{
  "abstract": "We study multi-product replenishment subject to aggregate operating limits imposed pathwise in place of expectation. On-hand and in-transit inventory, warehouse capacity and an emission allowance define a safe set, and the resulting multi-product lost-sales problem is posed as a safe stochastic optimal control problem. The Hamilton--Jacobi--Bellman (HJB) equation is represented as a system of forward--backward stochastic differential equations (FBSDEs) and solved by learning the value function and its gradient, with a quadratic program derived by control barrier function (CBF) is solved at each decision epoch. Because purchase orders enter the in-transit pipeline only, the in-transit constraints have relative degree one while the on-hand constraints have relative degree two. We extend the stochastic high-relative-degree construction of (Clark, 2021) from a single barrier to four coupled aggregate constraints of mixed relative degree, and we establish its initial-condition requirement by design.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11817",
  "author_total": 2,
  "authors": [
    {
      "name": "Keunwoo Lim"
    },
    {
      "name": "Michael Bloem"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92acan226dx29.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11817",
  "fulltext_url": null,
  "id": "92acan226dx29",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11817",
  "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.OC"
    ],
    [
      "openalex",
      "1404|Supply Chain and Inventory Management"
    ]
  ],
  "openalex": "W7212275183",
  "pdf_url": "https://arxiv.org/pdf/2609.11817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92acan226dx29.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Constrained Deep Inventory Management Using Forward-Backward SDEs",
  "title_text": "Constrained Deep Inventory Management Using Forward-Backward SDEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/92acan226dx29",
  "venue": "arXiv",
  "versions": []
}