This paper represents a generalization of the stability result on the
Euler-Maruyama solution, which is established in the paper M. Milosevic,
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 ?-Euler-Maruyama
solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential
equations with time-dependent delay. The existence and uniqueness of
solution of the approximate equation is proved by employing 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 ? ?(0, 1/2], while the
method is defined by employing the parameter ? 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 Euler-Maruyama case (? = 0) 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.
An application of Taylor series in the approximation of solutions to stochastic differential equations with time-dependent delay