  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 converge of the
state variables to an invariant distribution, which is significantly different
from the strong convergence used to obtain asymptotic results in "Filtering for
Fast Mean-Reverting Processes" (19).
