Neumann problem for backward SPDEs with singular terminal conditions and application in constrained stochastic control under target zone

Abstract

In this paper, a Neumann problem for the backward stochastic partial differential equation (BSPDE) with singular terminal condition is studied, which characterizes the value function for a constrained stochastic control problem (also called optimal liquidation problem) in target zone models. The existence and the uniqueness of strong solutions to such BSPDEs are addressed. Furthermore, the uniqueness and existence of strong solutions to the Neumann problem for general semilinear BSPDEs in finer function spaces, a comparison theorem, and a new link between forward–backward stochastic differential equations and BSPDEs are proved as well. In addition, we apply the strong solution theory to the associated optimal liquidation problems and derive the optimal feedback control.