{
  "id": "0806.1837",
  "version": "v1",
  "published": "2008-06-11T10:46:27.000Z",
  "updated": "2008-06-11T10:46:27.000Z",
  "title": "Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations",
  "authors": [
    "Marco Fuhrman",
    "Federica Masiero",
    "Gianmario Tessitore"
  ],
  "journal": "Siam Journal on Control and Optimization 48 Issue: 7 (2010) 4624-4651",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\\mathbf C$, with generator $\\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\\mathbf C$ driven by $\\mathcal L$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-11T10:46:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal control",
      "regular solutions",
      "stochastic equations",
      "hamilton-jacobi-bellman equations",
      "stochastic differential equation depending"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1837F"
    }
  }
}