Eq. (\[ave-eq\]) has a unique mild pathwise solution.

From now on, we assume $\beta\in(\frac{1}{2},1-\alpha)$ and to present our main averaging result, we need to impose another condition.

1.  $\sup_{x,y \in V}|b(x,y)| <\infty$.

\[mainthm\] Suppose that conditions [(A1)-(A5)]{} and [(B1)]{} hold. If $C_3< 2 \bar{\lambda}_1^2( 2+\bar{\lambda}_1)^{-1}$, then, we have $$\lim\limits_{\varepsilon \rightarrow 0}\mathbb{E}\big[\|X^{\varepsilon}-\bar{X}\|^2_{\alpha,T}\big]=0.$$

\[remthm\][ Suppose conditions [(A1)-(A5), (B1)]{} and $\sup_{x,y \in V}(|F(x,y)|+|G(x,y)|) <\infty$ hold. Without the condition $C_3< 2 \bar{\lambda}_1^2(2+\bar{\lambda}_1)^{-1}$, the conclusion of Theorem \[mainthm\] still holds.]{}

To prove Theorem \[mainthm\], firstly, following the discretization techniques inspired by Khasminskii in [@khas1966limit], we introduce an auxiliary process $(\hat{X}_{t}^{\varepsilon},\hat{Y}_{t}^{\varepsilon})$ and divide $[0,T]$ into intervals of size $\delta$, where $\delta \in(0,1)$ is a fixed number depending on $\varepsilon$ and $\delta > \varepsilon $, which will be chosen later. Then, we construct auxiliary processes $\hat{Y}^{\varepsilon}$ and $\hat{X}^{\varepsilon}$, by $$\begin{aligned}
\label{yj} \hat{Y}^{\varepsilon}_t&=S_{\frac{t}{\varepsilon}}Y_{0}+\frac{1}{\varepsilon}\int_0^tS_{\frac{t-s}{\varepsilon}}F(X^{\varepsilon}_{s(\delta)},\hat{Y}^{\varepsilon}_s)\mathrm{d}s+\frac{1}{\sqrt{\varepsilon}}\int_0^tS_{\frac{t-s}{\varepsilon}}G(X^{\varepsilon}_{s(\delta)},\hat{Y}^{\varepsilon}_s)\mathrm{d}W_s,\\
\label{xj} \hat{X}^{\varepsilon}_t&=S_{t}X_{0}+\int_{0}^{t}S_{t-s}b(X^{\varepsilon}_{s(\delta)},
\hat{Y}^{\varepsilon}_s)\mathrm{d}s+\int_{0}^{t}S_{t-s}g(X^{\varepsilon}_s)\mathrm{d}B^H_s,\end{aligned}$$ where $s(\delta)=\lfloor s / \delta\rfloor \delta$ is the nearest breakpoint preceding $s$. For $t\in [k\delta, \min\{(k+1)\delta,T\}]$, we assume the fast component $\hat{Y}^{\varepsilon}_{k\delta}$ is reset to equal ${Y}^{\varepsilon}_{k\delta}$ at each breakpoint $k\delta$. To proceed, we can derive uniform bounds $\|X^{\varepsilon}-\hat{X}^{\varepsilon}\|_{\alpha,T}^2 $. Next, based on the ergodic property of the frozen equation, we obtain appropriate control of $\|\hat{X}^{\varepsilon}-\bar{X}\|_{\alpha,T}^2 $. Finally, we can estimate $\|{X}^{\varepsilon}-\bar{X}\|_{\alpha,T}^2 