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

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.

1 Introduction

In the past years, 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 [1–6]. The impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time. Consequently, 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 [7–10]).

On the other hand, it is well known that the averaging technique represents a good mathematical tool that approximates complicated time varying differential equations to autonomous differential equations. 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 [12–16]. Up to now, there have been some works about stochastic averaging for dynamic problems with Gaussian random perturbation [17–19], Poisson noise [20, 21], Lévy motion [22–25], G-Brownian motion [26, 27], and fBm [28–31]. So far, no previous study has employed the periodic averaging technique to impulsive stochastic dynamical systems with fBm. Therefore, we make an attempt to 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.

We consider a class of ISDEs with fBm of the form

$$\begin{aligned} &dx(t) = a \bigl(t,x(t) \bigr)\,dt+b \bigl(t,x(t) \bigr)\,dW^{H}(t),\quad t\neq t_{j}, \\ &\triangle x(t_{j})=I_{j} \bigl(x \bigl(t^{-}_{j} \bigr) \bigr),\quad t= t_{j}, j \in\mathbb {N}, \\ &x(0) =x_{0}, \end{aligned}$$

(1)

where $\triangle x(t_{j})$ denotes the jump of x at $t= t_{j}$, for $0\leq t\leq T<\infty$, and $\triangle x(t_{j})=x(t^{+}_{j})-x(t^{-}_{j})$, such that $x(t^{+}_{j})= \lim_{t \rightarrow t^{+}_{j}}x(t)$ and $x(t^{-}_{j})= \lim_{t\rightarrow t^{-}_{j}}x(t)$. $x_{0}$ represents the initial data of the system with $E|x_{0}|^{2}<\infty$. The process $W^{H}(t)$ is fBm with Hurst index $H \in (\frac{1}{2},1)$ defined on the filtered probability space $(\varOmega ,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq{0}} , P)$. The coefficients $a(t,x(t)):[0, T ]\times R^{n}\rightarrow R^{n}$ and $b(t,x(t)):[0, T]\times R^{n}\rightarrow R^{n\times m}$ are measurable functions.

The outline of this manuscript is as follows. In Sect. 2, we provide some background about stochastic integral with respect to fBm. Section 3 is devoted to establishing the stochastic periodic averaging approach to Eq. (1) under non-Lipschitz condition. Finally, an example is presented to demonstrate the theoretical results in Sect. 4.

2 Framework

In this section, we introduce some basic notions and preliminaries on path-wise integrals with respect to fBm, and for more detailed discussion, we refer the reader to [6, 32–35].

Let $(\varOmega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq{0}} , P)$ be a complete probability space equipped with a natural filtration $\{ \mathcal{F}_{t}\}_{t\geq{0}}$, where $\mathcal{F}_{t}$ is the σ-algebra generated by $\{W^{H}(t), t\in[0,T]\}$ and $\mathcal{F}_{0}$ contains all P-null sets.

Definition 2.1

The process $\{W^{H}(t), 1/2< H<1\}$ is said to be a centered self-similar fBm if the following properties are satisfied:

• $W^{H}(0) =0$,

• $E[W^{H}(t)]=0$, $t\in[0,T]$,

• $E[W^{H}(t)W^{H}(s)]= \frac{1}{2} (|t|^{2H} +|s|^{2H} -|t-s|^{2H})$, $t,s\in[0,T]$.

Next, for the convenience of readers, we provide some basic properties on path-wise integrals. Firstly, we introduce the function $\varphi: \mathbb{R}_{+}\times\mathbb{R}_{+} \rightarrow\mathbb{R}_{+}$ defined as

$$\begin{aligned} \varphi(t,s)=H(2H-1) \vert t-s \vert ^{2H-2},\quad t,s\in \mathbb{R}_{+}, \end{aligned}$$

where $H \in(\frac{1}{2},1)$. Let $f:\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ be a Borel measurable function and define the space

$$\begin{aligned} L^{2}_{\varphi}(\mathbb{R}_{+})= \biggl\{ f: \Vert f \Vert ^{2}_{\varphi}= \int_{\mathbb {R}_{+}} \int_{\mathbb{R}_{+}}{f(t)f(s)\varphi(t,s)}\,ds\,dt < \infty \biggr\} , \end{aligned}$$

which becomes a separable Hilbert space under the inner product

$$\begin{aligned} \langle f_{1},f_{2}\rangle_{\varphi}= \int_{\mathbb{R}_{+}} \int_{\mathbb {R}_{+}}{f_{1}(t)f_{2}(s) \varphi(t,s)}\,ds\,dt,\quad f_{1}, f_{2} \in L^{2}_{\varphi}(\mathbb{R}_{+}). \end{aligned}$$

Now, consider the set $\mathcal{E}$ of smooth and cylindrical random variables of the form

$$\begin{aligned} F(\omega)=g \biggl( \int_{0}^{T}\psi_{1}(t)\,dW^{H}(t), \ldots, \int_{0}^{T}\psi _{n}(t)\,dW^{H}(t) \biggr), \end{aligned}$$

where $n\geq1$ and $g\in\mathcal{C}^{\infty}_{b}(\mathbb{R}^{n})$ (i.e., g and its partial derivatives are bounded). Moreover, let $\mathcal {H}$ be the family of measurable functions such that, for $\psi_{i}\in \mathcal{H}$, $i=1,\ldots,n$, $n\in\mathbb{N}$, we have $\langle\psi _{i},\psi_{j}\rangle_{\varphi}=\delta_{ij}$ and $\|\psi\|^{2}_{\varphi}<\infty$. The elements of $\mathcal{H}$ may not be functions but distributions of negative order. Thanks to this reason, it is convenient to introduce the space $|\mathcal{H}|$ of measurable functions h on $[0,T]$ satisfying

$$\begin{aligned} \Vert h \Vert ^{2}_{ \vert \mathcal{H} \vert } = \int_{0}^{T} \int_{0}^{T}{ \bigl\vert h(t) \bigr\vert \bigl\vert h(s) \bigr\vert \varphi (t,s)}\,ds\,dt < \infty, \end{aligned}$$

and it is easy to show that $|\mathcal{H}|$ is a Banach space under the norm $\|\cdot\|_{|\mathcal{H}|}$.

