Runge-Kutta Algorithm for the Numerical Integration of Stochastic Differential Equations

This paper presents a new Runge-Kutta (RK) algorithm for the numerical integration of stochastic differential equations. These equations occur frequently as a description of many mechanical, aerospace, and electrical systems. They also form the basis of modern control design using the linear quadratic regulator/Gaussian (LQR/LQG) method. It is convenient, and common practice, to numerically simulate these equations to generate sample random processes that approximate a solution of the system (often called Monte Carlo simulations). It is shown in the paper that the standard deterministic solution techniques are inaccurate and can result in sample sequences with covariances not representative of the correct solution of the original differential equation. A new set of coefficients is derived for a RK-type solution to these equations. Examples are presented to show the improvement in mean-square performance.