In this paper, we develop a new explicit scheme called modified truncated Milstein method which is motivated by truncated Milstein method proposed by Guo (2018) and modified truncated Euler-Maruyama method introduced by Lan (2018). We obtain the strong convergence of the scheme under local boundedness and Khasminskii-type conditions, which are relatively weaker than the existing results, and we prove that the convergence rate could be arbitrarily close to 1 under given conditions. Moreover, exponential stability of the scheme is also considered while it is impossible for truncated Milstein method introduced in Guo(2018). Three numerical experiments are offered to support our conclusions.
翻译:在本文中,我们制定了一个新的明确计划,称为经修改的短途Milstein方法,其动机是Guo提出的短途Milstein方法(2018年)和Lan提出的经修改的Euler-Maruyama方法(2018年),我们在当地边界和Khasminskii类型条件下,在相对弱于现有结果的条件下,实现了该计划的高度趋同,我们证明在特定条件下,趋同率可能任意接近于1。此外,还考虑了该计划的指数稳定性,而Guo(2018年)提出的短途Milstein方法则是不可能的,提出了三个数字实验来支持我们的结论。