{
  "abstract": "We establish functional quantization and empirical Wasserstein rates for continuous Itô processes under the supremum norm. We assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $ρ$-moment for some $ρ>1$. Under this assumption, the $n$-point $L^q$ quantization error is at most $C(\\log n)^{-1/2}$ for every $1\\leq q<ρ$. The known Brownian lower bound shows that the exponent $1/2$ is optimal over this class. Our proof combines an adaptive dyadic time partition with localization according to the size of the integrands. A general transfer principle yields the sharp mean rate $(\\log N)^{-1/2}$ for the $p$-Wasserstein distance between the empirical law of $N$ independent copies and their common path law, together with nonasymptotic deviation bounds, whenever $1\\leq p<ρ$. Applications include empirical path-law estimates for path-dependent SDEs and a path-space limit-theory estimate for path-dependent McKean--Vlasov interacting particle systems with common noise.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.07879",
  "author_total": 2,
  "authors": [
    {
      "name": "Xihao He"
    },
    {
      "name": "Fengyi Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4aw8bkvpya1t0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.07879",
  "fulltext_url": null,
  "id": "4aw8bkvpya1t0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.07879",
  "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.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7202156610",
  "pdf_url": "https://arxiv.org/pdf/2608.07879",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4aw8bkvpya1t0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2608.07879",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.07879"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp functional quantization and empirical Wasserstein rates for Itô processes",
  "title_text": "Sharp functional quantization and empirical Wasserstein rates for Itô processes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4aw8bkvpya1t0",
  "venue": "arXiv",
  "versions": []
}