I am asking an extension of the question here for SDEs of the Ito form.

Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}^{n\times d}$ is a bounded and sufficiently regular (such as Lipschitz continuous and differentiable) function. We assume $\sigma\sigma^\top$ is non-degenerate, i.e., its minimum eigenvalue is uniformly bounded away from zero.

I am wondering under which regularity condition of $\sigma$, one can show that for each $t>0$ and nonempty open ball $B\in \mathbb{R}^n$, we have that $\mathbb{P}(X_t\in B)>0$.

I could not find a precise reference for the result. I have tried to prove the solution $X_t$ has a strictly positive density. The uniformly bounded below assumption on $\sigma$ ensures the existence of a density. However, I am not sure how to argue the density is strictly positive. The result is argued for the one-dimensional setting with additive noises here. For multiplicative noises, a result has been given here for smooth $\sigma$ and SDEs of the Stratonovich form.