Equivalent Conditions for Weak Continuity of Nonlinear Filters

TL;DR

This paper establishes that semi-uniform Feller continuity of transition probabilities is both necessary and sufficient for the weak continuity of nonlinear filters, clarifying conditions under which belief-based processes preserve weak continuity.

Contribution

It proves the equivalence between weak continuity of nonlinear filters and semi-uniform Feller continuity of transition probabilities, extending previous results and reviewing related conditions.

Findings

Semi-uniform Feller continuity is necessary and sufficient for weak continuity.

The property is preserved when transitioning from original to belief-based processes.

The paper reviews various conditions for semi-uniform Feller continuity.

Abstract

This paper studies weak continuity of nonlinear filters. It is well-known that Borel measurability of transition probabilities for problems with incomplete state observations is preserved when the original discrete-time process is replaced with the process whose states are belief probabilities. It is also known that the similar preservation may not hold for weak continuity of transition probabilities. In this paper we show that the sufficient condition for weak continuity of transition probabilities for beliefs introduced by Kara, Saldi, and Yuksel (2019) is a necessary and sufficient condition for semi-uniform Feller continuity of transition probabilities. The property of semi-uniform Feller continuity was introduced in Feinberg, Kasyanov, and Zgurovsky (2021), and, if the original transition probability has this property, then the transition probability of the process, whose state is a pair consisting of the belief probability and observation, also has this property. Thus, this property implies weak continuity of nonlinear filters. This paper also reviews several necessary and sufficient conditions for semi-uniform Feller continuity.