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Strong convergence in the pth-mean of an averaging principle for two-time-scales SPDEs with jumps

Contributed equally
Advances in Difference Equations20172017:275

https://doi.org/10.1186/s13662-017-1333-9

Received: 12 May 2017

Accepted: 26 August 2017

Published: 8 September 2017

Abstract

The main goal of this work is to study an averaging principle for two-time-scales stochastic partial differential equations with jumps. The solutions of reduced equations with modified coefficients are derived to approximate the slow component of the original equation under suitable conditions. It is shown that the slow component can strongly converge to the solution of the corresponding reduced equation in the pth-mean. Our key and novel idea is how to cope with the changes caused by jumps and higher order moments.

Keywords

strong convergence in pth-meantwo-time-scalesSPDEsjumpsaveraging principle

1 Introduction

In practical science and engineering, many complex systems can be described as singularly perturbed systems with separated two-time-scales driven by random perturbations, for example, chemical reaction dynamics [1], electronic circuits [2] and laser systems [3]. In most cases, people are only interested in investigating the time evolution of the slow component, but that cannot be done directly, unless we solve the full two-time-scales equations. Although computers are now very advanced, they cannot deal with such a disparity of scales. Averaging methods can reduce the computational load. In view of this, the averaging principle, which is an effective tool to analyze the two-time-scales dynamical systems with random perturbations, becomes more and more important and popular to be applied to reduce the dimensions of the original systems.

The theory of the averaging principle has a long and rich history. Let us mention a few of them. Khasminskii [4] first proved the averaging principle of stochastic differential equations (SDEs) driven by Brownian noise. Since then, the averaging principle has been an active research field on which there is a great deal of literature. Freidlin and Wentzell [5] provided a mathematically rigorous overview of fundamental stochastic averaging methods. Golec and Ladde [6] and Xu et al. [714] proposed the averaging principle to stochastic dynamical systems in the sense of the mean-square, which implies the convergence in probability. Furthermore, \(\mathbb{L}^{2}\)-strong convergence (also called mean-square-strong convergence) in averaging principles for several types of slow-fast stochastic dynamical systems driven by Brownian noise has been investigated by Freidlin [5], Golec [15], Wang [16], and Fu et al. [17, 18].

In some circumstances, jump type perturbations can capture some large moves and unpredictable events in such diverse areas as mathematics, finance, statistical physics and life sciences [1934], while purely Brownian type perturbations cannot do so. It is well known that stochastic partial differential equations (SPDEs) driven by jump type perturbations may be more appropriate to model a great amount of complex systems, which are widely used to describe many interesting phenomena in the fields of physics, biology, chemistry, economics, finance and others [3541]. Up to now, many scholars have extensively investigated the existence and uniqueness for solutions of SPDEs driven by jump type perturbations. For example, Albeverio et al. [42] investigated the existence and uniqueness of mild solutions to stochastic heat equations driven by Poisson jumps. Hausenblas [43] considered the existence and uniqueness of mild solutions to SPDEs of the jump type. A series of useful theories and methods have been presented to explore SPDEs driven by jumps (see [19, 39]), and among them, the averaging method has been an important and useful tool to reduce SPDEs driven by jumps. Givon [44] established an averaging principle for two-time-scales jump-diffusion processes in the sense of the mean-square. Quite recently, Xu and Miao [45] established a \(\mathbb{L}^{2}\)-strong averaging principle for slow-fast SPDEs driven by Poisson random measures. Pei et al. [46] considered the averaging principle for stochastic hyperbolic- parabolic equations driven by Poisson random measures with slow and fast time-scales.

However, the work on the averaging principle mainly discussed \(\mathbb {L}^{2}\)-strong convergence for two-time-scales jump-diffusion processes, which does not involve \(\mathbb{L}^{p}\) (\(p>2\))-strong convergence in general. Generally, people need to estimate the higher order moments which possess a good robustness and can be applied in computations in statistics, finance and other fields. To the best of the authors’ knowledge, the averaging principle for two-time-scales SPDEs with jumps has not been considered in \(\mathbb {L}^{p}\) (\(p>2\))-strong convergence. Therefore, based on the above discussion, an attempt will be made to establish an averaging principle for two-time-scales SPDEs driven by jumps in \(\mathbb {L}^{p}\) (\(p>2\))-strong convergence. In this paper, our key and novelty is how to cope with the changes caused by jumps and higher order moments. It is drastically different because of the appearance of the jumps.

The paper is organized as follows. In Section 2, we present some notations and the formulation of the problem. In Section 3, the main result is stated. We derive the stochastic averaging principle for two-time-scales SPDEs driven by jumps in \(\mathbb{L}^{p}\) (\(p>2\))-strong convergence.

2 Preliminaries

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a complete probability space with a natural filtration \(\{\mathcal{F}_{t}\}_{t\geq0}\) satisfying the usual conditions. We fix \(l>0\) arbitrarily, and we denote \(D := (0, l)\), i.e., D is a fixed, open, bounded interval of the real line \(\mathbb{R}\). Let \(\mathbb{H}\) be a Hilbert space \(\mathbb{L}^{2}(D)\) equipped with the inner product \(\langle\cdot, \cdot\rangle _{\mathbb{H}}\) and the corresponding norm \(\|\cdot\|\). Let \(T>0\) be fixed arbitrarily. In this paper, we are concerned with the following SPDEs driven by both Brownian motions and Poisson random measures:
$$ \left \{ \textstyle\begin{array}{l} \frac{\partial X^{\varepsilon}_{t}(\xi)}{\partial t}= \varDelta X^{\varepsilon}_{t}(\xi)+f(X^{\varepsilon}_{t}(\xi),Y^{\varepsilon }_{t}(\xi))+g(X^{\varepsilon}_{t}(\xi))\dot{W}_{t}^{1}\\ \phantom{\frac{\partial X^{\varepsilon}_{t}(\xi)}{\partial t}=}{}+ \int_{\mathbb{Z}}h(X^{\varepsilon}_{t-}(\xi),z)\dot {\tilde{N}}_{1}(t,dz),\\ \frac{\partial Y^{\varepsilon}_{t}(\xi)}{\partial t}= \frac {1}{\varepsilon}\varDelta Y^{\varepsilon}_{t}(\xi)+\frac {1}{\varepsilon}F(X^{\varepsilon}_{t}(\xi),Y^{\varepsilon}_{t}(\xi ))+\frac{1}{\sqrt{\varepsilon}}G(X^{\varepsilon}_{t}(\xi ),Y^{\varepsilon}_{t}(\xi))\dot{W}_{t}^{2} \\ \phantom{\frac{\partial Y^{\varepsilon}_{t}(\xi)}{\partial t}=}{}+ \int_{\mathbb{Z}}H(X^{\varepsilon}_{t-}(\xi ),Y^{\varepsilon}_{t-}(\xi),z)\dot{\tilde{N}}^{\varepsilon }_{2}(t,dz),\\ X^{\varepsilon}_{t}(\xi)= Y^{\varepsilon}_{t}(\xi)=0, \quad(\xi ,t)\in\partial D\times(0,T],\\ X^{\varepsilon}_{0}(\xi)= X_{0}(\xi)\in\mathbb{H},\qquad Y_{0}^{\varepsilon}(\xi)=Y_{0}(\xi)\in\mathbb{H},\quad\xi\in D, \end{array}\displaystyle \right . $$
(2.1)
for \(\varepsilon>0\) and \((\xi,t)\in D\times[0,T]\), where the coefficients \(f(x,y):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb {R}\), \(g(x):\mathbb{R} \rightarrow\mathbb{R}\), \(h(x,z):\mathbb{R} \times\mathbb{Z} \rightarrow\mathbb{R}\), \(F(x,y):\mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}\), \(G(x,y):\mathbb{R}\times\mathbb {R}\rightarrow\mathbb{R}\), \(H(x,y,z) :\mathbb{R} \times\mathbb{R} \times\mathbb{Z} \rightarrow\mathbb{R}\) are all real-valued measurable functions. The detailed conditions for them will be given in the next section. \(\{ W_{t}^{1}\}_{t\geq0}\) and \(\{W_{t}^{2}\}_{t\geq0}\) are mutually independent real-valued \(\{\mathcal{F}_{t}\}_{t\geq0}\)-Wiener processes. Next, we explicate the Poisson random measures \(\tilde{N}_{1}(dt,dz)\) and \(\tilde {N}^{\varepsilon}_{2}(dt,dz)\). Let \((\mathbb{Z}, \mathcal {B}({\mathbb{Z}}))\) be a given measurable space and \(v(dz)\) be a σ-finite measure on it. Let \(D_{p^{i}_{t}}\), \(i=1,2\) be two countable subsets of \(\mathbb{R}_{+}\). Furthermore, let \(p^{1}_{t}\), \(t\in D_{p^{1}_{t}}\) be a stationary \(\mathcal {F}_{t}\)-adapted Poisson point process on \(\mathbb{Z}\) with characteristic v, and let \(p^{2}_{t}\), \(t\in D_{p^{2}_{t}}\) be a stationary \(\mathcal {F}_{t}\)-adapted Poisson point process on \(\mathbb{Z}\) with characteristic \(\frac{v}{\varepsilon}\). Denote by \(N^{i}(dt,dz)\) the Poisson counting measure associated with \(p_{t}^{i}\), i.e.,
$$N^{i}(t,A):=\sum_{s\in D_{p^{i}_{t}}, s\leq t}I_{A} \bigl(p^{i}_{t}\bigr),\quad i=1,2. $$
Let us denote the two corresponding compensated martingale measures
$$\tilde{N}_{1}(dt,dz):=N^{1}(dt,dz)-v(dz)\,dt $$
and
$$\tilde{N}_{2}^{\varepsilon}(dt,dz):=N^{2}(dt,dz)- \frac {1}{\varepsilon}v(dz)\,dt. $$
Let us define an abstract \(\mathbb{A}=\partial_{\xi\xi}\) with zero Dirichlet boundary conditions. Let \(\{e_{k}(\xi)\}_{k\in\mathbb{N}}\) be a complete orthonormal system of eigenvectors in \(\mathbb{H}\) such that, for \(k=1,2,\ldots\) ,
$$\mathbb{A}e_{k}=-\alpha_{k}e_{k},\quad e_{k}|_{\partial{D}=0}, $$
with \(0<\alpha_{1}\leq\alpha_{2}\leq\cdots\leq\alpha_{k}\leq\cdots\) .
Let \(\mathbb{V}\) be a Sobolev space \(H_{0}^{1}\) of order one with zero Dirichlet boundary conditions, which is densely and continuously injected in the Hilbert space \(\mathbb{H}\). Identifying \(\mathbb{H}\) with its dual space, we obtain the Gelfand triple
$$\mathbb{V}\subset\mathbb{H}\cong\mathbb{H}^{*}\subset \mathbb{V^{*}}. $$
Owing to Poincaré’s inequality, we obtain
$$ \langle\mathbb{A}u,u\rangle=-\|\nabla u\|^{2}\leq- \alpha_{1}\|u\|^{2}, $$
(2.2)
where \(\langle\cdot,\cdot\rangle\) denotes a dual pair of \((\mathbb {V},\mathbb{V^{*}})\).
Note that the Green’s function \(S(\xi,\zeta,t)\) for the deterministic equation \((\partial/\partial t-\mathbb{A})X(t,\xi)=0\) can be expressed as
$$S(\xi,\zeta,t)=\sum_{k=1}^{\infty}e^{-\alpha_{k}t}e_{k}( \xi )e_{k}(\zeta). $$
Recall that the associated Green’s operator is defined, for any \(\Lambda(\xi)\in\mathbb{H}\), by
$$S_{t}\Lambda(\xi)= \int_{D}S(\xi,\zeta,t)\Lambda(\zeta)\,d\zeta =\sum _{k=1}^{\infty}e^{-\alpha_{k}t}e_{k}(\xi) \langle e_{k}, \Lambda\rangle_{\mathbb{H}}. $$
It is straightforward that \(\{S_{t}\}_{t \geq0}\) are contractive semigroups on \(\mathbb{H}\) and \(\|S_{t}\Lambda(\xi)\| \leq\|\Lambda (\xi)\|\).
To give precise results, it is convenient to look at the equations in an abstract setting, where system (2.1) can be rewritten as
$$ \begin{aligned} \left\{\textstyle\begin{array}{l} dX^{\varepsilon}_{t}=[\mathbb{A}X^{\varepsilon }_{t}+f(X^{\varepsilon}_{t},Y^{\varepsilon}_{t})]\,dt+g(X^{\varepsilon }_{t})\,d{W}_{t}^{1}+ \int_{\mathbb{Z}}h(X^{\varepsilon }_{t-},z){\tilde{N}}_{1}(dt,dz),\\ dY^{\varepsilon}_{t}=\frac{1}{\varepsilon}[\mathbb {A}Y^{\varepsilon}_{t}+F(X^{\varepsilon}_{t},Y^{\varepsilon }_{t})]\,dt+\frac{1}{\sqrt{\varepsilon}}G(X^{\varepsilon }_{t},Y^{\varepsilon}_{t})\,d{W}_{t}^{2}\\ \phantom{dY^{\varepsilon}_{t}=}{} + \int_{\mathbb{Z}}H(X^{\varepsilon}_{t-},Y^{\varepsilon }_{t-},z){\tilde{N}}^{\varepsilon}_{2}(dt,dz),\\ X_{0}\in\mathbb{H}, \qquad Y_{0}\in\mathbb{H}. \end{array}\displaystyle \right.\end{aligned} $$
(2.3)

