Fractional Brownian motions (fBms) play a central role in the modeling and analysis of many complex phenomena in applications where the systems are subject to “rough” external forcing. An fBm with Hurst parameter (Hin(0,1)) is a zero-mean Gaussian process denoted by (W^{H}={W^{H}(t),tgeq0}). If (Hin(0,frac{1}{2})), then it is regarded as a short-memory process; if (H=frac{1}{2}), then it reduces to the standard Brownian motion; if (Hin(frac{1}{2},1)), then it is regarded as a long-memory process. It is easy to see that fBm is a generalization of Brownian motion, but it behaves different significantly from the standard Brownian motion. In particular, it is neither a semimartingale nor a Markov process. It is characterized by stationary increments and memory property, which make an fBm a potential candidate to model noise in biology and finance (see, e.g., [1–5], in geophysics ([6]), in communication networks ([7]), in electricity markets ([8]), and so on.
In recent years, stochastic differential equations driven by fBms have attracted increasing interest because of their applications in a variety of fields (see [4, 9–19] and references therein). However, most of the existing literature is focused on the existence and uniqueness of mild solutions for stochastic differential equations driven by fBms (see, e.g., [9–15]), but the existing results on the stability of mild solutions for stochastic differential equations driven by fBms are relatively few. We only found a few stories in the literature [4, 16–19]. In [4], the authors provided sufficient conditions to guarantee the exponential asymptotic behavior of solutions of general linear stochastic differential equations driven by fBms with time-varying delays. In [16], the authors provided the conditions for the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by an fBm. In [17], the authors provided conditions to ensure the exponential decay to zero in mean square of solutions of neutral stochastic differential equations driven by fBms in a Hilbert space. However, in [4, 16, 17], the impulses are not considered in the systems. In [18], the author gave asymptotic stability conditions for mild solutions of neutral stochastic differential equations driven by fBms with finite delays and nonlinear impulsive effects. In [19], the authors gave mean-square exponential stability conditions for mild solutions of neutral stochastic differential equations driven by fBms with infinite delays and impulses. However, the impulses in stochastic differential equations in [18, 19] only depend on the current states of the systems, which are supposed to be of the form (I_{k}(x(t_{k}))) at impulsive moments (t_{k}) ((k=1,2,ldots)). Here, the delayed impulses we consider describe the impulsive transients depending not only on their current states but also on historical states of the system. Delayed impulses exist in many practical problems, for example, in communication security systems based on impulsive synchronization. During the information transmission process, the sampling delay created from sampling the impulses at some discrete instances causes the impulsive transients depend on their historical states. There are some results ([20–25]) on delayed impulsive differential equation, where the delays in impulsive perturbations are fixed as constants or vary in a finite interval. However, most of these studies of stability for such delayed impulsive differential equations consider deterministic equations ([20–25]), whereas stochastic differential equations are rarely considered ([26]). In [26], the stochastic term was supposed to be of the form (g(x_{t},t),dw(t)), where (w(t)) is a standard Brownian motion, not a fractional one.
In view of the above discussion, we investigate the stability of impulsive stochastic delay differential equations driven by fBms with Hurst parameter (Hin(frac{1}{2},1)). Firstly, we assume that the impulse in such an equation depends on the current states and give mean-square exponential stability conditions. Secondly, we assume that the impulse in such an equation depends on the historical states and give mean-square asymptotic stability conditions.
The rest of this paper is organized as follows. In Section 2, we introduce some notation, concepts, and lemmas. In Section 3, we give mean-square exponential stability conditions for stochastic differential equations driven by fBms with impulses and time-varying delays, where the impulses only depend on the current states of the system. In Section 4, we study a delayed impulsive differential equation driven by an fBm with impulses and time-varying delay, where the impulses depend not only on the current states but also on the historical states of the system, and provide mean-square asymptotic stability conditions for such an equation. In Section 5, we give two illustrative examples. Conclusions are given in Section 6.