Filter Stability, Observability and Robustness for Partially Observed Stochastic Dynamical Systems

Abstract

Filter stability refers to the correction of an incorrectly initialized filter for a partially observed stochastic dynamical system with increasing measurements. In this thesis, we study the filter stability problem, develop new methods and results for both controlled and control-free stochastic dynamical systems, and study the implications of filter stability on robustness of optimal solutions for partially observed stochastic control problems. We introduce a definition of non-linear stochastic observability and through this notion of observability, we provide sufficient conditions for when a falsely initialized filter merges with the correctly initialized filter over time. We study stability under different notions such as the weak topology, total variation, and relative entropy. Additionally, we investigate properties of the transition kernel and measurement kernel which result in stability with an exponential rate of merging. We generalize our results to the controlled case, which is an unexplored area in the literature, to our knowledge. Stability results are then applied to stochastic control problems. Under filter stability, we bound the difference in the expected cost incurred for implementing an incorrectly designed control policy compared to an optimal policy and relate filter stability, robustness, and unique ergodicity of non-linear filters.