= Innovation exponential for Gaussian drift filtering
{title2=$dZ_t=-Z_tm_t^T\,d\widehat W_t$}
For the <Gaussian Brownian drift filter> with positive definite $V_0$, define $Z_t=\det(I+tV_0)^{1/2}\exp[-m_t^TV_t^{-1}m_t/2+m_0^TV_0^{-1}m_0/2]$. Because $V_t'=-V_t^2$ and $dm_t=V_t\,d\widehat W_t$, the <Itô formula> gives $d\log Z_t=-m_t^T\,d\widehat W_t-|m_t|^2dt/2$. Thus $Z$ is a positive <stochastic exponential>, hence a <nonnegative local martingale> and a <supermartingale>. For a singular prior, the inverse-matrix expression is undefined; the same process can instead be defined by $Z_t=(\mathbb E_{\lambda\sim N(m_0,V_0)}\exp[\lambda^TY_t-t|\lambda|^2/2])^{-1}$, with $Y_t$ treated as fixed inside this Gaussian integral.
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