{
  "abstract": "We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a $p$-th variation tensor along a sequence of partitions, for an arbitrary integer $p$; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite $p$-th variation and derive a change of variable formula for smooth functions of such paths. For $p=2$, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced $p$-jet of the test function, whose canonical highest-order component is determined by the $p$-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12225",
  "author_total": 1,
  "authors": [
    {
      "name": "Rama Cont"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cd1f6btdtem3z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12225",
  "fulltext_url": null,
  "id": "cd1f6btdtem3z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12225",
  "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.DG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7202353169",
  "pdf_url": "https://arxiv.org/pdf/2608.12225",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cd1f6btdtem3z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.12225",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12225"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Higher-order variation and pathwise Ito calculus on manifolds",
  "title_text": "Higher-order variation and pathwise Ito calculus on manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/cd1f6btdtem3z",
  "venue": "arXiv",
  "versions": []
}