We now introduce the definition of mild solutions of system (2.3).

Definition 2.1

A natural way to give a rigorous meaning to (2.3) is in terms of the following integral equations:
$$ \begin{aligned} \left\{\textstyle\begin{array}{l} X^{\varepsilon}_{t}=X_{0}S_{t}+\int_{0}^{t}S_{t-s}f(X^{\varepsilon }_{s},Y^{\varepsilon}_{s})\,ds+\int_{0}^{t}S_{t-s}g(X^{\varepsilon }_{s})\,dW_{s}^{1}\\ \phantom{X^{\varepsilon}_{t}=}{}+\int_{0}^{t}\int_{\mathbb{Z}}S_{t-s}h(X^{\varepsilon }_{s-},z)\tilde{N}_{1}(ds,dz),\\ Y^{\varepsilon}_{t}=Y_{0}S_{t/\varepsilon}+\frac{1}{\varepsilon }\int_{0}^{t}S_{(t-s)/\varepsilon}F(X^{\varepsilon }_{s},Y^{\varepsilon}_{s})\,ds+\frac{1}{\sqrt{\varepsilon}}\int _{0}^{t}S_{(t-s)/\varepsilon}G(X^{\varepsilon}_{s},Y^{\varepsilon }_{s})\,dW_{s}^{2}\\ \phantom{Y^{\varepsilon}_{t}=}{}+\int_{0}^{t}\int_{\mathbb{Z}}S_{(t-s)/\varepsilon }H(X^{\varepsilon}_{s-}, Y^{\varepsilon}_{s-},z)\tilde {N}^{\varepsilon}_{2}(ds,dz). \end{array}\displaystyle \right.\end{aligned} $$
(2.4)
Moreover, according to Itô’s formula [39, 47], for \(t\in [0,T]\), \(p>1\), the following equalities hold:
$$\begin{aligned}[b] \big\| X_{t}^{\epsilon}\big\| ^{2p}={}&\|X_{0} \|^{2p}+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle \mathbb{A}X_{s}^{\epsilon },X_{s}^{\epsilon} \bigr\rangle \,ds \\ &+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle f\bigl(X_{s}^{\epsilon },Y_{s}^{\epsilon} \bigr),X_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,ds \\ &+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle g\bigl(X_{s}^{\epsilon } \bigr),X_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,dW_{s}^{1} \\ &+2p(p-1) \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\big\| g\bigl(X_{s}^{\epsilon }\bigr)\big\| ^{2} \,ds \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s-}^{\epsilon }+h \bigl(X_{s-}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s-}^{\epsilon}\big\| ^{2p}\bigr]\tilde {N}_{1}(ds,dz) \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s}^{\epsilon }+h \bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz)\, ds \\ &-2p \int_{0}^{t} \int_{\mathbb{Z}}\big\| X_{s}^{\epsilon}\big\| ^{2p-2} \bigl\langle h\bigl(X_{s}^{\epsilon},z\bigr),X_{s}^{\epsilon} \bigr\rangle _{\mathbb{H}}v(dz)\,ds\end{aligned} $$
(2.5)
and
$$ \begin{aligned}[b] \big\| Y_{t}^{\epsilon} \big\| ^{2p}={}&\|Y_{0}\|^{2p}+\frac{2p}{\epsilon} \int _{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle \mathbb{A}Y_{s}^{\epsilon },Y_{s}^{\epsilon} \bigr\rangle \,ds \\ &+\frac{2p}{\epsilon} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle F\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr),Y_{s}^{\epsilon }\bigr\rangle _{\mathbb{H}} \,ds \\ &+\frac{2p}{\sqrt{\epsilon}} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle G\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr),Y_{s}^{\epsilon }\bigr\rangle _{\mathbb{H}} \,dW_{s}^{2} \\ &+\frac{2p(p-1)}{\epsilon} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p-2}\big\| G\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr)\big\| ^{2}\,ds \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| Y_{s-}^{\epsilon }+H \bigl(X_{s-}^{\epsilon},Y_{s-}^{\epsilon},z\bigr) \big\| ^{2p}-\big\| Y_{s-}^{\epsilon }\big\| ^{2p}\bigr] \tilde{N}_{2}^{\epsilon}(ds,dz) \\ &+\frac{1}{\epsilon} \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| Y_{s}^{\epsilon}+H \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z\bigr) \big\| ^{2p}-\big\| Y_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz) \,ds \\ &-\frac{2p}{\epsilon} \int_{0}^{t} \int_{\mathbb{Z}}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2} \bigl\langle H\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon },z \bigr),Y_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}}v(dz)\,ds. \end{aligned} $$
(2.6)

Convention

The letter C, with or without subscripts, will denote positive constants whose value may change in different occasions. We will write the dependence of constants on parameters explicitly if it is essential.

Now, we need to give some dissipative conditions [46] to ensure the ergodicity for the fast motion and global Lipschitz condition, and the growth condition to ensure the existence and uniqueness for (2.3).

Assumption 1

The coefficients of (2.3) are globally Lipschitz continuous in x, y, i.e., \(\forall x_{1},x_{2}, y_{1},y_{2}\in\mathbb{R}\), there exist six positive constants \(C_{f}\), \(C_{g}\), \(C_{h}\), \(C_{F}\), \(C_{G}\), \(C_{H}\). We have
$$ \begin{gathered} \big|f(x_{1},y_{1})-f(x_{2},y_{2})\big|^{2} \leq C_{f}\bigl(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2} \bigr), \\ \big|g(x_{1})-g(x_{2})\big|^{2} \leq C_{g}|x_{1}-x_{2}|^{2}, \\ \int_{\mathbb{Z}}\big|h(x_{1},z)-h(x_{2},z)\big|^{q}v(dz) \leq C_{h}|x_{1}-x_{2}|^{q},\quad q \geq2, \end{gathered} $$
and
$$ \begin{gathered} \big|F(x_{1},y_{1})-F(x_{2},y_{2})\big|^{2} \leq C_{F}\bigl(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2} \bigr), \\ \big|G(x_{1},y_{1})-G(x_{2},y_{2})\big|^{2} \leq C_{G}\bigl(|x_{1}-x_{2}|^{2}+|y_{1}-y_{2}|^{2} \bigr), \\ \int_{\mathbb {Z}}\big|H(x_{1},y_{1},z)-H(x_{2},y_{2},z)\big|^{q}v(dz) \leq C_{H}\bigl(|x_{1}-x_{2}|^{q}+|y_{1}-y_{2}|^{q} \bigr),\quad q \geq2. \end{gathered} $$

Remark 2.2

From Assumption 1, for all \(x_{1},y_{1}\in\mathbb{R}\), it immediately follows that
$$\begin{aligned}& \begin{gathered} \big\| f(x_{1},y_{1})\big\| ^{2}+\big\| g(x_{1}) \big\| ^{2}+\big\| F(x_{1},y_{1})\big\| ^{2}+\big\| G(x_{1},y_{1})\big\| ^{2} \\ \quad \leq2(C_{f}+C_{g}+C_{F}+C_{G}) \bigl(\| x_{1}\|^{2}+\|y_{1}\|^{2}\bigr) \\ \qquad{} +2\bigl(\big|f(0,0)\big|^{2}+\big|g(0)\big|^{2}+\big|F(0,0)\big|^{2}+\big|G(0,0)\big|^{2} \bigr),\end{gathered} \\& \int_{\mathbb{Z}}\big\| h(x_{1},z)\big\| ^{q}v(dz) \leq2^{q-1} \int_{\mathbb {Z}}\big\| h(0,z)\big\| ^{q}v(dz)+2^{q-1}C_{h} \|x_{1}\|^{q}, \\& \int_{\mathbb{Z}}\big\| H(x_{1},y_{1},z) \big\| ^{q}v(dz)\leq2^{q-1} \int _{\mathbb{Z}}\big\| H(0,0,z)\big\| ^{q}v(dz)+2^{q-1}C_{H} \bigl(\|x_{1}\|^{q}+\|y_{1}\| ^{q} \bigr),\quad q \geq2, \end{aligned}$$
so we set
$$\begin{gathered} K_{3}=\max \bigl\{ 2(C_{f}+C_{g}+C_{F}+C_{G}),2\big(\big|f(0,0)\big|^{2}+\big|g(0)\big|^{2}+\big|F(0,0)\big|^{2}+\big|G(0,0)\big|^{2}\big) \bigr\} , \\ K_{4}=\max \biggl\{ 2^{q-1} \int_{\mathbb{Z}}\big\| h(0,z)\big\| ^{q}v(dz),2^{q-1}C_{h} \biggr\} , \\ K_{5}=\max \biggl\{ 2^{q-1} \int_{\mathbb{Z}}\big\| H(0,0,z)\big\| ^{q}v(dz),2^{q-1}C_{H} \biggr\} ,\end{gathered} $$
and then we have
$$\begin{gathered} \big\| f(x_{1},y_{1})\big\| ^{2}+\big\| g(x_{1}) \big\| ^{2}+\big\| F(x_{1},y_{1})\big\| ^{2}+\big\| G(x_{1},y_{1})\big\| ^{2}\leq K_{3}\bigl(1+ \|x_{1}\|^{2}+\|y_{1}\|^{2}\bigr), \\ \int_{\mathbb{Z}}\big\| h(x_{1},z)\big\| ^{q}v(dz) \leq K_{4}\bigl(1+\|x_{1}\| ^{q}\bigr),\quad q \geq2, \\ \int_{\mathbb{Z}}\big\| H(x_{1},y_{1},z) \big\| ^{q}v(dz)\leq K_{5}\bigl(1+\|x_{1}\| ^{q}+\|y_{1}\|^{q}\bigr),\quad q \geq2.\end{gathered} $$

Assumption 2

f is globally bounded.

