Convergence and exponential stability of modified truncated Milstein method for stochastic differential equations

Abstract

In this paper, we develop a new explicit scheme called modified truncated Milstein method which is motivated by truncated Milstein method proposed by Guo et al. (2018) and modified truncated Euler–Maruyama method introduced by Lan and Xia (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, pth moment exponential stability of the scheme is also obtained Three numerical experiments are offered to support our conclusions.