{
  "abstract": "We model the dynamics of the liquidity profile in Uniswap~V3 and other concentrated-liquidity automated market makers with a stochastic partial integro-differential equation (SPIDE). The profile, the aggregate of all provider positions across tick space, evolves under a degenerate Sturm--Liouville operator, which encodes mean-reverting concentration around the market price, coupled with a Hilbert--Schmidt integral operator, which captures nonlocal redistribution driven by provider repositioning. We establish existence and uniqueness of the mild solution in a weighted $L^2$ space built on the Gegenbauer eigensystem, and reduce the SPIDE to a coupled system of linear stochastic differential equations whose low-order modes describe the level, slope, and curvature of the profile. Pairing the SPIDE with a log-price process yields tractable expressions for pool reserves, pool value, and loss-versus-rebalancing under Ornstein--Uhlenbeck-type and Black--Scholes-type noise. A finite-rank calibration in a fixed reference frame, applied to three Uniswap pools, gives interpretable estimates and lowers active-band forecast error relative to a martingale benchmark on the two pools compatible with the stationary specification. On the third pool, out-of-sample diagnostics are consistent with nonstationarity, which motivates price-aware and time-varying extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30539",
  "author_total": 4,
  "authors": [
    {
      "name": "Cheuk Yin Lee"
    },
    {
      "name": "Jimmy Risk"
    },
    {
      "name": "Shen-Ning Tung"
    },
    {
      "name": "Tai-Ho Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ebz0e4tgqxwpf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30539",
  "fulltext_url": null,
  "id": "ebz0e4tgqxwpf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30539",
  "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"
    ],
    [
      "arxiv",
      "q-fin.MF"
    ],
    [
      "arxiv",
      "q-fin.TR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7223412309",
  "pdf_url": "https://arxiv.org/pdf/2610.30539",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ebz0e4tgqxwpf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30539",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30539"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Dynamics of the Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach",
  "title_text": "Dynamics of the Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/ebz0e4tgqxwpf",
  "venue": "arXiv",
  "versions": []
}