A Note on Averaging Principles for Fractional Stochastic Differential Equations

: Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is ﬁrst proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements.


Introduction
As an example, this note considers Bogolyubov's averaging principle for the following Caputo fractional stochastic differential system with a fast oscillating right-hand side and Lévy noise: where D α t denotes Caputo fractional derivative; α ∈ ( 1 2 , 1], u(t) denotes an n-dimensional stochastic process; and f : are all measurable functions.w(t) denotes an m-dimensional Brownian motion in a complete probability space.Ñ(dt, dz) = N(dt, dz) − v(dz)dt is a d-dimensional compensated jump measure, in which N(dt, dz) is a real-valued Poisson counting measure with a σ-finite intensity measure v(dz) on a measurable set Z ∈ B(R d − 0).u 0 is a F 0 measurable R n -valued random variable satisfying E|u 0 | 2 < ∞.
Note that it is difficult to study Equation (1) directly; this is because f ( t ε , •), g( t ε , •), h( t ε , •, z) are fast time oscillating forces.It is generally known that a highly oscillating system can be "averaged" under suitable conditions.There are three different types of averaging principles, i.e., the so-called first Bogolyubov theorem, the second Bogolyubov theorem, and the global averaging principle [1].Up to now, all the results on averaging principles of fractional stochastic differential equations are first Bogolyubov theorems, and this work is no exception.
In recent years, averaging principles for fractional stochastic differential equations (FSDEs) have attracted great attention since they can provide approximate analytical methods to reduce such systems [2].Xu et al. [3,4] developed averaging results for various types of FSDEs early.Luo and Zhu [5] explored an averaging result for fractional stochastic delay differential equations (FSDDEs).Taking the correlated noise into account, an averaging theorem for FSDEs with fractional Brownian motion was demonstrated by Duan et al. [6].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].Introducing the neutral term, Shen et al. [10] presented the averaging result for neutral FSDEs.Xia and Yan et al. [11] derived an averaging theorem for a new kind of FSDE.Considering the different types of fractional derivatives, Makhlouf et al. [12] showed an averaging principle for Hadarmard Itô-Doob FSDEs.Liu et al. [13] established an averaging result for Caputo-Hadamard FSDEs.Recently, Yang et al. [14] developed an averaging principle for Hilfer FSDEs in the infinite-dimensional space.Li and Wang [15] presented an averaging theorem for Caputo FSDDEs driven by Gaussian white noise.Bai et al. [16] proved an averaging result for Caputo FSDDEs with Poisson noise.Overall, the studies of averaging principles for FSDEs gained rapid development with tremendous achievements.Specifically, Caputo fractional derivative can maintain the initial value condition of the original function and remain consistent with classical differentiation when the fractional order is an integer.Consequently, the studies on the average principles of Caputo FSDEs are the most extensive.But the biggest flaw in most of the existing related research studies is that they use the same standard form of the fractional differential equation as the integer-order case, which results in incorrect estimations of the convergence rate.
Accordingly, taking fractional stochastic differential equations with Lévy noise as an example, this note elaborates the standard form of fractional stochastic differential equations and the estimation of the convergence rate.Furthermore, the relationship between the averaging principles of fractional stochastic differential equations and integer-order stochastic differential equations is commented.

Preliminaries
Definition 1 ([4]).Suppose that f is a Lebesgue integrable function, the α-order integral of f is given by where 0 < α < 1 and Γ is a gamma function.

Definition 2 ([4]
).The α-order Caputo-type fractional derivative of f is determined by Next, we show the time scale change property for Caputo fractional derivative, which is a key tool to obtain the standard form of Caputo fractional differential equations.

Lemma 1. (Time scale change property). Let the time scale
Proof.Based on Definition 2, one can obtain and then the assertion of Lemma 1 is proved.
Suppose the following general assumptions hold throughout this paper.
Condition 1. Linear growth condition: where K is a positive constant.
Condition 2. The Lipschitz condition: there exists a constant L > 0 such that Then, following Ref.[17], we have no problem giving the existence and uniqueness result of system (1).Theorem 1.Under Conditions 1-2, there exists a unique solution to system (1) on [0, T].
At the end of this section, we recall the generalized Gronwall inequality proposed in Ref. [18], which is the key tool to prove our main result.Lemma 2. Assume that α > 0 and J = [0, S), S ≤ ∞.Let function a(t) be non-negative and locally integrable on J, while function b(t) be non-negative, nondecreasing, continuous and bounded on J.In case y(t) is non-negative and locally integrable, the following inequality holds: Furthermore, if a(t) is nondecreasing on J, then where E α (•) is the Mittag-Leffler function.

Main Results
This section is dedicated to presenting the deduction procedure of the correct standard form, the corresponding convergence interval and rate of Caputo fractional stochastic differential equations.At the beginning, we demonstrate the standard form of system (1).
Let t ε = τ, and with the aid of Lemma 1, system (1) can be rewritten as: without loss of generality, one can denote τ := t.In a word, with the natural time scaling t ε → t and Lemma 1, the standard form of fractional stochastic differential equations is given as then, we can rewrite Equation ( 2) in its integral form as Remark 1.The standard form of Caputo fractional stochastic differential equations under natural time scale should be obtained by the above derivation.It should not be the same as the case of integer-order stochastic differential equations unless α = 1.
In this way, we can prove that when the oscillating frequency speeds up (ε → 0 + ), the original solution u ε (t) converges to the solution x ε (t) of system (4).Remark 3. The conditions lim t→∞ γ j (t) = 0 guarantee the existence and uniqueness of the solution to the averaged system (4).This is because it can ensure the functions f (•), g(•), h(•) satisfy the Lipschitz condition and the linear growth condition.
For example, for ∀u, x ∈ R n , by Condition 2 and (H1), one can reach taking t → ∞, we demonstrate that f meets the Lipschitz condition.Moreover, Condition 1 and (H2) give that as t → ∞, it is shown that f meets the linear growth condition.Similarly, we can prove that g(•), h(•) also meet the Lipschitz condition and the linear growth condition.All the above assertions guarantee the existence and uniqueness of the averaged system (4).
Based on Refs.[1,17], the following lemma can be achieved.Lemma 3. Let the conditions of Theorem 2 hold; there exists a positive constant C such that Next, we briefly show the proof process of Theorem 2, and then demonstrate the impact of incorrect standard form on the convergence interval and rate.