{
  "id": "0904.0377",
  "version": "v4",
  "published": "2009-04-02T13:11:37.000Z",
  "updated": "2012-11-19T14:36:35.000Z",
  "title": "Backward stochastic dynamics on a filtered probability space",
  "authors": [
    "Gechun Liang",
    "Terry Lyons",
    "Zhongmin Qian"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/10-AOP588 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2011, Vol. 39, No. 4, 1422-1448",
  "doi": "10.1214/10-AOP588",
  "categories": [
    "math.PR"
  ],
  "abstract": "We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: \\[dY_t^j=-f_0^j(t,Y_t,L(M)_t) dt-\\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j\\] with $Y_T=\\xi$, on a general filtered probability space $(\\Omega,\\mathcal{F},\\mathcal{F}_t,P)$, where $B$ is a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution $(Y,M)$. In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-11-19T14:36:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "filtered probability space",
      "backward stochastic dynamics",
      "backward stochastic differential equations",
      "ordinary functional differential equations",
      "associated partial differential equations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.0377L"
    }
  }
}