ACID: A Low Dimensional Characterization of Markov-Modulated and Self-Exciting Counting Processes

TL;DR

This paper introduces the asymptotic conditional intensity distribution (ACID) as a new statistical tool for analyzing Markov-modulated and self-exciting counting processes, providing a simulation-free computational method for low-dimensional cases.

Contribution

It presents a novel approach to compute the ACID for certain counting processes using a backward recurrence time parametrization, applicable when the sufficient statistic is low-dimensional.

Findings

ACID effectively characterizes the statistical ensemble of the CI.

The method simplifies analysis for Markov-modulated and self-exciting processes.

Case studies demonstrate practical applications of the ACID approach.

Abstract

The conditional intensity (CI) of a counting process $Y_t$ is based on the minimal knowledge $\mathcal{F}_t^Y$, i.e., on the observation of $Y_t$ alone. Prominently, the mutual information rate of a signal and its Poisson channel output is a difference functional between the CI and the intensity that has full knowledge about the input. While the CI of Markov-modulated Poisson processes evolves according to Snyder's filter, self-exciting processes, e.g., Hawkes processes, specify the CI via the history of $Y_t$. The emergence of the CI as a self-contained stochastic process prompts us to bring its statistical ensemble into focus. We investigate the asymptotic conditional intensity distribution (ACID) and emphasize its rich information content. We assume the case in which the CI is determined from a sufficient statistic that progresses as a Markov process. We present a simulation-free method to compute the ACID when the dimension of the sufficient statistic is low. The method is made possible by introducing a backward recurrence time parametrization, which has the advantage to align all probability inflow in a boundary condition for the master equation. Case studies illustrate the usage of ACID for three primary examples: 1) the Poisson channels with binary Markovian input (as an example of a Markov-modulated Poisson process), 2) the standard Hawkes process with exponential kernel (as an example of a self-exciting counting process) and 3) the Gamma filter (as an example of an approximate filter to a Markov-modulated Poisson process).