Truncated Milstein method for non-autonomous stochastic differential equations and its modification

TL;DR

This paper extends the truncated Milstein method to non-autonomous stochastic differential equations with super-linear and H"older continuous components, proving convergence and improving step-size requirements using randomized techniques.

Contribution

It introduces an extension of the truncated Milstein method to more complex SDEs and employs randomized step-size to enhance convergence rate.

Findings

Proved convergence rate for the extended method

Reduced step-size restrictions compared to previous work

Enhanced convergence rate with randomized step-size

Abstract

The truncated Milstein method, which was initially proposed in (Guo, Liu, Mao and Yue 2018), is extended to the non-autonomous stochastic differential equations with the super-linear state variable and the H\"older continuous time variable. The convergence rate is proved. Compared with the initial work, the requirements on the step-size is also significantly released. In addition, the technique of the randomized step-size is employed to raise the convergence rate of the truncated Milstein method.