Assumption 3

\(\eta=\alpha_{1}-C_{F}-C_{G}-C_{H}>0\), where \(\alpha _{1}\) is the decay rate of \(\mathbb{A}\).

Remark 2.3

Assumption 3 is a strong dissipative condition, and it is very important to prove the ergodicity for the fast motion. The detailed proofs will be given in Appendix A.

It is easy to see that, complying with Assumption 1, and in terms of Remark 2.3, (2.3) has unique mild solutions [39, 40].

3 Averaging principle for two-time-scales SPDEs with jumps

In this section, we first prove two key lemmas and then present the main result of the paper.

Lemma 3.1

Let Assumptions 1-3 be satisfied. For any \(T>0\) there exists a positive constant \(C_{p,T}>0\), \(p>1\) such that for any \(\epsilon\in(0,1)\),
$$ \begin{aligned} \mathbb{E}\sup_{0 \leq t \leq T} \big\| X_{t}^{\epsilon}\big\| ^{2p} \leq C_{p,T}. \end{aligned} $$

Proof

For \(\|X_{t}^{\epsilon}\|^{2p}\), by the energy identities (2.5), we have
$$\begin{aligned} \big\| X_{t}^{\epsilon}\big\| ^{2p} =&\|X_{0} \|^{2p}+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle \mathbb{A}X_{s}^{\epsilon },X_{s}^{\epsilon} \bigr\rangle \,ds \\ &{}+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle f\bigl(X_{s}^{\epsilon },Y_{s}^{\epsilon} \bigr),X_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,ds \\ &{}+2p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\bigl\langle g\bigl(X_{s}^{\epsilon } \bigr),X_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,dW_{s}^{1} \\ &+2p(p-1) \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p-2}\big\| g\bigl(X_{s}^{\epsilon }\bigr)\big\| ^{2} \,ds \\ &{}+ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s-}^{\epsilon }+h \bigl(X_{s-}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s-}^{\epsilon}\big\| ^{2p}\bigr]\tilde {N}_{1}(ds,dz) \\ &{}+ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s}^{\epsilon }+h \bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz)\,ds \\ &{}-2p \int_{0}^{t} \int_{\mathbb{Z}}\big\| X_{s}^{\epsilon}\big\| ^{2p-2} \bigl\langle h\bigl(X_{s}^{\epsilon},z\bigr),X_{s}^{\epsilon} \bigr\rangle _{\mathbb{H}}v(dz)\,ds \\ =&\|X_{0}\|^{2p}+\sum_{i=1}^{7} \Pi_{t}^{i}. \end{aligned}$$
By Assumption 1 and Assumption 3, Young’s inequality and (2.2), we have
$$ \begin{aligned} \|X_{0}\|^{2p}+ \Pi_{t}^{1}+\Pi_{t}^{2}+ \Pi_{t}^{4} &\leq\|X_{0}\| ^{2p}-2 \alpha_{1}p \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p}\,ds+C_{p} \int _{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p}\,ds \\ &\leq C+C_{p} \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p}\,ds. \end{aligned} $$
For \(\Pi_{t}^{6}\), \(\Pi_{t}^{7}\), according to the binomial theorem, we calculate the coefficients in the expansion of \((a+b)^{2p}\),
$$ \begin{aligned}[b] (a+b)^{2p}={}&C^{2p}_{0}a^{2p}+C^{2p}_{1}a^{2p-1}b+C^{2p}_{2}a^{2p-2}b^{2} \\ &+\cdot\cdot\cdot +C^{2p}_{2p-2}a^{2}b^{2p-2}+C^{2p}_{2p-1}ab^{2p-1}+C^{2p}_{2p}b^{2p}, \end{aligned} $$
(3.1)
where \(C_{k}^{2p}=\frac{(2p)!}{(2p-k)!k!}\), \(k=0,1,2,\ldots,2p\). So, by Assumption 3 and Young’s inequality,
$$\begin{aligned} \Pi_{t}^{6}+\Pi_{t}^{7}={}& \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s}^{\epsilon}+h \bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz)\,ds \\ &-2p \int_{0}^{t} \int_{\mathbb{Z}}\big\| X_{s}^{\epsilon}\big\| ^{2p-2} \bigl\langle h\bigl(X_{s}^{\epsilon},z\bigr),X_{s}^{\epsilon} \bigr\rangle _{\mathbb{H}}v(dz)\,ds \\ ={}& \sum_{i=2}^{2p}C_{i}^{2p} \int_{0}^{t} \int_{\mathbb{Z}}\big\| X_{s}^{\epsilon}\big\| ^{2p-i} \big\| h\bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{i}v(dz)\,ds \\ \leq{}& C_{p} \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p}\,ds+C.\end{aligned} $$
Then we obtain
$$ \begin{aligned} \mathbb{E}\sup_{0 \leq s \leq t} \big\| X_{t}^{\epsilon}\big\| ^{2p} \leq C+C_{p} \int_{0}^{t}\mathbb{E}\sup_{0 \leq r \leq s} \big\| X_{r}^{\epsilon }\big\| ^{2p}\,ds+\mathbb{E}\sup _{0 \leq s \leq t} \Pi_{s}^{3}+\mathbb {E}\sup _{0 \leq s \leq t} \Pi_{s}^{5}. \end{aligned} $$
Now, by Young’s inequality and the BDG inequality, we find
$$\begin{aligned} \mathbb{E}\sup_{0 \leq s \leq t} \Pi_{s}^{5} \leq& C \mathbb{E} \biggl\{ \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| X_{s}^{\epsilon}+h \bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{2p}- \big\| X_{s}^{\epsilon}\big\| ^{2p}\bigr]^{2}v(dz)\,ds \biggr\} ^{\frac{1}{2}} \\ \leq& C \mathbb{E} \Biggl\{ \int_{0}^{t} \int_{\mathbb{Z}} \Biggl(\sum_{i=1}^{2p}C_{i}^{2p} \big\| X_{s}^{\epsilon}\big\| ^{2p-i}\big\| h\bigl(X_{s}^{\epsilon},z \bigr)\big\| ^{i} \Biggr)^{2}v(dz)\,ds \Biggr\} ^{\frac {1}{2}} \\ \leq& C \mathbb{E} \Biggl\{ \sum_{i=1}^{2p} \int_{0}^{t} \int _{\mathbb{Z}}\big\| X_{s}^{\epsilon}\big\| ^{4p-2i} \big\| h\bigl(X_{s}^{\epsilon},z\bigr)\big\| ^{2i}v(dz)\,ds \Biggr\} ^{\frac{1}{2}} \\ \leq& C \mathbb{E} \biggl\{ \int_{0}^{t}\big\| X_{s}^{\epsilon}\big\| ^{4p}\,ds \biggr\} ^{\frac{1}{2}}+C. \end{aligned}$$
Next, it is easy to see that
$$ \begin{aligned} \mathbb{E}\sup_{0 \leq s \leq t} \Pi_{s}^{3}\leq C \mathbb{E} \biggl\{ \int_{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{4p}\,ds \biggr\} ^{\frac{1}{2}}+C. \end{aligned} $$
Therefore,
$$\begin{aligned} \mathbb{E}\sup_{0 \leq s \leq t}\big\| X_{s}^{\epsilon} \big\| ^{2p} &\leq C+C_{p} \int_{0}^{t}\mathbb{E}\sup_{0 \leq r \leq s} \big\| X_{r}^{\epsilon }\big\| ^{2p}\,ds+C \mathbb{E} \biggl\{ \int_{0}^{t}\big\| X_{s}^{\epsilon}\big\| ^{4p}\,ds \biggr\} ^{\frac{1}{2}} \\ &\leq C+C_{p} \int_{0}^{t}\mathbb{E}\sup_{0 \leq r \leq s} \big\| X_{r}^{\epsilon}\big\| ^{2p}\,ds+C \mathbb{E} \biggl\{ \sup _{0 \leq s \leq t}\big\| X_{s}^{\epsilon}\big\| ^{2p} \int_{0}^{t}\big\| X_{s}^{\epsilon}\big\| ^{2p}\,ds \biggr\} ^{\frac{1}{2}} \\ &\leq C+C_{p} \int_{0}^{t}\mathbb{E}\sup_{0 \leq r \leq s} \big\| X_{r}^{\epsilon}\big\| ^{2p}\,ds+\frac{1}{2} \mathbb{E}\sup_{0 \leq s \leq t}\big\| X_{s}^{\epsilon} \big\| ^{2p}.\end{aligned} $$
Finally, by Gronwall’s inequality, we have
$$ \mathbb{E}\sup_{0 \leq s \leq t}\big\| X_{s}^{\epsilon} \big\| ^{2p} \leq C e^{C_{p}T}. $$
This is the proof of Lemma 3.1. □

Lemma 3.2

Let Assumptions 1-3 be satisfied. For any \(T>0\), there exists a positive constant \(C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma}>0\) such that for any \(\epsilon\in(0,1)\), \(\gamma>0\),
$$ \begin{aligned} \sup_{0 \leq t \leq T}\mathbb{E} \big\| Y_{t}^{\epsilon}\big\| ^{2p} \leq C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma}. \end{aligned} $$

