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 f̄, 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, f̄, 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.