Stability of delayed impulsive stochastic differential equations driven by a fractional Brown motion with time-varying delay

We study the stability problem of mild solutions of impulsive stochastic differential equations driven by a fractional Brown motion with finite time-varying delay. The Hurst parameter H of the fractional Brown motion belongs to (12,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\frac{1}{2},1)$\end{document}. In terms of fractional power of operators and semigroup theory, we obtain sufficient conditions that guarantee the stability of the mild solution of such a equation in two cases: the impulse depends on current states of the system and the impulse depends not only on current states but also on historical states of the system. We give two examples illustrating the theorems.