Abstract
ABSTRACTThe exponential mean-square stability of the θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. With some monotone conditions, the trivial solution of the equation is proved to be exponentially mean-square stable. If the drift coefficient and the parameters satisfy more strengthened conditions, for the constrained stepsize, it is shown that the θ-method can preserve the exponential mean-square stability of the trivial solution for θ ∈ [0, 1]. Since θ-method covers the commonly used Euler–Maruyama (EM) method and the backward Euler–Maruyama (BEM) method, the results are valid for the above two methods. Moreover, they can adapt to the NSDDEs and the stochastic delay differential equations (SDDEs) with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.
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