### Almost Sure Exponential Stability of the θ-Euler-Maruyama Method for Neutral Stochastic Differential Equations with Time-Dependent Delay when θ ∈ [0, 1 / 2 ]

#### Abstract

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper \textit{M. Milo\v sevi\'c, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 -- 899.} The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the $\theta$-Euler-Maruyama solution ($\theta\in[0,\frac{1}{2}]$), for a

class of neutral stochastic differential equations with time-dependent delay. The assumptions under which there exists unique solutions of the approximate equations are presented and they include the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption $\theta\in(0,\frac{1}{2}],$ while the method is defined by employing the parameter $\theta$ with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the case $\theta=0,$ that is, the Euler-Maruyama case is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

class of neutral stochastic differential equations with time-dependent delay. The assumptions under which there exists unique solutions of the approximate equations are presented and they include the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption $\theta\in(0,\frac{1}{2}],$ while the method is defined by employing the parameter $\theta$ with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the case $\theta=0,$ that is, the Euler-Maruyama case is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

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