{
  "abstract": "We investigate the nonparametric estimation of the diffusion matrix in stochastic differential equations featuring multidimensional, strong mixing covariate processes. We propose a flexible statistical framework based on general function classes that does not require a linear basis representation, rendering our results directly applicable to deep neural network estimators. Our approach employs a two-step estimation procedure: constructing a preliminary nonparametric quasi-likelihood estimator and subsequently regularizing it via a $β$-Hölder class approximation. We establish general risk comparison theorems between empirical and generalization risks for arbitrary estimators without relying on a specific probabilistic structure of the underlying process. In diffusion matrix learning based on $n + 1$ observations over the time interval $[0,T]$, the derived upper bounds capture the intrinsic interplay between the $T$-rate, associated with the mixing behavior of the covariate process, and the intrinsic $n$-rate governing the volatility estimation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13898",
  "author_total": 2,
  "authors": [
    {
      "name": "Arnaud Gloter"
    },
    {
      "name": "Nakahiro Yoshida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/agbq3nanm58ba.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13898",
  "fulltext_url": null,
  "id": "agbq3nanm58ba",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13898",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
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  ],
  "openalex": "W7213348500",
  "pdf_url": "https://arxiv.org/pdf/2609.13898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/agbq3nanm58ba.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.13898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Risk comparison theorems and application to deep learning of diffusion coefficients",
  "title_text": "Risk comparison theorems and application to deep learning of diffusion coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:51Z",
  "url": "https://research.draconitic.com/paper/agbq3nanm58ba",
  "venue": "arXiv",
  "versions": []
}