Definition 2.2

The Malliavin derivative $D^{H}_{t}$ of a smooth and cylindrical random variable F is defined as an $\mathcal {H}$-valued random variable such that

$$\begin{aligned} D^{H}_{t}F=\sum_{i=1}^{n}{ \frac{\partial g}{\partial x_{i}} \biggl( \int _{0}^{T}\psi_{1}(t)\,dW^{H}(t), \ldots, \int_{0}^{T}\psi_{n}(t)\,dW^{H}(t) \biggr)\psi_{i}(t)}, \end{aligned}$$

hence, $D^{H}_{t}$ represents a closable operator, so that $D^{H}_{t}:L^{p}(\varOmega) \mapsto L^{p}(\varOmega,\mathcal{H})$, $p\geq1$. The iteration of Malliavin derivative is denoted by $D^{H,k}_{t}$, $k\geq1$. For any $p \geq1$, the Sobolev space $\mathbb{D}^{k,p}$ represents the closer of $\mathcal{E}$ with respect to the norm

$$\begin{aligned} \Vert F \Vert ^{p}_{k,p} = E \vert F \vert ^{p}+E\sum_{i=1}^{k} \bigl\Vert D^{H,i}_{t}F \bigr\Vert ^{p}_{\mathcal {H}\otimes i}, \end{aligned}$$

where ⊗ denotes the tensor product.

Similarly, for a Hilbert space U, we denote by $\mathbb{D}^{k,p}(U)$ the corresponding Sobolev space of U-valued random variables, and for $p>0$, we denote by $\mathbb{D}^{1,p}(| \mathcal{H}|)$ the subspace of $\mathbb{D}^{1,p}(\mathcal{H})$ formed by the elements h of $|\mathcal{H}|$. According to [6], we introduce φ-derivative of F as follows:

$$\begin{aligned} D^{\varphi}_{t}F= \int_{\mathbb{R_{+}}} \varphi(t,s) D^{H}_{s}F \,ds. \end{aligned}$$

Definition 2.3

The space $\mathcal{L}_{\varphi}[0,T]$ of integrals is defined as the family of stochastic processes $V(t)$ on $[0,T]$ such that $E\|V(t)\|^{2}_{\varphi}< \infty$, $V(t)$ is φ-differentiable, the trace of the derivative $D^{\varphi}_{s} V(t)$ exists, and for $t,s \in[0,T]$,

$$\begin{aligned} E \biggl[ \int_{0}^{T} \int_{0}^{T}{ \bigl\vert D^{\varphi}_{t} V(s) \bigr\vert ^{2}}\,ds\,dt \biggr] < \infty. \end{aligned}$$

In addition, for each sequence of partitions $(\pi_{n}, n \in\mathbb {N})$ with $|\pi_{n}| \rightarrow0$ as $n\rightarrow\infty$, the following are satisfied:

$$\begin{aligned} \sum_{i=0}^{n-1} E \biggl[ \int_{t^{(n)}_{i}}^{t^{(n)}_{i+1}} \int _{t^{(n)}_{j}}^{t^{(n)}_{j+1}}{ \bigl\vert D^{\varphi}_{s} V^{\pi } \bigl(t^{(n)}_{i} \bigr)D^{\varphi}_{t} V^{\pi} \bigl(t^{(n)}_{j} \bigr)-D^{\varphi}_{s} V(t)D^{\varphi}_{t} V(s) \bigr\vert ^{2}}\,ds\,dt \biggr] \rightarrow0 \end{aligned}$$

and

$$\begin{aligned} E \bigl\Vert V^{\pi}- V \bigr\Vert ^{2}_{\varphi}\rightarrow0, \end{aligned}$$

as n tends to infinity, where $\pi_{n} = t^{(n)}_{0}< t^{(n)}_{1} < \cdots < t^{(n)}_{n-1} < t^{(n)}_{n}=T$, $|\pi|:= \max_{i}{(t_{i+1}-t_{i})}$ and $V^{\pi}=V_{t_{i}}$.

Now, define the space $\mathbb{H}^{1,2}_{\varphi}$, which represents the intersection of the spaces $\mathbb{D}^{1,2}(|\mathcal{H}|)$ and $\mathcal{L}_{\varphi}[0,T]$, such that $\mathbb{H}^{1,2}_{\varphi}=\mathbb{D}^{1,2}(|\mathcal{H}|)\cap\mathcal{L}_{\varphi}[0,T]$.

Definition 2.4

Let $V(t)$ be a stochastic process with integrable trajectories.

• The symmetric integral of $V(t)$ with respect to $W^{H}(t)$ is defined as follows:

  $$\begin{aligned} \lim_{\epsilon\rightarrow0}\frac{1}{2\epsilon} \int _{0}^{T}{V(s) \bigl[W^{H}(s+ \epsilon)-W^{H}(s-\epsilon) \bigr]}\,ds, \end{aligned}$$

  provided that the limit exists in probability, the symmetric integral is denoted by

  $$\int_{0}^{T}{V(s)}\,d^{\circ} W^{H}(s). $$

• The forward integral of $V(t)$ with respect to $W^{H}(t)$ is defined as follows:

  $$\begin{aligned} \lim_{\epsilon\rightarrow0}\frac{1}{\epsilon} \int_{0}^{T}{V(s) \biggl[\frac{W^{H}(s+\epsilon)-W^{H}(s)}{\epsilon} \biggr]}\,ds, \end{aligned}$$

  provided that the limit exists in probability, the forward integral is denoted by

  $$\int_{0}^{T}{V(s)}\,d^{-} W^{H}(s). $$

• The backward integral of $V(t)$ with respect to $W^{H}(t)$ is defined as follows:

  $$\begin{aligned} \lim_{\epsilon\rightarrow0}\frac{1}{\epsilon} \int_{0}^{T} {V(s) \biggl[\frac{W^{H}(s-\epsilon)-W^{H}(s)}{\epsilon} \biggr]}\,ds, \end{aligned}$$

  provided that the limit exists in probability, the backward integral is denoted by

  $$\int_{0}^{T}{V(s)}\,d^{+} W^{H}(s). $$

In order to establish our results, we need to introduce some lemmas. The next lemma follows (Remark 1 in [35]) and (Proposition 6.2.3 in [6]).

