Exact and approximate hidden Markov chain filters based on discrete observations

Abstract

We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically. The third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.

Discretely Observed Brownian Motion Goverened byTelegrapher's Process: Estimation and Applications to Finance

“…For discretely observed telegraph processes, several methods have been proposed such as pseudo-maximum likelihood (Iacus and Yoshida, 2009), method of moments (De Gregorio and Iacus, 2008), and least squares (De Gregorio and Iacus, 2011). For an underlying integrated continuous Markov chain perturbed by Brownian Motion and observed at discrete times, Bäuerle et al (2015) derived exact and approximate filters. For the BMT process, however, we face an additional challenge because we have two Brownian motions instead of two linear movements.…”