Characteristic functions of measures on geometric rough paths

By Ilya Chevyrev, Terry Lyons. 2013

107 citations recorded by the source corpus.

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 Levy, Gaussian and Markovian rough paths.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bff4310e8482c75106e3d084f9e92025200ee387

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Peter K. Friz
2. A. Shekhar
3. Nicolas Victoir
4. M. Folz

In Mathematical Physics

1. Ilya Chevyrev
2. Peter K. Friz
3. A. Shekhar
4. Nicolas Victoir
5. Massimiliano Gubinelli
6. Terry Lyons
7. Martin Hairer
8. Huilin Zhang

In Statistics and Probability

1. Peter K. Friz
2. Ilya Chevyrev
3. Martin Hairer

Citations in the corpus

• Canonical RDEs and general semimartingales as rough paths (2017, background): “, determines conditions under which the sequence of expectations ( E [ πmX̄0,T ]) m≥0 uniquely determines the law of the full signature of X̄ (see [15, Theorem 54] where the moment problem was discussed for the Lévy case, and [7, 10] for other families of random geometric rough paths).”
• General Rough integration, Levy Rough paths and a Levy--Kintchine type formula (2012, background): “As in [7], we need to show that exp(C), equivalently C = (C0,C1, C2, .”