Lemma 2.5

If the stochastic process $V(t)$satisfies

$$\begin{aligned} \int_{0}^{T} \int_{0}^{T}{ \bigl\vert D^{H}_{s}V(t) \bigr\vert \vert t-s \vert ^{2H-2}}\,ds\,dt < \infty,\quad V\in \mathbb{D}^{1,2} \bigl( \vert \mathcal{H} \vert \bigr), \end{aligned}$$

then the symmetric integral coincides with the forward and backward integrals.

Since fBm is neither semi-martingale nor Markov process, we definitely lost the use of Burkholder–Davis–Gundy inequality and Ito-isometry. Therefore, there is a pressing need to use the following two lemmas from [6] and [28].

Lemma 2.6

If $V(t)$is a stochastic process on $\mathbb {H}^{1,2}_{\varphi}$, then the symmetric integral is well defined and

$$\begin{aligned} \int_{0}^{T}{V(s)}\,d^{\circ} W^{H}(s)= \int_{0}^{T}{V(s)\diamond}dW^{H}(s)+ \int _{0}^{T}{D^{\varphi}_{s}V(s)}\,ds, \end{aligned}$$

where ⋄ denotes the Wick product.

We note that the forward and backward integrals are also well defined. Hence, by Lemma 2.5, the forward and backward integrals coincide with the symmetric integral under the condition of Lemma 2.6.

Lemma 2.7

Let $W^{H}(t)$be fBm with Hurst index $H\in(\frac {1}{2},1)$and $V(t)$be a stochastic process in $\mathbb {H}^{1,2}_{\varphi}$, then, for $0 \leq T < \infty$, there exists a constant $C >0$such that

$$ E \biggl\vert \int_{0}^{T}{V(s)}\,d^{\circ} W^{H}(s) \biggr\vert ^{2} \leq2HT^{2H-1}E \int _{0}^{T}{ \bigl\vert V(s) \bigr\vert ^{2}}\,ds+4CT^{2}. $$

The following requisite lemma is taken from [36].

Lemma 2.8

Let $T>0$, $x_{0}\geq0$, and $x(t)$, $y(t)$be two continuous functions on $[0,T]$. Assume that $\kappa:\mathbb{R}_{+} \rightarrow\mathbb{R}_{+}$is a concave continuous nondecreasing function such that $\kappa(v)>0$for $v>0$. If we have

$$\begin{aligned} x(t)\leq x_{0}+ \int_{0}^{t}{y(s)\kappa \bigl(x(s) \bigr)}\,ds\quad \forall t\in[0,T], \end{aligned}$$

then

$$\begin{aligned} x(t)\leq G^{-1} \biggl(G(x_{0})+ \int_{0}^{t}{y(s)}\,ds \biggr)\quad \forall t \in[0,T], \end{aligned}$$

where $(G(x_{0})+\int_{0}^{t}{y(s)}\,ds)\in \operatorname{Dom}(G^{-1})$, $G(v)=\int _{0}^{v}{\frac{ds}{\kappa(s)}}\,ds$, $v>0$. Moreover, if $x_{0}=0$and $\int _{0^{+}}{\frac{ds}{\kappa(s)}}\,ds = \infty$, then $x(t) = 0$for all $t\in[0,T]$.

Throughout this paper, the following assumptions are imposed.

Assumption A

For all $x,y \in\mathbb{R}^{n}$, $t\in[0,T]$, and $a(t,\cdot), b(t,\cdot) \in \mathbb{H}^{1,2}_{\varphi}$, there exists a function $\kappa(\cdot)$ such that

$$\begin{aligned} \bigl\vert a(t,x)-a(t,y) \bigr\vert ^{2}+ \bigl\vert b(t,x)-b(t,y) \bigr\vert ^{2}+ \bigl\vert D^{\varphi}_{t} \bigl(b(t,x)-b(t,y) \bigr) \bigr\vert ^{2} \leq\kappa \bigl( \vert x-y \vert ^{2} \bigr), \end{aligned}$$

where $\kappa(\cdot)$ is a concave continuous nondecreasing function such that $\kappa(0)=0$ and

$$\begin{aligned} \int_{0^{+}}{\frac{1}{\kappa(x)}}\,dx=\infty. \end{aligned}$$

Moreover, since $\kappa(\cdot)$ is a concave continuous nondecreasing function, then there must exist two constants $\lambda_{1}>0$ and $\lambda_{2}>0$ such that

$$ \kappa(x)\leq\lambda_{1}x+\lambda_{2}. $$

Remark 2.9

In view of Assumption A, we can see clearly, for a special case, if $\kappa(|x|)=K|x|$, then the Lipschitz condition is recovered. Therefore, Assumption A is much weaker than the usual Lipschitz condition.

