In this paper, we have considered two classes of second-order balanced stochastic Runge–Kutta methods to multidimensional Itô stochastic differential equations. The control functions in the proposed methods are used to improve and enhance the convergence and stability properties of the method. The strong convergence of the second-order balanced stochastic Runge–Kutta methods is analyzed. The preservation of sign of initial value to the second order balanced stochastic Runge–Kutta methods for a linear scalar test equations is discussed under suitable conditions. Moreover, the mean-square stability of the proposed methods is investigated using the properties of Kronecker product. Subsequently the stability region of the balanced stochastic Runge–Kutta methods for $s=4$ stages is compared with the other existing stochastic Runge–Kutta methods. From this comparison, we understood the balanced stochastic Runge–Kutta methods are more stable than the stochastic Runge–Kutta methods. Finally, the mean square stability for different step sizes $h$ is illustrated as well convergence results compared with other current methods are tabulated.

在本文中，我们考虑了二维Itô随机微分方程的两类二阶平衡随机Runge-Kutta方法。所提出的方法中的控制功能被用来改善和增强该方法的收敛性和稳定性。分析了二阶平衡随机Runge-Kutta方法的强收敛性。在适当的条件下，讨论了线性标量测试方程的二阶平衡随机Runge-Kutta方法的初始值符号的保存。此外，利用Kronecker产品的性质研究了所提出方法的均方稳定性。随后，平衡随机Runge-Kutta方法的稳定区域用于$s=4$将阶段与其他现有随机Runge-Kutta方法进行比较。通过这种比较，我们了解到平衡随机Runge–Kutta方法比随机Runge–Kutta方法更稳定。最后，不同步长的均方稳定性$H$ 图中显示了与其他当前方法相比的收敛结果。