristic functions of measures on geometric rough paths. Ann. Probab. 44 (2016), no. 6, 4049--4082. doi:10.1214/15-AOP1068. https://projecteuclid.org/euclid.aop/1479114270

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