On Hawkes Processes with Infinite Mean Intensity

5 Pages Posted: 9 Feb 2022

See all articles by Cecilia Aubrun

Cecilia Aubrun

Polytech'Savoie

Michael Benzaquen

Ecole Polytechnique, Palaiseau; Capital Fund Management

Jean-Philippe Bouchaud

Université Paris VI Pierre et Marie Curie - College de France

Date Written: December 28, 2021

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).

Keywords: Hawkes processes, Endogeneity ratio, Stationarity, QHawkes, ZHawkes

Suggested Citation

Aubrun, Cecilia and Benzaquen, Michael and Benzaquen, Michael and Bouchaud, Jean-Philippe, On Hawkes Processes with Infinite Mean Intensity (December 28, 2021). Available at SSRN: https://ssrn.com/abstract=3995505 or http://dx.doi.org/10.2139/ssrn.3995505

Cecilia Aubrun (Contact Author)

Polytech'Savoie ( email )

BP 80439
74944 ANNECY LE VIEUX Cedex
France

Michael Benzaquen

Capital Fund Management ( email )

23 rue de l'Université
Paris, 75007
France

Ecole Polytechnique, Palaiseau ( email )

Route de Saclay
Palaiseau, 91128
France

Jean-Philippe Bouchaud

Université Paris VI Pierre et Marie Curie - College de France ( email )

11 Place Marcellin Berthelot
Paris, 75005
France

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