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

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, which is of independent interest. We then establish a relation between RBSPDEs and optimal stopping of SPDEs, and apply the result to a risk minimizing stopping problem.