Stable approximation schemes for optimal filters

We explore a general truncation scheme for the approximation of (possibly unstable) optimal filters. In particular, let $\mS = (\pi_0,\kappa_t,g_t)$ be a state space model defined by a prior distribution $\pi_0$, Markov kernels $\{\kappa_t\}_{t\ge 1}$ and potential functions $\{g_t\}_{t \ge 1}$, and let $\sfc = \{C_t\}_{t\ge 1}$ be a sequence of compact subsets of the state space. In the first part of the manuscript, we describe a systematic procedure to construct a system $\mS^\sfc=(\pi_0,\kappa_t^\sfc,g_t^\sfc)$, where each potential $g_t^\sfc$ is truncated to have null value outside the set $C_t$, such that the optimal filters generated by $S$ and $S^\sfc$ can be made arbitrarily close, with approximation errors independent of time $t$. Then, in a second part, we investigate the stability of the approximately-optimal filters. Specifically, given a system $\mS$ with a prescribed prior $\pi_0$, we seek sufficient conditions to guarantee that the truncated system $\mS^\sfc$ (with {\em the same} prior $\pi_0$) generates a sequence of optimal filters which are stable and, at the same time, can attain arbitrarily small approximation errors. Besides the design of approximate filters, the methods and results obtained in this paper can be applied to determine whether a prescribed system $\mS$ yields a sequence of stable filters and to investigate topological properties of classes of optimal filters. As an example of the latter, we explicitly construct a metric space $(\mfS,D_q)$, where $\mfS$ is a class of state space systems and $D_q$ is a proper metric on $\mfS$, which contains a dense subset $\mfS_0 \subset \mfS$ such that every element $\mS_0 \in \mfS_0$ is a state space system yielding a stable sequence of optimal filters.