{
  "abstract": "In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T>0$. The optimal control of this type of problem can be obtained by standard methods; however, it is shown that the resulting optimal control is generally time-inconsistent. Thus, instead of seeking an optimal control, which is time-inconsistent, we aim to find a time-consistent, locally optimal, and time-invariant equilibrium strategy, by introducing a new and very interesting type of Riccati equation. Its main feature is that the generator depends on a delay term of the unknown. In other words, this Riccati equation is a backward ordinary differential equation (ODE) with delay, which is equivalent to a forward ODE with advanced terms. Such an equation is essentially a Fredholm integral equation, whose solvability is challenging. We overcome the difficulty by deriving a sharp a priori estimate and applying the Leray--Schauder fixed point theorem. To this end, we establish a comparison theorem between two matrix-valued nonlinear algebraic equations. The convergence behavior of the solution to the Riccati equation as $T\\to\\infty$ is also provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08225",
  "author_total": 3,
  "authors": [
    {
      "name": "Yutao Chen"
    },
    {
      "name": "Hongwei Lou"
    },
    {
      "name": "Hanxiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ej73veadkg9db.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08225",
  "fulltext_url": null,
  "id": "ej73veadkg9db",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08225",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7167919987",
  "pdf_url": "https://arxiv.org/pdf/2607.08225",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ej73veadkg9db.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2607.08225",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08225"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sub-Infinite Horizon Stochastic Linear-Quadratic Optimal Control Problems and Delayed Backward Riccati Equations",
  "title_text": "Sub-Infinite Horizon Stochastic Linear-Quadratic Optimal Control Problems and Delayed Backward Riccati Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/ej73veadkg9db",
  "venue": "arXiv",
  "versions": []
}