{
  "id": "1407.5031",
  "version": "v1",
  "published": "2014-07-18T15:16:51.000Z",
  "updated": "2014-07-18T15:16:51.000Z",
  "title": "Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients",
  "authors": [
    "Shanjian Tang"
  ],
  "comment": "16 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "We are concerned with the linear-quadratic optimal stochastic control problem with random coefficients. Under suitable conditions, we prove that the value field $V(t,x,\\omega), (t,x,\\omega)\\in [0,T]\\times R^n\\times \\Omega$, is quadratic in $x$, and has the following form: $V(t,x)=\\langle K_tx, x\\rangle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+\\int_0^t \\, dk_s+\\sum_{i=1}^d\\int_0^tL_s^i\\, dW_s^i, \\quad t\\in [0,T]$$ with $k$ being a continuous process of bounded variation and $$E\\left[\\left(\\int_0^T|L_s|^2\\, ds\\right)^p\\right] <\\infty, \\quad \\forall p\\ge 2; $$ and that $(K, L)$ with $L:=(L^1, \\cdots, L^d)$ is a solution to the associated backward stochastic Riccati equation (BSRE), whose generator is highly nonlinear in the unknown pair of processes. The uniqueness is also proved via a localized completion of squares in a self-contained manner for a general BSRE. The existence and uniqueness of adapted solution to a general BSRE was initially proposed by the French mathematician J. M. Bismut (1976, 1978). It had been solved by the author (2003) via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion, and gives the {\\it second but more comprehensive} adapted solution to a general BSRE via the DDP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-18T15:16:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "93E20",
      "49K45",
      "49N10",
      "60H10"
    ],
    "keywords": [
      "general linear quadratic optimal stochastic",
      "linear quadratic optimal stochastic control",
      "random coefficients",
      "dynamic programming",
      "optimal stochastic control problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5031T"
    }
  }
}