{
  "id": "2112.09639",
  "version": "v2",
  "published": "2021-12-17T17:27:17.000Z",
  "updated": "2022-01-21T16:48:18.000Z",
  "title": "Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations",
  "authors": [
    "Wilhelm Stannat",
    "Lukas Wessels"
  ],
  "comment": "29 pages; added a necessary assumption to Theorem 4.1 and improved the presentation; added Remark 4.3 regarding more general differential operators",
  "categories": [
    "math.PR",
    "math.OC"
  ],
  "abstract": "Using a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin's maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in non-smooth stochastic control problems is non-positive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton-Jacobi-Bellman equations in the framework of viscosity solutions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-21T16:48:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear stochastic partial differential equations",
      "optimal control",
      "sufficient conditions",
      "backward stochastic partial differential",
      "non-smooth stochastic control problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}