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Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. (English) Zbl 1207.34104
From the text: We discuss a class of stochastic differential delay equations with nonlinear impulsive effects of the form
\[ \begin{cases} dy(t) =\{-a_1(t)y(t)-a_2(t)y(t-\tau(t))\}\,dt\\ \qquad\;+\{-b_1(t)y(t)-b_2(t)y(t-\tau(t))]\,dw(t),\quad & t\neq t_k,\\ y(t_k^+)-y(t_k)=I_k(y(t_k)), & t=t_k,\;k\in\mathbb N,\end{cases} \]
where \(I_k\in C(\mathbb R,\mathbb R)\), \(k\in\mathbb N\) 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:
34K50 Stochastic functional-differential equations
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
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