{
  "abstract": "This paper studies differential properties of objective functions for discrete-time discounted Markov decision processes (MDPs) with finite state and action sets, when differentiation is considered with respect to the discount factor. Given a fixed discount factor for an MDP, we estimate the norms and seminorms of the objective functions, their derivatives, and the value function at a different discount factor. We then use these estimates to prove that objective functions, their derivatives, and value functions are Lipschitz continuous in discount factors. As a result, if an MDP is solved with a fixed discount factor, then we can estimate the value function at a different discount factor without solving it again.\\\\ %Following these estimations, we also reveal the relations between the seminorms of objective functions, their behavior when the discount factor approaches $1,$ and the discount factors at which the set of optimal policies changes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39326",
  "author_total": 2,
  "authors": [
    {
      "name": "Eugene A. Feinberg"
    },
    {
      "name": "Gaojin He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmzbmsz8ep8yf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39326",
  "fulltext_url": null,
  "id": "bmzbmsz8ep8yf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39326",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39326",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bmzbmsz8ep8yf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39326",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39326"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Differential Properties of Objective and Value Functions for Discounted Markov Decision Processes with Finite State and Action Sets",
  "title_text": "Differential Properties of Objective and Value Functions for Discounted Markov Decision Processes with Finite State and Action Sets",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/bmzbmsz8ep8yf",
  "venue": "arXiv",
  "versions": []
}