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

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].