= Finite-horizon pricing density for Gaussian drift learning
{title2=$D_t=e^{kY_t-k^2t/2}/L_t(Y_t)$}
For the <Gaussian Brownian drift filter> $Y_t=\alpha t+W_t$, with independent prior $\alpha\sim N(m_0,\tau_0^{-1})$, the observation law has density
$$
L_t(y)=\sqrt{\frac{\tau_0}{\tau_0+t}}\exp\left(\frac{(\tau_0m_0+y)^2}{2(\tau_0+t)}-\frac{\tau_0m_0^2}{2}\right)
$$
relative to driftless <Wiener measure>. The law having fixed observation drift $k$ has density $e^{kY_t-k^2t/2}$ relative to the same reference. Their ratio is therefore a positive mean-one density on every finite horizon. In the observation <filtration> it satisfies $dD_t=-D_t(m_t-k)d\widehat W_t$, where $m_t$ is the posterior mean and $\widehat W$ the <innovation process>. This density-ratio argument establishes a true <martingale> without an unsupported global <Novikov condition> for the unbounded posterior mean.
Back to article page
