{
  "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 $α^{2}$ in closed form and obtain their exact finite-sample joint law: $\\widehatθ$ is Gaussian, $N\\widehatα^{\\,2}/α^{2}$ 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 $N\\ge2$, whatever the Hurst vector. We also prove that the estimators are uniformly minimum variance unbiased with $\\widehatθ$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16976",
  "author_total": 2,
  "authors": [
    {
      "name": "Afrah Al-Harby"
    },
    {
      "name": "Ezzedine Mliki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3cws80a88ep20.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16976",
  "fulltext_url": null,
  "id": "3cws80a88ep20",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16976",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7213392077",
  "pdf_url": "https://arxiv.org/pdf/2609.16976",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3cws80a88ep20.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16976",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16976"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Exact finite-sample inference for multi-mixed fractional Brownian motion with drift",
  "title_text": "Exact finite-sample inference for multi-mixed fractional Brownian motion with drift",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/3cws80a88ep20",
  "venue": "arXiv",
  "versions": []
}