If I have a relationship:
$ y_t = a + \theta b_t \epsilon_t,$
where I observe $y_t$ and $b_t$. $a$ is a known parameter, $\theta$ is an unknown parameter with prior distribution at time $t$: $\theta \sim logN(\mu_t, \sigma_t^2)$, and $\epsilon_t$ is an IID random variable with distribution: $\epsilon_t \sim logN(0,\sigma_\epsilon^2)$.
Is it possible to obtain an analytic log-normal posterior for $\theta$? If not, do there exist other possible conjugate prior distributions with non-negative support (and whatever distribution on the noise term $v_t$ that's necessary)?