Assume a standard MIMO state-space \begin{align} x_{i+1}&= Fx_i + Gw_i\\ y_i&= Hx_i + v_i, \end{align} where the sequence $(w_i,v_i)$ is Gaussian i.i.d. with $\mathbb{E}[v_iw_i^T] = L$.
What is the corresponding transfer function?
This takes me to my main question whether we can "approximate" any continuous transfer function with a state-space with uncorrelated noise, i.e., $L=0$.
If we disregard the fact that the sequence $(w_i, v_i)$ is Gaussian or correlated. Then by considering $(w_i, v_i)$ as the input of the State-Space, one could rewrite it to the following: $$x_{i+1} = Fx_i + \begin{bmatrix}G & 0\end{bmatrix}\begin{bmatrix}w_i \\ v_i\end{bmatrix}$$ $$y_{i} = Hx_i + \begin{bmatrix}0 & I\end{bmatrix}\begin{bmatrix}w_i \\ v_i\end{bmatrix}$$
Which looks like a regular discrete-time state-space model. Therefore, the corresponding Transfer function would be: $$P(z) = \frac{Y(z)}{U(z)} = H(zI-F)^{-1}\begin{bmatrix}G & 0\end{bmatrix} + \begin{bmatrix}0 & I\end{bmatrix}$$ $$Y(z) = H(zI-F)^{-1}G W(z) + V(z)$$
Since I am not using any probabilistic properties, the fact that they are correlated is not important. This can however become troublesome if you try to control either $w$ or $v$, as the other one will change as well due to this correlation. With that, I should also note that this is also problematic if you try to identify a system with this structure, then we have to use alternative techniques (depending on what you can measure of course).