A numerical method for solving stochastic linear quadratic problem with a finance application

This paper is concerned with the stochastic linear quadratic regulator (LQR) optimal control problem in which dynamical systems have control-dependent diffusion coefficients. In fact, providing the solution to this problem leads to solving a matrix Riccati differential equation as well as a vector differential equation with boundary conditions. The present work mainly  proposes not only a novel method but also an efficient fixed-point scheme based on the spline interpolation for the numerical solution to the stochastic LQR problem. Via implementing the proposed method to the corresponding differential equation of the stochastic LQR optimal control problem, not only is the numerical solution gained, but also a suboptimal control law is obtained. Furthermore, the method application is illustrated by means of an optimal control example with the financial market problems, including two investment options.