I know that a generic 1-D SDE can be written in Ito form as:
$dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$.
I was curious as to how such an SDE is written when modelling time-independent processes. I was thinking that it must be driftless, so that $\mu = 0$, which leaves us with:
$dX_{t} = \sigma(X_t,t)dW_{t}$,
but, then must $\sigma = \sigma(X)$, i.e., no explicit dependence on time?
Thanks!