Electronic Communications in Probability

Concentration and exact convergence rates for expected Brownian signatures

Hao Ni and Weijun Xu

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The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\mathbb{R}^{d}$. In this article, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coefficients are of the leading order, and the convergence is concentrated on this subset of words. Moreover, under the choice of $l^{1}$ tensor norm, we give the explicit value of the leading term constant.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 8, 11 pp.

Accepted: 28 January 2015
First available in Project Euclid: 7 June 2016

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes

expected signature Brownian motion

This work is licensed under a Creative Commons Attribution 3.0 License.


Ni, Hao; Xu, Weijun. Concentration and exact convergence rates for expected Brownian signatures. Electron. Commun. Probab. 20 (2015), paper no. 8, 11 pp. doi:10.1214/ECP.v20-3636. https://projecteuclid.org/euclid.ecp/1465320935

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