Generalized two-step Milstein methods for stochastic differential equations

ABSTRACT

The main aim of this study is to propose a class of generalized two-step Milstein methods for solving Itô stochastic differential equations. We study their mean-square consistency and mean-square convergence and derive some conditions for the linear mean-square stability of the generalized two-step Milstein methods. Numerical examples are given to confirm the theoretical results.

KEYWORDS:

• Stochastic differential equation
• generalized two-step Milstein methods
• mean-square consistency
• mean-square convergence
• linear mean-square stability

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34F05
• 60H10

Acknowledgments

The authors are grateful to the referees for their helpful suggestions and valuable comments, which have greatly improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Quanwei Ren is supported by the High-Level Personal Foundation of Henan University of Technology (No. 2017BS023), The youth support project for basic research of Henan University of Technology (2018QNJH17), Foundation of Henan Educational Committee (No. 19A110013) and NSF of China (No. 11801146). The work of Hongjiong Tian is supported in part by E-Institutes of Shanghai Municipal Education Commission under Grant No. E03004, the National Natural Science Foundation of China under Grant Nos. 11671266 and 11871343, and the Natural Science Foundation of Shanghai under Grant No. 16ZR1424900.