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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.

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I feel like the other answer is providing a reference for existence of a density function and you are more interested in topological density of the support of $X_t$.

The support of the whole trajectory $X$ in $\mathcal C([0,\infty),\mathbb R^n)$ is described by the so-called Stroock-Varadhan support theorem. In On the support of diffusion processes with applications to the strong maximum principle, their Theorem 3.1 ensures that the (unique) solution to the martingale problem associated with $\frac12(\sigma\sigma^*)^{ij}\partial_i\partial_j$ has full support provided $\sigma$ is bounded continuous non-degenerate (possibly non-uniformly).

Since a solution to the SDE $\mathrm dX_t = \sigma(X_t)\mathrm dW_t$ will always be a solution to the martingale problem, it proves that $X$ has full support, which is much stronger than $X_t$ having full support. That said, if $\sigma$ is not regular enough (say locally Lipschitz) then you may have to resort to other results to prove existence of a solution (or just declare the solution to be the solution to the martingale problem; in the continuous non-denegerate case, we have existence and uniqueness).

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