{
  "id": "1110.4658",
  "version": "v2",
  "published": "2011-10-20T23:11:36.000Z",
  "updated": "2015-06-29T11:34:36.000Z",
  "title": "On well-posedness of forward-backward SDEs-A unified approach",
  "authors": [
    "Jin Ma",
    "Zhen Wu",
    "Detao Zhang",
    "Jianfeng Zhang"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1046 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 4, 2168-2214",
  "doi": "10.1214/14-AAP1046",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a \"User's Guide,\" for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-20T23:11:36.000Z",
      "title": "On Wellposedness of Forward-Backward SDEs --- A Unified Approach",
      "abstract": "In this paper we study the wellposedness of the forward-backward stochastic differential equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to overcome some fundamental difficulties that have been longstanding problems for non-Markovian FBSDEs. Our main devices are a {\\it decoupling random field} and its associated {\\it characteristic BSDE}, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions under which the characteristic BSDE is wellposed, which leads to the existence of the decoupling random field, and ultimately to the solvability of the original FBSDE. We show that all existing frameworks could be analyzed using our new criteria.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-29T11:34:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "35R60",
      "34F05"
    ],
    "keywords": [
      "unified approach",
      "forward-backward sdes",
      "wellposedness",
      "decoupling random field",
      "characteristic bsde"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.4658M"
    }
  }
}