Consider the following stochastic continuous-time state-space model:
$\dot{x} = Ax + Bu $
where $A$ and $B$ are constant matrices, and $u$ is stochastic control input with nonzero mean $\tilde{u}$ and power spectral density $Q$. If we suppose $u$ has a deterministic part $u^d$ or $\tilde{u}$ and a stochastic part $n$, then we can now construct the following model:
$\dot{x} = Ax + Bu^d + n $
where, now, $n$ is zero mean with power spectral density $Q$.
My question is how one can compute its discrete-time counterpart?
Normally discrete time models are obtained through zero-order hold (assume inputs are constant between time-steps).
The same can be applied to the stochastic input. Using the exact method for time-step $T_s$ we get
$$ \begin{bmatrix} F & G_u & G_n\\ 0 & I & 0\\ 0 & 0 & I \end{bmatrix} = \exp\left(T_s \begin{bmatrix} A & B & I\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}\right) $$
and the discrete model is given by
$$ x_{k+1} = F x_k + G_u u^d_k + G_n n_k $$
Also, note that the power spectral density of $ n $ is not $ Q $, but $ B Q B^T $ (to maintain consistency with previous use of variable) since you removed the B matrix from it.