Abstract This paper is concerned with a non-linear stochastic delay differential system with delay-dependent impulsive perturbations. In this work, the size of the jump is defined as a general non-linear delay-dependent state variable and the solution of the impulsive stochastic delay differential system corresponding to the system without impulsive perturbations is given. This work is based on the relation between the solution of the equivalent model of stochastic delay differential system without impulses corresponding to the solution of the system with impulses. Then the conditions of the exponential stability of the proposed impulsive system are obtained by deriving stability criteria of the corresponding system without impulses. The numerical approximation for the stochastic delay system without impulses is developed using the Runge-Kutta-Maruyama method and it is suitably applied for the corresponding impulsive system. Finally, the obtained theoretical results are illustrated graphically for a stochastic delay system with impulses.

中文翻译：

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具有广义时滞相关脉冲点的非线性随机时滞微分系统的指数稳定性

摘要 本文涉及具有延迟相关脉冲扰动的非线性随机延迟微分系统。在这项工作中，跳跃的大小被定义为一个一般的非线性延迟相关状态变量，并给出了对应于没有脉冲扰动的系统的脉冲随机延迟微分系统的解。这项工作基于无脉冲随机延迟微分系统等效模型的解与有脉冲系统的解之间的关系。然后通过推导相应系统的无冲量稳定性判据，得到所提出的冲量系统的指数稳定性条件。使用 Runge-Kutta-Maruyama 方法开发了无脉冲随机延迟系统的数值近似，它适用于相应的脉冲系统。最后，对于具有脉冲的随机延迟系统，以图形方式说明了所获得的理论结果。