Proof

Due to the energy identity (2.6), we find
$$\begin{aligned} \mathbb{E}\big\| Y_{t}^{\epsilon}\big\| ^{2p}={}& \|Y_{0}\|^{2p}+\frac {2p}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle \mathbb{A}Y_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr\rangle \,ds \\ &+\frac{2p}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle F\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr),Y_{s}^{\epsilon }\bigr\rangle _{\mathbb{H}} \,ds \\ &+\frac{2p(p-1)}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon } \big\| ^{2p-2}\big\| G\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr)\big\| ^{2}\,ds \\ &+\frac{1}{\epsilon}\mathbb{E} \int_{0}^{t} \int_{\mathbb{Z}}\bigl[\big\| Y_{s}^{\epsilon}+H \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z\bigr) \big\| ^{2p}-\big\| Y_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz) \,ds \\ &-\frac{2p}{\epsilon}\mathbb{E} \int_{0}^{t} \int_{\mathbb{Z}}\big\| Y_{s}^{\epsilon}\big\| ^{2p-2} \bigl\langle H\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon },z \bigr),Y_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}}v(dz)\,ds \\ ={}&\|Y_{0}\|^{2p}+\sum_{i=1}^{5} \Xi_{t}^{i}.\end{aligned} $$
In view of Assumption 1 and Assumption 3, we have
$$ \begin{gathered} \bigl\langle \mathbb{A}Y_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr\rangle \leq -\alpha_{1}\big\| Y_{s}^{\epsilon} \big\| ^{2}, \\ \bigl\langle F\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon} \bigr),Y_{s}^{\epsilon }\bigr\rangle _{\mathbb{H}}=\bigl\langle F \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon }\bigr)-F \bigl(X_{s}^{\epsilon},0\bigr),Y_{s}^{\epsilon}\bigr\rangle _{\mathbb {H}}+\bigl\langle F\bigl(X_{s}^{\epsilon},0 \bigr),Y_{s}^{\epsilon}\bigr\rangle _{\mathbb {H}}, \\ \big\| G\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon}\bigr) \big\| ^{2}\leq K_{3} \bigl(1+\big\| Y_{s}^{\epsilon} \big\| ^{2}+\big\| X_{s}^{\epsilon}\big\| ^{2}\bigr). \end{gathered} $$
(3.2)
Then by taking (3.2) and \(\gamma>0\) small enough for Young’s inequality in the form \(|ab|\leq\gamma|b|^{m}+C_{r,m}|a|^{\frac {m}{m-1}}\), we have
$$\begin{aligned} \|Y_{0}\|^{2p}+\sum_{i=1}^{3} \Xi_{t}^{i}\leq{}& \|Y_{0}\|^{2p}- \frac {2p\alpha_{1}}{\epsilon} \mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p}\,ds+\frac{p(C_{F}+1)}{\epsilon} \mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p}\,ds \\ &+\frac{p}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon}\big\| ^{2p} \,ds+\frac{p}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon } \big\| ^{2p-2}C_{F} \bigl(1+\big\| X_{s}^{\epsilon} \big\| ^{2}\bigr)\,ds \\ &+\frac{2p(p-1)}{\epsilon} K_{3}\mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p-2}\bigl(1+\big\| Y_{s}^{\epsilon}\big\| ^{2}+\big\| X_{s}^{\epsilon}\big\| ^{2}\bigr)\,ds \\ \leq{}& \|Y_{0}\|^{2p}-\frac{C_{p,\alpha_{1},C_{F},K_{3},\gamma}}{\epsilon } \mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p}\,ds \\ &+\frac{C'_{p}}{\epsilon}\mathbb{E} \int_{0}^{t}\big\| X_{s}^{\epsilon } \big\| ^{2p}\,ds+\frac{C'_{p}t}{\epsilon}.\end{aligned} $$
By the binomial theorem (3.1), Young’s inequality and Assumption 3, we have
$$\begin{aligned} \Xi_{t}^{4}+\Xi_{t}^{5}&\leq \frac{1}{\epsilon} \sum_{i=2}^{2p}C_{i}^{2p} \mathbb{E} \int_{0}^{t} \int_{\mathbb{Z}}\big\| Y_{s}^{\epsilon}\big\| ^{2p-i} \big\| H\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z\bigr)\big\| ^{i}v(dz)\,ds \\ &\leq \frac{1}{\epsilon} K_{5}\sum_{i=2}^{2p}C_{i}^{2p} \mathbb {E} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p-i}\bigl(1+\big\| X_{s}^{\epsilon}\big\| ^{i}+ \big\| Y_{s}^{\epsilon}\big\| ^{i}\bigr)\,ds \\ &\leq \frac{C_{p,K_{5},\gamma}}{\epsilon} \mathbb{E} \int_{0}^{t}\big\| Y_{s}^{\epsilon} \big\| ^{2p}\,ds+\frac{C'_{p}}{\epsilon} \mathbb{E} \int _{0}^{t}\big\| X_{s}^{\epsilon} \big\| ^{2p}\,ds+\frac{C'_{p}t}{\epsilon}.\end{aligned} $$
With the help of Gronwall’s inequality (see reference [17], p.74), we know there exists a positive constant \(C_{p,\alpha _{1},C_{F},K_{3},K_{5},\gamma}>0\). We have
$$\begin{aligned} \sup_{0 \leq s \leq T}\mathbb{E}\|Y_{s}^{\epsilon} \|^{2p} &\leq \| Y_{0}\|^{2p}-\frac{C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma}}{\epsilon } \mathbb{E} \int_{0}^{t}\sup_{0 \leq r \leq s} \big\| Y_{r}^{\epsilon}\big\| ^{2p}\,dr+\frac{C'_{p}}{\epsilon}t \\ &\leq \|Y_{0}\|^{2p}e^{-\frac{C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma }}{\epsilon}T}+C'_{p} \bigl(e^{-\frac{C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma }}{\epsilon}T}-1\bigr) \\ &\leq C_{p,\alpha_{1},C_{F},K_{3},K_{5},\gamma}.\end{aligned} $$
This is the proof of Lemma 3.2. □

Theorem 3.3

Let Assumptions 1-3 be satisfied. \(\overline {X}_{t}\) denotes the stochastic process determined by the SPDE
$$ d\overline{X}_{t}=\bigl[\mathbb{A}\overline{X}_{t}+ \bar{f}(\overline {X}_{t})\bigr]\,dt+g(\overline{X}_{t}) \,dW_{t}^{1}+ \int_{\mathbb {Z}}h(\overline{X}_{t-},z) \tilde{N}_{1}(dt,dz). $$
Then for \(T>0\), \(p > 1\), we have
$$ \mathbb{E}\sup_{0\leq t \leq T}\big\| X_{t}^{\epsilon}- \overline{X}_{t}\big\| ^{2p}\rightarrow0, $$
as \(\epsilon\rightarrow0\).

Proof

In order to prove the above theorem Theorem 3.3, we divide the course of the proof in three steps. In Step 1, \(\|X_{t}^{\epsilon}-\hat{X}_{t}^{\epsilon}\|^{2p}\) will be estimated. We prove the other estimate \(\|\hat{X}_{t}^{\epsilon }-\overline{X}_{t}\|^{2p}\) in Step 2. Finally, through Step 1 and Step 2, Theorem 3.3 will be obtained.

Step 1. We consider a partition of \([0,T]\) into intervals of the same length δ (\(\delta<1\)). Then, for \(t\in[k\delta,\min\{(k+1)\delta,T\} ]\), \(k=0,1,\ldots,\lfloor{T}/{\delta}\rfloor\), we construct auxiliary processes \(\hat{Y}^{\epsilon}_{t}\) and \(\hat{X}^{\epsilon}_{t}\), by means of the relations
$$ \begin{aligned}[b] \hat{Y}_{t}^{\epsilon}={}&Y^{\epsilon}_{k\delta}+ \frac{1}{\epsilon } \int_{k\delta}^{t}\bigl[\mathbb{A}\hat{Y}_{s}^{\epsilon}+F \bigl(X_{k\delta }^{\epsilon},\hat{Y}_{s}^{\epsilon}\bigr) \bigr]\,ds+\frac{1}{\sqrt{\epsilon }} \int_{k\delta}^{t}G\bigl(X_{k\delta}^{\epsilon}, \hat{Y}_{s}^{\epsilon }\bigr)\,dW_{s}^{2} \\ &+ \int_{k\delta}^{t} \int_{\mathbb{Z}}H\bigl(X_{k\delta}^{\epsilon }, \hat{Y}_{s-}^{\epsilon},z\bigr)\tilde{N}^{\epsilon}_{2}(ds,dz) \end{aligned} $$
(3.3)
and
$$ \begin{aligned}[b] \hat{X}_{t}^{\epsilon}={}&X_{0}+ \int_{0}^{t}\mathbb {A}X_{s}^{\epsilon} \,ds+ \int_{0}^{t}f\bigl(X_{[s/\delta]\delta}^{\epsilon }, \hat{Y}_{s}^{\epsilon}\bigr)\,ds+ \int_{0}^{t}g\bigl(X_{s}^{\epsilon } \bigr)\,dW_{s}^{1} \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}h\bigl(X_{s-}^{\epsilon},z\bigr)\tilde {N}_{1}(ds,dz),\quad t\in[0,T]. \end{aligned} $$
(3.4)
To proceed, by the mild solution \(X_{t}^{\epsilon}\) of (2.3), we make the following estimation:
$$ \begin{aligned}[b] \big\| X_{t}^{\epsilon}-X_{k\delta}^{\epsilon} \big\| ^{2p}\leq{}& 4^{2p-1}\big\| X_{k\delta}^{\epsilon}(S_{t-k\delta}- \mathbb{I})\big\| ^{2p}+4^{2p-1} \biggl\Vert \int_{k\delta}^{t}S_{t-s}f \bigl(X_{s}^{\epsilon },Y_{s}^{\epsilon}\bigr)\,ds \biggr\Vert ^{2p} \\ &+4^{2p-1} \biggl\Vert \int_{k\delta}^{t}S_{t-s}g \bigl(X_{s}^{\epsilon }\bigr)\,dW_{s}^{1} \biggr\Vert ^{2p} \\ &+4^{2p-1} \biggl\Vert \int_{k\delta}^{t} \int_{\mathbb {Z}}S_{t-s}h\bigl(X_{s-}^{\epsilon},z \bigr)\tilde{N}_{1}(ds,dz) \biggr\Vert ^{2p} \\ ={}&I_{1}+I_{2}+I_{3}+I_{4}, \end{aligned} $$
(3.5)
where \(\mathbb{I}\) denotes the identity operator.
First of all, since f is globally bounded, by Hölder’s inequality and Assumption 3, detailed computation leads to
$$ \begin{aligned}[b] \mathbb{E}I_{2}&=4^{2p-1} \biggl\Vert \int_{k\delta }^{t}S_{t-s}f \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon}\bigr)\,ds \biggr\Vert ^{2p} \\ &\leq C \|t-k\delta\|^{2p-1} \int_{k\delta}^{t}\big\| f\bigl(X_{s}^{\epsilon },Y_{s}^{\epsilon} \bigr)\big\| ^{2p}\,ds \\ &\leq C \|t-k\delta\|^{2p}. \end{aligned} $$
(3.6)
Second, from the BDG inequality, Hölder’s inequality and Lemma 3.1, it follows that
$$ \begin{aligned}[b] \mathbb{E}I_{3}&=4^{2p-1} \mathbb{E} \biggl[ \int_{k\delta}^{t}\big\| S_{t-s}g \bigl(X_{s}^{\epsilon}\bigr)\big\| ^{2}\,ds \biggr]^{p} \\ &\leq C \|t-k\delta\|^{p-1}\mathbb{E} \int_{k\delta}^{t}\big\| g\bigl(X_{s}^{\epsilon} \bigr)\big\| ^{2p}\,ds \\ &\leq C \|t-k\delta\|^{p-1} \int_{k\delta}^{t}\bigl(1+\mathbb{E}\big\| X_{s}^{\epsilon}\big\| ^{2p}\bigr)\,ds \\ &\leq C \|t-k\delta\|^{p}. \end{aligned} $$
(3.7)
Next, by Kunita’s inequality [19, Theorem 4.4.23], we have
$$ \begin{aligned}[b] \mathbb{E}I_{4}\leq{}& C \mathbb{E} \int_{k\delta}^{t} \int_{\mathbb {Z}}\big\| S_{t-s}h\bigl(X_{s}^{\epsilon},z \bigr)\big\| ^{2p}v(dz)\,ds +C \mathbb{E} \biggl\{ \int_{k\delta}^{t} \int_{\mathbb{Z}}\big\| S_{t-s}h\bigl(X_{s}^{\epsilon},z \bigr)\big\| ^{2}v(dz)\,ds \biggr\} ^{p} \\ \leq{}& C \mathbb{E} \int_{k\delta}^{t}\bigl(1+\big\| X_{s}^{\epsilon} \big\| ^{2p}\bigr)\,ds \\ \leq{}& C \|t-k\delta\|. \end{aligned} $$
(3.8)
Finally, we will estimate the first term \(I_{1}\) of (3.5). To proceed, we define three functions and establish a key lemma.
Define
$$ \begin{gathered} \Upsilon_{t}^{\epsilon}:= \int_{0}^{t}S_{t-s}f \bigl(X_{s}^{\epsilon },Y_{s}^{\epsilon}\bigr)\,ds, \\ \Phi_{t}^{\epsilon}:= \int_{0}^{t}S_{t-s}g \bigl(X_{s}^{\epsilon }\bigr)\,dW_{s}^{1}, \\ \Psi_{t}^{\epsilon}:= \int_{0}^{t} \int_{\mathbb {Z}}S_{t-s}h\bigl(X_{s-}^{\epsilon},z \bigr)\tilde{N}_{1}(ds,dz). \end{gathered} $$
Since the semigroup \(\{S_{t}\}_{t \geq0}\) is analytic, the trajectories of \(\Upsilon_{t}^{\epsilon}\), \(\Phi_{t}^{\epsilon}\) and \(\Psi_{t}^{\epsilon}\) are Hölder continuous-valued. We will give some estimations of the slow component \(X^{\epsilon}_{t}\) as a value process in \(D((-\mathbb{A})^{\alpha})\), \(\alpha\in(0,\frac {1}{8})\).

