The averaging method for a class of stochastic differential equations

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

“…In Khasminskii [10], the author initiated to study a stochastic averaging method for SDEs with Gaussian random fluctuations. Stoyanov and Kolomiets [12,20] investigated the stochastic averaging method for SDEs driven by Poisson noises. Xu [24,[26][27][28] proved the stochastic averaging for SDEs driven by Lévy noise or by fBm, where an averaged system is presented to replace the original one both in the sense of convergence in mean square and in probability.…”

Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

“…Since Krylov and Bogolyubov [11] put forward the cornerstone of the averaging principles for deterministic dynamical systems, averaging method has received considerable attention, and it has been found available and useful for exploring dynamical systems in many fields [12][13][14][15][16]. Up to now, 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]. So far, no previous study has employed the periodic averaging technique to impulsive stochastic dynamical systems with fBm.…”

An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion

“…In Gaussian random fluctuations case, this analytic technique has been developed by Stratonovich [5,6] and Khasminskii [7,8]. Shortly afterwards, researchers began to study the averaging principle of SDEs driven by Poisson noises [9,10]. Recently, Zhu and his coworkers also investigated this averaging principle for a class of nonlinear systems with Poisson noises [11][12][13].…”