Asymptotically Efficient Runge-Kutta Methods for a Class of Itô and Stratonovich Equations

In certain applications of stochastic differential equations, approximate solutions must be found that depend only on samples of the driving process. It is known that the order of convergence of such approximations is limited, and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean-square errors as power series in the sample step size. This article develops asymptotically efficient Runge-Kt tta methods that involve evaluations either of the coefficients of an Itô equation or of the coefficients of the corresponding Stratonovich equation. Simpler approximations, which converge with the maximum possible order but which are not asymptotically efficient, are also defined. The Runge-Kutta methods for Itô equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size. The approximations are tested along with the classical Euler method on five examples. The simulations suggest that in many cases the improved accuracy of the asymptotically efficient methods is worth the extra computational burden that they involve.