Perturbation bounds when the observation dimension is smaller than the state dimension

Establish whether the perturbation bounds developed for the Feynman–Kac representations of the specified state-space models remain valid when the observation dimension \(r\) is strictly smaller than the state-space dimension \(d\), despite the paper’s bi-Lipschitz observation-function assumption requiring \(r\geq d\).

Background

For the class of state-space models treated in the examples, the observation functions $H_t$ are assumed to be bi-Lipschitz. Such maps from $\mathbb{R}^d$ to $\mathbb{R}^r$ can exist only when $r\geq d$, so the main perturbation theory does not cover models with fewer observation coordinates than state variables.

The unresolved question concerns extending the perturbation estimates beyond this observability regime to models with $r<d$. This would broaden the applicability of the theory to partially observed systems in which the observations do not identify the full state instantaneously.