Characteristic functions of measures on geometric rough paths

Understand

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.