{
  "abstract": "We study a correlated-noise Zakai equation for nonlinear filtering when the signal and observation share a Brownian component. Under bounded coefficients, with bounded-Lipschitz state regularity for the second-order, correlated-noise, and observation coefficients, and under uniform ellipticity of the private signal noise, the equation is well posed in the variational L 2 framework. The main result is a quantitative inhomogeneous H -1 stability estimate based on (I -∆) -1/2 : the second-order drift and the correlated stochastic divergence combine through the parabolicity matrix 2a-γγ T to produce time-integrated L 2 dissipation controlled by the initial H -1 distance. This mechanism also yields stability under L ∞ coefficient perturbations within uniformly bounded-Lipschitz classes, without convergence of coefficient derivatives. A weighted L 2 propagation estimate gives an explicit no-escape criterion, allowing weak L 2 convergence of initial densities to imply the strong negative-topology convergence required by the stability theorem. Under uniform control bounds, the same estimates imply H -1 continuity of separated linear-in-density costs and of the associated value function. We further establish a controlled approximation chain from whole-space localization to stable Galerkin projection, fixed-dimensional Euler-Maruyama time stepping, and quadrature, under the explicit additional temporal and spatial regularity assumptions stated for those numerical steps. Reduced finite-dimensional crowd and traffic models provide theorem-compatible numerical benchmarks and a comparison with a positive weighted-particle Kallianpur-Striebel representation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "C. Dogbe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07j2dvnv9kdmq.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "07j2dvnv9kdmq",
  "issns": [],
  "landing_url": "https://hal.science/hal-05747042v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05747042/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07j2dvnv9kdmq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:34Z",
      "native_id": "hal-05747042",
      "source": "hal",
      "url": "https://hal.science/hal-05747042v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Well-Posedness and Negative-Sobolev Stability of a Correlated-Noise Zakai Equation",
  "title_text": "Well-Posedness and Negative-Sobolev Stability of a Correlated-Noise Zakai Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/07j2dvnv9kdmq",
  "venue": "HAL",
  "versions": []
}