A new explicit stochastic scheme of order 1 is proposed for solving commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. The proposed method is a semi-tamed version of Milstein scheme to solve SDEs with the drift coefficient consisting of non-Lipschitz continuous term and globally Lipschitz continuous term. It is easily implementable and achieves higher strong convergence order. A stability criterion for this method is derived, which shows that the stability condition of the numerical methods and that of the solved equations keep uniform. Compared with some widely used numerical schemes, the proposed method has better performance in inheriting the mean square stability of the exact solution of SDEs. Numerical experiments are given to illustrate the obtained convergence and stability properties.
翻译:为了用非全球Lipschitz持续系数解决通货性随机差异方程式(SDEs),建议了一个新的第1号命令的明确随机方案,以非全球Lipschitz持续系数(SDEs)解决通货性随机差异方程式(SDEs),提议的方法是用非Lipschitz连续术语和全球Lipschitz连续术语(Lipschitz连续术语)的漂移系数解决SDEs的半堆版的Milstein方案,该方法容易实施,并达到更强的趋同顺序。该方法的稳定性标准可以得出,该方法表明数字方法和已解决方程式的稳定性保持统一。与一些广泛使用的数字法相比,该拟议方法在继承SDEs确切解决方案的平均平方形稳定性方面表现更好。做了数值实验,以说明获得的趋同性和稳定性。