Remark 3.4

In this paper, we assume that \(\frac{1}{1-4\alpha}< p<\frac {1}{4\alpha}\) and \(\alpha\in(0,\frac{1}{8})\).

Lemma 3.5

For any \(t\in[0,T]\) and \(\frac{1}{1-4\alpha}< p<\frac{1}{4\alpha }\), \(\alpha\in(0,\frac{1}{8})\), there exists a constant \(C_{\alpha ,p,T}\) such that
$$\mathbb{E}\big\| X_{t}^{\epsilon}\big\| ^{2p}_{\alpha} \leq C_{\alpha,p,T}. $$

Proof

The estimations of \(\Upsilon_{t}^{\epsilon}\), \(\Phi_{t}^{\epsilon}\) can be obtained from [17]. Here, we give the proof of the third term \(\Psi_{t}^{\epsilon}\). For the third term, by the factorization formula, we have
$$ \Psi_{t}^{\epsilon}=C_{\alpha} \int_{0}^{t}(t-s)^{\alpha -1}S_{t-s}U_{\alpha}^{\epsilon}(s) \,ds, $$
with \(U_{\alpha}^{\epsilon}(s)=\int_{0}^{s}\int_{\mathbb {Z}}(s-r)^{-\alpha}S_{s-r}h(X_{r-}^{\epsilon},z)\tilde{N}_{1}(dr,dz)\).
Note that, for any \(\frac{1}{1-4\alpha}< p<\frac{1}{4\alpha}\), \(\alpha \in(0,\frac{1}{8})\), we have
$$\begin{aligned} \big\| \Psi_{t}^{\epsilon}\big\| _{\alpha}^{2p}&\leq C_{\alpha} \biggl[ \int _{0}^{t}(t-s)^{\alpha-1} \big\| U_{\alpha}^{\epsilon}(s)\big\| _{\alpha }\,ds \biggr]^{2p} \\ &\leq C_{\alpha} \sup_{0\leq s \leq t} \big\| U_{\alpha}^{\epsilon }(s) \big\| _{\alpha}^{2p} \biggl[ \int_{0}^{t}(t-s)^{\alpha-1}\,ds \biggr]^{2p} \\ &\leq C_{\alpha,p,T} \sup_{0\leq s \leq t} \big\| U_{\alpha}^{\epsilon }(s) \big\| _{\alpha}^{2p}.\end{aligned} $$
Next, for any \(t > 0\), the operator \((-\mathbb{A})^{\alpha}S_{t}\) is bounded and its operator norm \(\|(-\mathbb{A})^{\alpha}S_{t}\|\leq M_{\alpha} t^{-\alpha}\) [48]. Then, by Kunita’s first inequality [19, Theorem 4.4.23], Hölder’s inequality and Lemma 3.2, we have
$$\begin{aligned} \mathbb{E}\big\| \Psi_{t}^{\epsilon}\big\| _{\alpha}^{2p} \leq{}& C_{\alpha ,p,T}\mathbb{E}\sup_{0\leq s \leq t} \biggl\Vert \int_{0}^{s} \int _{\mathbb{Z}}(s-r)^{-\alpha}(-\mathbb{A})^{\alpha }S_{s-r}h \bigl(X_{r-}^{\epsilon},z\bigr) \tilde{N}_{1}(dr,dz) \biggr\Vert ^{2p} \\ \leq{}& C_{\alpha,p,T}\mathbb{E} \int_{0}^{t} \int_{\mathbb {Z}}(s-r)^{-4p\alpha}\big\| h\bigl(X_{r}^{\epsilon},z \bigr)\big\| ^{2p}v(dz)\,dr \\ &+ C_{\alpha,p,T}\mathbb{E} \biggl[ \int_{0}^{t} \int_{\mathbb {Z}}(s-r)^{-4\alpha}\big\| h\bigl(X_{r}^{\epsilon},z \bigr)\big\| ^{2}v(dz)\,dr \biggr]^{p} \\ \leq{}& C_{\alpha,p,T}\mathbb{E} \biggl[\sup_{0 \leq r\leq t} \int _{\mathbb{Z}}\big\| h\bigl(X_{r}^{\epsilon},z\bigr) \big\| ^{2p}v(dz) \biggr] \int _{0}^{t} \int_{\mathbb{Z}}(s-r)^{-4p\alpha}\,dr \\ &+C_{\alpha,p,T} \biggl[ \int_{0}^{t}(s-r)^{\frac{4p\alpha }{1-p}}\,dr \biggr]^{p-1}\mathbb{E} \int_{0}^{t} \biggl[ \int_{\mathbb {Z}}\big\| h\bigl(X_{r}^{\epsilon},z\bigr) \big\| ^{2}v(dz) \biggr]^{p}\,dr \\ \leq{}& C_{\alpha,p,T} \int_{0}^{T}\mathbb{E}\bigl[1+ \big\| X_{r}^{\epsilon}\big\| ^{2p}\bigr]\,dr \\ \leq{}& C_{\alpha,p,T}.\end{aligned} $$
Then, by \(\|S_{t}X_{0}\|^{2p}_{\alpha} \leq\|X_{0}\|^{2p}_{\alpha}\), we have
$$\mathbb{E}\big\| X_{t}^{\epsilon}\big\| ^{2p}_{\alpha} \leq C_{\alpha,p,T}. $$
This is the proof of Lemma 3.5. □
To proceed, we give the estimation of \(I_{1}\). According to [48], there exists a constant \(C_{\alpha}>0\) such that for all \(x\in D((-\mathbb{A})^{\alpha})\),
$$ I_{1}=\big\| X_{k\delta}^{\epsilon}(S_{t-k\delta}-\mathbb{I}) \big\| \leq C_{\alpha}\|t-k\delta\|^{\alpha}\big\| X_{k\delta}^{\epsilon} \big\| _{\alpha}, $$
and then, according to Lemma 3.5, we deduce
$$ \begin{aligned}[b] \mathbb{E}I_{1}&=4^{2p-1} \big\| X_{k\delta}^{\epsilon}(S_{t-k\delta }-\mathbb{I})\big\| ^{2p} \\ &\leq 4^{2p-1} C_{\alpha}\|t-k\delta\|^{2p\alpha}\mathbb{E}\big\| X_{k\delta}^{\epsilon}\big\| ^{2p}_{\alpha} \\ &\leq C_{\alpha,p,T} \|t-k\delta\|^{2p\alpha}. \end{aligned} $$
(3.9)
It then follows from (3.6)-(3.9) that
$$ \begin{aligned}[b] \mathbb{E}\big\| X_{t}^{\epsilon}-X_{k\delta}^{\epsilon} \big\| ^{2p}&\leq C_{\alpha,p,T} \|t-k\delta\|^{2p\alpha}+C_{\alpha,p,T} \|t-k\delta\| \\ &\leq C_{\alpha,p,T} \|t-k\delta\|^{2p\alpha} \\ &\leq C_{\alpha,p,T}\delta^{2p\alpha}. \end{aligned} $$
(3.10)
Note that the result also holds for \(p=1\) [45]:
$$ \begin{aligned} \mathbb{E}\big\| X_{t}^{\epsilon}-X_{k\delta}^{\epsilon} \big\| ^{2}\leq C_{\alpha,p,T}\delta^{2\alpha}. \end{aligned} $$
Next, from the definitions of \(Y_{t}^{\epsilon}\), (2.3) and \(\hat{Y}_{t}^{\epsilon}\) (3.3), by energy identities (2.6), for \(t\in[k\delta,\min\{(k+1)\delta,T\}]\), \(\mathbb{E}\| Y_{t}^{\epsilon}-\hat{Y}_{t}^{\epsilon}\|^{2p}\) will be estimated:
$$ \begin{gathered} \mathbb{E}\big\| Y_{t}^{\epsilon}- \hat{Y}_{t}^{\epsilon}\big\| ^{2p} \\ \quad =\frac{2p}{\epsilon}\mathbb{E} \int_{k\delta}^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle \mathbb {A}Y_{s}^{\epsilon}-\mathbb{A}\hat{Y}_{s}^{\epsilon },Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\bigr\rangle \,ds \\ \qquad{} +\frac{2p}{\epsilon}\mathbb{E} \int_{k\delta }^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle F \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon}\bigr)-F \bigl(X_{k\delta}^{\epsilon},\hat {Y}_{s}^{\epsilon} \bigr),Y_{s}^{\epsilon}-\hat{Y}_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,ds \\ \qquad{} +\frac{2p(p-1)}{\epsilon}\mathbb{E} \int_{k\delta }^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-2}\big\| G\bigl(X_{s}^{\epsilon}, \hat{Y}_{s}^{\epsilon}\bigr)-G\bigl(X_{k\delta}^{\epsilon },Y_{s}^{\epsilon} \bigr)\big\| ^{2}\,ds \\ \qquad{} +\frac{1}{\epsilon}\mathbb{E} \int_{k\delta }^{t} \int_{\mathbb{Z}}\bigl[\big\| \bigl(Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon }\bigr)+\bigl(H\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z \bigr)-H\bigl(X_{k\delta}^{\epsilon },\hat{Y}_{s}^{\epsilon},z \bigr)\bigr)\big\| ^{2p} \\ \qquad{} -\big\| Y_{s}^{\epsilon}-\hat{Y}_{s}^{\epsilon}\big\| ^{2p}\bigr]v(dz)\,ds \\ \quad\quad{} -\frac{2p}{\epsilon}\mathbb{E} \int_{k\delta }^{t} \int_{\mathbb{Z}}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle H \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z\bigr) \\ \qquad{} -H\bigl(X_{k\delta}^{\epsilon},\hat{Y}_{s}^{\epsilon },z \bigr),Y_{s}^{\epsilon}-\hat{Y}_{s}^{\epsilon}\bigr\rangle _{\mathbb {H}}v(dz)\,ds \\ \quad =J_{1}+J_{2}+J_{3}+J_{4}+J_{5}. \end{gathered} $$
First of all, from Assumptions 1-3 and Young’s inequality, it is easy to get
$$\begin{aligned} J_{1}+J_{2}+J_{3}&\leq \frac{C}{\epsilon} \mathbb{E} \int_{k\delta }^{t}\bigl(\big\| X_{s}^{\epsilon}-X_{k\delta}^{\epsilon} \big\| ^{2p}+\big\| Y_{s}^{\epsilon}-\hat{Y}_{s}^{\epsilon} \big\| ^{2p}\bigr)\,ds \\ &\leq \frac{C}{\epsilon}\mathbb{E} \int_{k\delta}^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p}\,ds+\frac{C_{\alpha ,p,T}\delta^{2p\alpha+1}}{\epsilon}.\end{aligned} $$
Then, by equality (3.1), we have
$$\begin{aligned} J_{4}+J_{5}&=\sum_{i=2}^{2p}C_{i}^{2p} \mathbb{E} \int_{k\delta }^{t} \int_{\mathbb{Z}}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-i}\big\| H\bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon},z \bigr)-H\bigl(X_{k\delta }^{\epsilon},\hat{Y}_{s}^{\epsilon},z \bigr)\big\| ^{i}v(dz)\,ds \\ &\leq \frac{C}{\epsilon}\sum_{i=2}^{2p} \mathbb{E} \int_{k\delta }^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p-i} \bigl(\big\| X_{s}^{\epsilon}-X_{k\delta}^{\epsilon}\big\| ^{i}+ \big\| Y_{s}^{\epsilon }-\hat{Y}_{s}^{\epsilon} \big\| ^{i}\bigr) \,ds \\ & \leq \frac{C}{\epsilon}\mathbb{E} \int_{k\delta}^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p}\,ds+\frac{C_{\alpha ,p,T}\delta^{2p\alpha+1}}{\epsilon}.\end{aligned} $$
Therefore, for \(t\in[k\delta,\min\{(k+1)\delta,T\}]\), we obtain
$$ \begin{aligned}[b] \mathbb{E}\big\| Y_{t}^{\epsilon}- \hat{Y}_{t}^{\epsilon}\big\| ^{2p} & \leq \frac{C}{\epsilon} \mathbb{E} \int_{k\delta}^{t}\big\| Y_{s}^{\epsilon}- \hat{Y}_{s}^{\epsilon}\big\| ^{2p}\,ds+\frac{C_{\alpha ,p,T}\delta^{2p\alpha+1}}{\epsilon} \\ &\leq \frac{C_{\alpha,p,T}\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}. \end{aligned} $$
(3.11)
To proceed, we give another key lemma to complete the proof of Step 1.

