A numerical method to simulate the stochastic linear-quadratic optimal control problem with control constraint in higher dimensions

We propose an {\em implementable} numerical scheme for the discretization of linear-quadratic optimal control problems involving SDEs in higher dimensions with {\em control constraint}. For time discretization, we employ the implicit Euler scheme, deriving discrete optimality conditions that involve time discretization of a backward stochastic differential equations. We develop a recursive formula to compute conditional expectations in the time discretization of the BSDE whose computation otherwise is the most computationally demanding step. Additionally, we present the error analysis for the rate of convergence. We provide numerical examples to demonstrate the efficiency of our scheme in higher dimensions.