B-convergence of split-step one-leg theta methods for stochastic differential equations

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?Crspa.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 parameter satisfies $\frac{1}{2}\leq\theta \leq1$ . 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.