随机变延迟微分方程平衡方法的均方收敛性与稳定性

针对一类变延迟微分方程，应用全隐式方法—平衡方法，研究了其收敛性和稳定性.结果表明平衡方法以$\frac{1}{2}\gamma，\gamma\in（0，1]$阶收敛到精确解；并且强平衡方法和弱平衡方法都能保持解析解的均方稳定性；进一步数值实验验证了算法理论分析的正确性，并且表明全隐式的平衡方法比显式方法—Euler方法具有更好的稳定性.

MEAN SQUARE CONVERGENCE AND STABILITY OF BALANCED METHODS FOR STOCHASTIC VARIABLE DELAY DIFFERENTIAL EQUATIONS

The convergence and stability of a class of variable delay differential equations are studied by using a fully implicit method balanced methods. The results show that the balanced methods converges to the exact solution of order $\frac{1}{2}\gamma, \gamma\in(0, 1]$; Moreover, both the strong balanced methods and the weak balanced methods can reproduce the mean-square stability of the system with sufficiently small stepsize $h$; Further, some numerical experiments included in the paper illustrate the theoretical results, and show that the fully implicit balanced methods has better stability than the explicit—Euler methods.