Next, according to Lemma 3.1 in [37], the solution of impulsive stochastic dynamical system (1) can be given by the following integral equation:

$$\begin{aligned} x(t)= x_{0}+ \int_{0}^{t} {a \bigl(s,x(s) \bigr)}\,ds+ \int_{0}^{t} {b \bigl(s,x(s) \bigr)}\,d^{\circ }W^{H}(s)+ \sum_{0 < t_{j} < t}I_{j} \bigl(x(t_{j}) \bigr). \end{aligned}$$

(2)

Now, consider the standard ISDE with fBm

$$\begin{aligned} x_{\epsilon}(t)= x_{0}+\epsilon^{2H} \int_{0}^{t} {a \bigl(s,x_{\epsilon }(s) \bigr)}\,ds +\epsilon^{H} \int_{0}^{t} {b \bigl(s,x_{\epsilon}(s) \bigr)}\,d^{\circ }W^{H}(s)+\epsilon^{H}\sum _{0 < t_{j} < t}I_{j} \bigl(x_{\epsilon }(t_{j}) \bigr), \end{aligned}$$

(3)

where $\epsilon\in(0,\epsilon_{0}]$ is a positive small parameter and $\epsilon_{0}$ is a fixed number. Moreover, the averaged SDE of the standard ISDE (3) is

$$ z_{\epsilon}(t)= x_{0}+\epsilon^{2H} \int_{0}^{t} { \bigl[\bar{a} \bigl(z_{\epsilon }(s) \bigr)+\bar{I} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,ds+\epsilon^{H} \int_{0}^{t} {\bar {b} \bigl(z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s), $$

(4)

where the functions $\bar{a}(x): \mathbb{R}^{n} \rightarrow\mathbb {R}^{n}$, $\bar{b}(x): \mathbb{R}^{n} \rightarrow\mathbb{R}^{n}$ and $\bar{I}(x): \mathbb{R}^{n} \rightarrow\mathbb{R}^{n}$ are measurable functions satisfying

$$\begin{aligned}& \bar{a}(x)=\frac{1}{T} \int_{0}^{T} {a(t,x)}\,dt, \\& \bar{b}(x)=\frac{1}{T} \int_{0}^{T} {b(t,x)}\,dt, \\& \bar{I}(x)=\frac{1}{T}\sum_{j=1}^{k} {I_{j}(x)}. \end{aligned}$$

Assumption B

For any $x,y \in\mathbb{R}^{n}$, there exist positive constants $N_{1}$ and $N_{2}$ such that

$$ \bigl\vert I_{j}(x) \bigr\vert ^{2} \leq N_{1}, \qquad\bigl\vert I_{j}(x) - I_{j}(y) \bigr\vert ^{2} \leq N_{2} \vert x-y \vert ^{2}. $$

Assumption C

For all $t\in[0,T]$, $x \in\mathbb{R}^{n}$, the coefficients of Eq. (3) and Eq. (4) are bounded. Then there exists a positive constant M such that

$$\begin{aligned} \bigl\vert a(t,x) \bigr\vert ^{2} \leq M,\qquad \bigl\vert b(t,x) \bigr\vert ^{2} \leq M, \qquad\bigl\vert \bar{a}(x) \bigr\vert ^{2} \leq M, \qquad\bigl\vert \bar{b}(x) \bigr\vert ^{2} \leq M. \end{aligned}$$

Now, the existence and uniqueness result for Eq. (2) is given by the following theorem.

Theorem 2.10

Assume that Assumptions A–Care satisfied. Then, for every initial value $x_{0} \in\mathbb{R}^{n}$, there exists a unique solution $x(t)$to Eq. (2) on $[0,T]$.

Proof

The proof is a special case of the proof of Theorem 3.1 in Abouagwa et al. [38] and easy to be derived. So, we omit the proof here. □

3 Periodic averaging principle

In this section, we study the periodic averaging principle of ISDEs driven by fBm under non-Lipschitz condition.

In order to provide the periodic averaging results, we assume that the functions a and b are T-periodic in the first argument and the impulses $I_{j}$ are periodic in the sense that there exist $k\in\mathbb {N}$ such that $0 \leq t_{1} < t_{2} < \cdots< t_{k} < T$, and for every $j>k$, we have $t_{j} = t_{j-k} +T$, $I_{j}=I_{j-k}$.

Following Theorem 3.6 in Mao et al. [39], we now establish our main result which is used for revealing the relationship between the processes $x_{\epsilon}(t)$ and $z_{\epsilon}(t)$.

Theorem 3.1

Consider standard ISDE (3) and averaging SDE (4) if Assumptions A–Chold. Then, for $T > 0$, the following equality is satisfied:

$$\begin{aligned} \lim_{\epsilon\rightarrow0} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon }(t) \bigr\vert ^{2} =0. \end{aligned}$$

(5)

Proof

From Eqs. (3) and (4), taking expectation and employing the basic inequality $|a+b+c|^{2} \leq3|a|^{2} + 3|b|^{2} + 3|c|^{2}$, we obtain

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}\leq{}&3 \epsilon ^{4H}E \biggl\vert \int_{0}^{t} {a \bigl(s,x_{\epsilon}(s) \bigr)- \bar{a} \bigl(z_{\epsilon }(s) \bigr)}\,ds \biggr\vert ^{2} \\ &+3\epsilon^{2H}E \biggl\vert \int_{0}^{t} {b \bigl(s,x_{\epsilon}(s) \bigr)- \bar {b} \bigl(z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s) \biggr\vert ^{2} \\ &+3\epsilon^{2H}E \Biggl\vert \sum _{j=1}^{\infty}{I_{j} \bigl(x_{\epsilon }(t_{j}) \bigr)}- \int_{0}^{t} {\bar{I} \bigl(z_{\epsilon}(s) \bigr)}\,ds \Biggr\vert ^{2}\\ ={}&\sum_{l=1}^{3}{Q_{l}}. \end{aligned}$$

