{
  "abstract": "We study finite-state approximations of Brownian-driven Markovian backward stochastic differential equations when the same one-step increment is used both for weak propagation and for martingale representation. For a centered isotropic state-independent increment with vanishing third moment, we prove first-order convergence of the Markov value field, its spatial gradient, and the predictable $Z$-field under suitable smoothness assumptions. The corresponding orthogonal martingale has maximal $L^2$ size $O(h^{1/2})$. We identify its quadratic representation defect and prove a joint functional limit theorem for the state process and the normalized orthogonal martingale, which in particular yields a mixed-normal terminal limit with path-dependent conditional variance. On the trace-free quadratic sector, the representation defect has the sharp universal operator lower bound $d/[2(d+2)]$; equality holds exactly for normalized laws that are weighted spherical 4-designs. In dimension $d\\geq 2$, vanishing third moment is incompatible with exact scalar one-step predictable representation for any finite-state centered isotropic increment. Consequently, every such exact-representation law has a nonzero cubic direction, and a smooth Fourier terminal condition converts this direction into a sharp global $h^{1/2}$ weak error. For the centrally symmetric regular-simplex rule we also identify the leading quadratic GKW component, compute the exact first-order asymptotic of the accumulated orthogonal-martingale variance, and recover it from the general functional limit. Thus the same third-order tensor links weak accuracy and predictable representation, yielding an accuracy-representation tradeoff for finite-state BSDE approximations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20880",
  "author_total": 1,
  "authors": [
    {
      "name": "Akihiro Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3g18cy60zh3tn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20880",
  "fulltext_url": null,
  "id": "3g18cy60zh3tn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20880",
  "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": "W7222520353",
  "pdf_url": "https://arxiv.org/pdf/2610.20880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3g18cy60zh3tn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Third-moment cancellation and predictable representation in finite-state BSDE approximation",
  "title_text": "Third-moment cancellation and predictable representation in finite-state BSDE approximation",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:45Z",
  "url": "https://research.draconitic.com/paper/3g18cy60zh3tn",
  "venue": "arXiv",
  "versions": []
}