Consider the stochastic differential equation
$$dX_t = {\bf 1}_{\{0<X_t<1\}} a(t,X_t)dW_t, \quad \forall t\in [0,T],$$
where $a$ is continuous on $[0,T)\times [0,1]$ and is Holder continuous with respect to $x$, uniformly in $t\in [0,T']$ for all $T'<T$. Assume further for any $T'<T$ there exist $c,C>0$ such that
$$c\le a(t,x)\le C,\quad \forall (t,x)\in [0,T']\times [0,1].$$
In addition, it holds
$$\int_0^T a(t,x)^2dt=\infty,\quad \forall x\in (0,1).$$
For any $x_0\in (0,1)$, can we show the uniqueness in law of the solutions to the above SDE? If not, what additional condition is needed?
PS: I found by chance some slides https://ymsc.tsinghua.edu.cn/__local/A/9A/2A/46FE1992A338F1BAFA7FC117BA5_1C533A7B_32F37.pdf where the theorem of Stroock-Varadhan (page 3) is applied to ensure the uniqueness in law. The only crucial point is the continuity/boundedness of $a$ and its uniform ellipticity. Nevertheless, I'm unable to find the original of this theorem.
