Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection

In Section 2 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. In Section 3, existence and uniqueness of solutions of RBSPDEs are established. In Section 4 we prove a relation between RBSPDEs and optimal stopping of SPDEs, and in Section 5 we apply this result to a risk minimizing optimal stopping problem.