Numerical analysis of a Neumann boundary control problem with a stochastic parabolic equation

This paper analyzes the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where an additive noise occurs in the Neumann boundary condition. The convergence is established for general filtrations, and the convergence rate $ O(\tau^{1/4-\epsilon} + h^{3/2-\epsilon}) $ is derived for the natural filtration of the Q-Wiener process. A Monte-Carlo method to solve the discrete stochastic optimal control problem is proposed.