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Zbl 1207.34104
Li, Chunxiang; Sun, Jitao; Sun, Ruoyan
Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. (English)
[J] J. Franklin Inst. 347, No. 7, 1186-1198 (2010). ISSN 0016-0032; ISSN 1879-2693/e

From the text: We discuss a class of stochastic differential delay equations with nonlinear impulsive effects of the form $$\cases dy(t) =\{-a_1(t)y(t)-a_2(t)y(t-\tau(t))\}\,dt\\ \qquad\ +\{-b_1(t)y(t)-b_2(t)y(t-\tau(t))]\,dw(t),\quad & t\ne t_k,\\ y(t_k^+)-y(t_k)=I_k(y(t_k)), & t=t_k,\ k\in\Bbb N,\endcases$$ where $I_k\in C(\Bbb R,\Bbb R)$, $k\in\Bbb N$ are continuous functions with $I_k(0)\equiv 0$. The purpose of this paper is to build a bridge between the given stochastic impulsive delay equation and a corresponding stochastic delay equation without impulsive effects, and to establish some stability criteria for these systems. Furthermore, the desired conditions are given explicitly.

MSC 2000:: *34K50 Stochastic delay equations
34K20 Stability theory of functional-differential equations
34K45 Equations with impulses

Keywords: stability; stochastic differential equations; delay; impulsive effects

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