An error corrected Euler-Maruyama method for stiff stochastic differential equations

In this paper, we propose an error corrected Euler-Maruyama method, which is constructed by adding an error correction term to the Euler-Maruyama scheme. The correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the Euler-Maruyama's continuous-time extension. The method is proved to be mean-square convergent with order 12 and is as easy to implement as standard explicit schemes but much more efficient for solving stiff stochastic problems. For a linear scalar test equation with a scalar noise term, it is shown that the mean-square stability domain of the method is much bigger than that of the Euler-Maruyama method. It is proved the method preserves the mean-square stability and asymptotic stability of the linear scalar equation without any constraint on the numerical step size. Finally, numerical examples are reported to show the accuracy and effectiveness of the method.