 Gaussian Brownian drift filter

Observe $Y_t=\lambda t+W_t$, with an independent Gaussian random vector $\lambda$ of mean $m_0$ and covariance matrix $V_0$. Its posterior is Gaussian with mean $m_t=(I+t{V}_0{)}^{-1}(m_0+V_0{Y}_t)$ and covariance matrix $V_t=V_0(I+t{V}_0{)}^{-1}$. These formulas also hold for a singular positive semidefinite matrix $V_0$. When $V_0$ is invertible, completing the square in the Bayes' theorem gives $V_t^{-1}=V_0^{-1}+tI$ and $m_t=V_t(V_0^{-1}{m}_0+Y_t)$. The observed innovation process ${\hat{W}}_t=Y_t-{\int }_0^t{m}_{s}ds$ is a Brownian motion in the observation filtration, and $d{m}_t=V_{t}d{\hat{W}}_t$.