On Hawkes Processes with Infinite Mean Intensity

TL;DR

This paper demonstrates that quadratic Hawkes processes can remain stationary with infinite mean intensity even when the total endogeneity ratio exceeds one, challenging traditional stability conditions.

Contribution

It introduces the idea that quadratic Hawkes processes can be stable with infinite mean intensity despite a total endogeneity ratio greater than one.

Findings

Quadratic Hawkes processes are always stationary if the linear component's endogeneity ratio is less than one.

Stationarity can occur with infinite mean intensity when the total endogeneity ratio exceeds one.

A balance between mean-reversion and trend excitation explains the stability despite infinite mean intensity.

Abstract

The stability condition for Hawkes processes and their non-linear extensions usually relies on the condition that the mean intensity is a finite constant. It follows that the total endogeneity ratio needs to be strictly smaller than unity. In the present note we argue that it is possible to have a total endogeneity ratio greater than unity without rendering the process unstable. In particular, we show that, provided the endogeneity ratio of the linear Hawkes component is smaller than unity, Quadratic Hawkes processes are always stationary, although with infinite mean intensity when the total endogenity ratio exceeds one. This results from a subtle compensation between the inhibiting realisations (mean-reversion) and their exciting counterparts (trends).