Exponential stability of non-linear stochastic delay differential system with generalized delay-dependent impulsive points

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.