{
  "id": "2406.00286",
  "version": "v1",
  "published": "2024-06-01T03:32:57.000Z",
  "updated": "2024-06-01T03:32:57.000Z",
  "title": "Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution $(Y,Z)$",
  "authors": [
    "Juan Li",
    "Zhanxin Li",
    "Chuanzhi Xing"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of $Z$-component of the solution process $(Y, Z)$. A natural question is whether general mean-field BSDEs whose coefficients depend on the law of $Z$ have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process $(Y,Z)$. With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component $Y$ of the solution $(Y,Z)$ for such mean-field BSDEs, but also the second component $Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-01T03:32:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "comparison theorem",
      "mean-field backward stochastic differential equations",
      "generators",
      "general mean-field backward stochastic differential",
      "solution process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}