Correlated noise in Kalman filter Assume a standard state-space setting:
\begin{align}
x_{t+1} &= Ax_t + w_t\\
y_t &= Cx_t+v_t,
\end{align}
where the noise sequences are i.i.d. Gaussian with $w_i\sim N(0,W)$ and $v_i\sim N(0,V)$.
I am interested in the pre- and post- error covariance matrices recursions when there is a correlation $\mathbb{E}[w_tv_t^\top]= S$.
Specifically, using the notation
\begin{align}
\Sigma_{t|t} &= \mathbf{cov}(x_{t}|y^t)\\
\Sigma_{t+1|t} &= \mathbf{cov}(x_{t+1}|y^t),
\end{align}
how does the standard propagations:
\begin{align}
\Sigma_{t|t}&= \Sigma_{t|t-1} - \Sigma_{t|t-1} C^\top (C\Sigma_{t|t-1}C^\top+V)^{-1}C \Sigma_{t|t-1}\\
\Sigma_{t+1|t} &= A\Sigma_{t|t}A^\top+W
\end{align}
change. Thanks.
 A: I will use your notation, and also define the conditional mean $\hat x_{t|s} = \mathbf{E}(x_t \mid y^{s})$.
The way you wrote the KF updates in your question have them split in two steps, the "measurement update" $(t|t-1) \to (t|t)$ and the "time update" $(t|t) \to (t+1|t)$.
In general, you can just combine the two updates into a single update $(t|t-1) \to (t+1|t)$. Here is what you get for the case $S\neq 0$ when you do that; these are the standard KF update equations.
$$
\begin{aligned}
\Sigma_{t+1|t} &= \left(A \Sigma_{t|t-1} A^\top + W\right) - \left( A \Sigma_{t|t-1} C^\top + S \right) \left( C \Sigma_{t|t-1} C^\top + V \right)^{-1} \left( C \Sigma_{t|t-1}A^\top + S^\top\right) \\
L_t &= \left( A \Sigma_{t|t-1} C^\top + S \right) \left( C \Sigma_{t|t-1} C^\top + V \right)^{-1} \\
\hat x_{t+1|t} &= A \hat x_{t|t-1} + L_t (y_t - C\hat x_{t|t-1})
\end{aligned}
$$
If you want to look at the measurement and time updates separately, that's also possible. The measurement update when $S\neq 0$ is the same as in the case $S=0$. For completeness sake, here is the measurement update:
\begin{align}
\hat x_{t|t} &= \hat x_{t|t-1} + \Sigma_{t|t-1} C^\top (C\Sigma_{t|t-1}C^\top +V)^{-1}(y_t - C \hat x_{t|t-1})\\
\Sigma_{t|t} &= \Sigma_{t|t-1} - \Sigma_{t|t-1} C^\top (C\Sigma_{t|t-1}C^\top+V)^{-1}C \Sigma_{t|t-1}
\end{align}
The time update is different when $S\neq 0$. Actually, you can't write $(t+1|t)$ neatly in terms of just $(t|t)$. I'll try to give some insight. When $S=0$, you can take the state update equation $x_{t+1} = Ax_t + w_t$ and take conditional expectations given $(y_0,\dots,y_t)$ and directly obtain $\hat x_{t+1|t} = A \hat x_{t|t}$ and $\Sigma_{t+1|t} = A\Sigma_{t|t}A^\top + W$. This works because $x_t$ and $w_t$ are independent. But when $S\neq 0$, although $x_t$ and $w_t$ are still independent by assumption, they become correlated when you condition given $(y_0,\dots,y_t)$, since $y_t$ depends on $x_t$ and $v_t$ (which is correlated with $w_t$), i.e. $y_t$ is correlated with both $x_t$ and $w_t$. Therefore, your terms involving $(w_t|y^t)$ will need to be expressed in terms of $(w_t|y^{t-1})$, which is simply $w_t$, but this introduces dependency on $(x_t|y^{t-1})$, which we were trying to avoid in the first place. Working through this, you get:
\begin{align}
\hat x_{t+1|t} &= A\hat x_{t|t} + S (C\Sigma_{t|t-1}C^\top +V)^{-1}(y_t - C \hat x_{t|t-1})\\
\Sigma_{t+1|t} &= A\Sigma_{t|t}A^\top + W - \left( A \Sigma_{t|t-1} C^\top + S \right) \left( C \Sigma_{t|t-1} C^\top + V \right)^{-1} \left( C \Sigma_{t|t-1}A^\top + S^\top\right)\\
&\hspace{1cm}+A\Sigma_{t|t-1} C^\top (C\Sigma_{t|t-1}C^\top+V)^{-1}C \Sigma_{t|t-1}A^\top
\end{align}
Note that combining the measurement and time updates recovers the standard KF update equations at the top of the post.
