{
  "abstract": "We introduce a notion of viscosity solution for Hamilton--Jacobi--Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \\[ \\left(\\partial_t+\\frac12Δ+b\\cdot\\nabla\\right)h(t,x) =-H(t,x,h(t,x),\\nabla h(t,x)), \\] where $b$ is singular in the sense of \\cite{paradistrib} and has regularity $\\mathcal C^{-α}$ for $α\\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward--backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13828",
  "author_total": 4,
  "authors": [
    {
      "name": "Dirk Becherer"
    },
    {
      "name": "Nicolas Perkowski"
    },
    {
      "name": "Yuchen Sun"
    },
    {
      "name": "Carlos Villanueva Mariz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0gb7kzf8a1exc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13828",
  "fulltext_url": null,
  "id": "0gb7kzf8a1exc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13828",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213324616",
  "pdf_url": "https://arxiv.org/pdf/2609.13828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0gb7kzf8a1exc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2609.13828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control",
  "title_text": "Viscosity Solutions for Singular HJB Equations: BSDE Representations and Stochastic Control",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/0gb7kzf8a1exc",
  "venue": "arXiv",
  "versions": []
}