On hypoellipticity of degenerate operators in testing and detection problems

Abstract

We study a class of degenerate diffusion generators arising 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 $k<n$. After transforming to posterior likelihood coordinates, we analyze Hörmander's condition both in the absence of state switching (testing) and in its presence (detection). We characterize hypoellipticity in the testing case and give explicit sufficient conditions in the detection case. We further study the stationary posterior operator, its parabolic extension, the joint observation-posterior operator, and its parabolic extension; their Hörmander conditions need not coincide. We characterize the relationships among these operators under our main structural regimes and discuss their probabilistic consequences and the regularity of the associated optimal stopping problem.