University of TabrizComputational Methods for Differential Equations2345-39825320170701Stability of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type2012136263ENOmidFarkhondeh RouzFaculty of Mathematical Sciences, University of Tabriz, Tabriz, IranDavoodAhmadianFaculty of Mathematical Sciences, University of Tabriz, Tabriz, IranJournal Article20170404This paper examines stability analysis of two classes of improved backward Euler methods, namely split-step $(\theta, \lambda)$-backward Euler (SSBE) and semi-implicit $(\theta,\lambda)$-Euler (SIE) methods, for nonlinear neutral stochastic delay differential equations (NSDDEs). It is proved that the SSBE method with $\theta, \lambda\in(0,1]$ can recover the exponential mean-square stability with some restrictive conditions on stepsize $\delta$, drift and diffusion coefficients, but the SIE method can reproduce the exponential mean-square stability unconditionally. Moreover, for sufficiently small stepsize, we show that the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. Finally, numerical experiments are included to confirm the theorems. https://cmde.tabrizu.ac.ir/article_6263_5ce76ae675ed3f591df2c191f12c3d50.pdf