{
  "id": "1606.03836",
  "version": "v1",
  "published": "2016-06-13T07:05:33.000Z",
  "updated": "2016-06-13T07:05:33.000Z",
  "title": "Path-differentiability of BSDE driven by a continuous local martingale",
  "authors": [
    "Kihun Nam"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the existence and uniqueness, and the path-differentiability of solution for backward stochastic differential equation (BSDE) driven by a continuous local martingale $M$ with $[M,M]_{t}=\\int_{0}^{t}m_{s}m_{s}^{*}d{\\rm tr}[M,M]_{s}$: \\[ Y_{t}=\\xi(M_{[0,T]})+\\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\\rm tr}[M,M]_{s}-\\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t} \\] Here, for $t\\in[0,T]$, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$, and $\\xi(\\gamma_{[0,T]})$ and $f(t,\\gamma_{[0,t]},y,z)$ are deterministic functions of $(t,\\gamma,y,z)\\in[0,T]\\times D\\times\\mathbb{R}^{d}\\times\\mathbb{R}^{d\\times n}$. The path-derivative is defined as a directional derivative with respect to the path-perturbation of $M$ in a similar way to the vertical functional derivative introduced by Dupire (2009), and Cont and Fournie (2013). We first prove the existence and uniqueness of solution in the case where $f(t,\\gamma_{[0,t]},y,z)$ is Lipschitz in $y$ and $z$. After proving $Z$ is a path-derivative of $Y$, we extend the results to locally Lipschitz $f$. When the BSDE is one-dimensional, we could show the existence and uniqueness of solution. On the contrary, when the BSDE is multidimensional, we show the existence and uniqueness only when $[M,M]_{T}$ is small enough: we provide a counterexample which the solution blows up otherwise. Lastly, we investigate the applications to the control of SDE driven by $M$ and optimal portfolio selection problem under power and exponential utility function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-13T07:05:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60H07",
      "93E20"
    ],
    "keywords": [
      "continuous local martingale",
      "bsde driven",
      "path-differentiability",
      "uniqueness",
      "optimal portfolio selection problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}