Lemma 3.6

For any \(t\in[0,T]\), and \(\frac{1}{1-4\alpha}< p<\frac{1}{4\alpha }\), \(\alpha\in(0,\frac{1}{8})\), we have
$$ \begin{aligned} \mathbb{E}\sup_{0\leq t\leq T} \big\| X_{t}^{\epsilon}-\hat {X}_{t}^{\epsilon} \big\| ^{2p} \leq C_{\alpha,p,T}\biggl(\frac{\delta ^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+\delta ^{2p\alpha}\biggr)e^{C_{p}T}. \end{aligned} $$

Proof

We begin with
$$\begin{aligned} \big\| X_{t}^{\epsilon}-\hat{X}_{t}^{\epsilon} \big\| ^{2p}={}&2p \int_{0}^{t}\big\| X_{s}^{\epsilon}- \hat{X}_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle \mathbb {A}X_{s}^{\epsilon}-\mathbb{A}\hat{X}_{s}^{\epsilon },X_{s}^{\epsilon}- \hat{X}_{s}^{\epsilon}\bigr\rangle \,ds \\ &+2p \int_{0}^{t}\big\| X_{s}^{\epsilon}- \hat{X}_{s}^{\epsilon}\big\| ^{2p-2}\bigl\langle f \bigl(X_{s}^{\epsilon},Y_{s}^{\epsilon}\bigr)-f \bigl(X_{[s/\delta ]\delta}^{\epsilon},\hat{Y}_{s}^{\epsilon} \bigr),X_{s}^{\epsilon}-\hat {X}_{s}^{\epsilon}\bigr\rangle _{\mathbb{H}} \,ds.\end{aligned} $$
Thanks to Lemma 3.6 and (3.11), for any \(u\in[0,T]\), we get
$$\begin{aligned} \mathbb{E}\sup_{0\leq t \leq u}\big\| X_{t}^{\epsilon}-\hat {X}_{t}^{\epsilon}\big\| ^{2p}\leq{}& C_{p} \int_{0}^{t}\mathbb{E}\sup_{0\leq r \leq s} \big\| X_{r}^{\epsilon}-\hat{X}_{r}^{\epsilon}\big\| ^{2p}\,ds \\ &+C_{p} \int_{0}^{u}\mathbb{E}\sup_{0\leq r \leq s} \big\| X_{r}^{\epsilon }-X_{[r/\delta]\delta}^{\epsilon} \big\| ^{2p}\,ds \\ &+C_{p} \int_{0}^{t}\mathbb{E}\big\| Y_{s}^{\epsilon}- \hat {Y}_{s}^{\epsilon}\big\| ^{2p}\,ds \\ \leq{}& C_{p} \int_{0}^{u}\mathbb{E}\sup_{0\leq r \leq u} \big\| X_{r}^{\epsilon}-\hat{X}_{r}^{\epsilon} \big\| ^{2p}\,ds +C_{\alpha,p,T}\delta^{2p\alpha} \\ &+\frac{C_{\alpha,p,T}\delta^{2p\alpha+1}}{\epsilon} e^{\frac {C}{\epsilon}\delta}.\end{aligned} $$
With the help of Gronwall’s inequality, we have
$$ \mathbb{E}\sup_{0\leq t \leq T}\big\| X_{t}^{\epsilon}-\hat {X}_{t}^{\epsilon}\big\| ^{2p} \leq C_{\alpha,p,T} \biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+\delta^{2p\alpha} \biggr)e^{C_{p}T}. $$
This is the proof of Lemma 3.6. □
Step 2. In this step, we will estimate \(\mathbb{E}\sup_{0\leq t\leq T}\|\hat {X}_{t}^{\epsilon}-\overline{X}_{t}\|^{2p}\). It follows from the definitions of \(\overline{X}_{t} \) and \(\hat{X}^{\epsilon}_{t}\) that
$$\begin{aligned} \hat{X}_{t}^{\epsilon}-\overline{X}_{t} =& \int_{0}^{t}S_{t-s}\bigl[f \bigl(X_{k\delta}^{\epsilon},\hat {Y}_{s}^{\epsilon} \bigr)-\bar{f}\bigl(X_{s}^{\epsilon}\bigr)\bigr]\,ds \\ &+ \int_{0}^{t}S_{t-s}\bigl[\bar{f} \bigl(X_{s}^{\epsilon}\bigr)-\bar{f}\bigl(\hat {X}_{s}^{\epsilon} \bigr)\bigr]\,ds+ \int_{0}^{t}S_{t-s}\bigl[\bar{f}\bigl( \hat {X}_{s}^{\epsilon}\bigr)-\bar{f}(\overline{X}_{s}) \bigr]\,ds \\ &+ \int_{0}^{t}S_{t-s}\bigl[g \bigl(X_{s}^{\epsilon}\bigr)-g\bigl(\hat{X}_{s}^{\epsilon } \bigr)\bigr]\,dW_{s}^{1}+ \int_{0}^{t}S_{t-s}\bigl[g\bigl( \hat{X}_{s}^{\epsilon }\bigr)-g(\overline{X}_{s})\bigr] \,dW_{s}^{1} \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}S_{t-s}\bigl[h\bigl(X_{s-}^{\epsilon },z \bigr)-h\bigl(\hat{X}_{s-}^{\epsilon},z\bigr)\bigr] \tilde{N}_{1}(ds,dz) \\ &+ \int_{0}^{t} \int_{\mathbb{Z}}S_{t-s}\bigl[h\bigl(\hat{X}_{s-}^{\epsilon },z \bigr)-h(\overline{X}_{s-},z)\bigr]\tilde{N}_{1}(ds,dz) \\ =&\sum_{i=1}^{7}\Xi_{i}(t). \end{aligned}$$
Using Hölder’s inequality, the contractive property of semigroup \(S_{t}\), and the globally Lipschitz continuity of , for any \(u\in[0,T]\), we obtain
$$ \begin{aligned}[b] \mathbb{E}\sum _{i=2,4}\sup_{0 \leq t \leq u}\big\| \Xi_{i}(t)\big\| ^{2p}&\leq C_{T} \int_{0}^{u}\mathbb{E} \sup_{0 \leq r \leq s} \big\| X_{r}^{\epsilon}-\hat{X}_{r}^{\epsilon} \big\| ^{2p}\,ds \\ &\leq C_{\alpha,p,T}\biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+ \delta^{2p\alpha}\biggr)e^{C_{p}T}. \end{aligned} $$
(3.12)
Similarly, it is also easy to derive that the estimate for any \(u\in[0,T]\),
$$ \mathbb{E}\sum_{i=3,5}\sup _{0 \leq t \leq u}\big\| \Xi_{i}(t)\big\| ^{2p}\leq C_{T} \int_{0}^{u} \mathbb{E}\sup_{0 \leq r \leq s} \big\| \hat{X}_{r}^{\epsilon}-\overline{X}_{r} \big\| ^{2p}\,ds. $$
(3.13)
Now, by Kunita’s first inequality [19, Theorem 4.4.23] and Hölder’s inequality, we have
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq u}\big\| \Xi_{6}(t)\big\| ^{2p}={}&\mathbb {E}\sup_{0 \leq t \leq u} \biggl[ \int_{0}^{t} \int_{\mathbb {Z}}S_{t-s}\bigl[h\bigl(X_{s-}^{\epsilon},z \bigr)-h\bigl(\hat{X}_{s-}^{\epsilon },z\bigr)\bigr] \tilde{N}_{1}(ds,dz) \biggr]^{2p} \\ \leq{}& C_{p} \mathbb{E} \int_{0}^{u} \int_{\mathbb{Z}}\big\| S_{t-s}\big[h\bigl(X_{s}^{\epsilon},z \bigr)-h\bigl(\hat{X}_{s}^{\epsilon},z\bigr)\big]\big\| ^{2p}v(dz) \,dt \\ &+ C_{p} \mathbb{E} \biggl[ \int_{0}^{u} \int_{\mathbb{Z}}\big\| S_{t-s}\big[h\bigl(X_{s}^{\epsilon},z \bigr)-h\bigl(\hat{X}_{s}^{\epsilon},z\bigr)\big]\big\| ^{2}v(dz) \,dt \biggr]^{p} \\ \leq{}&C_{p,T} \int_{0}^{u} \mathbb{E}\sup_{0 \leq r \leq s} \big\| \hat {X}_{r}^{\epsilon}-X^{\varepsilon}_{r} \big\| ^{2p}\,ds \\ \leq{}& C_{\alpha,p,T}\biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+ \delta^{2p\alpha}\biggr)e^{C_{p}T} \end{aligned} $$
(3.14)
and
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq u}\big\| \Xi_{6}(t)\big\| ^{2p}={}&\mathbb {E}\sup_{0 \leq t \leq u} \biggl[ \int_{0}^{t} \int_{\mathbb {Z}}S_{t-s}\bigl[h(\overline{X}_{s-},z)-h \bigl(\hat{X}_{s-}^{\epsilon },z\bigr)\bigr]\tilde{N}_{1}(ds,dz) \biggr]^{2p} \\ \leq{}& C_{p} \mathbb{E} \int_{0}^{u} \int_{\mathbb{Z}}\big\| S_{t-s}\big[h(\overline{X}_{s-},z)-h \bigl(\hat{X}_{s-}^{\epsilon},z\bigr)\big]\big\| ^{2p}v(dz)\,dt \\ &+ C_{p} \mathbb{E}\biggl[ \int_{0}^{u} \int_{\mathbb{Z}}\big\| S_{t-s}\big[h(\overline{X}_{s-},z)-h \bigl(\hat{X}_{s-}^{\epsilon},z\bigr)\big]\big\| ^{2}v(dz)\,dt \biggr]^{p} \\ \leq{}&C_{p,T} \int_{0}^{u} \mathbb{E}\sup_{0 \leq r \leq s} \big\| \hat {X}_{r}^{\epsilon}-\overline{X}_{r} \big\| ^{2p}\,ds. \end{aligned} $$
