If a signal is assumed to evolve in time like the OU process
$$ dX_t = -a X_t dt + \sigma dV_t $$
with observations
$$ dY_t = h X_t + dW_t $$
I am told that, the posterior distribution, $\pi_t = \mathcal{N}( \mu_t, \nu_t ) $
$$\mu_t = 2 \nu_t \left( x \frac{\overline{a}}{\sigma^2 \sinh(t\overline{a})}+h \int_0^t \frac{\sinh(s\overline{a})}{\sinh(t\overline{a})} dY_s \right) $$
$$ \nu_t = \frac{\sigma^2}{a+\overline{a} \coth(\overline{a})} $$
and $\overline{a} = \sqrt{a^2 + h^2 \sigma^2}$.
How exactly would I compute this?
I have tried using Kallianpur-Striebel formula but it did not work for me and the workings are very messy. What would be nice is a direction to go in and perhaps a reason for doing so. Thank you