Equivalent conditions for weak continuity of nonlinear filters
Eugene Aleksandrovich Feinberg, Pavlo O. Kasyanov · Systems & Control Letters · 2023
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 et al. (2019) is a necessary and sufficient condition for semi-uniform Feller continuity of transition probabilities. The property of semi-uniform Feller continuity was introduced recently by Feinberg et al. (2022), and the original transition probability for a Markov decision processes with incomplete information has this property if and only if 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.