Concentration and exact convergence rates for expected Brownian signatures

TL;DR

This paper establishes the precise convergence rates of expected signatures for piecewise linear approximations of Brownian motion, identifying dominant terms and providing explicit constants under the projective tensor norm.

Contribution

It introduces exact convergence rates for expected Brownian signatures and explicitly computes the leading term constants under the projective tensor norm.

Findings

Convergence rates are explicitly derived for expected signatures.

Identification of dominant words in the signature expansion.

Explicit leading term constants are provided under the projective tensor norm.

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.