# Stochastic Observability and Filter Stability under Several Criteria

## TL;DR

This paper introduces a new definition of stochastic observability for non-linear systems, linking it to filter stability and robustness, with theoretical analysis and multiple illustrative examples.

## Contribution

It proposes an explicit functional definition of stochastic observability and demonstrates its implication for filter stability under various convergence criteria.

## Key findings

- Filter stability is guaranteed under the new observability definition.
- The work connects observability to robustness and approximation in stochastic control.
- Numerous examples illustrate the theoretical results.

## Abstract

Despite being a foundational concept of modern systems theory, there have been few studies on observability of non-linear stochastic systems under partial observations. In this paper, we introduce a definition of observability for stochastic non-linear dynamical systems which involves an explicit functional characterization. To justify its operational use, we establish that this definition implies filter stability under mild continuity conditions: an incorrectly initialized non-linear filter is said to be stable if the filter eventually corrects itself with the arrival of new measurement information. Numerous examples are presented and a detailed comparison with the literature is reported. We also establish implications for various criteria for filter stability under several notions of convergence such as weak convergence, total variation, and relative entropy. These findings are connected to robustness and approximations in partially observed stochastic control.