{
  "abstract": "We study the Euler-Maruyama approximation of the degenerate stochastic differential equation $dX_t = |X_t|^α\\,dW_t$, $0 < α< 1/2$, $X_0 = 0$, for which uniqueness in law fails. The Euler scheme started exactly from zero is trapped by the absorbing solution, whereas a nonzero initialization $x_h \\to 0$ can select a different weak solution. We identify the initialization-dependent limiting regimes. The critical spatial scale is $r_h = h^{1/[2(1-α)]}$, and the relevant depth parameter is $Θ_h = h\\log(1+\\log^+(r_h/|x_h|))$. If $Θ_h \\to 0$, the scheme converges to the non-sticky weak solution. If $Θ_h \\to θ\\in (0,\\infty)$, it converges to the solution that stays at zero for the deterministic time $θ/|\\logα|$ and then evolves as the non-sticky solution. If $Θ_h \\to \\infty$, the scheme converges to the absorbing solution. The proof is based on an exact scale-invariant Euler chain, a logarithmic exit-time law of large numbers, a discrete Tanaka occupation estimate, and a compact-uniform restart argument. Every finite deterministic delay can be obtained by a suitable vanishing initialization, while oscillating depth parameters yield process-level nonconvergence. Thus a vanishing initialization can survive in the limit as a macroscopic deterministic delay and thereby act as a solution selector for the Euler-Maruyama approximation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13800",
  "author_total": 1,
  "authors": [
    {
      "name": "Akihiro Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/djsyyjtz1y67m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13800",
  "fulltext_url": null,
  "id": "djsyyjtz1y67m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13800",
  "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": "W7221561764",
  "pdf_url": "https://arxiv.org/pdf/2610.13800",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/djsyyjtz1y67m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13800",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13800"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE",
  "title_text": "Initialization-Dependent Solution Selection by the Euler-Maruyama Scheme for a Degenerate SDE",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/djsyyjtz1y67m",
  "venue": "arXiv",
  "versions": []
}