On estimation for Brownian motion governed by telegraph process with multiple off states

TL;DR

This paper develops a likelihood estimation method for a Brownian motion with variance governed by a three-state Markov chain, including a telegraph process with multiple off states, and applies it to real animal movement data.

Contribution

It introduces a new likelihood estimation approach for Brownian motion driven by a three-state Markov process with multiple off states, using hidden Markov model techniques.

Findings

Distribution of occupation time of the on state derived.

Estimation procedure validated through simulations.

Applied to mountain lion position data successfully.

Abstract

Brownian motion whose infinitesimal variance changes according to a three-state continuous time Markov Chain is studied. This Markov Chain can be viewed as a telegraph process with one on state and two off states. We first derive the distribution of occupation time of the on state. Then the result is used to develop a likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with the forward algorithm in the general framework of hidden Markov models. The analytic results are confirmed with simulation studies. The estimation procedure is applied to analyze the position data from a mountain lion.