{
  "abstract": "We develop a martingale-residual framework for continuous-time reinforcement learning in high-dimensional financial problems, linking policy evaluation (PE), policy gradient (PG), and entropy-regularized Actor–Critic (AC) learning. First, continuous-time PE and PG are often developed through separate constructions. We formulate a common martingale residual for the Critic and Actor, and show that, as the local time step vanishes, its rescaled policy derivative recovers the classical continuous-time PG. We also quantify how Critic approximation errors affect policy-gradient accuracy. Second, martingale-orthogonality temporal-difference (TD) methods require prescribed test functions, and their performance can depend on that choice. We introduce a weak-martingale adversarial formulation in which the test function is learned rather than prescribed. The resulting minimax problem avoids the sensitivity to hand-chosen test functions and provides a unified computational representation of the associated moment conditions. Third, direct high-dimensional Hamilton–Jacobi–Bellman (HJB) implementations can require explicit Hessians and other higher-order differential operators. Our algorithms work directly with sampled martingale residuals and first-order automatic differentiation, yielding Hessian-free implementations for both offline training from complete trajectories and online updating from sequential transitions. Numerical experiments on high-dimensional basket-option pricing and mean–variance portfolio selection demonstrate accurate PE and effective PG. In high-dimensional mean–variance portfolio-selection experiments, the proposed method achieves competitive or superior performance on several key metrics relative to benchmarks: the offline variant achieves a mean annualized return of 16.223%, the online variant attains a mean Sharpe ratio of 2.337, and fixed-budget training time increases by 19.3% as the asset dimension grows from 10 to 1000.",
  "abstract_shown": true,
  "affiliations": [
    "School of Mathematical Sciences, Tongji University, Shanghai 200092, China",
    "School of Mathematics and Information of Science, Guangzhou University, Guangzhou 510006, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Hang Qi",
      "orcid": "0009-0007-2727-2734"
    },
    {
      "affs": [
        0
      ],
      "name": "Dingli Xia"
    },
    {
      "affs": [
        1
      ],
      "name": "Mingxiang Deng"
    },
    {
      "affs": [
        0
      ],
      "name": "Xiaofei Guan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9p2snkf1bs9vh.bib",
  "cited_by": 0,
  "doi": "10.3390/math14214012",
  "fulltext_url": null,
  "id": "9p2snkf1bs9vh",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/4012",
  "language": null,
  "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": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/9p2snkf1bs9vh.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T00:23:28Z",
      "native_id": "10.3390/math14214012",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/4012"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Martingale Deep Reinforcement Learning for High-Dimensional Financial Problems",
  "title_text": "Martingale Deep Reinforcement Learning for High-Dimensional Financial Problems",
  "type": "peer-reviewed",
  "updated_at": "2026-11-05T00:20:49Z",
  "url": "https://research.draconitic.com/paper/9p2snkf1bs9vh",
  "venue": "Mathematics",
  "versions": []
}