Filters and smoothers for self-exciting Markov modulated counting processes

TL;DR

This paper develops a closed-form, finite-dimensional optimal filter and smoother for a self-exciting counting process influenced by a hidden Markov chain, with applications demonstrated through simulation and analysis of the 2010 flash crash.

Contribution

It introduces a novel closed-form filter and smoother for self-exciting Markov modulated counting processes, enhancing analysis of complex jump data.

Findings

Filter performs well on simulated data

Effective analysis of 2010 flash crash

Finite-dimensional filter simplifies computations

Abstract

We consider a self-exciting counting process, the parameters of which depend on a hidden finite-state Markov chain. We derive the optimal filter and smoother for the hidden chain based on observation of the jump process. This filter is in closed form and is finite dimensional. We demonstrate the performance of this filter both with simulated data, and by analysing the `flash crash' of 6th May 2010 in this framework.