A Palm Space Approach to Non-Linear Hawkes Processes

TL;DR

This paper characterizes stationary non-linear Hawkes processes using operator methods, relaxes classical Lipschitz conditions, and establishes existence and uniqueness results under certain growth constraints.

Contribution

It introduces a novel operator framework for Hawkes processes, relaxes Lipschitz conditions, and provides new existence and uniqueness results for stationary solutions.

Findings

Existence of stationary Hawkes processes under linear growth conditions.

Uniqueness of stationary Hawkes process with exponential memory function.

Scaling behavior of Hawkes process starting from empty state for polynomial activation functions.

Abstract

A Hawkes process on $\R$ is a point process whose intensity function at time $t$ is a functional of its past activity before time $t$. It is defined by its activation function $\Phi$ and its memory function $h$. In this paper, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if $\Phi$ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function $\Phi$ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on $\R_-$ plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion. If the memory function $h$ is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial $\Phi$ with degree ${>}1$, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.