Consider a time-varying linear system
\begin{align*}
\dot x(t)=A(t)x(t)
\end{align*}
and its average system (the existence is assumed)
\begin{align*}
\dot x(t)=\bar A x(t),  \quad \bar A=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}A(s)ds}{t-t_0}.
\end{align*}
By the classic averaging theory, if the average system  is asymptotically stable, then there exists an $\varepsilon^* >0$ such that for all $\varepsilon \in (0, \varepsilon^*)$,  the following fast time-varying system
\begin{align*}
\dot x(t)=A(t/\varepsilon)x(t)
\end{align*}
is asymptotically stable.

Can this result be extended to stochastic case as follows?  For the stochastic linear system
\begin{align*}
d x(t)=A(t)x(t)dt+B(t)x(t)dB(t),
\end{align*}
suppose its time-average system
\begin{align*}
d x(t)=\bar A x(t)dt+\bar Bx(t)dB(t), \quad \bar A=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}A(s)ds}{t-t_0},
\bar B=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}B(s)ds}{t-t_0}
\end{align*}
exists and it is asymptotically stable in a certain sense.
Then can we conclude that the fast time-varying stochastic system
\begin{align*}
d x(t)=A(t/\varepsilon)x(t)dt+B(t/\varepsilon)x(t)dB(t),
\end{align*}
for all $\varepsilon \in (0, \varepsilon^*)$ is asymptotically stable in the same sense as above?

Any pointer will be helpful and be appreciated. 

--- 

In fact, the exact question I am considering is motivated by the following fact which has been reported in for example [Dirk Aeyels and Joan Peuteman, On exponential stability of nonlinear time-varying
differential equations, Automatica 35 (1999) 1091-1100]. There exists an $\varepsilon^*>0$ such that for any $\varepsilon \in (0, \varepsilon^*)$ the system $\dot x(t)=A(t/\varepsilon)x(t)$ has an exponentially stable null solution if:

1. there exists an increasing sequence of times $t_k, k\in \mathbb{Z}$ with $t_k\rightarrow \infty$ as $k\rightarrow \infty$ and  $t_k\rightarrow -\infty$ as $k\rightarrow -\infty$;

2. there exists a finite $T>0$ such that $t_{k+1}-t_k \leq T, \forall k\in \mathbb{Z}$;

3. the sample average value of $A(t)$, $\bar A_k=\frac{\int_{t_k}^{t_{k+1}}A(s)ds}{t_{k+1}-t_k}$ is Hurwitz for all $k\in \mathbb{Z}$, i.e., there exist a positive definite matrix $P$ and a constant  $v>0$ such that $P\bar A_k+\bar A_k^T P <-vI$.

Putting this issue in a stochastic case, I am now wondering whether the stability of the sample average stochastic systems
 $d x(t)=\bar A_k x(t)dt+\bar B_k x(t)dB(t)$  implies the stability of the fast time-varying stochastic system 
 $d x(t)=\bar A(t/\varepsilon) x(t)dt+\bar B(t/\varepsilon) x(t)dB(t)$ for all $\varepsilon \in (0, \varepsilon^*)$?
  
I am posting it here hoping to get some answers or references. Thanks a lot.