Abstract
This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.
Highlights
1 Introduction In the past years, stochastic dynamical systems driven by fractional Brownian motion (fBm) have became an active area of investigation due to their applications in telecommunications networks, finance markets, biology, and other fields [1,2,3,4,5,6]
The impulsive differential equations have a wide range of applications in numerous branches of sciences such as finance, economics, medicine, biology, electronics, and telecommunications
There have been some works about stochastic averaging for dynamic problems with Gaussian random perturbation [17,18,19], Poisson noise [20, 21], Lévy motion [22,23,24,25], G-Brownian motion [26, 27], and fBm [28,29,30,31]
Summary
Stochastic dynamical systems driven by fBm have became an active area of investigation due to their applications in telecommunications networks, finance markets, biology, and other fields [1,2,3,4,5,6]. The impulsive differential equations have a wide range of applications in numerous branches of sciences such as finance, economics, medicine, biology, electronics, and telecommunications (see [7,8,9,10]). No previous study has employed the periodic averaging technique to impulsive stochastic dynamical systems with fBm. we make an attempt to Khalaf et al Advances in Difference Equations (2019) 2019:526 establish the periodic averaging principle to ISDEs with fBm, which allows the averaged systems without impulses to replace the original ISDEs both in mean square sense and probability.
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