Starting with the first term $Q_{1}$, we have

$$\begin{aligned} Q_{1}\leq{}&6\epsilon^{4H}E \biggl\vert \int_{0}^{t} {a \bigl(s,x_{\epsilon }(s) \bigr)-a \bigl(s,z_{\epsilon}(s) \bigr)}\,ds \biggr\vert ^{2}+ 6\epsilon^{4H}E \biggl\vert \int_{0}^{t} {a \bigl(s,z_{\epsilon}(s) \bigr)- \bar {a} \bigl(z_{\epsilon}(s) \bigr)}\,ds \biggr\vert ^{2}\\ ={}&Q_{11}+Q_{12}. \end{aligned}$$

For $Q_{11}$, by applying the Cauchy–Schwarz inequality, Jensen’s inequality, and Assumption A, one can get

$$\begin{aligned} Q_{11}\leq{}&6\epsilon^{4H}E \biggl(t \int_{0}^{t} \bigl\vert a \bigl(s,x_{\epsilon }(s) \bigr)-a \bigl(s,z_{\epsilon}(s) \bigr) \bigr\vert ^{2}\,ds \biggr) \\ \leq{}&6\epsilon^{4H}\sup_{0 \leq t \leq u}t \biggl(E \int_{0}^{t} {\kappa \bigl( \bigl\vert x_{\epsilon}(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds \biggr) \\ \leq{}&6\epsilon^{4H}u \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon }(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds. \end{aligned}$$

Now, to deal with $Q_{12}$, let m be the largest positive integer such that $mT\leq t$. Then, for every $i={1,\ldots,m}$,

$$\begin{aligned} Q_{12}\leq{}&12\epsilon^{4H}E \Biggl\vert \sum_{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[a \bigl(s,z_{\epsilon}(s) \bigr)-\bar{a} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,ds \Biggr\vert ^{2} \\ &+12\epsilon^{4H}E \biggl\vert \int_{mT}^{t}{ \bigl[a \bigl(s,z_{\epsilon}(s) \bigr)-\bar {a} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,ds \biggr\vert ^{2} \\ \leq{}&36\epsilon^{4H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[a \bigl(s,z_{\epsilon}(s) \bigr)-a \bigl(s,z_{\epsilon}(iT) \bigr) \bigr]}\,ds \Biggr\vert ^{2} \\ &+36\epsilon^{4H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[a \bigl(s,z_{\epsilon}(iT) \bigr)-\bar{a} \bigl(z_{\epsilon}(iT) \bigr) \bigr]}\,ds \Biggr\vert ^{2} \\ &+36\epsilon^{4H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[\bar {a} \bigl(z_{\epsilon}(iT) \bigr)-\bar{a} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,ds \Biggr\vert ^{2} \\ &+12\epsilon^{4H}E \biggl\vert \int_{mT}^{t}{ \bigl[a \bigl(s,z_{\epsilon}(s) \bigr)-\bar {a} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,ds \biggr\vert ^{2}. \end{aligned}$$

(6)

Note that, by the definition of ā, we have

$$\begin{aligned} &E \Biggl\vert \sum_{i=1}^{m} \int_{(i-1)T}^{iT}{ \bigl[a \bigl(s,z_{\epsilon}(iT) \bigr)-\bar {a} \bigl(z_{\epsilon}(iT) \bigr) \bigr]}\,ds \Biggr\vert ^{2} \\ &\quad\leq m \sum_{i=1}^{m}E \biggl\vert \int_{0}^{T}{a \bigl(s,z_{\epsilon }(iT) \bigr)}\,ds-T\bar{a} \bigl(z_{\epsilon}(iT) \bigr) \biggr\vert ^{2}\\ &\quad=0, \end{aligned}$$

thus, by the Jensen inequality and Assumptions A, C, Eq. (6) becomes

$$\begin{aligned} Q_{12}\leq72\epsilon^{4H}mT \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+48 \epsilon^{4H}uMT. \end{aligned}$$

Then we can deduce that $Q_{1}$ has the following approximation:

$$\begin{aligned} Q_{1}\leq{}&6\epsilon^{4H}u \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon }(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds \\ &+72\epsilon^{4H}mT \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+48 \epsilon^{4H}uMT \\ :={}&\epsilon^{H}K_{1} \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon }(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds \\ &+ \epsilon^{H}K_{2} \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+ \epsilon^{H}O_{1}. \end{aligned}$$

(7)

Now, to estimate $Q_{2}$, we have

$$\begin{aligned} Q_{2}\leq{}&6\epsilon^{2H}E \biggl\vert \int_{0}^{t} {b \bigl(s,x_{\epsilon }(s) \bigr)-b \bigl(s,z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s) \biggr\vert ^{2} \\ &+6\epsilon^{2H}E \biggl\vert \int_{0}^{t} {b \bigl(s,z_{\epsilon}(s) \bigr)- \bar {b} \bigl(z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s) \biggr\vert ^{2}\\ ={}&Q_{21}+Q_{22}. \end{aligned}$$

(8)

Thanks to Lemma 2.7 and Assumption A, we can obtain

$$\begin{aligned} Q_{21}\leq12\epsilon^{2H}u^{2H-1}H \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon}(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds+24 \epsilon^{2H}u^{2}C. \end{aligned}$$

