I am interest in proving that the transition probability density function $p(t,x,y)$ for the process $(X_t,Y_t)$ is STRICTLY positive on all of $\mathbb{R}^2$, where $(X_t,Y_t)$ is the solution to the SDE
\begin{align} dX_t&=\mu_1(t,X_t,Y_t)dt+\sigma_{11}(t,X_t,Y_t)dW^1_t+\sigma_{12}(t,X_t,Y_t)dW^2_t \\ dY_t&=\mu_2(t,X_t,Y_t)dt+\sigma_{21}(t,X_t,Y_t)dW^1_t+\sigma_{22}(t,X_t,Y_t)dW^2_t \end{align}
and the functions $\mu_i$ and $\sigma_{ij}$ satisfy the usual conditions to guarantee a solution, in addtion $\sigma_{i,j}>0$ for $i,j=1,2$.
I have seen this being proved with a pde approach using Hormander's theorem. I am more interested in whether there is a way to prove this by approximating the stochastic integrals and using the fact that convergence in $L^2$ implies convergence in distribution.