On hypoellipticity of degenerate operators in testing and detection problems

We study a class of degenerate diffusion generators that arise in sequential testing and quickest detection problems with partial information. The observation process is driven by $k$ independent Brownian motions, while the hidden state takes $n+1$ values with $k<n$. By moving to the posterior likelihood coordinates, we analyze the H\"omander's condition of the operator both without state switching (testing) and with switching (detection). We characterize the cases where the operator is hypoelliptic for the former, give two different sufficient conditions for the latter, and discuss their consequences.