{
  "id": "2404.04781",
  "version": "v1",
  "published": "2024-04-07T01:54:16.000Z",
  "updated": "2024-04-07T01:54:16.000Z",
  "title": "The convergence of the EM scheme in empirical approximation of invariant probability measure for McKean-Vlasov SDEs",
  "authors": [
    "Cui Yuanping",
    "Li Xiaoyue"
  ],
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "Based on the assumption of the existence and uniqueness of the invariant measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a self-interacting process that depends only on the current and historical information of the solution is constructed for MV-SDEs. The convergence rate of the weighted empirical measure of the self-interacting process and the invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric. Furthermore, under the condition of linear growth, an EM scheme whose uniformly 1/2-order convergence rate with respect to time is obtained is constructed for the self-interacting process. Then, the convergence rate between the weighted empirical measure of the EM numerical solution of the self-interacting process and the invariant measure of MV-SDEs is derived. Moreover, the convergence rate between the averaged weighted empirical measure of the EM numerical solution of the corresponding multi-particle system and the invariant measure of MV-SDEs in the W2-Wasserstein metric is also given. In addition, the computational cost of the two approximation methods is compared, which shows that the averaged weighted empirical approximation of the particle system has a lower cost. Finally, the theoretical results are validated through numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-07T01:54:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariant probability measure",
      "em scheme",
      "empirical approximation",
      "convergence rate",
      "invariant measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}