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Mathematical Problems in Engineering
Volume 2018, Article ID 1682513, 13 pages
Research Article

Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematics, Faculty of Science, Menoufia University, Menoufia 32511, Egypt

Correspondence should be addressed to Haiying Zhang; nc.ude.tih@yhhz

Received 9 September 2017; Revised 16 December 2017; Accepted 24 December 2017; Published 24 January 2018

Academic Editor: Fiorenzo A. Fazzolari

Copyright © 2018 Mahmoud A. Eissa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSM methods is investigated. Furthermore, the stability regions of the DSSM methods are compared with those of test equation, and it is proved that the methods with are stochastically A-stable. Second, the nonlinear stability of DSSM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.