Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms

TL;DR

This paper introduces a new Kalikow decomposition for continuous-time multivariate counting processes, enabling efficient simulation algorithms for complex networks with various intensity conditions, and demonstrates applications on several Hawkes process models.

Contribution

It develops a novel Kalikow decomposition for multivariate counting processes and derives new simulation algorithms applicable to infinite and finite networks with different intensity bounds.

Findings

Decomposition exists in various cases including infinite networks.

Simulation algorithms work for stationary processes with bounded intensities.

Algorithms are applicable to multiple Hawkes process variants.

Abstract

We propose a new Kalikow decomposition for continuous time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before 0. The Kalikow decomposition is not unique and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods on several examples: linear Hawkes process, age dependent Hawkes process, exponential Hawkes process, Galves-L\"ocherbach process.