Weak and strong discretization errors for signatures of Gaussian processes
Abstract
Let X be a continuous Gaussian process in ℝd, d≥2, with independent components of roughness H∈(1/4,1), possibly with a linear drift, and let Xn be its piecewise-linear interpolant on a mesh of width δ. We study how well the truncated signature of Xn approximates that of the natural lift of X, in mean (the weak error) and in L2 (the strong error). Over a class of drivers defined by quantitative regularity conditions, the weak error at every fixed truncation level is O(δγ(H)) with γ(H)=min(2H,4H−1), uniformly over the class, and for truncation levels M≥4 fractional Brownian motion attains this rate. For centred drivers the weak error vanishes at levels $1$ to $3$, Gaussian or not. It first appears at level~$4$, where we compute its leading constant for independent fractional Brownian motion in d=2: on (1/2,1) it is 1/(2√3), independent of H, and on (1/4,1/2) it is given by a ζ-series. For fractional Brownian motion with drift, a level-$3$ error of the same order appears for H≥1/2. Below 1/2 the drift reaches the rate only at levels ≥5 (proved for d≥3). The strong error is of exact order δ2H−1/2 for fractional Brownian motion with H∈(1/2,1), so in mean square the weak error lies a full power of δ below it. For H≤1/2 the strong error is O(δϑ) for every ϑ<2H−1/2, uniformly over the class, and the exponent 2H−1/2 cannot be improved. Finally, we construct pairs of laws in the class whose mesh data have the same law but whose expected signatures differ by order δγ(H). Hence no quantity computed from the mesh data has mean within o(δγ(H)) of the expected signature uniformly over the class.
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BibTeX
@misc{huber2026weak,
title = {{Weak and strong discretization errors for signatures of Gaussian processes}},
author = {Florian Huber},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08588},
eprint = {2611.08588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08588},
}
RIS
TY - UNPB
TI - Weak and strong discretization errors for signatures of Gaussian processes
AU - Florian Huber
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08588
UR - https://arxiv.org/abs/2611.08588
AB - Let X be a continuous Gaussian process in ℝ^d, d≥2, with independent components of roughness H∈(1/4,1), possibly with a linear drift, and let X^n be its piecewise-linear interpolant on a mesh of width δ. We study how well the truncated signature of X^n approximates that of the natural lift of X, in mean (the weak error) and in L^2 (the strong error). Over a class of drivers defined by quantitative regularity conditions, the weak error at every fixed truncation level is O(δ^γ(H)) with γ(H)=min(2H,4H−1), uniformly over the class, and for truncation levels M≥4 fractional Brownian motion attains this rate. For centred drivers the weak error vanishes at levels $1$ to $3$, Gaussian or not. It first appears at level~$4$, where we compute its leading constant for independent fractional Brownian motion in d=2: on (1/2,1) it is 1/(2√3), independent of H, and on (1/4,1/2) it is given by a ζ-series. For fractional Brownian motion with drift, a level-$3$ error of the same order appears for H≥1/2. Below 1/2 the drift reaches the rate only at levels ≥5 (proved for d≥3). The strong error is of exact order δ^2H−1/2 for fractional Brownian motion with H∈(1/2,1), so in mean square the weak error lies a full power of δ below it. For H≤1/2 the strong error is O(δ^ϑ) for every ϑ<2H−1/2, uniformly over the class, and the exponent 2H−1/2 cannot be improved. Finally, we construct pairs of laws in the class whose mesh data have the same law but whose expected signatures differ by order δ^γ(H). Hence no quantity computed from the mesh data has mean within o(δ^γ(H)) of the expected signature uniformly over the class.
ER -