On the averaging principle for stochastic differential equations involving Caputo fractional derivative

Abstract

In this paper, we investigate the averaging principle for Caputo-type fractional stochastic differential equations driven by Brownian motion. Different from the approach of integration by parts or decomposing integral interval to deal with the estimation of integral involving singular kernel in the existing literature, we show the desired averaging principle in the sense of mean square by using Hölder inequality via growth conditions on the nonlinear stochastic term. Finally, a simulation example is given to verify the theoretical results.

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

“…By developing the approach in [8], sufficient conditions are established to guarantee an averaging principle for Caputo type FSDEs with non-Gaussian Lévy noise in the sense of pth moment. Different from previous work [13], the method presented in this paper can successfully deal with the singular integral kernel caused by fractional integration, and we prove the desired averaging principle for FSDEs in the sense of pth moment.…”

“…In addition, [5,6,7] used the similar methods to derive the averaging principle for different type fractional order derivative or equations under the different assumptions. Recently, Xiao et al [8] explored a new approach, which is different from [1], to deal with the singular integrals term raised from fractional integrals.…”

“…Based on the previous work [8,13], we consider the averaging principle for the following standard Caputo type FSDEs with Lévy noise…”

Relatively exact controllability of fractional stochastic delay system driven by Lévy noise

“…In the real world, many practical systems affected by random disturbances from the external environment are usually modeled as stochastic differential equations (SDEs). In many literatures [31][32][33], we can find that the noise process many researchers assume in stochastic questions is the Wiener process. For instance, the relatively exact controllability of FSDS with Wiener noise was discussed by adopting delayed matrix functions of sine and cosine in the work of wang et al [34] In addition, for many other researched results about stochastic systems with Wiener noise, one can see the papers [35,36].…”

Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions of fractional stochastic evolution equations

“…Recently, Li and Wang [25] studied the existence, uniqueness and continuous dependence of mild solutions to the fractional stochastic evolution equations by means of fractional resolvent operator and the Schauder fixed point theorem. For more recent works of existence and stability results, one can see the contributions of [37,38,41] and reference therein.…”