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

Document Type : Research Paper

Authors

Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

Abstract

This 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.

Keywords

### History

• Receive Date: 04 April 2017
• Revise Date: 16 May 2017
• Accept Date: 01 July 2017