{
  "id": "2211.05676",
  "version": "v1",
  "published": "2022-11-10T16:28:17.000Z",
  "updated": "2022-11-10T16:28:17.000Z",
  "title": "Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth",
  "authors": [
    "Tao Hao",
    "Ying Hu",
    "Shanjian Tang",
    "Jiaqiang Wen"
  ],
  "comment": "40 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for one-dimensional mean-field BSDEs when the generator $g\\big(t,Y,Z,\\mathbb{P}_{(Y,Z)}\\big)$ grows in $Z$ quadratically and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth and give the rate of convergence. Finally, in this framework, we use the mean-field BSDE to give a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short), as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-10T16:28:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60H30"
    ],
    "keywords": [
      "mean-field backward stochastic differential equations",
      "quadratic growth",
      "mean-field bsde",
      "nonlocal pdes",
      "general mean-field backward stochastic differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}