{
  "abstract": "We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06593",
  "author_total": 2,
  "authors": [
    {
      "name": "Mao Fabrice Djete"
    },
    {
      "name": "Mattia Martini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cz8re4ch5qpf7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06593",
  "fulltext_url": null,
  "id": "cz8re4ch5qpf7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06593",
  "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.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
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  "openalex": "W7220662694",
  "pdf_url": "https://arxiv.org/pdf/2610.06593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cz8re4ch5qpf7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences",
  "title_text": "W-BSDEs with Common Noise: Well-posedness, Characterization, and Consequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/cz8re4ch5qpf7",
  "venue": "arXiv",
  "versions": []
}