{
  "abstract": "We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for $γ-$Hölder controls with $\\tfrac{1}{2}\\geq γ>\\sqrt{2}-1$, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals $G_0,\\ldots,G_p$ which satisfy compatible higher-order controlled Taylor estimates along $γ$-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\\nabla_ω^jG_0,\\qquad j=1,\\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06228",
  "author_total": 2,
  "authors": [
    {
      "name": "Anna Ananova"
    },
    {
      "name": "Rama Cont"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0kzbjqy9npx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06228",
  "fulltext_url": null,
  "id": "f0kzbjqy9npx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06228",
  "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.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7212071053",
  "pdf_url": "https://arxiv.org/pdf/2609.06228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0kzbjqy9npx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Functional representation and functional calculus for controlled paths",
  "title_text": "Functional representation and functional calculus for controlled paths",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/f0kzbjqy9npx6",
  "venue": "arXiv",
  "versions": []
}