Exact finite-sample inference for multi-mixed fractional Brownian motion with drift
Abstract
In this paper we study a linear drift perturbed by a superposition of m independent fractional Brownian motions with known Hurst parameters and a common scale, observed at N equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift θ and of the scale α2 in closed form and obtain their exact finite-sample joint law: θ̂ is Gaussian, Nα̂ 2/α2 is chi-square with N−1 degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every N≥2, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with θ̂ attaining the Cramér--Rao bound at every N, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.
Related
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BibTeX
@misc{alharby2026exact,
title = {{Exact finite-sample inference for multi-mixed fractional Brownian motion with drift}},
author = {Afrah Al-Harby and Ezzedine Mliki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16976},
eprint = {2609.16976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16976},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact finite-sample inference for multi-mixed fractional Brownian motion with drift
AU - Afrah Al-Harby
AU - Ezzedine Mliki
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16976
UR - https://arxiv.org/abs/2609.16976
AB - In this paper we study a linear drift perturbed by a superposition of m independent fractional Brownian motions with known Hurst parameters and a common scale, observed at N equidistant times. Inference for such models is usually asymptotic; we show that here it is exact. We derive the maximum likelihood estimators of the drift θ and of the scale α^2 in closed form and obtain their exact finite-sample joint law: θ̂ is Gaussian, Nα̂^ 2/α^2 is chi-square with N−1 degrees of freedom, and the two are independent. As this law is free of every model parameter, we deduce Student and chi-square confidence intervals and tests of exact level for every N≥2, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with θ̂ attaining the Cramér--Rao bound at every N, that both are strongly consistent and asymptotically normal, and that the drift estimators form, in law, a Brownian motion run along their own variance scale. A sharp non-asymptotic bound shows that the accuracy of the drift is governed by the length of the observation window and not by the mesh, and a Monte Carlo study confirms exact coverage, even at small sample sizes, and quantifies what is lost when the Hurst vector is misspecified.
ER -