{
  "abstract": "This paper deals with a nonparametric estimator of the drift function $b$ in a delayed differential equation perturbed by a differentiable approximation $W^{\\varepsilon}$ ($\\varepsilon > 0$) of the Brownian motion. Unlike diffusion processes, the solution $Y^{\\varepsilon}$ of this equation has continuously differentiable paths and a memory controlled by $\\varepsilon$. For these reasons, $Y^{\\varepsilon}$ is well-suited for applications in pharmacokinetics or mathematical finance. The first part of the paper deals with probabilistic properties of the delayed equation, which are crucial for our statistical purposes. In particular, the absolute continuity of the probability distribution of $Y_{t}^{\\varepsilon}$ ($t\\in (0,T]$) with respect to Lebesgue's measure on $\\mathbb R$ is investigated via the Malliavin calculus. Thanks to these probabilistic preliminaries, the second part of the paper deals with a (nonparametric) projection least squares estimator $\\widehat b_m$ of $b$ computed from $N$ independent copies of $Y^{\\varepsilon}$ ($0 < m\\leqslant N$). Precisely, a risk bound is established on $\\widehat b_m$, and then a model selection procedure is investigated.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Marie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5zhd8nf71bsex.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "5zhd8nf71bsex",
  "issns": [],
  "landing_url": "https://hal.science/hal-05712907v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-st"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05712907/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5zhd8nf71bsex.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05712907",
      "source": "hal",
      "url": "https://hal.science/hal-05712907v2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-st",
      "subfield": "statistics"
    }
  ],
  "title": "Nonparametric Estimation in Delayed Equations Driven by a Differentiable Approximation of the Brownian Motion",
  "title_text": "Nonparametric Estimation in Delayed Equations Driven by a Differentiable Approximation of the Brownian Motion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/5zhd8nf71bsex",
  "venue": "HAL",
  "versions": []
}