For the linear Gaussian system $dX_t = AX_t dt + dW_t$, $dY_t = CX_t dt + dV_t$, the conditional mean $\hat{X}_t = \mathbb{E}[X_t \mid \mathcal{F}_t^Y]$ satisfies:

$d\hat{X}_t = A\hat{X}_t dt + \Sigma_t C^T (dY_t - C\hat{X}_t dt),$

where $\Sigma_t$ solves the Riccati equation. The filter is linear, recursive, and optimal (minimum MSE).