Singular Control and Optimal Stopping of SPDEs, and Backward SPDEs with Reflection

Abstract : We consider general singular control problems for random fields given by a stochastic partial differential equation (SPDE). We show that under some conditions the optimal singular control can be identified with the solution of a coupled system of SPDE and a reflected backward SPDE (RBSPDE). As an illustration we apply the result to a singular optimal harvesting problem from a population whose density is modeled as a stochastic reaction-diffusion equation. Existence and uniqueness of solutions of RBSPDEs are established, as well as comparison theorems. We then establish a relation between RBSPDEs and optimal stopping of SPDEs, and apply the result to a risk-minimizing stopping problem.
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Mathematics of Operations Research, INFORMS, 2013, 〈http://pubsonline.informs.org/doi/abs/10.1287/moor.2013.0602〉
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Bernt Øksendal, Agnès Sulem, Tusheng Zhang. Singular Control and Optimal Stopping of SPDEs, and Backward SPDEs with Reflection. Mathematics of Operations Research, INFORMS, 2013, 〈http://pubsonline.informs.org/doi/abs/10.1287/moor.2013.0602〉. 〈hal-00919136〉

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