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This study shows that quadratic Hawkes processes can be stable even with an endogeneity ratio over unity. This stability arises from a balance between mean reversion and trend-following behaviors.

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Area of Science:

  • Stochastic Processes
  • Time Series Analysis
  • Mathematical Finance

Background:

  • Hawkes processes are crucial for modeling self-exciting phenomena.
  • Stability typically requires finite mean intensity and an endogeneity ratio below unity.
  • Nonlinear extensions present unique challenges to stability conditions.

Purpose of the Study:

  • To challenge the conventional stability condition for Hawkes processes.
  • To investigate the stability of quadratic Hawkes processes with an endogeneity ratio greater than unity.
  • To identify the conditions under which such processes remain stationary.

Main Methods:

  • Analysis of quadratic Hawkes processes.
  • Mathematical derivation of stability conditions.
  • Examination of the interplay between linear and quadratic components.

Main Results:

  • Quadratic Hawkes processes can be stationary with an endogeneity ratio exceeding unity.
  • Stability is maintained if the linear Hawkes component's endogeneity ratio is less than unity.
  • Infinite mean intensity is observed when the total endogeneity ratio surpasses 1.

Conclusions:

  • The traditional stability criterion for Hawkes processes is not universally applicable.
  • A subtle compensation between inhibiting and exciting factors ensures stationarity in quadratic Hawkes processes.
  • This finding expands the theoretical understanding of self-exciting point processes.