To spike or not to spike: the whims of the Wonham filter in the strong noise regime

• 成果类型: Article
• 署名作者: Bernardin, Cedric; Chhaibi, Reda; Najnudel, Joseph; Pellegrini, Clement
• 署名单位: HSE University (National Research University Higher School of Economics); Communaute d'universites et etablissements de Toulouse (Comue); Universite de Toulouse (EPE); Centre National de la Recherche Scientifique (CNRS); Institut National des Sciences Appliquees de Toulouse; CNRS - National Institute for Mathematical Sciences (INSMI); University of Bristol
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01450-0
• 发表日期: 2026-08
• 页码: 1823-1875
• 关键词: Exponential stability; state

摘要

We study the celebrated Shiryaev-Wonham filter (Wonham, W.M., in J. Soc. Ind. Appl. Math. 347-369, 1964) in its historical setup, where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of the authors show the appearance of spikes in the filtered process, akin to a metastability phenomenon. This paper is aimed at understanding the smoothed optimal filter, which is relevant for any system with feedback. In particular, we exhibit a sharp phase transition between a spiking regime and a regime with perfect smoothing.