{
  "id": "1510.08439",
  "version": "v1",
  "published": "2015-10-28T19:59:08.000Z",
  "updated": "2015-10-28T19:59:08.000Z",
  "title": "Stochastic control for a class of nonlinear kernels and applications",
  "authors": [
    "Dylan Possamaï",
    "Xiaolu Tan",
    "Chao Zhou"
  ],
  "comment": "44 pages",
  "categories": [
    "math.PR",
    "math.OC",
    "q-fin.MF"
  ],
  "abstract": "We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are non-linear generalizations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic pro- gramming principle for this control problem in an abstract setting, which we then use to provide a semimartingale characterization of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [76]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a non-linear optional decomposition in a robust setting, extending recent results of [63], which we then use to obtain a superhedging duality in uncertain, incomplete and non-linear financial markets. Finally, we relate, under addi- tional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-28T19:59:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic control",
      "nonlinear kernels",
      "applications",
      "appropriate path-dependent partial differential equation",
      "value function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151008439P"
    }
  }
}