Concentration and exact convergence rates for expected Brownian signatures

2012/08/06 by Hao Ni, Ni, Hao, Weijun Xu +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.1208.1067

openalex publication_date 2012/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \RR^d. In this note, 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 projective tensor norm, we give the explicit value of the leading term constant.

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• Expected signatures via partial integration, coordinate change and symmetrization

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