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2015 Concentration and exact convergence rates for expected Brownian signatures
Hao Ni, Weijun Xu
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Electron. Commun. Probab. 20: 1-11 (2015). DOI: 10.1214/ECP.v20-3636


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.


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Hao Ni. Weijun Xu. "Concentration and exact convergence rates for expected Brownian signatures." Electron. Commun. Probab. 20 1 - 11, 2015.


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

zbMATH: 1307.60110
MathSciNet: MR3304414
Digital Object Identifier: 10.1214/ECP.v20-3636

Primary: 60G15

Keywords: Brownian motion , Expected signature

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