Convergence Analysis for an Implementable Scheme to Solve the Linear-Quadratic Stochastic Optimal Control Problem with Stochastic Wave Equation

Abstract We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin’s maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By using a new technical approach, we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus . For practical computation, we develop a gradient descent algorithm based on artificial iterates that employs an exact computation of the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency.