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.