(3.15)
Next, to deal with the first term, by the boundedness of the functions f, , we have
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq T} \big\| \Xi_{1}(t)\big\| ^{2p}&\leq \mathbb {E} \sup _{0 \leq t \leq T} \biggl\Vert \int_{0}^{t}S_{t-s}\bigl[f \bigl(X_{[s/\delta ]\delta}^{\epsilon},\hat{Y}_{s}^{\epsilon} \bigr)-\bar {f}\bigl(X_{s}^{\epsilon}\bigr)\bigr]\,ds \biggr\Vert ^{2p} \\ &\leq C_{T}\mathbb{E} \sup_{0 \leq t \leq T} \biggl\Vert \int _{0}^{t}S_{t-s}\bigl[f \bigl(X_{[s/\delta]\delta}^{\epsilon},\hat {Y}_{s}^{\epsilon} \bigr)-\bar{f}\bigl(X_{s}^{\epsilon}\bigr)\bigr]\,ds \biggr\Vert ^{2}. \end{aligned} $$
(3.16)
For \(t\in[k\delta,\min\{(k+1)\delta,T\}]\), we write
$$ \begin{aligned} \Xi_{1}(t)={}&\sum _{k=0}^{k-1} \int_{k\delta}^{(k+1)\delta }S_{t-s}\bigl[f \bigl(X_{k\delta}^{\epsilon},\hat{Y}_{s}^{\epsilon} \bigr)-\bar {f}\bigl(X_{k\delta}^{\epsilon}\bigr)\bigr]\,ds \\ &+\sum_{k=0}^{k-1} \int_{k\delta}^{(k+1)\delta}S_{t-s}\bigl[\bar {f} \bigl(X_{k\delta}^{\epsilon}\bigr)-\bar{f}\bigl(X_{s}^{\epsilon} \bigr)\bigr]\,ds \\ &+ \int_{k\delta}^{t}S_{t-s}\bigl[f \bigl(X_{k\delta}^{\epsilon},\hat {Y}_{s}^{\epsilon} \bigr)-\bar{f}\bigl(X_{s}^{\epsilon}\bigr)\bigr]\,ds \\ ={}&\sum_{i=1}^{3}\Xi_{1i}(t). \end{aligned} $$
From (3.10), it follows that
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq u} \big\| \Xi_{12}(t)\big\| ^{2}&=\mathbb {E}\sup_{0 \leq t \leq u} \Biggl\Vert \sum_{k=0}^{k-1} \int_{k\delta }^{(k+1)\delta}S_{t-s}\bigl[\bar{f} \bigl(X_{k\delta}^{\epsilon}\bigr)-\bar {f}\bigl(X_{s}^{\epsilon} \bigr)\bigr]\,ds \Biggr\Vert ^{2} \\ &\leq C_{T} \int_{0}^{T}\mathbb{E}\big\| X_{s}^{\epsilon}-X_{k\delta }^{\epsilon} \big\| ^{2}\,ds \\ &\leq C_{\alpha,T}\delta^{2\alpha}, \end{aligned} $$
(3.17)
and by Assumption 3, we have
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq u} \big\| \Xi_{13}(t)\big\| ^{2}&=\mathbb {E}\sup_{0 \leq t \leq u} \biggl\Vert \int_{k\delta }^{t}S_{t-s}\bigl[f \bigl(X_{k\delta}^{\epsilon},\hat{Y}_{s}^{\epsilon} \bigr)-\bar {f}\bigl(X_{s}^{\epsilon}\bigr)\bigr]\,ds \biggr\Vert ^{2} \\ & \leq C_{T}\delta. \end{aligned} $$
(3.18)

Lemma 3.7

Suppose that Assumptions 1-3 hold. Then there is a constant \(C>0\) such that we have
$$\begin{aligned} \mathbb{E}\sup_{0 \leq t \leq T}\big\| \Xi_{11}(t) \big\| ^{2}&= \mathbb {E}\sup_{0 \leq t \leq T} \Biggl\Vert \sum _{k=0}^{k-1} \int_{k\delta }^{(k+1)\delta}S_{t-s}\bigl[f \bigl(X_{k\delta}^{\epsilon},\hat {Y}_{s}^{\epsilon} \bigr)-\bar{f}\bigl(X_{k\delta}^{\epsilon}\bigr)\bigr]\,ds \Biggr\Vert ^{2} \\ & \leq C \frac{\delta}{\epsilon},\end{aligned} $$
where C is independent of \((\delta, \epsilon)\).

Proof

See Appendix B. □

Now, using the above estimation (3.16)-(3.18) and Lemma 3.7, we obtain
$$ \begin{aligned}[b] \mathbb{E}\sup_{0 \leq t \leq u} \big\| \Xi_{1}(t)\big\| ^{2p}&\leq C \delta + C_{\alpha,T} \delta^{2\alpha} +C_{T} \frac{\epsilon}{\delta} \\ &\leq C_{\alpha,T}\delta^{2\alpha} +C_{T} \frac{\epsilon}{\delta}. \end{aligned} $$
(3.19)
As regards the above discussion, from (3.12), (3.13) and (3.19), through Gronwall’s inequality, it is easy to see that
$$\begin{aligned} \mathbb{E}\sup_{0\leq t\leq u}\big\| \hat{X}_{t}^{\epsilon}- \overline {X}_{t}\big\| ^{2p} \leq{}& C_{\alpha,T} \delta^{2\alpha} +C_{T} \frac {\epsilon}{\delta} +C_{\alpha,p,T} \biggl(\frac{\delta^{2p\alpha +1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+\delta^{2p\alpha } \biggr)e^{C_{p}T} \\ &+C_{T} \int_{0}^{u} \mathbb{E}\sup_{0 \leq r \leq s} \big\| \hat {X}_{r}^{\epsilon}-\overline{X}_{r} \big\| ^{2p}\,dr.\end{aligned} $$
Therefore, by Gronwall’s inequality, we have
$$ \mathbb{E}\sup_{0\leq t\leq u}\big\| \hat{X}_{t}^{\epsilon}- \overline {X}_{t}\big\| ^{2p} \leq \biggl[C_{\alpha,T} \delta^{2\alpha} +C_{T} \frac{\epsilon }{\delta} +C_{\alpha,p,T} \biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+\delta^{2p\alpha} \biggr)e^{C_{p}T}\biggr] e^{C_{p,T}}. $$
Step 3. According to Step 1 and Step 2, we have
$$\begin{aligned} \mathbb{E}\sup_{0\leq t\leq T}{\big\| {X_{t}^{\varepsilon}- {{ \overline{X}}_{t}}} \big\| ^{2p}} ={}& \mathbb{E}\sup _{0\leq t\leq T}{\big\| {X_{t}^{\varepsilon}- \hat{X}_{t}^{\varepsilon}+ \hat{X}_{t}^{\varepsilon}- {{ \overline{X}}_{t}}} \big\| ^{2p}} \\ \le{}& 2^{2p-1}\mathbb{E}\sup_{0\leq t\leq T}{\big\| {X_{t}^{\varepsilon}- \hat{X}_{t}^{\varepsilon}} \big\| ^{2p}} + 2^{2p-1}\mathbb{E}\sup_{0\leq t\leq T}{\big\| { \hat{X}_{t}^{\varepsilon}- {{\overline{X}}_{t}}} \big\| ^{2p}} \\ \leq{}& \biggl[C_{\alpha,T}\delta^{2\alpha} +C_{T} \frac{\epsilon }{\delta} +C_{\alpha,p,T}\biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac{C}{\epsilon}\delta}+ \delta^{2p\alpha}\biggr)e^{C_{p}T}\biggr] e^{C_{p,T}} \\ &+C_{\alpha,p,T}\biggl(\frac{\delta^{2p\alpha+1}}{\epsilon} e^{\frac {C}{\epsilon}\delta}+ \delta^{2p\alpha}\biggr)e^{C_{p}T} \\ \leq{}& C_{\alpha,p,T}\biggl[\delta^{2\alpha}+\delta^{2p\alpha}+ \frac {\epsilon}{\delta} + \frac{\delta^{2p\alpha+1}}{\epsilon}e^{\frac {C}{\epsilon}\delta}\biggr].\end{aligned} $$
Thus, for \(t\in[0,T]\), selecting \(\delta=\epsilon\sqrt{-\ln \epsilon}\), we obtain
$$ \begin{aligned} \mathbb{E}\sup_{0\leq t\leq T}\big\| \hat{X}_{t}^{\epsilon}-\overline {X}_{t} \big\| ^{2p}\rightarrow0, \end{aligned} $$
as \(\epsilon\rightarrow0\), \(t\in[0,T]\).

