Ito diffusion crosses initial value infinitely many times?

Question

Consider the stochastic process $Y_t=y+\int_0^t\sigma(Y_s)dW_s$ where $W$ is a standard Brownian motion. It is well known that if $\sigma(y)=\sigma >0$ constant we have that $Y_t$ crosses $y$ infinitely many times in any arbitrarily small interval of time $(0,\varepsilon]$. I think that if $|\sigma(y)|>0$ and $\sigma$ satisfies the linear growth (i.e. $|\sigma(z)|\leq D(1+|z|)$ for some $D>0$) and locally Lipschitz (i.e. $|\sigma(x)-\sigma(z)|\leq C_n|x-z|$ for all $|x|,|z|\leq n$ for some sequence $C_n$) conditions, then it is true that $Y_t$ crosses $y$ infinitely many times as above, like a Brownian motion. Intuitively, I would say that if $\varepsilon$ is small enough we have $$Y_t\approx y+\sigma(y)W_t,\,t \in [0,\varepsilon]$$ and the property holds. Is there a reference for this that provides a formal proof (if it is true, of course)?