And, similar to Eq. (6),

$$\begin{aligned} Q_{22}\leq{}&12\epsilon^{2H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[b \bigl(s,z_{\epsilon}(s) \bigr)-\bar{b} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,d^{\circ }W^{H}(s) \Biggr\vert ^{2} \\ &+12\epsilon^{2H}E \biggl\vert \int_{mT}^{t} { \bigl[b \bigl(s,z_{\epsilon}(s) \bigr)-\bar {b} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,d^{\circ}W^{H}(s) \biggr\vert ^{2} \\ \leq{}&36\epsilon^{2H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[b \bigl(s,z_{\epsilon}(s) \bigr)-b \bigl(s,z_{\epsilon}(iT) \bigr) \bigr]}\,d^{\circ}W^{H}(s) \Biggr\vert ^{2} \\ &+36\epsilon^{2H}E \Biggl\vert \sum_{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[b \bigl(s,z_{\epsilon}(iT) \bigr)-\bar{b} \bigl(z_{\epsilon}(iT) \bigr) \bigr]}\,d^{\circ }W^{H}(s) \Biggr\vert ^{2} \\ &+36\epsilon^{2H}E \Biggl\vert \sum _{i=1}^{m} \int_{(i-1)T}^{iT} { \bigl[\bar {b} \bigl(z_{\epsilon}(iT) \bigr)-\bar{b} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,d^{\circ}W^{H}(s) \Biggr\vert ^{2} \\ &+12\epsilon^{2H}E \biggl\vert \int_{mT}^{t}{ \bigl[b \bigl(s,z_{\epsilon}(s) \bigr)-\bar {b} \bigl(z_{\epsilon}(s) \bigr) \bigr]}\,d^{\circ}W^{H}(s) \biggr\vert ^{2}, \end{aligned}$$

employing Lemma 2.7 and Assumptions A, C implies

$$\begin{aligned} Q_{22}\leq{}&144\epsilon^{2H}u^{2H-1}mTH \int_{0}^{T}{\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+288\epsilon ^{2H}Cu^{2} \\ &+72\epsilon^{2H}u^{2H-1}mTHE \int_{0}^{T}{ \bigl\vert b \bigl(s,z_{\epsilon }(s) \bigr)-\bar{b} \bigl(z_{\epsilon}(s) \bigr) \bigr\vert ^{2}}\,ds+144 \epsilon^{2H}Cu^{2} \\ &+24\epsilon^{2H}u^{2H-1}HE \int_{0}^{T}{ \bigl\vert b \bigl(s,z_{\epsilon }(s) \bigr)-\bar{b} \bigl(z_{\epsilon}(s) \bigr) \bigr\vert ^{2}}\,ds+48 \epsilon^{2H}Cu^{2} \\ \leq{}&144\epsilon^{2H}u^{2H-1}mTH \int_{0}^{T}{\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+288\epsilon^{2H}Cu^{2} \\ &+288\epsilon^{2H}u^{2H-1}mHT^{2}M+144 \epsilon^{2H}Cu^{2}\\ &+96\epsilon ^{2H}u^{2H-1}mHT^{2}M+48 \epsilon^{2H}Cu^{2}. \end{aligned}$$

Consequently, taking $Q_{21}$ and $Q_{22}$ into account, we conclude that

$$\begin{aligned} Q_{2}\leq{}& 12\epsilon^{2H}u^{2H-1}H \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon}(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds+24 \epsilon^{2H}u^{2}C \\ &+ 144\epsilon^{2H}u^{2H-1}mTH \int_{0}^{T}{\kappa \bigl(E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+288\epsilon^{2H}Cu^{2} \\ &+288\epsilon^{2H}u^{2H-1}mHT^{2}M+144 \epsilon^{2H}Cu^{2}+96\epsilon ^{2H}u^{2H-1}mHT^{2}M+48 \epsilon^{2H}Cu^{2} \\ :={}&\epsilon^{H}K_{3} \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon }(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds \\ &+\epsilon^{H}K_{4} \int_{0}^{T}{\kappa \bigl(E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds+ \epsilon^{H}O_{2}. \end{aligned}$$

(9)

Arriving at the last term $Q_{3}$, we apply Assumption B to obtain

$$\begin{aligned} Q_{3}\leq{}& 6\epsilon^{4H}k(m+1)E \sum_{j=1}^{k}{ \bigl\vert I_{j} \bigl(x_{\epsilon}(t_{j}) \bigr) \bigr\vert ^{2}} \\ &+6\epsilon^{4H}k(m+1)\frac{t}{T^{2}}E\sum _{j=1}^{k} \int_{0}^{t} { \bigl\vert I_{j} \bigl(z_{\epsilon}(s) \bigr) \bigr\vert ^{2}}\,ds \\ \leq{}&6\epsilon^{4H}k^{2}(m+1)N_{1}+6\epsilon ^{4H}k^{2}(m+1)^{2}N_{1}:= \epsilon^{H}O_{3}. \end{aligned}$$

(10)

Now, combining (7), (9), and (10) together, we get

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}\leq{}& \epsilon ^{H}\tilde{O}+ \epsilon^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds \\ &+\epsilon^{H}(K_{1}+K_{3}) \int_{0}^{t} {\kappa \bigl(E \bigl\vert x_{\epsilon }(s)-z_{\epsilon}(s) \bigr\vert ^{2} \bigr)}\,ds, \end{aligned}$$

(11)

where $\tilde{O}=O_{1}+ O_{2}+ O_{3}$. Obviously, the function $\kappa(x)$ is nondecreasing on $\mathbb{R_{+}}$ and $\kappa(0)=0$. Then, for any $t_{0} >0$, by setting $G(t)=\int_{t_{0}}^{t}\frac{ds}{\kappa(s)}$, it follows from Lemma 2.8 that

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}\leq{}& G^{-1} \biggl(G \biggl[\epsilon ^{H} \tilde{O}+ \epsilon^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds \biggr] \\ &+\epsilon^{H}(K_{1}+K_{3})T \biggr). \end{aligned}$$

Note that

$$\begin{aligned} \biggl\{ \epsilon^{H}\tilde{O}+\epsilon^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds \biggr\} \rightarrow0, \end{aligned}$$

as ϵ converges to zero. Recalling the condition $\int _{0^{+}}\frac{ds}{\kappa(s)}=\infty$, we can conclude that

$$\begin{aligned} G \biggl[\epsilon^{H}\tilde{O}+\epsilon^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds \biggr]+\epsilon^{H}(K_{1}+K_{3})T \rightarrow- \infty. \end{aligned}$$

