On hypoellipticity of degenerate operators in testing and detection problems

TL;DR

This paper investigates the hypoellipticity of degenerate diffusion operators in sequential testing and detection problems with partial information, providing conditions for hypoellipticity and analyzing their implications.

Contribution

It characterizes when these operators are hypoelliptic in testing and detection scenarios, offering new sufficient conditions and insights into their properties.

Findings

Identifies conditions for hypoellipticity without state switching

Provides two sufficient conditions for hypoellipticity with switching

Discusses implications for sequential testing and detection

Abstract

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.