Say that there is an unknown state $\theta\in \{0,1\}$ with prior $p_t$ (that state is 1) and we observe a "discrete" realization of the signal $$ dY_t = \sigma_\theta dW_t$$ meaning that we observe $dY$ for $dt$ units of time. By standard bayes rule $$ p_{t+dt} = \frac{p_t \frac{1}{\sigma_1} \phi(\frac{dY}{\sigma_1 dt^{1/2}})}{p_t \frac{1}{\sigma_1} \phi(\frac{dY}{\sigma_1 dt^{1/2}}) + (1-p_t) \frac{1}{\sigma_0} \phi(\frac{dY}{\sigma_0 dt^{1/2}})}$$
where $\phi(x)=\frac{1}{\sqrt{2\pi}} \exp(\frac{1}{2}x^2)$ is a standard normal density. With some manipulations we get the loglikelyhood $\ell_{t}= \log(\frac{p_t}{1-p_t})$ satifies
$$d \ell_t = \log\left(\frac{\sigma_0}{\sigma_1}\right) + \left(\frac{1}{\sigma_0^2}-\frac{1}{\sigma_1^2} \right)\frac{(dY_t)^2}{dt} $$
Does this make sense? I was expecting to get some SDE for the posterior/likelihood but I feel there is something fishy in the expression I get. Is it possible to do optimal filtering of wiener process of unknown variance in some (other?) way?