We know that for if $X$ is a stochastic integral of the form below -

$X_t = \int_0^t v(s,\omega) db(s,\omega)$. then we can use time change formula to claim that $X_t = W_{\alpha(t)}$ where $W$ is a different brownian motion with a changed clock.

This holds true, under mild conditions, even when $b$ is a d-dimensional brownian motion.

I was wondering if I can use the same result to say that if $X$ is a stochastic integral of the form - $X_t = \int_0^t v(s,X_s) db(s,\omega)$.

Intuitively I don't see why it might be an issue as $v(s,X_s)$ is essentially a function of $\omega$ only. So I wanted to know if I can write $X_t = W_{\alpha(t)}$ for some other brownian motion.