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Abstract

This work develops the Milstein scheme for commutative stochastic differential equations with piecewise continuous arguments (SDEPCAs), which can be viewed as stochastic differential equations with time-dependent and piecewise continuous delay. As far as we know, although there have been several papers investigating the convergence and stability for different numerical methods on SDEPCAs, all of these methods are Euler-type methods and the convergence orders do not exceed 1/2. Accordingly, we first construct the Milstein scheme for SDEPCAs in this work and then show its convergence order can reach 1. Moreover, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several numerical examples to verify the theoretical results.

Details

Title
Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments
Author
Zhang, Yuhang 1 ; Song, Minghui 1 ; Liu, Mingzhu 1 ; Zhao, Bowen 2 

1  Harbin Institute of Technology, School of Mathematics, Harbin, China (GRID:grid.19373.3f) (ISNI:0000 0001 0193 3564) 
2  Harbin Institute of Technology, Center for Control Theory and Guidance Technology, Harbin, China (GRID:grid.19373.3f) (ISNI:0000 0001 0193 3564) 
Pages
417-448
Publication year
2024
Publication date
May 2024
Publisher
Springer Nature B.V.
ISSN
10171398
e-ISSN
15729265
Source type
Scholarly Journal
Language of publication
English
DOI
https://doi.org/10.1007/s11075-023-01652-4
ProQuest document ID
3033730412
Copyright
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.