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

We propose an implementable numerical scheme for the discretization of linear-quadratic optimal control problems involving stochastic differential equations in higher dimensions with control constraints . For time discretization, we employ the implicit Euler scheme, deriving discrete optimality conditions that involve time discretization of a backward stochastic differential equation. We develop a recursive formula to compute conditional expectations in the time discretization of the backward stochastic differential equation whose computation otherwise is the computationally most 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.