= Gaussian Brownian drift filter
{c}
{title2=$m_t=(I+tV_0)^{-1}(m_0+V_0Y_t)$}
= Gaussian static drift filter
{c}
{synonym}
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 $m_t=(I+tV_0)^{-1}(m_0+V_0Y_t)$ and <covariance matrix> $V_t=V_0(I+tV_0)^{-1}$. 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> $\widehat W_t=Y_t-\int_0^t m_sds$ is a <Brownian motion> in the observation <filtration>, and $dm_t=V_t\,d\widehat W_t$.
Back to article page
