JAMC J Appl Math Comput (2012) 38:489–503 DOI 10.1007/s12190-011-0492-1 B-convergence of split-step one-leg theta methods for stochastic differential equations Xiaojie Wang · Siqing Gan Received: 9 October 2010 / Published online: 8 June 2011 © Korean Society for Computational and Applied Mathematics 2011 Abstract For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the explicit schemes fail to converge strongly to the exact solution (see, Hutzenthaler, Jentzen and Kloeden in Proc. R. Soc. A, rspa.2010.0348v1–rspa.2010.0348, 2010). In this article a class of implicit methods, called split-step one-leg theta methods (SSOLTM), are introduced and are shown to be mean-square convergent for such SDEs if the method param- eter satisfies 1 2 ≤ θ ≤ 1. This result gives an extension of B-convergence from the theta method for deterministic ordinary differential equations (ODEs) to SSOLTM for SDEs. Furthermore, the optimal rate of convergence can be recovered if the drift coefficient behaves like a polynomial. Finally, numerical experiments are included to support our assertions. Keywords Split-step one-leg theta methods · One-sided Lipschitz condition · Boundedness of moments · B-convergence · Strong convergence Mathematics Subject Classification (2000) 65C20 · 60H35 · 65L20 This work was supported by NSF of China (No. 10871207) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. The first author is supported by Hunan Provincial Innovation Foundation For Postgraduate (No. CX2010B118) and would like to express his gratitude to Prof. P.E. Kloeden for his kind help during the author’s stay in University of Frankfurt am Main. X. Wang ( ) · S. Gan School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410075, Hunan, China e-mail: x.j.wang7@gmail.com S. Gan e-mail: siqinggan@yahoo.com.cn