Nonlinear filters for hidden Markov models of regime change with fast mean-reverting states

Andrew Papanicolaou

Research output: Contribution to journalArticlepeer-review


We consider filtering for a hidden Markov model that evolves with multiple time scales in the hidden states. In particular, we consider the case where one of the states is a scaled Ornstein-Uhlenbeck process with fast reversion to a shifting-mean that is controlled by a continuous time Markov chain modeling regime change. We show that the nonlinear filter for such a process can be approximated by an averaged filter that asymptotically coincides with the true nonlinear filter of the regime-changing Markov chain as the rate of mean-reversion approaches infinity. The asymptotics exploit weak convergence of the state variables to an invariant distribution, which is significantly different from the strong convergence used to obtain asymptotic results in [A. Papanicolaou, Asymptot. Anal., 70 (2010), pp. 155-176].

Original languageEnglish (US)
Pages (from-to)906-935
Number of pages30
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2012


  • Ergodic theory
  • Filtering
  • Hidden Markov models
  • Homogenization

ASJC Scopus subject areas

  • General Chemistry
  • Modeling and Simulation
  • Ecological Modeling
  • General Physics and Astronomy
  • Computer Science Applications


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