{
  "id": "1212.2170",
  "version": "v4",
  "published": "2012-12-10T19:15:33.000Z",
  "updated": "2013-09-24T15:37:13.000Z",
  "title": "Stochastic Perron's method for Hamilton-Jacobi-Bellman equations",
  "authors": [
    "Erhan Bayraktar",
    "Mihai Sirbu"
  ],
  "comment": "Final version. To appear in the SIAM Journal on Control and Optimization. Keywords: Perron's method, viscosity solutions, non-smooth verification, comparison principle",
  "categories": [
    "math.PR",
    "cs.SY",
    "math.AP",
    "math.OC"
  ],
  "abstract": "We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman (HJB) equation, completely avoiding the proof of the so-called dynamic programming principle (DPP). Using Stochastic Perron's method we construct a super-solution lying below the value function and a sub-solution dominating it. A comparison argument easily closes the proof. The program has the precise meaning of verification for viscosity-solutions, obtaining the DPP as a conclusion. It also immediately follows that the weak and strong formulations of the stochastic control problem have the same value. Using this method we also capture the possible face-lifting phenomenon in a straightforward manner.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-09-24T15:37:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49L20",
      "49L25",
      "60G46",
      "60H30",
      "35Q93",
      "35D40"
    ],
    "keywords": [
      "stochastic perrons method",
      "hamilton-jacobi-bellman equations",
      "stochastic control problem",
      "value function",
      "dynamic programming principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.2170B"
    }
  }
}