This is the proof of Theorem 3.6.  □

Remark 3.8

To compare with the work of Xu and Miao [45] that the \(\mathbb{L}^{2}\)-strong averaging principle for slow-fast SPDEs with Poisson random measures was established, in this paper, we cope with high order moments which possess a good robustness and can be applied in computations in statistics, finance and other aspects.

Notes

Declarations

Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and for the clarifying comments, which lead to an improvement of the presentation of the paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Aeronautics, Northwestern Polytechnical University, Xi’an, China
(2)
Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China

References

  1. Larter, R, Steinmetz, C, Aguda, B: Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction. J. Chem. Phys. 89, 6504-6514 (1988) Google Scholar
  2. Krupa, M, et al.: Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron. Chaos 18, Article ID 015106 (2008) MathSciNetView ArticleMATHGoogle Scholar
  3. Dubbeldam, J, Krauskopf, B: Self-pulsations in lasers with saturable absorber: dynamics and bifurcations. Opt. Commun. 159, 325-338 (1999) View ArticleGoogle Scholar
  4. Khasminskii, R: On the averaging principle for stochastic differential Itô equations. Kybernetika 4, 260-279 (1968) MathSciNetGoogle Scholar
  5. Freidlin, M, Wentzell, A: Random Perturbations of Dynamical Systems. Springer, New York (1998) View ArticleMATHGoogle Scholar
  6. Golec, J, Ladde, G: Averaging principle and systems of singularly perturbed stochastic differential equations. J. Math. Phys. 31, 1116-1123 (1990) MathSciNetView ArticleMATHGoogle Scholar
  7. Xu, Y, Duan, J, et al.: An averaging principle for stochastic dynamical systems with Lévy noise. Physica D 240, 1395-1401 (2011) MathSciNetMATHGoogle Scholar
  8. Xu, Y, Pei, B, Li, Y: Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise. Math. Methods Appl. Sci. 30, 2120-2131 (2015) View ArticleMATHGoogle Scholar
  9. Xu, Y, Pei, B, Wu, J: Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion. Stoch. Dyn. 17(2), Article ID 1750013 (2017) MathSciNetView ArticleMATHGoogle Scholar
  10. Xu, Y, Guo, R, Liu, D, Zhang, H, Duan, J: Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst., Ser. B 19(4), 1197-1212 (2014) MathSciNetView ArticleMATHGoogle Scholar
  11. Xu, Y, Pei, B, Guo, R: Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete Contin. Dyn. Syst., Ser. B 20(7), 2257-2267 (2015) MathSciNetView ArticleMATHGoogle Scholar
  12. Xu, Y, Pei, B, Li, Y: An averaging principle for stochastic differential delay equations with fractional Brownian motion. Abstr. Appl. Anal. 2014, Article ID 479195 (2014) MathSciNetGoogle Scholar
  13. Pei, B, Xu, Y, Yin, G: Averaging principles for SPDEs driven by fractional Brownian motions with random delays modulated by two-time-scale Markov switching processes. Stoch. Dyn. 18, Article ID 1850023 (2018) Google Scholar
  14. Pei, B, Xu, Y, Yin, G: Stochastic averaging for a class of two-time-scale systems of stochastic partial differential equations. Nonlinear Anal., Theory Methods Appl. 160, 159-176 (2017) MathSciNetView ArticleMATHGoogle Scholar
  15. Golec, J: Stochastic averaging principle for systems with pathwise uniqueness. Stoch. Anal. Appl. 13, 307-322 (1995) MathSciNetView ArticleMATHGoogle Scholar
  16. Wang, W, Roberts, A: Average and deviation for slow-fast stochastic partial differential equations. J. Differ. Equ. 253, 1265-1286 (2012) MathSciNetView ArticleMATHGoogle Scholar
  17. Fu, H, Liu, J: Strong convergence in stochastic averaging for two-time-scales stochastic partial differential equations. J. Math. Anal. Appl. 384, 70-86 (2011) MathSciNetView ArticleMATHGoogle Scholar
  18. Fu, H, Wang, L, Wang, Y, Liu, J: Strong convergence rate in averaging principle for stochastic FitzHugh-Nagumo system with two-time-scales. J. Math. Anal. Appl. 416, 609-628 (2014) MathSciNetView ArticleMATHGoogle Scholar
  19. Applebaum, D: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009) View ArticleMATHGoogle Scholar
  20. Xu, Y, Feng, J, et al.: Lévy noise induced switch in the gene transcriptional regulatory system. Chaos 23, Article ID 013110 (2013) MathSciNetView ArticleGoogle Scholar
  21. Xu, Y, Li, JJ, et al.: Lévy noise-induced stochastic resonance in a bistable system. Eur. Phys. J. B 86, Article ID 198 (2013) View ArticleGoogle Scholar
  22. Li, Y, Xu, Y, et al.: Lévy-noise-induced transport in a rough triple-well potential. Phys. Rev. E 94(4), Article ID 042222 (2016) Google Scholar
  23. Xu, Y, Li, Y, et al.: The switch in a genetic toggle system with Lévy noise. Sci. Rep. 6, Article ID 31505 (2016) View ArticleGoogle Scholar
  24. Wang, Z, Xu, Y, Yang, H: Lévy noise induced stochastic resonance in an FHN model. Sci. China, Technol. Sci. 59(3), 371-375 (2016) View ArticleGoogle Scholar
  25. Xu, Y, Li, H, et al.: The estimates of the mean first exit time of a bistable system excited by Poisson white noise. J. Appl. Mech. 84(9), Article ID 091004 (2017) View ArticleGoogle Scholar
  26. Yin, C, Wen, Y, Zhao, Y: Optimal dividends problem with a terminal value for spectrally positive Lévy processes. Insur. Math. Econ. 53, 769-773 (2013) View ArticleMATHGoogle Scholar
  27. Yin, C, Wen, Y, Zhao, Y: On the optimal dividend problem for a spectrally positive Lévy process. ASTIN Bull. 44, 635-651 (2014) MathSciNetView ArticleGoogle Scholar
  28. Wen, Y, Yin, C: Exit problems for jump processes having double-sided jumps with rational Laplace transforms. Abstr. Appl. Anal. 2014, Article ID 747262 (2014) MathSciNetGoogle Scholar
  29. Lu, Y, Wu, R: The differentiability of dividends function on jump-diffusion risk process with a barrier dividend strategy. Front. Math. China 9, 1073-1088 (2014) MathSciNetView ArticleMATHGoogle Scholar
  30. Dong, H, Liu, Z: The ruin problem in a renewal risk model with two-sided jumps. Math. Comput. Model. 57, 800-811 (2013) MathSciNetView ArticleMATHGoogle Scholar
  31. Yin, C, Shen, Y, Wen, Y: Exit problems for jump processes with applications to dividend problems. J. Comput. Appl. Math. 245, 30-52 (2013) MathSciNetView ArticleMATHGoogle Scholar
  32. Lu, Y, Wu, R, Xu, R: The joint distributions of some actuarial diagnostics for the jump-diffusion risk process. Acta Math. Sci. 30(3), 664-676 (2010) MathSciNetView ArticleMATHGoogle Scholar
  33. Yin, C, Wen, Y, Zong, Z, Shen, Y: The first passage time problem for mixed-exponential jump processes with applications in insurance and finance. Abstr. Appl. Anal. 2014, Article ID 571724 (2014) MathSciNetGoogle Scholar
  34. Xu, Y, Pei, B, Guo, G: Existence and stability of solutions to non-Lipschitz stochastic differential equations driven by Lévy noise. Appl. Math. Comput. 263, 398-409 (2015) MathSciNetGoogle Scholar
  35. Pei, B, Xu, Y: Mild solutions of local non-Lipschitz neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching. Stoch. Anal. Appl. 35, 391-408 (2017) MathSciNetView ArticleMATHGoogle Scholar
  36. Pei, B, Xu, Y: Mild solutions of local non-Lipschitz stochastic evolution equations with jumps. Appl. Math. Lett. 52, 80-86 (2016) MathSciNetView ArticleMATHGoogle Scholar
  37. Bouchaud, J, Georges, A: Anomalous diffusion in disordered media: statistic mechanics, models and physical applications. Phys. Rep. 195, 127-293 (1990) MathSciNetView ArticleGoogle Scholar
  38. Duan, J: An Introduction to Stochastic Dynamics. Cambridge University Press, Cambridge (2015) MATHGoogle Scholar
  39. Peszat, S, Zabczyk, J: Stochastic Partial Differential Equations with Lévy Noise. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2007) View ArticleMATHGoogle Scholar
  40. DaPrato, G, Zabczyk, J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) View ArticleGoogle Scholar
  41. Duan, J, Wang, W: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, Amsterdam (2014) MATHGoogle Scholar
  42. Albeverio, S, Wu, J-L, Zhang, T: Parabolic SPDEs driven Poisson white noise. Stoch. Process. Appl. 74, 21-36 (1998) MathSciNetView ArticleMATHGoogle Scholar
  43. Hausenblas, E: Existence, uniqueness and regularity of SPDEs driven by Poisson random measures. Electron. J. Probab. 10, 1496-1546 (2005) MathSciNetView ArticleMATHGoogle Scholar
  44. Givon, D: Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems. SIAM J. Multiscale Model. Simul. 6, 577-594 (2007) MathSciNetView ArticleMATHGoogle Scholar
  45. Xu, J, Miao, Y, et al.: Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete Contin. Dyn. Syst., Ser. B 20, 2233-2256 (2015) MathSciNetView ArticleMATHGoogle Scholar
  46. Pei, B, Xu, Y, Wu, J-L: Two-time-scales hyperbolic-parabolic equations driven by Poisson random measures: existence, uniqueness and averaging principles. J. Math. Anal. Appl. 447, 243-268 (2017) MathSciNetView ArticleMATHGoogle Scholar
  47. Chow, P: Stochastic Partial Differential Equations. Chapman & Hall/CRC, Boca Raton (2014) Google Scholar
  48. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (2012) MATHGoogle Scholar
  49. Cerrai, S, Freidlin, M: Averaging principle for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 144(1-2), 137-177 (2009) MathSciNetView ArticleMATHGoogle Scholar
  50. Cerrai, S: A Khasminskii type averaging principle for stochastic reaction-diffusion equations. Ann. Appl. Probab. 19(3), 899-948 (2009) MathSciNetView ArticleMATHGoogle Scholar

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