Stability of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type

This paper examines stability analysis of two classes of improved backward Euler methods, namely splitstep $(theta, lambda)-backward Euler (SSBE) and semiimplicit $(theta,lambda)$Euler (SIE) methods, for nonlinear neutral stochastic delay differential equations (NSDDEs). It is proved that the SSBE method with $theta, lambdain(0,1]$ can recover the exponential meansquare stability with some restrictive conditions on stepsize $delta$, drift and diffusion coefficients, but the SIE method can reproduce the exponential meansquare 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.