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

TL;DR

This paper introduces a modified truncated Milstein method for stochastic differential equations, demonstrating strong convergence, near-first-order rate, and exponential stability, supported by numerical experiments.

Contribution

It proposes a new explicit scheme with weaker conditions for convergence and stability, improving upon previous truncated Milstein methods.

Findings

Strong convergence under weaker conditions

Convergence rate arbitrarily close to 1

Exponential stability of the scheme

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 (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.