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On Hawkes Processes with Infinite Mean Intensity

Author

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  • Cecilia Aubrun

    (LadHyX - Laboratoire d'hydrodynamique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Michael Benzaquen
  • Jean-Philippe Bouchaud

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).

Suggested Citation

  • Cecilia Aubrun & Michael Benzaquen & Jean-Philippe Bouchaud, 2022. "On Hawkes Processes with Infinite Mean Intensity," Post-Print hal-03518579, HAL.
    • Handle: RePEc:hal:journl:hal-03518579
    DOI: 10.1103/PhysRevE.105.L032101
    Note: View the original document on HAL open archive server: https://hal.science/hal-03518579
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