On the other hand, because the function G is strictly increasing, we obtain that G has an inverse function which is strictly increasing too, and $G^{-1}(- \infty)= 0$. Namely,

$$\begin{aligned} G^{-1} \biggl(G \biggl[\epsilon^{H}\tilde{O}+ \epsilon ^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon }(iT) \bigr\vert ^{2} \bigr)}\,ds \biggr]+\epsilon^{H}(K_{1}+K_{3})T \biggr) \rightarrow0 \end{aligned}$$

as $\epsilon\rightarrow0$. Finally, we get

$$\begin{aligned} &\lim_{\epsilon\rightarrow0} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon }(t) \bigr\vert ^{2} \\ &\quad\leq \lim_{\epsilon\rightarrow0} \biggl(G^{-1} \biggl(G \biggl[\epsilon ^{H}(\tilde{O})+ \epsilon^{H}(K_{2}+K_{4}) \int_{0}^{T} {\kappa \bigl(E \bigl\vert z_{\epsilon}(s)-z_{\epsilon}(iT) \bigr\vert ^{2} \bigr)}\,ds \biggr] \\ &\qquad{}+\epsilon^{H}(K_{1}+K_{3})T \biggr) \biggr)\\ &\quad=0. \end{aligned}$$

This completes the proof. □

Remark 3.2

In Theorem 3.1, we establish the strong convergence (in the moment sense) of the processes $x_{\epsilon}$ and $z_{\epsilon}$ under non-Lipschitz condition. In other words, we have proved that, for a sufficiently small ϵ, the solutions of $x_{\epsilon}$ and $z_{\epsilon}$ are close enough.

For the sake of establishing the periodic stochastic averaging of Eq. (4) in finite time interval, we need the following auxiliary lemma.

Lemma 3.3

Let (4) be averaged SDE of standard ISDE (3). If Assumption Cholds, then, for $\epsilon_{1}\in(0,\epsilon _{0}]$, there exist $\epsilon\in(0,\epsilon_{1}]$and a positive constant $D>0$such that

$$\begin{aligned} E \bigl\vert z_{\epsilon}(t)-z_{\epsilon}(iT) \bigr\vert ^{2} \leq D \end{aligned}$$

for all $t\in[(i-1)T,iT]$, $i=1,2,\ldots,m$, $m \in\mathbb{N}$.

Proof

By Eq. (4), taking expectation and using the simple inequality $|a+b|^{2} \leq2|a|^{2}+2|b|^{2}$ yield

$$\begin{aligned} E \bigl\vert z_{\epsilon}(t)-z_{\epsilon}(iT) \bigr\vert ^{2} \leq2\epsilon^{4H} E \biggl\vert \int _{iT}^{t} {\bar{a} \bigl(z_{\epsilon}(s) \bigr)}\,ds \biggr\vert ^{2}+2\epsilon^{2H}E \biggl\vert \int _{iT}^{t} {\bar{b} \bigl(z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s) \biggr\vert ^{2}. \end{aligned}$$

Now, let $0\leq t\leq u\leq T$, then by the Cauchy–Schwarz inequality, Lemma 2.7, and Assumption C we get

$$\begin{aligned} E \biggl\vert \int_{iT}^{t} {\bar{a} \bigl(z_{\epsilon}(s) \bigr)}\,ds \biggr\vert ^{2} \leq u E \int _{iT}^{t} { \bigl\vert \bar{a} \bigl(z_{\epsilon}(s) \bigr) \bigr\vert ^{2}}\,ds \leq(m+1)T^{2}M \end{aligned}$$

and

$$\begin{aligned} E \biggl\vert \int_{iT}^{t} {\bar{b} \bigl(z_{\epsilon}(s) \bigr)}\,d^{\circ}W^{H}(s) \biggr\vert ^{2} \leq{}& 2Hu^{2H-1}E \int_{iT}^{t} { \bigl\vert \bar{b} \bigl(z_{\epsilon}(s) \bigr) \bigr\vert ^{2}}\,ds +4Cu^{2} \\ \leq{}& 2H(m+1)^{2H-1}T^{2H}M+4C(m+1)^{2}T^{2}, \end{aligned}$$

where m is the largest integer such that $mT \leq t$.

Finally, one can deduce that

$$\begin{aligned} E \bigl\vert z_{\epsilon}(t)-z_{\epsilon}(iT) \bigr\vert ^{2} \leq{}& 2\epsilon ^{4H}(m+1)T^{2}M+4 \epsilon^{2H}H(m+1)^{2H-1}T^{2H}M \\ &+8\epsilon^{2H}C(m+1)^{2}T^{2}\\ :={}&D. \end{aligned}$$

Hence, proved. □

Theorem 3.4

Suppose that Assumptions A–Care fulfilled for standard ISDE (3) and for averaged SDE (4), then, for given $\beta> 0$, $\alpha\in(0,1)$, and ${\epsilon_{1}} \in (0,{\epsilon_{0}}]$, there exist $\gamma>0$and $\epsilon\in (0,\epsilon_{1}]$such that

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2} \leq\gamma \epsilon^{H} \end{aligned}$$

(12)

for all $t\in[0,\beta\epsilon^{-\alpha H}]$.

