Probability that SDE visits any point

Question

This is a reference request question.

Statement: I am interested in an SDE of the form

\begin{equation}\fbox{1}~~~ {\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t \end{equation}
Where we assume that $g(t)\in (0,\infty)$ (notice that g(t) is a determistic function), and $f(\cdot)$ satisfies all the conditions required such that $\fbox{1}$ admits a strong continuous solution for all $t\geq 0$.

Result: I am interested in a reference of a result of the type $\mathbb{P}( \text{for some }t>0,\,X_t=x)>0$ for all $x \in \mathbb{R}$, in other words $X_t$ can visit any point with some positive probability.

Remark$^1$: The aforemnetioned conditions imposed on f and g can be changed, the spirit of the question is given an SDE with some necessary regularity conditions the previously described result holds.

Remark$^2$: I had posted this question in math.stachexchange and I had not received an answer after an expired bounty and a passage of two weeks. I have now deleted the post on math.stackexchange.

Answer 1

How about changing the time? Write $d\sigma(t) = (g(t))^2 dt$ and $dY_t = g(t) dB_t$. Then $\tilde{Y}_s = Y_{\sigma^{-1}(s)}$ is a Wiener process (provided that $g$ is nice enough; more precisely, we require $\sigma(t)$ to be finite for all $t > 0$ and unbounded), and if $\tilde{X}_s = X_{\sigma^{-1}(s)}$, then $$d\tilde{X}_s = \tilde{f}(s, \tilde{X}_s) ds + d\tilde{Y}_t,$$ where $\tilde{f}(s, x) = f(x) / (g(\sigma^{-1}(s)))^2$.

Now you should be able to use the Girsanov theorem, provided that $\tilde{f}$ is not too rough.

(This is more a comment than an answer, but I am not privileged enough to add comments).