Averaging principle for a type of Caputo fractional stochastic differential equations

Abstract

Averaging principle for Caputo fractional stochastic differential equations attracted much attention recently. In this paper, we investigate the averaging principle for a type of Caputo fractional stochastic differential equations. Comparing with the existing literature, we shall use different estimate methods to investigate the averaging principle, which will enrich the development of the theory for Caputo fractional stochastic differential equations.

Averaging Principle for a Class of Time-Fractal-Fractional Stochastic Differential Equations

“…It is noteworthy that other academics have used similar methods to analyze the averaging principle for such classic SDEs (see, e.g., [9,10]). Caputo fractional SDEs and the averaging condition for Caputo fractional SDEs [7] are consistent with (1) and (2) if α = 1 and β ∈ (0, 1). Additionally, the condition lim T→∞ ρ(T) = 0 is crucial to the method and conclusion of our article, which has been cited in several works (see, e.g., [8,14,19]), but it has not been utilized in the proofs.…”

“…Khasminskii [4] initially established the stochastic differential equations' averaging principle in the 1960s. Next, [5][6][7][8][9][10][11][12][13][14][15][16][17], authors including Besjes [11], Bogoliubov [2], Gikhman [13], and Volosov [17] as well as references herein conducted research on the averaging principle of ordinary differential equations(ODEs). However, in reality, various forms of perturbations are ubiquitous.…”

“…In recent years, Guo, Shen, and other scholars have given the averaging principle for such equations; see [5,7,16] for more details.…”

A Note on Averaging Principles for Fractional Stochastic Differential Equations

“…After considering the external excitation with jumps, an averaging principle for Hilfer FSDDEs driven by Poisson noise was subsequently proven by Ahmed and Zhu [7]. Later, Guo et al [8] obtained an averaging result for FSDEs by applying distinct techniques. After that, with the aid of an alternative processing skill, an averaging principle for FSDEs was developed by Xiao et al [9].…”

The averaging principle for Caput type fractional stochastic differential equations with Lévy noise

“…The averaging principle for fractional stochastic differential equations with Gaussian noise was first proposed by Xu et al [1], and then Guo et al [2] revisited this averaging principle by leveraging the idea of integration interval decomposition. Thereafter, Gao Peng [3] studied the averaging principle for multiscale stochastic reaction-diffusion-advection equations.…”