Proof

By Assumption A, the concave function κ satisfies

$$ \kappa(x)\leq\lambda_{1}x+\lambda_{2}, $$

where $\lambda_{1}$ and $\lambda_{2}$ are positive constants. Applying this property for Eq. (11) yields

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}\leq{}& \epsilon ^{H}K_{5}\lambda_{1} \int_{0}^{t} {E \bigl\vert x_{\epsilon}(s)-z_{\epsilon }(s) \bigr\vert ^{2}}\,ds+\epsilon^{H}K_{5} \lambda_{2}t \\ &+\epsilon^{H}K_{6}\lambda_{1} \int_{0}^{T} {E \bigl\vert z_{\epsilon }(s)-z_{\epsilon}(iT) \bigr\vert ^{2}}\,ds+\epsilon^{H}K_{6} \lambda_{2}T+ \epsilon ^{H}\tilde{O}. \end{aligned}$$

Thanks to Lemma 3.3, we can obtain

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}\leq{}& \epsilon ^{H}K_{5}\lambda_{1} \int_{0}^{t} {E \bigl\vert x_{\epsilon}(s)-z_{\epsilon }(s) \bigr\vert ^{2}}\,ds+\epsilon^{H}K_{5} \lambda_{2}t \\ &+\epsilon^{H}K_{6}\lambda_{1}DT+ \epsilon^{H}K_{6}\lambda_{2}T+\epsilon ^{H}\tilde{O} \\ :={}& \epsilon^{H}K \int_{0}^{t} {E \bigl\vert x_{\epsilon}(s)-z_{\epsilon }(s) \bigr\vert ^{2}}\,ds +\epsilon^{H}O. \end{aligned}$$

Finally, applying Gronwall’s inequality implies

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}&\leq \epsilon ^{H}Oe^{\epsilon^{H}Kt}. \end{aligned}$$

Now, choose $\alpha\in(0,1)$ and $\beta> 0$, we can select $\epsilon _{1}\in(0,\epsilon_{0}]$ such that, for every $\epsilon\in(0,\epsilon _{1}]$, $t\in[0, \beta\epsilon^{-\alpha H }] \subseteq[0, \infty)$. And let $\gamma=Oe^{K \beta}$, we conclude

$$\begin{aligned} E \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert ^{2}&\leq \gamma\epsilon^{H}. \end{aligned}$$

Therefore, Theorem 3.4 is proved. □

Remark 3.5

Theorem 3.4indicates that the order of convergence of the processes $x_{\epsilon}$and $z_{\epsilon}$in finite time is about $\epsilon^{-\alpha H }$for $\alpha\in(0,1)$.

Next, we shall use the previous results to establish the convergence in probability between the solutions of Eq. (3) and Eq. (4).

Corollary 3.6

Let Assumptions A–Chold, for arbitrary small number $\delta>0$, there exist $\epsilon_{1}\in(0,\epsilon_{0}]$, $\beta >0$, and $0< \alpha< 1$such that, for all $\epsilon\in (0,\epsilon_{1}]$, we have

$$\begin{aligned} \lim_{\epsilon\rightarrow0} P \Bigl(\sup_{0\leq t \leq\beta\epsilon ^{-\alpha H }} \bigl\vert x_{\epsilon}(t)-z_{\epsilon}(t) \bigr\vert > \delta \Bigr) = 0. \end{aligned}$$

Proof

By Theorem 3.4 and employing the Chebyshev–Markov inequality, for any given number $\delta>0$, one can obtain that

$$\begin{aligned} P \Bigl(\sup_{0\leq t \leq\beta\epsilon^{-\alpha H }} \bigl\vert x_{\epsilon }(t)-z_{\epsilon}(t) \bigr\vert > \delta \Bigr) \leq{}& \frac{1}{\delta^{2}}E \Bigl(\sup _{0\leq t \leq\beta\epsilon^{-\alpha H }} \bigl\vert x_{\epsilon }(t)-z_{\epsilon}(t) \bigr\vert ^{2} \Bigr) \\ \leq{}& \frac{\epsilon^{H}Oe^{\epsilon^{H}Kt}}{\delta^{2}}, \end{aligned}$$

letting $\epsilon\rightarrow0$. Then the required result follows. □

4 Example

In this section, we provide an example to illustrate the foregoing averaging principle results.

Consider the following impulsive stochastic dynamical system:

$$ \begin{gathered} dx_{\epsilon}(t)= -\epsilon^{2H} \,dt+ \epsilon^{H} \cos^{2}(t)\lambda \,d^{\circ}W^{H}(t),\quad t\neq t_{j} \\ \triangle x_{\epsilon}(t)=\epsilon^{2H}j^{3} x_{\epsilon}\bigl(t^{-}_{j} \bigr),\quad t= t_{j}, j \in \mathbb{N}, \\ x_{\epsilon}(0)=x_{0},\end{gathered} $$

(13)

where $a(t,x)=-1$, $b(t,x)=\lambda \cos^{2}(t)$, and $I_{j}(x)=j^{3}x$. Let $T=1$ and $\lambda=3$. Then, by the definitions of $\bar{a}(\cdot)$, $\bar{b}(\cdot)$, and $\bar{I}(\cdot)$ in Sect. 2, we have

$$\begin{aligned} &\bar{a}(z_{\epsilon}) = \frac{1}{T} \int_{0}^{T}{a(t,z_{\epsilon})}\,dt=-1, \\ &\bar{b}(z_{\epsilon})= \frac{1}{T} \int_{0}^{T}{b(t,z_{\epsilon})}\,dt= \lambda \int_{0}^{1}{\cos^{2}(t)}\,dt=3 \times0.73=2.19, \\ &\bar{I}(z_{\epsilon})= \frac{1}{T}\sum _{j=1}^{k} j^{3}z_{\epsilon}=z_{\epsilon}\sum_{j=1}^{k} j^{3}=\frac{k^{2}(k+1)^{2}}{4}z_{\epsilon}. \end{aligned}$$

So then, the solution to averaged SDE for the impulsive dynamical system (13) can be interpreted as follows:

$$\begin{aligned} z_{\epsilon}(t)&=x_{0}+\frac{k^{2}(k+1)^{2}}{4} \epsilon^{2H} \int_{0}^{t} \bigl(z_{\epsilon}(s)-1 \bigr) \,ds+ 2.19\epsilon^{H} \int_{0}^{t} \,d^{\circ}W^{H}(t). \end{aligned}$$

(14)

It is easy to verify that the conditions of Theorem 3.1, Theorem 3.4, and Corollary 3.6 are satisfied. Then the solution of averaged SDEs (14) converges to that of standard Eq. (13) in the sense of mean square and in probability.