Quadratic Hawkes Processes Can Be Stationary with Infinite Mean Intensity

Summary

The note examines stability in Hawkes processes and their nonlinear extensions. Standard stability arguments often assume a finite, constant mean intensity, which leads to a requirement that the total endogeneity ratio remain below one. The paper challenges that requirement for Quadratic Hawkes processes, distinguishing the behavior of the linear component from the total endogeneity ratio.

It argues that a Quadratic Hawkes process remains stationary when the linear component’s endogeneity ratio is below one, even if the total ratio exceeds one. In that case, the stationary process has infinite mean intensity. The proposed explanation is a balance between inhibiting realizations, associated with mean reversion, and exciting realizations, associated with trends. The excerpt states the theoretical conclusion but gives no derivation, empirical test, parameter guidance, or practical trading application, so those details cannot be assessed from the available text.

Key ideas

• A finite mean intensity is a common assumption in Hawkes process stability conditions.
• The total endogeneity ratio can exceed one without necessarily making a Quadratic Hawkes process unstable.
• Stationarity is claimed when the linear Hawkes component has an endogeneity ratio below one.
• In the stated regime, the process can be stationary while its mean intensity is infinite.
• The explanation balances inhibiting, mean-reverting realizations against exciting, trend-like realizations.

Tags

# On Hawkes Processes with Infinite Mean Intensity


# On Hawkes Processes with Infinite Mean Intensity









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).

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