Nonlinear Filters for Hidden Markov Models of Regime Change with Fast Mean-Reverting States

TL;DR

This paper develops an approximation for nonlinear filters in hidden Markov models with fast mean-reverting states, showing that the averaged filter converges to the true filter as the mean reversion rate increases.

Contribution

It introduces an averaged filtering approach for regime-switching models with fast mean-reverting states, extending previous methods by using weak convergence techniques.

Findings

Averaged filter approximates the true nonlinear filter as mean reversion rate increases.

Asymptotic analysis relies on weak convergence to an invariant distribution.

The approach differs from strong convergence methods used in prior work.

Abstract

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).